P O RO ELAS TI C STRUCTURES
P O RO ELAS TI C STRUCTURES
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P 0 RO ELAS TI C STRUCTURES Gabriel Cederbaum Department of Mechanical Engineering Ben-Gurion University of the Negev Beer-Sheva, Israel
LePing Li Biomedical Engineering Institute Ecole Polytechnique of Montreal Montreal, Canada
Kalman Schulgasser Department of Mechanical Engineering Ben-Gurion University of the Negev Beer-Sheva, Israel
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To our beloved children
Jonathan and Daniel Cederbaum Hon g Yi Li Joshua, Yael, Daniel and Noam Schulgasser
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Contents
CHAPTER 1. Introduction
1
CHAPTER 2. Modeling of poroelastic beams 2.1. Basic equations 2.2. Characteristic times 2.3. Note on a beam impermeable at both ends 2.4. Equations in non-dimensional form
9 9 15 16 17
CHAPTER 3. Analytical solutions for quasi-static beams 3.1. Simply-supported beams with permeable ends 3.2. Beams subjected to loads suddenly applied and constant thereafter
19 19 20
CHAPTER 4. Finite element formulation and solutions for quasi-static beams 4.1. Introduction 4.2. Variational principles 4.3. Finite element formulation 4.4. Examples and discussion
33 33 34 37 41
CHAPTER 5. Vibrations of poroelastic beams 5.1. Initial value problems 5.2. Forced harmonic vibrations 5.3. Closure
53 53 63 65
CHAPTER 6. Large deflection analysis of poreolastic beams 6.1. Governing equations 6.2. Equations in non-dimensional form when F ~ = O 6.3. Numerical formulation 6.4. Numerical procedure for the finite difference method 6.5. Examples and discussion
67 67 71 72 77 79
...
\ 111
Contents
CHAPTER 7. Stability of poroelastic columns 7.1. Buckling of columns 7.2. Limits of critical load 7.3. Time-dependence of critical load and deflections 7.4. Post-buckling: formulation 7.5. Post-buckling: results and discussion 7.6. Imperfection sensitivity 7.7. Dynamic stability of poroelastic columns 7.8. Stability boundaries and critical load amplitude
89 89 90 92 96 99 102 104 107
CHAPTER 8. Analysis of poroelastic plates 8.1. Basic equations for thin plates 8.2. Bending equations in non-dimensional form 8.3. Analytical solutions for quasi-static bending 8.4. Transverse vibrations of simply supported plates
111 111 118 119 128
CHAPTER 9. Closure
135
REFERENCES
141
APPENDIX A. Proof of the variational principles
145
APPENDIX B. A finite element for poroelastic beams
149
APPENDIX C. Several useful Laplace inverse transformations
151
APPENDIX D. Coefficients in difference formulas
153
APPENDIX E. Determination of boundary values at x1 for the finite difference method
155
SUBJECT INDEX
157
Chapter 1 INTRODUCTION
Poroelasticity is a continuum theory for the analysis of a porous media consisting of an elastic matrix containing interconnected fluid-saturated pores. In physical terms the theory postulates that when a porous material is subjected to stress, the resulting matrix deformation leads to volumetric changes in the pores. Since the pores are fluid-filled, the presence of the fluid not only acts as a stiffener of the material, but also results in the flow of the pore fluid (diffusion) between regions of higher and lower pore pressure. If the fluid is viscous the behavior of the material system becomes time dependent. The basic phenomenological model for such a material was proposed by Biot (Biot, 1941a,b, 1955, 1956a,b, 1962, 1964; Biot and Willis, 1957). His motivation (and the application of the theory over the years) was concerned with soil consolidation (quasi-static) and wave propagation (dynamic) problems in geomechanics. The literature dealing with poroelasticity based on the now classical model is voluminous. It cannot possibly be reviewed within the scope of the present work. Most of this literature deals with very specific aspects of the theory or with specific physical problems. We cite a selection of general and fundamental references in which numerous other citations can be found: Bear (1972, 1990), Cheng et al. (1993), Cleary (1977), Coussy (1995), Detourney and Cheng (1993), Kumpel (1991), Rice and Cleary (1976), Rudnicki (1985) and Thomson and Willis (1991). We point out that simultaneous with the development of poroelasticity, literature dealing with thermoelasticity has evolved. Here it is assumed that there is a coupling between the thermal diffusion equations and the equations of mechanical equilibrium. The complete mathematical analogy of the poroelastic and thermoelastic problems was noted by Biot (1956c). A comprehensive treatment from this latter point of view is given by Nowacki (1986). A parallel literature has been produced, but it is far less voluminous, probably due to the lesser occurrence of instances when it is physically significant to consider coupling in the thermoelastic case. Poroelastic theories were originally motivated by problems in soil and geomechanics. This is the point of departure of most of the literature cited above. These problems generally concern massive structures and are by nature three-dimensional. Consolidation problems, seismic wave propagation, crustal dynamics and seabed mechanics are some examples. This application of poroelastic theory is relatively mature. In the past two decades the poroelastic model has also been extensively and successfully applied in biomechanics. There are similarities in mechanical properties and behaviors between
2
Ch. 1
Introduction
geomechanical and certain biological structures. For example, it has been found that certain porous rocks, marbles and granites have material properties that are similar to those of bone (e.g. Cowin, 1999). The pores in such biological structures are interconnected so that the pore fluid can transport nutrients to, and take waste away from, the cells. One of the most important functions of the fluid in articular cartilage is to lubricate the joints and thus protect them from wearing. The mechanism of the fluid flow in bone and cartilage is so 'designed' that it also protects the biological structure from damage resulting from dynamic loading. Application of poroelastic theory in biomechanics is presently very vigorous. Therefore it is appropriate to review some of this work. This gives some indication of the potential fertility of this material model for other than the original application for which the idea was developed. Poroelastic models of bone were first reported 30 years ago (e.g. Nowinski and Davis, 1970, 1972; Nowinski, 1971, 1972; Johnson et al., 1982). The bulk modulus of the matrix of bone is usually several times higher than that of the fluid in the bone. Thus the pore fluid does not share much of the the mechanical loading when loaded slowly. However, this is not true in the case of dynamic loading - the transient stiffness can be much higher than the stiffness of the drained bone. The importance of such studies using poroelasticity in bone mechanics also lies in understanding the fluid motion, which induces the streaming potentials (electrical potentials produced by ionic motion) and transports nutrients. A survey of the application of poroelasticity in bone mechanics has been given by Cowin (1999). As opposed to bone, soft tissue has a much lower elastic modulus and thus the pore fluid carries a significant portion of the mechanical load. For articular cartilage, the transient stiffness may be ten times as high as the drained stiffness under physiological loadings. Poroelastic theories have been extensively employed for the study of cartilage and other soft tissue (e.g. Mow et al., 1984, 1993; Eisenberg and Grodzinsky, 1985, 1987). Uniaxial compression tests (e.g. Armstrong et al., 1984; Lanir, 1987), indentation tests (e.g. Mak et al., 1987; Spilker et al., 1992b), joint contact analyses (e.g. Ateshian and Wang, 1995; Wu et al., 1996), determination of streaming potentials (e.g. Gu et al., 1993, 1998), and other situations have been considered. Finite element methods have been generally used to extract solutions (e.g. Goldsmith et al., 1996; Simon, 1992; Spilker and Suh, 1990). Due to the complicated behavior of the biological fluids and the sophisticated matrix structure different models have been offered (e.g. Mow et al., 1980; Mak, 1986a,b; Lai et al., 1991; Setton et al., 1993; Almeida and Spilker, 1998; Suh and Bai, 1998; Li et al., 1999b, 2000; Suh and DiSilvestro, 1999). Other applications of poroelastic models include analyses of spinal motion segments (Simon et al., 1985a,b; Laible et al., 1993; Argoubi and Shirazi-Adl, 1996), meniscus of knee joints (Spilker et al., 1992a), blood flow through soft tissue (Vankan et al., 1996, 1997) and orthopedic implants (e.g. Prendergast, 1997). Relatively few papers have thus far investigated the poroelastic beams or plates, the light structures, for which the boundary conditions and the type of loading, and thus the behavior of the structure, are quite different from those for large formations. When such elements are subjected to bending, the stress gradients would generally be expected to be much greater in the perpendicular direction than in the axial or in-plane directions. Thus if the bulk material can be considered to be isotropic, the diffusion in the transverse direction is dominant. Hence, in the studies reported in the literature the diffusion in the axial or in-plane directions is, justifiably, considered negligible and the fluid movement
Introduction
3
in the perpendicular direction has been taken as the prevailing diffusion effect. Among such papers available is that of Nowinski and Davis (1972), who modeled a beam as an anisotropic poroelastic body subjected to a uniform bending moment or a uniform torsion moment. This is a case where no diffusion occurs in the direction of the beam axis since the stress gradients are zero in that direction. Taber (1992) and Theodorakopoulos and Beskos (1993, 1994) formulated the Kirchhoff plate, assuming that the fluid-velocity gradients within the plate plane relative to the solid are negligible. Zhang and Cowin (1994) considered combinations of pure bending and axial compression for rectangular beams, again for no diffusion in the direction of the beam axis. Biot (1964) discussed the buckling problem for a plate with special deformation: in one direction within the plate, the normal strain is zero; in the perpendicular direction within the plate, the fluid displacement relative to the solid is zero. Thus the fluid can flow only in the transverse direction. This book is devoted to the analysis of fluid-saturated poroelastic beams, columns and plates made of materials for which diffusion in the longitudinal direction(s) is viable, while in the perpendicular direction(s) the flow can be considered negligible because of the microgeometry of the solid skeletal material. Our initial motivation to investigate such structures is mainly related to plant stems and petioles. These elements of plants serve the dual functions of providing structural strength and stiffness, and also contain the vascular tissue, which conducts water from the root system to transpiring leaves. Living herbaceous stems and woody stem tissue are water saturated, the former often containing as much as 85% free water by weight, the latter as much as 60%. Such structural plant material is highly anisotropic. The axial stiffness is some 20 times greater than transverse stiffness for woody tissue, and for other plant tissue the anisotropy is probably much greater (Schulgasser and Witztum, 1992). The crucial attribute of such plant elements is that their microstructure is designed to transport water axially. Plants native to non-desert areas transport enormous quantities of water daily from the root systems through stems which are transpired from the leaves (Weier et al., 1982). We are thus led to model a living plant stem as a beam consisting of a poroelastic material for which water movement in the axial direction is dominant. While the overwhelming bulk of literature on poroelasticity in general, and of the references already cited in particular, deals with media which are taken to be statistically isotropic, the theory has also been developed for the anisotropic case (e.g. Carroll, 1979; Rudnicki, 1985; Thomson and Willis, 1991). The underlying physical assumptions are the same. However, the increased mathematical complexity in obtaining solutions, the profusion of material constants whose values must be determined, and also the fact that for most applications previously considered, isotropy is a reasonable model, has resulted in the concentration on the isotropic case. Here we take the anisotropy to the extreme and consider elements whose dimension in the direction perpendicular to fluid motion is small compared to dimensions in the flow direction(s). On the one hand, no new physical assumptions are involved, and on the other hand, for this special case, the number of material parameters is eminently tractable. The material constituting the beam elements will be taken to be transversely isotropic in the cross-sectional plane. The study provides a methodology and a theoretical basis for investigating the mechanical behaviors of the structural elements made of such materials. These are not limited to the materials of plant
4
Ch. 1
Introduction
stems, which was the original motivation for the material model; artificial materials with similar behavior are of greater interest and potential. Many microstructures and fabrication schemes could be imagined, which would produce bulk materials with the postulated behavior. It is not the purpose of this book to deal with micromechanics or with the relationship of microstructure to effective bulk properties. Nor is it our purpose to investigate appropriate fabrication procedures. We simply point out that such materials could be produced with a wide range of bulk behaviors, and could be tailored to specific situations. We only demonstrate how easily any closed pore foam (i.e. the pores are not interconnected) can be converted to the type of material under consideration. This is illustrated in Fig. 1.1. We have simply pierced the (previously closed pore) model with a battery of parallel needles. The behavior of light structures (beams, columns, plates) constituted of such material was first investigated by the present authors in a series of articles (Li et al., 1995, 1996, 1997a,b,c, 1999a; Cederbaum et al., 1998; Cederbaum, 2000a,b). See also Li (1997). The governing equations for a transversely isotropic poroelastic beam (transversely isotropic in the cross-section) subjected to transverse and/or axial loads, as obtained within the small deflection theory, are presented in Chapter 2, including the inertia of the bulk material. Biot's theory, with relative motion between the solid and fluid governed by Darcy's law, is adapted for the case considered. The governing differential equations can be separated into two groups, one for bending and another for extension; since they are not coupled, they can be solved independently. Each group includes three equations for three unknown time-dependent functions: the total stress resultant, the pore pressure resultant and the displacement. The conditions for determination of solutions include the geometrical boundary conditions, the load boundary conditions and the diffusion boundary conditions, as well as the initial conditions. Chapter 3 presents analytical solutions for the quasi-static bending problem of beams. The formulation is derived by deleting the inertia term from the partial differential equation, which governs the equilibrium of the beam. The elastic solutions, i.e. the solutions for the corresponding drained beams, are introduced in order to simplify the solution procedure so that various closed form solutions for the poroelastic beams can be found. Series solutions are found for normal loading with various mechanical and diffusion boundary conditions. Due to the complexity of the boundaries and the governing differential equations, it is often difficult to get analytical solutions for general cases, especially when the boundary
Fig. 1.1. 'Hand-made' poroelastic material with axial diffusion.
Introduction
5
conditions are not homogenous or when they cannot be decoupled. Therefore, finding suitable numerical methods for respective problems is an important part of the present work. The finite element method is employed for the quasi-static beams and columns under small deflection in Chapter 4. Variational principles are first developed. The variational functional is expressed in terms of integrals of the unknown time-dependent functions with respect to position on the beam and convolution integrals with respect to time. Two types of variables, the displacements and pore pressure resultants, are involved in the time-dependent functionals. The method of Lagrange multipliers is employed in order to include the flow equations (generalized Darcy's law) in the Euler-Lagrange equations of the functionals. Two functionals are given: one includes the initial values of the unknown functions and is more convenient for the interpolation of the displacement velocities; another functional is more convenient for the interpolation of the displacements. Both functionals are found to be equivalent to each other in terms of their stationary conditions, which give the governing differential equations and boundary conditions. A mixed finite element scheme is then presented based on one of the variational functionals obtained. Numerical solution examples for both types of variables are presented in order to test the finite element model, and good coincidence with the previously found analytical solutions is shown. The results also demonstrate some unique features of poroelastic beams and columns, which cannot be shown by examples for which analytical solutions can be found. In Chapter 5 solutions are found for free and forced vibration situations of the poroelastic beams. Closed form solutions of the initial value problems are obtained for simply supported beams with general loading by use of Laplace transformation. It turns out that the fluid works as a damper. Similar to the classic vibration theory of damped elastic beams, the responses to initial deviations can be classified into three types: light damping, critical damping and over damping. The vibration patterns are also dependent on the nature of the initial conditions; observed behavior of this sort cannot be explained by the classical vibration theory. Computations for forced harmonic vibrations are carried out for different boundaries. The amplitude response versus the frequency of the loading, and the resonance areas, are investigated. Chapter 6 deals with large deflections of beams. While in the previous chapters the deflection was considered to be small, and thus linear theories are sufficient when the constitutive law is linear, for some situations it may be necessary to employ a large deflection theory in order to correctly describe the behavior of the structure. On the other hand, the deformation can be still small and the skeletal material yet behaves elastically; the large deflection is made possible by the slenderness of the beam. Therefore, it is modeled as geometrically non-linear and constitutively linear. Biot's constitutive law and Darcy's law are adopted as in the linear theory, while new geometrical relations and equilibrium equations are necessarily introduced. In the large deflection case the stretching and bending problems are coupled. The non-linear boundary value problem is solved numerically by using the finite difference method with respect to the spatial coordinate and using a simple successive implicit formula (the trapezoid formula) to deal with the time variable. Several types of geometrical and diffusion boundary conditions are investigated by means of numerical solutions. Results are presented, for which observations are made, and some interesting features are found which do not occur when the problem is modeled as linear (i.e. small deflections).
6
Ch. 1 Introduction
The stability of poroelastic columns is investigated in Chapter 7. Three problems are considered: buckling, post-buckling and dynamic stability. For the buckling problem, the time dependent behaviors of the critical loads and deflections are considered for various diffusion and geometrical boundaries. Upper (short time) and lower (long time) limits for the time-dependent critical load are found for the case of a time-dependent load. It is also shown that buckling can be avoided during a loading procedure by properly choosing the loading path, even when the load at finite time is greater than the lower limit of the critical load. For the post-buckling problem, the time-dependent behavior of the columns, governed by three coupled equations, is obtained by using the large deflection theory. These equations are transformed into a single one, enabling the analytical derivation of the initial and the final responses. It is shown that unlike the quasi-static response obtained by using the small deflection theory, the long time response derived here is bounded. The imperfection sensitivity of these columns is also investigated. For the dynamic stability problem, stability conditions and boundaries are derived. It is shown that the stability regions are expanded with respect to the elastic (drained) case. The critical (minimum) loading amplitude for which instability occurs is also given. Formulations are found in Chapter 8 for fluid-saturated poroelastic plates consisting of a material for which the diffusion is possible in the in-plane directions only, both for bending and for in-plane loading. The plates considered are isotropic in the plate plane, and the Kirchhoff hypotheses are assumed. Again Biot's constitutive law is adopted and Darcy's law is used to describe the fluid flow in pores. The basic equations are so derived that they could be easily extended for the situation of an orthotropic poroelastic plate. Closed form solutions are extracted for quasi-static problems and for vibrations. Observations are made on the types of deflection/vibration patterns, which are obtained. Finally in Chapter 9 we present a beam model composed of a discrete elastic structure containing a fluid which together will behave in an identical manner to the beam composed of a poroelastic continuum used in the formulation of Chapter 2. This discrete model has heuristic value in appreciating the phenomena. We show how adjusting the physical parameters of the materials and the geometric parameters influence the extent of the poroelastic effect. Throughout the book some very unique features of the proposed model are shown (in some cases a comparison is made with another time-dependent model, viscoelasticity). First, the mechanical behaviors of the structural elements are shown to be greatly dependent on the diffusion patterns which are in turn dependent on the loading, on the geometrical and diffusion boundary conditions, as well as on the material parameters. This is in contrast to the case of problems of the same kind of structures (as far as geometry and loading) with diffusion in the transverse direction, where the transverse diffusion patterns for different positions along the beam are all similar. Second, three time scales are required to describe the vibration system, as compared to two in viscoelasticity. Moreover, the vibration patterns of the present system are determined not only by the parameters of the material and the geometry of the structure, but also by the initial pore pressure conditions. This implies that even in the case of 'light damping' oscillatory motion may not occur for some initial pore pressure conditions, which would not be explainable if the structures were modeled as viscoelastic. Third, the pore pressure at a given position does not necessarily decay monotonically after a suddenly applied and then constant loading; it may
Introduction
7
increase for some time and then decay toward the final value in some cases. This phenomenon is similar to the so-called Mandel-Cryer effect. Moreover, the pore pressure at other positions can possibly increase monotonically for all time; in some cases the sign of pore pressure can even change twice during the diffusion process. Fourth, the pore fluid works also as a damper. This damping mechanism is useful in reducing vibrations. By changing the properties of the fluid, which for instance can result from a temperature increment, or by altering the diffusion boundaries, a resonance can be avoided or an oscillatory motion could be made to disappear when so desired. It is also possible to avoid oscillatory motions by choosing initial pore pressure conditions, without changing the properties of the material and the geometry of the structural element. Again, this is quite different from a viscoelastic damping system. As a result of the features mentioned above, it is recognized that the response of the poroelastic structural element to loading is sensitive to the properties of the fluid and to the diffusion boundaries, which can be easily altered in practice. Therefore, such structural elements and thus their features are potentially controllable. In other words, it could be possible to convert such elements into intelligent or smart structures. If this is so, it would be interesting that such structural elements could work as both sensors and actuators, e.g. the fluid can 'feel' the change of the temperature by changing its viscosity and this results in a change of the behavior of the structure. This book attempts to constitute a reasonably self-contained presentation of a wide spectrum of problems related to the analysis of the type of poroelastic structure considered. It is hoped that the book will serve as an inspiration, guide, and reference for applying such elements in mechanical, biomechanical, civil and aerospace engineering, as well as a textbook for graduate studies.
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Chapter 2 MODELING OF POROELASTIC BEAMS
In this chapter formulations are established for poroelastic beams subjected to axial and transverse loads, when the fluid diffusion is axial by virtue of the material microgeometry. The material is taken to be transversely isotropic in the cross-sectional plane. The basic assumptions employed in formulating the geometrical relations are those of classical beam theory. Following the basic formulations, general features of the system are demonstrated. Finally the non-dimensional equations are obtained.
2.1. Basic Equations Consider the beam shown in Fig. 2.1. The area moment of inertia of the cross-section with respect to the y-axis is I; cross-sectional area is A, which is taken to be symmetric with respect to the z-axis. The mass per unit length of the beam is p. The beam is subjected to a distributed normal load qn(X, t) and a distributed axial load qs(X, t). As noted above, the elastic porous matrix is saturated with a fluid, and the microstructure is such that the beam material is transversely isotropic and permits fluid movement in the axial direction only (or at least predominantly). The classical beam theory is suitable for a beam whose crosssectional dimensions are small compared to its length. The basic assumptions and implications of this theory are: 9 plane cross-sections of the beam remain plane after deformation 9 upon the application of load the cross-section undergoes, at most, a translation and a rotation with respect to the original coordinate system 9 a normal to the cross-section remains normal to this surface 9 the rotational inertia is negligible, and 9 the only non-negligible normal stress is the x component of normal stress. In the present situation this stress is averaged over both solid and fluid phases. Thus %c = Ox - ~bpf
(2.1)
where Ox is the partial stress acting on the solid skeleton, pf is the pore pressure in the fluid, and ~b is the porosity, i.e. the pore volume fraction. The internal axial force and moment shown in Fig. 2.1 are the resultants of the normal stress, and are thus given by
10
Ch. 2
N- i
!
Modeling o f Poroelastic B e a m s
t
Fig. 2.1. Beam subjected to distributed normal and distributed axial loads, showing also the internal forces. The inertia moment of the cross-sectionwith respect to the y-axis is I and the mass per unit length of the beam is p.
N - [ z "rx dA,
M -- ~A "rxZ d A
(2.2a, b)
The equilibrium equations for the small deflection problem of an elastic beam are still valid for the poroelastic beam; these are ON 0x --
32u qs + P 0t--T
oZM
OZw
o~X2
- - qn + P O t 2
OM
0x
(2.3a)
(2.3b)
-- Q
(2.3c)
where u and w are the axial and transverse displacements, respectively, of the beam axis (at z = 0) and Q is the shear force. It has been assumed in writing the last term on the fight of Eq. (2.3a) that the acceleration of the fluid relative to the solid skeleton is small. This is justified from a practical point of view by the low permeability to be expected in such materials. As a result no inertia terms reflecting the motion of the fluid with respect to the solid are included. The rotational inertia (of both the fluid and the solid) may also be neglected. The kinematic assumption above permits the writing of the following relationships for the horizontal displacement and strain at any point in the solid skeleton of the material: u s -- u - ZW, x,
~~ -- U,x - ZW, xx
(2.4a, b)
where the subscript comma denotes x-direction derivatives, and W,x is the rotation of the cross-section. The constitutive equations for a transversely isotropic poroelastic material, as given by Biot (1962) and using his notation, are
Basic Equations
ry
=
rz
11
B3
B2 + 2B1
82
4
B3
B2
B 2 + 2B 1
8~
+
B6 ~
(2.5a)
B6 (2.5b)
p f - B7gSx -t- B6( 4 -+- 8~) -+- B8~
Here ~" is the increment of fluid content in the pores (or pore volume change); it is the porosity times the difference in traces of the strain in the solid and in the fluid, i.e. ~" = 4~(es - e e)
(2.6)
where es-
eSx + @ + gsz
and
~f-
efx + eyf + ,s'z f
(2.7a, b)
The superscripts s and f are used to refer to the solid and fluid phases, respectively. There are seven independent material constants in the constitutive equations; due to the third assumption of classical beam theory, those involving shear stress need not be and hence are not included. Eqs. (2.5) are rewritten in the form "1"x
'ry
I Cll -- C12
C12 C22
C12
C23
"1z
pf-
<21
C23] Sy _ C22 ~sz
(2.8a)
(2.8b)
ol 1 e~x - a2( 4 + e~)]
F[~-
a2 pf a2
Here the Cij are the stiffness coefficients of the solid skeleton (drained solid) alone; o/i and F depend on the elastic properties of the solid skeletal material and on the compressibility of the fluid, as well as on the microgeometry. As mentioned above, within the classical assumptions of beam theory ry = r z = 0, so that the constitutive law applicable to the transversely isotropic poroelastic beam is obtained as "rx - - E o ~
-
rlp f ,
~ -
rl eSx
+
jSp f
(2.9a, b)
where E = Cll --
T] = OL1 -1
/3 --
F
2C22 C22 -+- C23 2C12 Og2 C22 -k- C23
(2.10a)
(2.10b)
2~ 2
+
C22 -+- C23
(2.10c)
The physical meaning of the three constants in Eqs. (2.9) is as follows. Consider the case where pf -- 0, which implies that the material under consideration is drained. Thus it
Ch. 2
12
Modeling of Poroelastic Beams
is immediately recognized that E is the axial Young's modulus of the solid skeleton (drained). r/ is the constant by which eSx must be multiplied in order to find the change in pore volume for the drained material stressed in the x direction. It is not at all clear that ~7 must be positive, but it is to be expected that for practical materials 77 will be positive. Finally,/3 is the ratio of pore fluid increment to pore pressure when there is no axial strain. Clearly,/3 must be positive. For convenience later, instead of using/3, a new parameter is introduced A --
(2.11)
E/3
This parameter can also be interpreted as follows. Consider a sample of the material for which the fluid is trapped, i.e. the material is 'jacketed'. Then ~"= 0, and the following expression can be derived from Eqs. (2.9): ~'x = (1 + Arl)Ee s,
sr = 0
(2.12)
In other words, it is noticed that (1 + Ar/)E is the Young' s modulus of the bulk material when the fluid is trapped. This relationship is then valid in relating stress to strain immediately after the application of suddenly applied loads. By Eq. (2.1 1), A and 7/must have the same sign, so, as expected, (1 + Ar/)E > E. It is further noted that E and 77 are independent of the only possible relevant mechanical property of the fluid, i.e. its compressibility, while A depends both on the mechanical properties and microstructure of the skeleton, as well as on the fluid compressibility. When ~"= 0 we further record, for later use, another relationship which is valid, A P f - - -- ~ ~'x, I+A~/
if=0
(2.13)
Now, using Eqs. (2.4b) and (2.9a), the internal forces as defined in Eqs. (2.2) can be written as
N = EAu, x + tINp,
M = -EIw, xx + rlMp
(2.14a, b)
Mp-- -- I Apfz dA
(2.1 5a, b)
where Np= -;
A
pfdA,
are the pore pressure resultant and moment resultant over the cross section, respectively. Eqs. (2.14) enable one to express the equilibrium equations, (2.3a) and (2.3b), as
02U ONp 02u EA-~x2 + rl---~x + qs- P--~ = 0
04w
o~Mp
o~w
EZ ~x 4 -- rl Ox2 + qn + P - - ~ - - 0
(2.16a)
(2.16b)
The above partial differential equations, (2.14) and (2.16), are not sufficient for determining the six unknown variables, u, w, Np, Mp, N and M, and relations involving fluid flow must be added. Thus the generalized Darcy law which has been accepted as the
Basic Equations
13
benchmark description of fluid motion in completely saturated porous materials (Biot, 1962; Cleary, 1977; Burridge and Keller, 1981; Kingsbury, 1984; Kumpel, 1991; Detourney and Cheng, 1993) is introduced
[k]
= - ~gradpe /zf
(2.17)
where w = (h(u f - uS), and u e and u s are the displacement vectors of fluid and solid material particles, respectively. The dot denotes derivative with respect to time. Here /zf is the fluid viscosity, and [k] is the permeability matrix with components kij. Since fluid flow is taken to be possible in the axial direction only, its only non-zero component is kll, which, by noting that ~"= - d i v w, yields
~ _ kll oq2pf /d,f o~X2
(2.18)
Then by using Eqs. (2.9b) and (2.4b) one obtains
K o2pf
O2w
- - ]gf - -
Oft
(2.19)
hEz--ff~-x2 + hg-~xx
where k,, K - - /.Lf13
(_ kllhE)
(2.20)
/ZfT]
In order to express the fluid pressure in terms of the global quantities Np and Mp, Eq. (2.19) is first integrated over the cross section, and is then multiplied by z and integrated over the same area. Noting that w and u are not dependent on the integral variable, these yield
32Np
-K
hE
-- -iVp3x 2
--
oq2fi; f j
~
a2a//p - K
e2w --
ax 2
-
l~p
-
AEI
Oft
A
zdA + A E A ~
3X
aa f
+ hE ~
~
ax
j
A
z dA
(2.21a)
(2.21
b)
where it has been assumed that the order of operations of integration over the area and derivatives with respect to time and axial coordinate can be interchanged. Note that the above two equations are coupled. In order to uncouple the response to axial loads from the response to transverse loads, the y-axis is chosen to be a centroidal axis of the cross section. Thus the integrals above are zero and Eqs. (2.21) are reduced to
K
o72Np
aft
-]~[p 3x 2
q- A E A ~ - - 0
a2a4p K ~X2
(2.22a)
3x
a2w -
](/lp -
hEI
=
0
(2.22b)
Now the boundary and initial conditions are considered. The boundary conditions on displacement for the axial problem are that u is given at certain points on the beam,
Ch. 2
14
Modeling of Poroelastic Beams
possibly as a function of time. For the bending problem the displacement w and/or the (rotation) angle 0 may be specified, i.e.
Ule= /i,
Wle-- 1~',
0[e=
0
(2.23a, b, c)
where the subscript e refers to a given point, and an overbar refers to a given function. As mentioned above, for the linear theory used here, 0 - W,x. The mechanical boundary conditions for the axial problem are that N is given at certain points, and for the bending problem M and/or Q are specified
Nle--/V,
Mle--/IS-'/,
Qle-
0
(2.24a, b, c)
The diffusion boundary condition for a permeable end surface is that Pf[e--/3f, while for an impermeable end surface the boundary condition is Pf,x[e- 0. In terms of pore pressure resultants, these are
Np[e= ]Vp,
Mp[e--/~ p
(2.25a, b)
for a permeable boundary, and
ONe[3xe--0,
(2.26a, b)
3MPl3xe- - 0
for an impermeable boundary. The initial conditions can be determined by considering separately the case of suddenly applied loads, i.e. a jump at t = 0-. Then at t = 0, we have ~"= 0 and the relationships (2.12) and (2.13) apply. Eq. (2.12) reveals immediately that the instantaneous response of the beam is that for an elastic beam of Young's modulus (1 + h r/)E. Then u(x, 0) and w(x, 0) are calculated and N(x, 0) and M(x, 0) can be found. (If the problem is statically determinate these would be known a priori.) Then integrating Eq. (2.13) over the area, and also after first multiplying by z, we have h Np -- h-------~ 1+ N,
h Mp = 1 + h~ M,
t -- 0
(2.27a, b)
These can be given in terms of the displacements as
Ou Np -- A E A - -
O~X'
OZw Mp -- - A E I ~
O~X2'
t -- 0
(2.28a, b)
Thus the pore pressure resultant and moment resultant at t = 0 can be determined. Hence all necessary equations for the problem have been found. In summary, for the axial part of the problem the system of Eqs. (2.16a) and (2.22a), having two unknowns u and Np, must be considered. For the bending part of the problem the system of equations, (2.16b) and (2.22b), having the two unknowns w and Mp, are to be considered. Further, Eq. (2.14a), which additionally contains N, or Eq. (2.14b), which contains M, can be included in the above problems, respectively, if this is required by the type of boundary conditions in the problem considered. Finally, it is noted that if h and r/are zero (if h = 0 then r/-- 0, and vice versa), the corresponding equations will degenerate to those for an elastic beam, or a drained poroelastic beam.
Characteristic Times
15
2.2. Characteristic Times Before the equations governing the problem are systematically solved, it is important to consider the nature of the behavior, with respect to time, for certain combinations of the physical and geometrical parameters of the beam. Consider a segment of the beam of length L, onto which is suddenly imposed an axial displacement and thus an initial pore pressure is produced. Now, suppose the segment is restrained in order to maintain the displacement unchanged (i.e. ti - 0), while permitting diffusion thereafter through both ends of the beam (at x -- 0 and x = L). For such a case, it is found from Eq. (2.22a), with the last term on the left taken as zero, that the pore pressure resultant of the segment is given by Np -- ~ . b~sinn~rx exp -
n=l
L
LZ/K
t
(2.29)
where the Fourier coefficients bn are determined by the initial pore pressure resultant,
Np(x, 0). Hence L2 ~'D = -~-
(2.30)
is a measure of the diffusion time of a beam of length L. This clearly applies also to the situation of bending, i.e. a suddenly applied curvature since from Eq. (2.22b) it is immediately seen that Eq. (2.29) will be valid with Np replaced by Mp. Next, consider a drained beam (i.e. rt = 0) of length L. Taking qn to be zero, Eq. (2.16b) gives the governing equation for the free vibration of the drained beam which is elastic
04w
o~w
E1 ~x4 + p - ~
--0
(2.31)
If the beam is simply supported and no moments are applied at the ends, the deflection can be given in the form oo
w = ~" c~sin ~n'nx sin(ant +/3~) L n=l
(2.32)
where a~ = (nTr)2x/EI/pL 4 are then the natural (circular) frequencies of the drained beam. The period of the first bending mode is (2/Tr)~/pL4/EI. Thus the characteristic time for the drained beam is introduced as L4
~'s --
(2.33)
E1
which is the period of the first bending mode times 7r/2. If the fluid is trapped, this must be
pL4 TT-
(1 + Art)E1
(2.34)
16
Ch. 2 Modeling of Poroelastic Beams
For later use, a non-dimensional parameter is defined as follows:
y _ K _E_i~/P
(_
rs)rD
(2.35)
It should be further pointed out that the actual time required for completing the diffusion produced by an initial pore pressure in a beam not completely restrained is dependent on At/ as well as on 7D. For example, if a simply supported beam with permeable ends is subjected to a sinusoidally varying load qn = sin(rrx/L), which is suddenly applied, the deflection and pore pressure moment resultant prescribed by Eqs. (2.16b) and (2.22b) and the related boundary and initial conditions are found to be W-
-- 77.4---~
and
1 --
1 q- aT]
qnL2(
Mp-
exp(
rr2(1 + At/) exp - ( 1 +
(1 qt-
I~.'o)L2/Kt
ayl)t2/g t
/]
(2.36)
(2.37)
Thus another characteristic time is recorded rf =
(1 +
Ar/)L2 K
(2.38)
Consequently, three time scales are required to define the poroelastic beam model: rD, the diffusion time when the beam is restrained from further deformation; rf, the diffusion time when the beam is free to deform after a sinusoidal load is suddenly applied; and rf, the characteristic time when the fluid is trapped. On the other hand, rs is a characteristic time for an elastic beam (the drained beam). In practice, however, any three of the four time scales can be taken to describe the system, in which three independent quantities, (1 + At/), LZ/K and pL4/EI, are involved in measuring time.
2.3. Note on a B e a m I m p e r m e a b l e at Both Ends
As is seen from Eqs. (2.27), at t = 0, Np or Mp can be expressed in terms of N or M in a simple way. This of course does not apply to the situation when t > 0. In general, the relationships between Mp and M (Np and N) are prescribed through the differential equations involving the displacements. However, certain relationships can be found in an obvious manner if the beam is impermeable at both ends; these are derived below. According to Eqs. (2.9), the pore fluid increment is given as -
E
a
Pf + rx
(2.39)
Now, when the beam is sealed at both ends and thus no fluid is lost (and since there is no fluid movement perpendicular to the beam axis) it must be that at any time ~ L ~'dx = 0 for any position on the cross section. Hence
Equations in N o n - D i m e n s i o n a l F o r m
f
v
CdV -- 0
f 1
and
.] v
17
~'z dV - 0
(2.40a, b)
Noting that the material parameters, E, A and r/, are at most x-dependent, then by substituting Eq. (2.39) into Eqs. (2.40), the following integral relations are found: LE
A
Np-N
dx-0
(2.41a)
LE
A
Mp - M d x - 0
)
(2.41b)
at any t. When t approaches infinity, both Np and Mp will approach constants and thus Np --
~ L[ ~TU/E] dx fL[( 1 + Ar/)r//AE] dx'
t -- oo
(2.42a)
Mp --
f L[rlM/E] dx IL[( 1 + Ar/)~'ffAE] d x '
t -- co
(2.425)
and when the material parameters are constant along the beam length, the above equation are reduced to
Up-
A 1 f N dx, 1 +Ar/L L
A lfMdx Mp-- 1 + A ~ / L L
t - co
'
(2.43a)
t = co
(2.43b)
2.4. Equations in Non-Dimensional Form For the sake of convenience for use in the following chapters, in which solutions will be sought, the relevant equations are presented here in their non-dimensional forms. The quantities involved are normalized as follows: x*--
,_ w
x L ' w L'
t* .
Kt L2,
, Np--
.
Np EA'
, qs
qs .L EA ' N*--
., qn
N EA'
qn L3
E1 ' , Mp--
MpL E1 '
,
u
u
L ' M*
(2.44)
ML -- E1
It is required that EA, E1 and K are constant in order that it is meaningful to do this. Then the partial differential equations, (2.14), (2.16) and (2.22), take the forms given below: N*--
O~U* Ox*
+ ~?Np
'
092W*
M*= - ~
Ox.2
which represent the constitutive law,
+ r/Mp
(2.45a, b)
18
Ch. 2
On2/,/* 0X .2
o~g;
,
o~X*
Oq4W*
o M;
O~X.4
0X,2
..
072U*
+ qs - ~
+ rl~
Modeling of Poroelastic Beams
. +qn
= 0
Ot*2
+ T2
(2.46a)
2w* Ot*2 = 0
(2.46b)
which refer to the equilibrium of the beam, where the dimensionless constant Yu is defined as Yu-
-s
(2.47)
EA
y is given in Eq. (2.35), and
02N -
0X.2
Ot*
~02 +A _u~
Ox* Ot*
_0,
0X.2
-
~ 3a -A _w~ =0 Ot* 0X.20t*
(2.48a, b)
The dimensionless forms of the boundary conditions, Eqs. (2.23)-(2.26), and the initial conditions, Eqs. (2.27), remain unchanged. However, the initial conditions, Eqs. (2.28), must be rewritten as follows: Ou* Np = A Ox---7 ,
OZw * Mp = - A ~
t* -- 0
09X,2 "
(2.49a, b)
The relations for the sealed beam are converted next. It is supposed that E is constant. Then Eqs. (2.41) takes the following form for 0 -< t* -< c~:
0
(2.50b)
and Eqs. (2.42) can be written as ,
,
Np =
i01[(1 + Ar/)~//A]dx*'
t -- oo
(2.51a)
, Mp =
f~ ~TM*dx* i l [ ( 1 + Ar/)r//A]dx*'
, t - e~
(2.51b)
Finally, by Eqs. (2.43) one obtains
=14
Mp =
An A
I+A~
dx ,
1M* dx*, o
t - oo
t = c~
which requires that A and 77 as well as E are constant.
(2.52a)
(2.52b)
Chapter 3 ANALYTICAL SOLUTIONS FOR QUASI-STATIC BEAMS
In this chapter we consider quasi-static solutions for the case of bending only, i.e. qs = 0 and with inertia neglected. All quantities are considered in their non-dimensional forms except when otherwise noted. However, for the sake of convenience we omit the superscript * and also the subscript n on the loading function qn. Further, 'pore pressure' is used to refer to Mp when no confusion is thus produced.
3.1. Simply Supported Beams with Permeable Ends Consider a beam subjected to the load q(x,t) only. At t < 0- the beam is unloaded. For the problem considered, the boundary conditions are then (based on the section title specification)
w(O,t) = w(1, t) = 0
(3.1a)
W,xx(O, t) = W, xx(1, t) = 0
(3.1b)
Mp(O,t) = Mp(1, t) = O
(3.1c)
where the second equation follows from Eq. (2.45b) and the third from Eq. (2.27b). The initial conditions are
w(x,O) -- wO(x),
Mp(x, O) = M~
(3.2a, b)
If there is no jump in q(x, t) at t = 0- then the right-hand sides above are zero; if there is a jump then w~ is taken from the elastic response with Young' s modulus taken as (1 + Ar/)E, and M~p is found from Eq. (2.49b). The assumed solutions are of the form oo
w-
E Wn(t)sin(n~rx), n=l
0o
Mp = E mn(t)sin(n~rx)
(3.3a, b)
n=l
which satisfy the boundary conditions (3.1). The functions % ( 0 and mn(t) are to be determined. The load q(x, t) is also expanded in a Fourier sine series in x (assuming this
20
Ch. 3
Analytical Solutions for Quasi-Static Beams
to be possible) q(x, t) = Z bn(t)sin(n~rx),
n=l
bn(t) = 2
(3.4a, b)
q(x,t)sin(nTrx) dx
Then, by substituting Eqs. (3.3) and (3.4a) into (2.46b) and (2.48b), the following pair of equations for determining the tOn(t) and mn(t) are obtained (the inertia is zero): rhn(t) + n 2 ~ m n ( t ) - An2~ ~bn(t) = 0
(3.5a)
rln27r2mn(t) + n47r4tOn(t) + b n ( t ) - - 0
(3.5b)
The initial conditions that (3.5) are required to fulfill are obtained by inverting (3.3) and (3.2) for t -- 0, i.e.
tOn(O)-2~iw~ mn(O ) --
(3.6a)
fl
2 J0 M~p(X)sin(n~x)dx
(3.6b)
The system (3.5) together with (3.6) constitutes an initial value problem for each n; the complete solution is then mn(t)=
[
A
mn(O)- (l+An)n 2 ~
ft bn(tt)exp( n27r2 ) ) dt tt] o (1 ~- A - r / ) e x p
('
n27r2 1 ~ At/t (3.7a)
tOn(t) --
bn(t) rl n4 7r4 -- n277.2mn(t )
(3.7b)
3.2. Beams Subjected to Loads Suddenly Applied and Constant Thereafter
We will limit ourselves to cases when M is not time dependent for t > 0, as is obviously the case for statically determinate beams. We have already pointed out that at t -- 0 the problem is solved by finding the solution to the elastic problem with Young's modulus taken as (1 + Art)E. Consider now the long time solution (t ~ co). Fluid motion in the axial direction ceases and derivatives with respect to time approach zero. pf, and by its definition also Mp, necessarily become independent of position along the length of the beam. If at least one of the beam-ends is permeable Mp approaches zero and the equations simply reduce to the elastic beam equations with Young's modulus being that of the drained beam, E. So the long time deflection is easily found. If neither end is permeable then the pore pressure will become constant, and can be determined by Eq. (2.52b), oo
MP = 1 - / A n
0
M(x, co) dx
Beams Subjected to Loads Suddenly Applied and Constant Thereafter
21
Even if both beam-ends are impermeable, for the special condition that the integral above is zero, i.e. ~~0M(x, oo) dx = 0
(3.8)
there is no long time pressure in the pores and the elastic solution with Young's modulus taken under the drained conditions is the solution for long times. For all cases when Mp -0 we can generally find series solutions for the transition behavior between the initial response and the long time response. Let the solution for deflection be written as
w(x, t) = wE(x) + Aw(x, t)
(3.9)
where w E is the elastic solution of the drained beam. Then M - - w E , XX " Substituting this relation into (2.45b) and rewriting (2.48b) yields
02Aw r/Mp --
o3x2 ,
~Mp _ ~tp + A 02~ * o3x2
( 3 . 1 0 a , b)
o~x~
By separating variables, a general solution is obtained in the form
( 1
Aw = [cl + c2x + c3cos(cox) + c4sin(cox)]exp -
Arlt
)
(3.11)
Thus, if the constants ci can be determined so that the boundary and initial conditions are satisfied, the solution is found. In the following, Aw will be determined for various cases. Aw will take the form of sums of solutions of the type given in (3.11) in order to satisfy the initial conditions of a particular problem. This is justified since the partial differential equation for Aw is homogenous. Since w E necessarily satisfies the boundary conditions required for w, the boundary conditions for Aw will always be homogeneous. The initial condition on Aw is simply the difference between w(x, 0) and w E, i.e. AwIt=0~--- -- A'-----~-~wE
(3.12)
I+A~
A simply supported beam with both ends permeable The boundary conditions for this case are Awls-0-- Awl~=l- Aw, x~lx=O-- A W ,
xxlx=l = 0
The last two conditions follow from Eq. (3.10a) by noting that Mp(0, t) = Mp(1, t) = 0. The various ci are obtained as cl = c2 = c3 = 0, so that
Co = nor
(n = 1,2, 3 .... ),
c4 non-zero
Ch. 3 Analytical Solutions for Quasi-Static Beams
22
Aw = Z
bnsin(nTrx)exp -
1 + hr I
n=l
t
(3.13)
is an acceptable form for the solution. Comparing this form at t = 0 to the initial condition on Aw we have co
bnsin(nerx)--
Z n--1
I~~wE
(3.14)
1 + At/
and the constants bn can be found by expanding wE into a Fourier sine series.
A simply supported beam with
one end
impermeable and
the other end
permeable
The boundary conditions here are
Awlx=o-AWlx=l-
03• ~
_ ~Xw[ x=0--
-0
~X-"----~x=l,
The third condition follows from Eq. (3.10a), when noting that Mp,~(0, t) = 0. Aw is found to be of the form
[
(2m 1
~ . bm - 1 + x + c o s ~ 7 r x
Aw-
m-1
exp -
2
4(1+A~/)
t
(3.15)
Thus, if the series below is uniformly convergent, we can write
OAw (x, O) -- mZ1b m 1 - 2 m -
Ox
=
2
1 zrsin
2m-1 2
\-I 7rxl[ )'_1
(3 16)
and therefore oo
Z
bm
OAw Ox (0, O)
--
(3.17)
m=l
For the determination of b m a n odd function ~O(x) with period 4 is introduced, which is defined in the interval [0,2] by ~x)
--
f - - ~OAw x ( x , 0 ) + OAw (0,0), 3x
~2-x),
if0<-x-<
1
(3.18)
if 1 ~ x _ 2
Now Eq. (3.16) can be rewritten and extended by Eqs. (3.17) and (3.18) as follows:
qKx)-- ~.bm 2 m - 1 m=l
T
2m- 1
7r sin ~
7rx
On the other hand, qJ(x) can be expanded into a Fourier sine series as
(3.19)
Beams Subjected to Loads Suddenly Applied and Constant Thereafter ~x)=
y.
23
n~
ansin --~--x
(3.20)
n=l
After operations on the integrals, the coefficients can be calculated in the interval [0,1] an =
o
~(x) sin
Z-
x dx
{2f x sin(2m 1 ) 0
----~Trx
dx,
0
if
n=2m -1
ifn=2m
(3.21)
(m=1,2,3 .... )
Therefore b m c a n be determined by comparing Eqs. (3.19) and (3.20) 2
bm ---
(2m - 1)Tr a m - -
8 OAw (2m - 1)2 77.2 0X (0, 0)
4
(2m - 1)Tr
floOAW(x,O) s i n ( 2 m - 1 ) Ox ~ 7rx dx
(3.22)
and after the initial condition (3.12) is introduced, we have 4At/ [ 2 dw E ~(0) bm-- (1 +Ar/)(2m - 1)Tr - (2m - 1)Tr dx
+
~
1 dw E
0 ---~--
sin
( 2m -
1
2 (3.23)
A simply supported beam with both ends impermeable The boundary conditions in this case are
~Xwlx=0=~Xwlx--1--
03Aw
~Aw I
Ox3 x=O
0
O~X3 x= 1
Then Aw will have the form Aw -- if'. b,, { - 1 + [ 1 - ( - 1)n]x + cos(nTrx) }exp ~=l
t
(3.24)
1 + A~
Thus, if the series below is uniformly convergent, then by applying the initial condition (3.12), we have oo
Z
n2 77-2bnc~
77"x) --
n=l
A TI
d2w E
1 + At/ dx 2
(3.25)
The right-hand side above can be expanded into such a series if its integral over the interval [0,1] is zero. This will be so if dw E
dw E
- - ~ (0) - ~
(1) -- 0
(3.26)
Ch. 3 Analytical Solutions for Quasi-Static Beams
24
TTTTTTTTTTTTTTT q=l
z///////~
(1)
~ A
z///////~ I_.1"
0.5
i~.
~//////~
P-1
I "~ ~ p-I-~.
0.5
o
z///////g "~_1 F- I
(2) q=20
TTTTTTTT 0.5
,,..<
0.5
>
(3) Fig. 3.1. Load cases for simply supportedbeams. which in turn is true if the condition (3.8) is fulfilled. One obvious instance when the condition is fulfilled is if the load distribution q(x), and hence M, is antisymmetric with respect to x -- 0.5. Then the coefficients are determined by expanding the right-hand side of Eq. (3.25) into a Fourier series _ 2A~7 fl0 d2wE bn -- (1 + Ar/)n 2 ~ dx 2 cos(nTrx) dx
(3.27)
We now consider several examples of the above three cases. In order to illustrate the unique aspects of the time dependent behaviors possible for poroelastic simply supported beams exhibiting axial diffusion, the numerical calculations for the loadings shown in Fig. 3.1 were performed using the series solutions developed above. All beams are of unit length, and all quantities have been normalized as per Eqs. (2.44). The diffusion boundary conditions (DBCs) in these figures are labeled as follows: (1) permeable at both ends, P-P; (2) impermeable at x - 0 and permeable at x -- 1, I-P; (3) impermeable at both ends, I-I. For all cases A = ~ / - 1 was chosen. The important role of the diffusion in determining the deformation history of the beam is shown and discussed. A comparison with the timedependent response of beams whose material is modeled as viscoelastic is also given. Consider first the case of a simply supported beam with uniformly distributed or central load with permeable/permeable and impermeable/permeable end conditions (Fig. 3.2).
Beams Subjected to Loads Suddenly Applied and Constant Thereafter -0.006
oLoad I / DBC 1 o Load I / DBC 2
~N. -0.010 7
25
\'~
"Load 2 ] DBC 1
~~"o...~
+ Load 2 / DBC2
-0 014
-0.018
-0.022
. . . .
I
. . . .
0.5
0.0
I
'
'
'
'
I
1.0
. . . .
1.5
2.0
t Fig. 3.2. Deflection vs. time for Loads 1 and 2 at x = 0.5 for DBC 1 (permeable/permeable) and DBC 2 (impermeable/permeable).
T h e c e n t r a l d e f l e c t i o n varies m o n o t o n o u s l y f r o m the i n s t a n t a n e o u s (t = 0) d e f l e c t i o n to the final d e f l e c t i o n (t = c~). T h e s e curves, at least q u a l i t a t i v e l y , c o u l d j u s t as w e l l h a v e b e e n t h o s e for a l i n e a r v i s c o e l a s t i c m a t e r i a l . N o w c o n s i d e r Fig. 3.3; the l o a d i n g is a n t i s y m m e t r i c . A n i n t e r e s t i n g p h e n o m e n o n is o b s e r v e d . T h e b e h a v i o r for D B C 2 ( i m p e r m e a b l e / p e r m e a b l e ) is u n i q u e l y different f r o m
0
0.016
n
o a f -
0.01z 7 / /
~
o DBc / ==0.75
l l / ~
o DBC 2 / z=0.75
lie--
~ DBC 3 / x=0.75 +DBC 2 / z=O 25 X(-l) "i
0.008
. . . . 0.0
i . . . . 0.5
i
1.0
'
,
,
1.5
t Fig. 3.3. Deflection vs. time for Load 3 at x = 0.75 for DBCs 1 (permeable/permeable), 2 (impermeable/permeable) and 3 (impermeable/impermeable) - which for DBCs 1 and 3 are identical to minus the deflections at x -- 0.25" and deflection times ( - 1) at x -- 0.25 for DBC 2.
Ch. 3 Analytical Solutions for Quasi-Static Beams
26
0.3 "i o DBC I / o DBC 2 / ADBC 3 / + DBC 2 /
!~ 0.2 I |
==0.75 ==0.75 ==0.75 ==0.25
0.I
0.0 -~
\
~
o
~
o
~
-0.1 0.0
O.5
1.0
1.5
t Fig. 3.4. Decay of pore pressure moment resultant for Load 3 at x = 0.75 for DBCs i, 2 and 3 - which for DBCs 1 and 3 are identical to minus the resultants at x -- 0.25; and pore pressure moment resultant times ( - l) at x = 0.25 for DBC 2.
that for the o t h e r t w o diffusion b o u n d a r y conditions. F o r the point x -- 0.25 there is a p e r i o d o f t i m e w h e n the deflection at s o m e points on the b e a m r e a c h e s a plateau. F o r the point x -- 0.75 there is an ' o v e r s h o o t ' - the m a x i m u m deflection, a p p r o x i m a t e l y at t - 0.3, is about 2 5 % g r e a t e r than for the long time solution w h e n the b e a m is d r a i n e d (t ---, c~). S u c h b e h a v i o r is i n c o n c e i v a b l e for a viscoelastic b e a m . A better u n d e r s t a n d i n g o f w h a t is h a p p e n i n g can be a c h i e v e d by c o n s i d e r i n g the pore pressure m o m e n t d e c a y for this l o a d i n g at the quarter and t h r e e - q u a r t e r points (Fig. 3.4). T h e pore p r e s s u r e m o m e n t
0.02 0
0.01 -
.
0.00
o DBC 1 o DBC 2 DBC 3
oj
-O.O1
txJ oJ
-0.02
'
0.00
'
'
'
I
'
0.25
'
'
'
I
'
0.50
'
'
'
I
'
0.75
'
'
'
1.00
Fig. 3.5. Deflection vs. position for Load 3 at time t -- 0.15 for DBCs 1, 2 and 3.
Beams Subjected to Loads Suddenly Applied and Constant Thereafter
27
0.00
-0.02
o DBC 1
-O.04
-0.06
-0.08
~---------o
'
'
0.00
'
'
i
. . . .
0.25
I
. . . .
0.50 X
I
.
.
.
0.75
.
1.00
Fig. 3.6. Pore pressure m o m e n t resultant vs. position for Load 2 at time t -- 0.15 for DBCs 1 and 2.
immediately after load application ( t - - 0 ) has no relationship to the diffusion boundary conditions. But pore pressure moment, as expected, decays much faster at the three-quarter point (close to the permeable boundary) than at the quarter point (close to the impermeable boundary). The implication is that the left half of the beam is effectively 'stiffer' than the fight half. Clearly under the present loading conditions if the left half is constrained to be relatively straight the deflection in the right half is increased. This is further clarified by Fig. 3.5. For a given time (t -- 0.15), we see that the deformation response of the fight half (x > 0.5) is greater than that of the left half. The influence of the diffusion boundary conditions on the structure behavior is demon0.14
d
0.07-
0.00~
-0.07
-0.14
. . . .
0.00
I
. . . .
0.25
I
. . . .
0.50
I
. . . .
0.75
1.00
Fig. 3.7. Pore pressure m o m e n t resultant vs. position for Load 3 at time t -- 0.15 for DBCs 1, 2 and 3.
Ch. 3
28
Analytical Solutions for Quasi-Static Beams
strated also in Figs. 3.6 and 3.7, where distributions of the pore pressure moment resultant vs. position along the beam are shown (again for t = 0.15). The large differences between the distributions are clearly seen.
A cantilever beam with impermeable fixed end and permeable free end The fixed end is at x = 0. The boundary conditions are
03aw
~W
awlx=0=-~x x=O
-
O~X3 x=O
~
I -o
tgX2 x=l
Then Aw will have the form
Aw -- Z bm - 1 + cos
(2ml
m--1
7i'X
exp(
2
1,2 t)
4(1 + AT/)
(3.28)
Considering the initial condition (3.12), we have 130
~. bm-- ~ A T / w E ( 1 )
(3.29)
1 + At/
m=l
A function qJ(x) is introduced as defined below in [0,2] and extended as an even function with period 4 elsewhere ~(x) =
f
AT] [wE(x) 1 + At/
wE(l)] '
- ~ ( 2 - x),
if 0 --<x --< 1
(3.30)
i f l --<x --<2
Substituting Eqs. (3.12) and (3.29) into (3.28) (which is valid for 0 <-- x --< 1) and introducing ~(x) yields t//(X) = m=l Z bmc~
()2m 1
(3.31)
TTi'X
which has been extended to be valid for - c o < x < co. On the other hand, q~(x) can be expanded into a Fourier cosine series with period 4. Denoting the coefficients by an ( n - - 0 , 1 , 2 .... ) w e h a v e
an -- I20 qt(x)cos -nTrx - ~ dx
=
f2f~x)cos( O,
and a0 = 0. Thus,
\ 2m - 1 7rx] dx, ] 2
if n - 2m - 1 (3.32) i f n = 2 m ( m = l , 2 , 3 .... )
Beams Subjected to Loads Suddenly Applied and Constant Thereafter
bm--2f10
~x)c~
2 m - 1 2 7rx) dx
[, l,m 1 +A'r/
29
(2m-1)Tr
WE ( 1 ) +
2
o
W E(x)cOS
~
1 ) ] 2
7rX (iX
(3.33)
A cantilever beam with both ends impermeable The boundary conditions for this case are
~Xwlx=o=
~aw
~W
03Awl
m
x=0
O~X3
x=0
-0
o~X3 x--1
Then as long as condition (3.8) is fulfilled, Aw can take the form Aw = ~ . bn[- 1 + cos(nTrx)]exp -
t
(3.34)
1 + A'q
n=l
If this series is uniformly convergent, then by applying the initial condition (3.12), we find Z n=l
bn --
"~TI
(3.35)
WE dx
1 + At/
Substituting the above and Eq. (3.12) into (3.34) and letting t be zero yields
Z bncos(nTrx) n=l
(
At/ -w E + 1 + A~
wE dx 0
)
(3.36)
Now the coefficients bn can certainly be found by Fourier series theory since the integral of the right-hand side above over the interval [0,1] equals zero for any wE,
bn -- 2 f l I +A'qA rI [ -- wE(x) + Io1 WE(x t) dxl ]cos(nTrx ) dx 2At/ floWE(X)COS(nTrx)dx l+A~7
(3.37)
The loading cases shown in Fig. 3.8 are now considered for cantilever beams. In Fig. 3.9 we see another interesting response. This is the case of a cantilever beam loaded at the midpoint by a concentrated load; the DBC is impermeable/permeable. At intermediate times (t = 0.2 is shown) the fight half of the beam is not straight (the deviation for the parameters used is small, but perceptible, in the figure). This is due to the fact that even though no loads are imposed on this portion of the beam, the pore pressure moment resultant is not zero until the diffusion process has been completed. It is interesting to note that the implications of the present material model are that in the case of
impermeable~impermeable end conditions even at infinite time an unloaded portion of the beam need not remain straight. Consider again the cantilever beam loaded by a concentrated load at the midpoint. From Eq. (2.52b), we readily find that Mp -- 1/16. Referring to
Ch. 3 Analytical Solutions for Quasi-Static Beams
30
q-1
(4) p-1
84
(5) P-1
L
i-~.
05
._1_
,.I
I - i-~.
F- I
(6) Fig. 3.8. Load cases for cantilever beams. Eq. (2.45b) w e see that the free right h a l f o f the b e a m will h a v e a c u r v a t u r e ( d i m e n s i o n less) o f 1/16; this c o m p a r e s w i t h a c u r v a t u r e o f 7/16 at the fixed e n d (of o p p o s i t e sign). In Fig. 3.3 the ' o v e r s h o o t i n g ' o f the d e f l e c t i o n w a s p r e s e n t e d . It is n o w p o i n t e d out that a p p a r e n t l y a n o m a l o u s b e h a v i o r o f the p o r e p r e s s u r e d e c a y is also possible, as c a n be seen
0.000
-0.025
-0.050
3 )
-0.075
ot=0.z
\
I
i .... 0.75
1.00
,, t=2.0 . . . . . . . -0.100
.... 0.00
I .... 0.25
i .... 0.50
Fig. 3.9. Deflection vs. position for Load 6 at times t = 0, 0.2 and 2.0 for DBC 2.
Beams Subjected to Loads Suddenly Applied and Constant Thereafter
31
o Load 4 o Load 5
0.20 -] ~
0.15
0.10
0.00
. . . .
0.0
0.5
1.0
1.5
t Fig. 3.10. Decay of pore pressure moment resultant for Loads 4, 5 and 6 at x = 0.5 for DBC 2. in Fig. 3.10. This p h e n o m e n o n is similar to the so-called M a n d e l - C r y e r effect, which is considered and discussed in more detail in Chapter 4. The p h e n o m e n o n in our case can be explained with reference to Load 6 as follows. There is no pore pressure for the right half
0.25 i ~ o t=0
0.20~~ .
ot=0.04
~ ~
\
a t=0.20 + t=l.20
0.15
0.10
0.05
0.00
.
.
0.00
.
.
I
'
0.25
'
'
'
I
.
0.50
.
.
.
I
'
0.75
'
'
'
1.00
Fig. 3.11. Pore pressure moment resultant vs. position for Load 4 at times t -- 0, 0.04, 0.2 and 1.2 for DBC 2.
32
Ch. 3 Analytical Solutions for Quasi-Static Beams
(x > 0.5) immediately after load application, but there should be pore pressure for this segment during the diffusion process as the fluid is forced out from the wall to the open right end. Consider further the case of the uniformly loaded cantilever with impermeable/ permeable diffusion boundary conditions (see Fig. 3.11). Pore pressure moment variations along the uniformly loaded cantilever beam, impermeable/permeable, are shown. At t = 0 the distribution is simply that for M times the constant A/(1 + A~) (see Eq. (2.27b)). It is noted that there may be more than one instant of time when the pressure at a given point is the same. For example, this is true for Load 4 at about x = 0.45 at times t = 0.04 and t = 0.2.
Chapter 4 FINITE ELEMENT FORMULATION AND SOLUTIONS FOR QUASI-STATIC BEAMS
The finite element method is used for the fluid-saturated poroelastic beams modeled in Chapter 2, for the quasi-static case. Variational principles are developed first for this purpose. Two types of variables, the displacements and pore pressure resultants, are involved in the time-dependent functionals. The method of Lagrange multipliers is employed in order to include the flow equations (generalized Darcy's law) in the Euler-Lagrange equations of the functionals. A mixed finite element scheme is then presented based on one of the variational functionals obtained. Numerical solutions for both types of variables are found to coincide well with the previously found analytical solutions. Some interesting results are demonstrated which are not available from analytical methods.
4.1. Introduction
Analytical solutions were presented in Chapter 3 for the quasi-static beam problem. However, it is not possible to solve all cases analytically due to the complexity of the partial differential equations and the relevant boundary conditions governing the problem. For further investigations, a numerical scheme is required. Thus, in this chapter, the finite element method is applied to the present time-dependent problem. The (generalized) Galerkin method (see e.g. Washizu, 1982) is popularly used in the finite element approximations, which is a particular form of the weighted residual method. When applied to the poroelastic problem in the general Biot formulation, spurious oscillations may occur in the pore pressure results for early times, and the accuracy for the pore pressure is usually lower than that for the displacements; this has been surveyed by Murad and Loula (1992, 1994), where a post-processing technique was employed in order to improve the pore pressure results. Some investigators suggested also techniques of reduced integration. A finite element model for the poroelastic problem based on a variational principle will be established here. The variational functionals include convolutions following Sandhu and Wilson (1969). These functionals are analogous to the potential energy functional for a classical extremum principle in elasticity, except that modification terms are introduced by the method of Lagrange multipliers in order that the flow mechanism can be reflected in the functional. The multipliers are identified by the stationary
Ch. 4
34
Finite Element Formulation and Solutions for Quasi-Static Beams
condition of the functional itself so that the number of field variables is not increased due to use of the method. Two different variational functionals are developed which are found to be identical in terms of the Euler-Lagrange equations derived from the stationary conditions. Each represents a generalized variational principle with two types of field variables, the displacements and the pore pressure resultants. A finite element formulation is then obtained from one of the variational principles. Unlike Murad and Loula (1994), who define such a variational problem in three stages (i.e. for t -- 0, 0 < t < co and t = co) with different natures, from considerations of practicality only the middle stage is considered. The algebraic equations of the discrete system for the initial stage are obtained by considering the limit of the algebraic equations for the middle stage when the discrete forms of the initial conditions are introduced. It is not necessary to formulate the third stage since in practice the long time solution can be used in order to obtain the solution for infinite time. Full integration is adopted for numerical computation of the coefficient matrices and no post-processing on the results is performed. Numerical solution examples are presented to test the finite element model and to demonstrate some features of the present poroelastic beam model, which are not shown by the analytical solutions.
4.2. Variational Principles Before considering a finite element procedure for the quasi-static case, a variational functional for the problem is sought. The displacement boundary conditions (2.23) and the initial conditions (2.28) are considered as variational constraints, i.e., they are required to be satisfied in the functional in order to guarantee the validity of the corresponding variational principle. On the other hand, Eqs. (2.14) are considered as non-variational constraints. In other words, N and M are included in the functional only for writing convenience; they are dependent on u, w, Np and Mp as shown by Eqs. (2.14). Since the problem is time dependent and a classical potential functional does not exist, the convolution, whose operation is denoted by *, will be introduced into the functional. The definition and relations of the convolution used here are as follows: A convolution of two time-dependent functions A(t) and B(t), recorded as A 9 B, is defined here as t
A(t) * B(t) --
A(t - ~')B(T) dT
(4.1)
0
According to this definition, A 9 B -- B 9A. Further, it can be proved that (A * B) 9 C = A * (B * C). These will simply be written as A * B 9 C. When a derivative with respect to time is involved, the definitions are ,4,B--
B * A -
f t o d A ( t - ~ ' ) B ( ~ . ) d ~. d(t - ~-)
;
~
da(~') B(t -
~')
0
so that ii 9B ----B * A.
d(r)
d~"
(4.2)
Variational Principles
35
The relationship between A */) and A * B is derived as follows:
a 9 [~ = A 9 B + a(O)B(t) - a(t)B(O)
(4.3)
For the problem under consideration, a functional involving tension and bending is defined as f
U -- | [N * it x + Nu, x(X, O) - M * W, xx - Mw, xx(X, 0)]dx dL
(4.4)
where L refers to the domain a ~< x ~< b. The second and fourth terms in the integral are produced by the initial deformation. This functional is analogous to strain energy in the elastic problem. Note that U is actually U(t) ; here and in all further notation this implicit dependence is suppressed in the notation. The external forces include all the loads and boundary forces. These are involved in a second functional, analogous to potential energy, defined as
g -- 2 f L qn * fV dx -- 2 ~L qs * it dx - [2N * it]ba+ [21fl * w,x] ba
- [20 * W]ab -
~
Np * g p ,
x
-
~
Mp * Mp, x
(4.5)
of which the third to fifth terms are due to the prescribed forces at the boundaries (boundary loads). If no such force exists at the boundary, the corresponding term is taken to be zero. The last two terms refer to the case of permeable boundaries. If a boundary is impermeable, then Np,x = Mp,x = O. The stationary condition of U + V (i.e. 6(U + V ) = 0) will not produce the motion equations (2.22). To obtain these equations from the variation of the functional, the method of the Lagrange multipliers is used. The following items need to be added to the functional: "B'mP
-- f
L
Al (X,t) * (KNp,xx - lVp + AEAti, x )dx
+ f L A 2 ( x , t ) * (KMp,xx- l~/Ip- AEI1Jv,xx)dx
(4.6)
where the multipliers, A1 and A 2, are identified by the stationary condition of the functional 7r = U + V + 3"fmp (i.e. 6~ = 0) as
a l --
~q Np(x, t), AEA
a 2 - ~ Mp(x, t) AEI
If 6~r = 0, it is shown in Appendix A that the following equations are obtained: equilibrium equations (2.16) (after the inertia is neglected), motion equations (2.22), mechanical boundary conditions (2.24) and diffusion boundary conditions (2.25) and/or (2.26). Therefore, 7r is the required variational functional and its stationary condition refers to a generalized variational principle. Before the functional is used in a finite element formulation, some mathematical manipulations need to be fulfilled in order to achieve better numerical results. Integrating by
36
Ch. 4
Finite Element Formulation and Solutions f o r Quasi-Static Beams
parts with respect to those terms of Eq. (4.6) that contain the second derivative of Np or Mp, the variational functional then becomes 77 AEA Np * iVp -
+ (EIw, xx - 2"oMp) * W,9 xx _
"OK Npa * Npa + Nu x(X, O) AEA
,7 AEI Mp * Mp -
nKMp,x,Mp,
x
- M w , xx(X, 0) + 2q,, * w - 2q, * ti ]dx - [2/V */i]ab + [2/17/* W x]ab
- [ 2 0 * W]ab + ~
(Np -/'~/p) * Npoc -k- - ~
(Mp -- AT'/p)* Mpoc
(4.7)
In a similar way, a different variational functional could have been obtained * 1 ; [(N,u,x_M,w=)+ "n" - - - ~ L '
"o * Np * (KNp,xx - ]Wp q- hEAft, x ) 2hEA
"o * Mp* (KMp,xx- M p 2AEI
- [Q * w]ba - IN * U]ba-
2AEA
AEICv, xx)+ q n * W -
9tip 9Npa
-
qs* u]dx + [/~'/* W,x]ba
2AEI 9hT'/p 9Mp,x
(4.8)
which also corresponds to a variational principle (see Appendix A). It is found that the two variational principles are equivalent, namely they give the same Euler-Lagrange equations. We emphasize that either could be used as the basis for a finite element procedure; we choose to use the latter for reasons of computational convenience. The functional (4.7) has the dimension of 'energy', and thus its apparent physical meaning may be more immediate; the dimension of (4.8) is 'energy-time'. But the only justification that is really necessary for the choice of a functional is that it produces the required Euler-Lagrange equations. Applying integration by parts to those terms of Eq. (4.8) involving the second derivative of Np or Mp, gives 7r* -- 1
--2 L
N *ux - M *w
'
"oK * Mp,x * Mp,x hEl
, xx
-
AEA
"o *Np */Vp -k "o*Np *t~ AEA ,x
AEI "o * M p * M p - "o*Mp*w, xx + 2q. * w - 2qs * u]dx
+[l(/l * W'x]ba--[Q * W]ba--[~l" * U]ba+
+
* Np,x * Np, x -
2AEI * (Mp -/if/p) 9Mp,x
"OK 2AEA
,(Up-tip),Np]i (4.9)
37
Finite E l e m e n t F o r m u l a t i o n
If Mp and Np are required to satisfy the permeable boundary conditions (2.25), i.e. Eqs. (2.25) are considered as variational constraints, the last two terms in Eq. (4.9) vanish. Further, eliminating the derivatives with respect to time and using the initial condition (2.28) when necessary, a simpler form of 7r* is obtained: T o, -
~1
EAu x + 2~Np
L
rl -- AEA Np * Np
+
9 W,x]
* u x +
E I w , xx - 2 ~ M p
rlK A E I , Mp,x , Mp, x -
[Q 9
* w , xx -
AEA * Np, x * Np,x
] A "q E I Mp * Mp + 2qn * w - 2qs * u dx
(4.10)
9
Both Eqs. (4.7) and (4.10) are suitable for finite element formulations.
4.3. Finite Element Formulation
The finite element formulas will be derived based on the variational functional (4.10) in which only four field variables, u, w, Np and Mp, are involved. Since the boundaries for N and M are natural boundaries, i.e. the boundary conditions (2.24) are not variational constraints, in no case are N and M required to be determined simultaneously with the other four variables. Thus they will not be considered in the finite element computation. Let {Yu}, {fiN} and {tiM} represent respectively the nodal values for u, Np and Mp; {yw} denotes the nodal values of w and its derivative with respect to x, the rotation of the crosssection of the beam. The four unknown functions can now be calculated by the nodal parameters via the following interpolations: u = [Nu]{Yu},
w-
[Nw]{Yw},
Np---[Su]{[~U},
Mp
=
[SM]{[~M}
(4.11)
where {Yw} is arranged as
{~/~}- [w~, (W,x)l, w2, (W, x h . . . . ]T
(4.12)
[Nu], [Nw], [SN] and [SM] are interpolation matrices. The interpolation matrices to be used in the actual calculations are developed and displayed in Appendix B; these are for a 3-node element. Then those derivatives included in the variational functional can be calculated by the interpolation formulas (4.11) in the element domain
u,z = [Bu]{ Yu },
Np~ = [CN]{#N },
W,x -- [B0]{ Yw },
W, xz = [Bw]{ ~'w},
Mp,z = [CM]{#M }
(4.13)
where the matrices are defined as [Bu] -
[CN]
--
O[SN] Ox
O[N~]
O2[Nw]
O[N,] ax '
[Bw] -
,
[CM]
a~x-
--
O[SM] Ox
,
[Bo]
-
Ox (4.14)
Ch. 4 Finite Element Formulation and Solutions for Quasi-Static Beams
38
It is observed that all matrices in Eqs. (4.11) and (4.14) are independent of time, while the nodal parameters { Yu}, { Yw}, {fiN} and {tiM} are dependent on time. Thus the functional (4.10) in a matrix form for an element denoted by e is as follows:
1
' W e - --2 { ")/u
}T
* [k uu ] { ~u } + { ~ N
}T
1
* [k Nu ] { "Yu } + -~{Tw
1
--{/3M}T* [kMw]{'yw } - -~ * {[3N
}T
}T
* [kww ] { "Yw }
c 1 }T * [k~NN]{[~N} -- ~{[3 N * [kNN]{[~N}
1 }T c 1 T -- -- * { ~M * [kMM] { tiM } -{ tiM } * [kMM]{ tiM } -{~'u} T * {p.}
-
{~,w} T *
{pw}
(4.15)
where the matrices are given as
[kuu]--EA I [kww] -- E11
Le
Le
[kNu]= "o f
[Bw]T[Bw]dx,
[kMw] -- "o ~t e [SM]T[Bw]dx
c "oK f Le [CN]T[CN]dX, [k~NN]AEA
c
[k~MM]- ~"oK
[SN]T[Bu]dx
[Bu]T[Bu]dx,
f
Le
[kNN] =
[CM]T[CM]dX,
which are constant. The load vectors
Le
[SN]T[SN]dx
n 1"
AEA JL
[kMM]- ~
e
n f Le [SM]T[SM]~
{Pu} and {Pw} are
(4.16)
obtained by
{Pu} - fE e [gu]Tqsdx + ([Nu]Wlv)lba, (4.17)
{pw} - - f'~e [Nw]Tqndx + (--[B0]T/17/+ [Nw]T{))]: which are dependent on time if and only if the loads are dependent on time. Here Le refers to the domain of the element, i.e. [a,b]. The variation of "we with respect to the nodal parameters is given as {~'we = ~{~/u} T * ([kuu]{'Yu} + [kNu]T{j~N} -- {Pu})
+8{/3N} T 9 ([kN,]{~.} -[k~vN] * {/3N} -[kNN]{/3N})
--8{]3M}T * ([kMwl{'Yw} + [kCMM]* {tiM} + [kMM]{[3M})
(4.18)
Finite Element Formulation
39
Letting 67re = 0, the governing equations for the finite element method at the element level are obtained as [kuu]{ Tu } + [kNu] y { [~U } = {Pu } [kNu]{'Yu} -- [kuu]{[~N} -- [kCNN] • {/3N}
(4.19)
[kww] { ~w } - [kMw] T { I3M } = {Pw }
[~Mw] { Vw } + [~MM] {/3M } = -- [~hM] * {/3M } The matrix equations can then be extended to the global level by a standard superposition procedure used in a conventional finite element method. Using the capital letters to denote the corresponding global quantities, we write [K..] {Fu }
+ [gNu] T { BN
}
=
{ Pu }
[gNu]{ 1-'u } -- [KNN] { ON } -- [K~qN] * { ON }
[Kww] { Fw }
-
[KMw] T { BM
}
=
(4.20)
{ Pw }
[gMw] { I-'w } + [KMM] { BM } = -- [K~M] * { BM } These matrix equations govern the discrete structural system. The first and third equations of (4.20) represent equilibrium of the system and the others prescribe the fluid flow. If {BN} and {BM} a r e set to be zero, the first and third equations will decay to the corresponding equations for the elastic case. [Kuu] and [Kww] are the stiffness matrices for the elastic system, i.e. the poroelastic system when the pore fluid is drained. Eqs. (4.20) include the time integrals. Hence, in order to get the nodal unknowns, the time domain must also be discretized. Consider first the discrete forms of the initial conditions (2.28), and introduce them into the governing equations of the finite element method for time to = 0. Observing Eqs. (4.11) and (4.13), Eqs. (2.28) have the discrete form at the element level as follows:
[Sg] { fig } = AEA[Bu] { Tu },
[SM] { fiM } = - hEI[Bw] { Yw }
(4.21)
Let Eqs. (4.2 l) be evaluated at each node of the element, and noting the interpolation rules of the matrices [SN] and [SM], it can be written in a more practical form
{ ~3N } -- [Ou]{ ")/u},
{/3M } = -- [Ow] { "Yw}
(4.22)
where the element dependent matrices are
[Bu]l [ o , ] - AEA
[Bu]2 , 9
[Bw]l
[Ow]- AEI
[Bw]2
(4.23)
o
in which [Bu]l denotes that [Bu] is evaluated at the node l, and so on. Eqs. (4.22) can be extended to the global level in an obvious way, recorded as
{BN} = [O~]{/-'~},
{BM} = --[Ow]{I-'w}
(4.24)
Ch. 4
40
Finite Element Formulation and Solutions for Quasi-Static Beams
Further, after Eqs. (4.21) are applied and Eqs. (4.16) are considered, the first and the third equations of (4.19) take the form (1 + Art)[k~]{~,u} = {Pu},
(1 + Art)[kww]{~/w} = {Pw}
(4.25)
which can be simply extended to the global level as [K~
= {Pu},
[K~
= {Pw}
(4.26)
Thus the required equations for to have been found; they are Eqs. (4.24) and (4.26). The geometrical boundary conditions (2.23) must be considered when Eqs. (4.26) are solved. In all formulations the material parameters are considered as element dependent except when specifically noted. For the special case when Art = constant, [K~ and [K~ are as follows: [K~] = (1 + Art)[guu],
[g~
= (1 -t- Art)[gww ]
Then the equations in (4.26) are the same as the stiffness equations of the finite element method for an elastic case except that the factor (1 + Art) is now multiplied on the lefthand side of each equation. Hence the nodal displacements for to can be obtained from the corresponding elastic solutions if Art takes the same value for all elements. This conclusion was shown analytically in Chapter 2. Now, supposing that all the nodal parameters are known at t - tj_ 1; we need to determine the parameters at t - tj. Here the time interval N t - tj - tj-1 is taken to be small. For convenience, the following symbols are introduced: {r.lJ = {F.(tj)},
A J I F . } - {Fu} j - {r.} j-1 . . . .
The convolutions in Eq. (4.20) are now calculated in the time interval AJt by the trapezoid formula. Noting that [K~N] and [K~tM] are constant matrices, this gives
(
1
z~J([K~cN] * {BN} ) -- /kJt[K~cN ] {BN}J-1 + -~ AJ{BN} •
* {BM}) --
ZXit[KhM] {BMy-I+
)
1 AJ{BM} )
(4.27)
Finally, computing (4.20) over the time interval N t or simply applying the operator Aj to (4.20) and using (4.27), the discrete forms with respect to time are derived
[Kuu]Z~ { I-'u } + [KNu]Tz~ { BN } -- Z~ { pu } [KNulAJ {1-'u} -
( [KNN ] + -~1AJt[K~vN] ) AJ{BN}
-- z~Jt[K~cN]{BN} j-1 (4.28)
[Kww]Z~ {1-'w } - [KMwlTZ~ { BM } -- z~ { pw } [gMw] N{F~} +
[KMM] + -~2~t[K~UM] ZY{BMI - --ZYt[K~M]{BM} j-1
From these equations, the increments of the nodal parameters can be obtained, on the condition that the geometrical boundary conditions (2.23) and the permeable diffusion
Examples and Discussion
41
boundary conditions (2.25) are considered. The integrals are calculated by the trapezoid formula (being good enough to provide satisfactory accuracy for the b e a m problem). The above finite element formulas are based on the functional in Eq. (4.10). For the functional in Eq. (4.7), interpolations to the velocities of the displacements can be directly applied.
4.4. Examples and Discussion The finite element formulation has been programmed for computation of all types of boundary and loading conditions. The numerical results are carried out by using the 3node element, which is presented in Appendix B. The results obtained for certain combinations of boundary and loading conditions, for which the closed solutions are presented in Chapter 3, have been tested and no significant numerical differences have been found between the solutions extracted in the two different ways (one example is shown). Additional interesting results, for which the closed form solutions are not available, are presented here. For convenience all quantities appear in dimensionless form normalized as per Eqs. (2.44). Thus only two material parameters, A and r/, will be involved. For the sake of convenience, the superscript * is omitted. And further, 'pore pressure' refers to Mp or N o since these have the same distributions as for pore pressure in terms of x. For infinite time, Eqs. (2.51) determine the pore pressures for beams impermeable at both ends. As mentioned above, all numerical results presented in Chapter 3 can be obtained by the finite element scheme, without producing significant differences. The example of case (2) of Chapter 3 is presented in order to show the convergence and accuracy of the numerical method. The beam is simply supported with one end impermeable and the other permeable. The material parameters are A = r / = 1. A point load is applied at the center of the beam. For convenience of listing the results, the loading value is taken to be - 100 (downward). Some results are shown in Table 4.1. It is observed that the finite element solutions converge very quickly to the analytical solutions. It is also seen that the long time solution will converge to the corresponding elastic solution, as it should in this case. We emphasize that both w and Mp are of high accuracy. We will now apply the finite element scheme described above to investigate a number of
Table 4.1. Comparison of finite element solution with closed form solution for a simply supported beam which is subjected to a point load at the center and impermeable at one end and permeable at another end (results are shown for the loading point) Variable type
w Mp
t= 0
t = 0.01
t = 0.40
a
9 nodes
1.0417 12.500
1 . 0 9 5 0 1.0942 1 0 . 5 1 1 10.507
41 nodes
t = 4.00
Closed
9 nodes
41 nodes
Closed
1.0942 10.506
1.6498 3.6683
1.6497 3.6687
1.6497 2.0783 3 . 6 6 8 9 0.0427
a
a The closed form solution and the finite element solutions of 9 and 41 nodes (4 and 20 elements, respectively) are the same for t =- 0 and t - 4.00 as far as five digits is concerned. The solution for infinite time or the elastic solution for the observed point should be w -- 25/12 ~ 2.0833 and Mp -- 0.
42
Ch. 4
Finite Element Formulation and Solutions for Quasi-Static Beams
cases for which solutions are not feasible using analytical methods. Our purpose is to illustrate a number of very unusual effects, which are observed when poroelastic structural elements of the type considered respond to suddenly applied loads. Before presenting these behaviors, we point out that the classical instance of 'deviant' behavior in the mechanics of poroelastic media was apparently first reported by Mandel (1953). He considered an isotropic poroelastic specimen (see Fig. 4.1a), infinitely long in one direction and sandwiched between two rigid, frictionless plates, which is subjected to equal and opposite forces applied suddenly to the rigid plates. The lateral faces of the specimen are permeable to fluid. It turns out that the pore pressure at the center of the specimen, after attaining an initial value immediately on application of the load, continues
Fig. 4.1. The Mandel-Cryer effect.
Examples and Discussion
43
to rise to a peak before gradually decaying to zero (see Fig. 4.1b). This non-monotonic pore pressure response to a step loading has been called the Mandel-Cryer effect. It is easy to understand the reason for the non-monotonic pressure response; as the fluid drains out of the lateral edges of the specimen, a larger portion of the applied load is transferred towards the effectively stiffer (temporarily) central region of the specimen. Cryer (1963) observed a similar effect for a spherical specimen under hydrostatic pressure. The Mandel-Cryer effect is in contrast to the response, which would be obtained with the older models of fluid saturated media (Terzaghi, 1943). A more general version of Mandel's problem, in which the material is transversely isotropic, was examined by Abousleiman et al. (1996). For light poroelastic structures (beams) a Mandel-Cryer type behavior was first observed by Li et al. (1995). In Chapter 3 we presented some of these results (see Figs. 3.3 and 3.10). Because of the limitations of analytical procedures only relatively simple cases could be considered. We can now investigate more general situations, using the finite element code developed above, and we will find a panoply of unusual behavior patterns. Consider first the case of a cantilever beam subjected to a uniformly distributed load. Fig. 4.2 shows the variation of pore pressure with time at the quarter, half, and threequarter points along the beam. All possible combinations of diffusion boundary conditions are considered, i.e. permeable both at the wall and at the free end (P-P), permeable at the 0.16
_] ~ ~" ~176 I~
o.oo
0.16
a) P - P
0
ox=o.z5 I o,=O.SO I
o.oe
o.oo
0.5
1
0.16
b) P - I
-
0
0.5
1
0
0.5 t
1
0.16 m
~:~' 0.08
0.00
0.08
0
0.5 t
1
0.00
Fig. 4.2. Variation of pore pressure with time at the quarter, half, and three-quarterpoints along a cantileverbeam subjected to a uniformly distributed load, for the various diffusion boundary conditions.
44
Ch. 4
Finite Element Formulation and Solutions for Quasi-Static Beams
wall and impermeable at the free end (P-I), impermeable at the wall and permeable at the free end (I-P), and both ends impermeable (I-I). For most positions and diffusion boundary conditions the pore pressure, after taking on its initial value immediately after application of the load (Eq. 2.27b), rises to some maximum value before decaying to zero. This is the pattern of pore pressure response at the center of the specimen in Mandel's problem. For the I-I case the effect is discernible, but only barely so, for the quarter position. Of course, in this case the long time pore pressure is not zero; it approaches the value obtained from Eq. (2.52b). Note that the pore pressure approaches its final value from above or from below depending on the position on the beam being examined. A physical explanation for a Mandel-Cryer type pore pressure pattern in the present problem is as follows. Consider the mid-point curve in Fig. 4.2a. At t = 0 pore pressure is higher to the left of the mid-point than to the right of it. Thus the pressure differential is forcing fluid towards the right, and the immediate response is a rise in pore pressure at the mid-point. This is seen in all four cases of diffusion boundary conditions; the initial rates of pore pressure rise are identical. However, in cases (a) and (b), for which the left end (higher initial pore pressure) is permeable, the high pressure is quickly dissipated and the rise is quickly halted. In case (c), on the other hand, where the left end is impermeable, no loss of fluid on the left is possible and the pressure differential continues driving the process for a longer time, until finally fluid loss on the right end dominates the process, and the pressure decays towards zero. In case (d), where both ends are impermeable, no fluid loss on the right is possible and an asymptote is quickly reached. A higher level of complexity, which the pore pressure-time behavior can reach, is shown in the next example. Fig. 4.3 presents such behavior for a beam fixed at both ends; again the loading is uniform, and all possible combinations of diffusion boundary conditions are considered. Let us examine the behavior at the three-quarter point. In all cases, the pore pressure initially falls from its value at t = 0; then in the three cases when at least one end is permeable there is a subsequent pressure rise before the pressure again decreases and is ultimately completely dissipated. But when both ends are impermeable the 'strangeness' exhibited above is not present. The pore pressure continues its initial steep descent and very quickly vanishes. It is not at all obvious that the long time pressure should be zero for a beam with two impermeable ends, but for the special case when both ends are fixed it is possible to prove that for any loading whatsoever the long time pressure must indeed go to zero. Consider Eq. (2.48b). Integrating on x from 0 to 1 gives c~t 0 M P + A - ~ x 2
dx-
Ox
Now, if the beam is impermeable at both ends, the right-hand side is zero, and the integral then equals a constant. But this constant must be zero, otherwise relationship (2.49b) would be contradicted. Thus
M. +
d
-O
Examples and Discussion
45
41~
/I
o
P-I I-a
\\
0.4
0.3
0.2
0.1
0.0
-0.1
'
0.0
'
I
0.3
'
'
I
0.6
'
'
I
0.9
'
'
1.2
Fig. 4.3. Variation of pore pressure with time at the three-quarter point of a beam fixed at both ends for all possible combinations of diffusion boundary conditions; the loading is uniform.
Further, if the slope of the beam is the same at both ends (as is the case for both ends fixed), then w,x(0) - w x(1) and we have
at all times! Now, since for long times Mp must necessarily approach a constant then it is clear that in this case it approaches zero. That the anomalous behavior observed above (non-monotonic) can extend also to the deflection of the beam is shown in our next example. Consider a simply supported beam with a suddenly applied concentrated moment at the mid-point. All four diffusion boundary conditions are considered. Fig. 4.4a shows mid-point deflection. When the diffusion boundary conditions are symmetric (P-P or I-I) and the loading is antisymmetric, the midpoint remains stationary. But when they differ (I-P or P-I) anomalous behavior is observed. Clearly at t = 0 and at t -- co the deflection must be zero; but at intermediate times there is a relatively quick rise to a maximum and then a decay to zero. Away from the mid-point even stranger behavior is observed. At x = 0.7, with the P-I diffusion condition, the deflection reaches a maximum, then drops, and then rises again to its asymptotic value. For the other asymmetric diffusion condition, I-P, the deflection behavior is always non-monotonic, but with only a single extremum. With symmetric diffusion boundary conditions, the deflection is always monotonic, approaching its asymptotic value
Ch. 4
46
Finite Element Formulation and Solutions for Quasi-Static Beams
50 ]
a) =:-(i15o
10.0".
" z x ~
oP-P oP-I al-P
+I-I
5.0-
2.5
b)
. . . . . . . .
0.0
0.00
0.75 t
1.50
0.0
. . . .
o.oo
|
x=O.70
oP-P oP-I AI-P § [
|
0.75 t
i
'
|"
1.50
5.0
2.5
"~ /
e) x=O.90
I
"P-P
oP-i AI-p
I i
0
. 0.00
0
~ 0.75 t
1 50
Fig. 4.4. Variation of deflection with time for a simply supported b e a m to which is suddenly applied a concentrated m o m e n t at the mid-point: (a) at the mid-point, x -- 0.5" (b) at x -- 0.7; (c) at x -- 0.9; all four diffusion boundary conditions are shown.
from below. At x = 0.9 the phenomenon of two local extrema does not exist, but otherwise the behavior is similar to that at x -- 0.7.
A cantilever beam, uniformly loaded at the left half only We illustrate again an irregularity first observed in Section 3.2, which is outside of the scope of phenomena observed for the elastic or viscoelastic counterpart. Consider a cantilever as in Fig. 4.2, except that now only the left half of the beam (nearer the wall) is loaded. For the elastic or viscoelastic case it is obvious that the unloaded portion of the beam remains straight (no curvature). This is not the case in the present instance. After the initial response, for which the unloaded portion indeed remains straight, curvature begins to appear in this portion due to the existence of pore pressure throughout the entire length of the beam (see Fig. 4.5). If at least one of the beam-ends is permeable, then the unloaded portion will finally be straight again since the pore pressure ultimately vanishes. In case (d), where both ends are sealed and the final value of the pore pressure is non-zero (Eq.
Examples and Discussion 0.000
:~
0.000
."
-0.009
-0.018
47
o t=o.3o , t=o.9o + t=3.00
"~
a) P-P 0
0.009
NN\ \\\
0.5
b) P-I .0.018
1
'
'
'
0
0.000
0.000
-0.009
-0.009
-
I
'
0.5
'
'
'
1
d I-I
-0.018
-0.018
0
'
0.5
X
1
I
0
0.5
1
X
Fig. 4.5. Deflectionvs. position for various times for a cantilever beam loaded by a uniformly distributed load on its left half (closer to fixed end); all four-diffusion boundary conditions are shown. 2.52b), some curvature remains for all times in the unloaded portion, and its sign is opposite in sense to that in most of the loaded portion. Thus the beam tip deflection is for all times less in this case (I-I) than for the instances when at least one end is permeable. Note also that cases (a) and (b), for which the clamped end is permeable, behave quite differently from case (c) for which the clamped end is impermeable. In (a) and (b), after the initial response, the deflection quickly approaches very nearly to its final value, while in (c) the change is more equally distributed in time.
A cantilever beam subjected to a unit deflection at the free end The above loadings might be considered analogous to examining the creep behavior of a viscoelastic structure. In the following case the analogy of the relaxation behavior is examined; i.e. what happens if a deflection is suddenly imposed at some point along the beam, which is then held constant. Consider an unloaded cantilever beam, at the free end of which a unit deflection is suddenly applied. The subsequent deflections along the beam for all diffusion boundary conditions are shown in Fig. 4.6. We see that the gross shape of the beam itself varies with time. This is qualitatively different from the behavior in the viscoelastic case; in that case the shape remains unchanged and only the internal stress is time-dependent.
48
Ch. 4
1.0
Finite Element Formulation and Solutions for Quasi-Static Beams
0.5
0.5
/ /
/ j"
0.0
t.O
T
, ~
0.0
'
'
f
/
'
'
I
ot=o
ot=o.o6 .,t=o.la I
0.5
'
'
+ t=2.00 '
9
1.0
// / /
,@
ot=o ot=o.oe o t=o.24
//j"
.~I"" 0.0 v . . . .
, ....
0.0
1.0
c) I - P
0.5
*
|=4.00
0.5
d) I-I
1.0
§
0.5
//)~
,I// /j"
0.0
b) P-
1.0
a) P-
ot=o o t=o.2 o ~ t=o.7o
, , ~ v , , , , ?tr4.~176 0.0
0.5
x
1.0
////
o t=o.lo , t=o. 6o + |=2.60
////
0.0
.~~" "I"
0.0
'
'
'
'
I
0.5
'
'
'
'
1.0
x
Fig. 4.6. Deflection vs. position for various times for an unloaded cantilever beam to whose free end a sudden unit deflection is applied and then held constant; all four diffusion boundary conditions are shown. The initial (t = 0) shapes for all cases are the same and are identical to the shape for the elastic and viscoelastic cases, and these are in turn identical to the long time (t = oo) shape for all cases for which at least one end is permeable. But the long time behavior for the I - I case is again deviant. Here at all points along the beam, the deflection increases monotonically with time. In fact for x > 0.85 the final deflection is even greater than unity, resulting in a truly surprising shape. Fig. 4.7a shows the deflection angle (slope), 0, vs. time at x = 0.5 (and a magnified version for the shorter times, 0 ~< t ~< 1.2) for the same loading case. Here again, nonmonotonic curves are obtained; these are in the complicated form of a double variation. W h e n the clamped end is permeable there is a reduction in the angle at short times, then an increase, followed by a slow reduction towards the initial value obtained at t = 0. The I - P case has the same tendency, but the angle change is more moderate. Thus a Mandellike behavior with respect to another (third) variable is noted. The I - I case seems to show the simplest behavior here, monotonic. But if the point x - 0.95 is considered, i.e. very close to the displaced end, then again a manifestation of the truly unusual behavior in this case is seen. The angle actually changes sign before finally reaching its asymptotic value. For points even closer to the displaced end the long time slope would be even more negative.
49
Examples and Discussion A column subjected to a uniformly distributed vertical load
All of the above examples involved bending. We now present one instance of the seemingly simpler situation of purely axial loading. Even here we find unexpected behavior patterns. Consider the axial deflection of a column subjected to a uniformly distributed vertical unit load (downward) suddenly applied, as would occur due to a uniform upward acceleration. In Fig. 4.8 the vertical deflection, u, vs. the vertical location, x, is shown for all diffusion boundary conditions, at several times. When the top (x = 1) is permeable (cases (a) and (c)) the axial deflection is monotonic both with respect to location and with respect to time; this is as expected. When both ends are permeable the long time downward deflection of 0.5 at the top is very very nearly achieved at t = 1. This is unlike the case when the column base is impermeable, case (c), for which the final deflection (essentially that shown for t = 1 in case (a)) is not achieved until much longer times. Now when the top end is impermeable (cases (b) and (d)), then while u is monotonic with respect to time it is not monotonic with respect to position. We see that in the upper 1.8
1.8
1.6
~:~
"
1.4
1.6
oP-P oP-I aI-P
1.4
1.2
1.2
1.0
1.0
, , , , [ , , , , l , , . . [ , . , ,
1.0
0.0
2.0
3.0
4.0
~
'
I
'
'
0.4
.....
\
\ /
/
/~"
l
0.8
1.6
'
'
'
1.2
,
1.2
1.2
~-/\ \ \
'
t
1.6
0.4"
'
0.0
t
a) x=0.50
0.8
A
A
0.8
oP-I *[-P
0.4
+ [-I
§
0.0
-0.4
0.0
b) x=0.95
1.0
2.0
t
3.0
4.0
-0.4 0.0
0.4
0.8
1.2
t
Fig. 4.7. Deflection angle vs. time for the conditions of Fig. 4.6: (a) at x -- 0.5" (b) at x = 0.95. All four diffusion boundary conditions are shown. The short time portions are also shown expanded.
50
Ch. 4
Finite Element Formulation and Solutions f o r Quasi-Static Beams 0.00
0.00
-0.25
-0.25 at=O o t=O.l 0
,t=o3o
"~,~ +~-~,,~
~ ~
+ t=-l.O0
-0.50
. . . . 0.0
0.00
i 0.5
.
.
.
-0.50
.
l.o
-
0.00
e)
,
,
|
/
!
,
1
|
0.5
1.0
i . . . . 0.5
1.0
""
I-P
-0.25
-o.25
-0.50
i
0.0
0.0
. . . .
i . . . . 0.5
1.0
X
-0.50
0.0
. . . .
X
Fig. 4.8. Axial deflection vs. position, at various times, for a column subjected to a uniformly distributed load, all four diffusion boundary conditions are shown.
portion of the column Ou/Ox may be positive; higher material points in the column deflect less than lower material points. Thus the skeletal material of the column is under tension in some regions. Moreover, when both ends are sealed, case (d), the immediate (at t = 0) axial deflection at the top of the column does not change at all with time. That this must be so is shown as follows. Eliminating Np from Eq. (2.41a) using (2.14a), we have for all time i~ --~ [N - (1 + A'rl)EAu, x]dx -- 0
(4.29)
Now, if the material parameters A, ~/and E are constant along the length of the column u(L, t) = u(O, t) + EA (1 + A ~)
N dx
(4.30)
However, u(0, t), the deflection at the base, is zero, and clearly the integral on the right is a constant in time; so u(L, t) - constant. In fact, since for any load distribution the integral on the right is independent of time, we can conclude that for any such loading the top of the column remains motionless.
Examples and Discussion
51
0.00 ."
O.OOI
-0.05
-0.05-
-0.10
-0.10-
-0.15
-0.15 -
o t=O
-0.20
0.25
-0.20 -
ot=0.26 " t=O.80 + t=2.00
a) , , , ,
,.... . 0.5
0.0
.
,
b)
,
1.0
Z
0.00
-0.25
i
0.0
|
| .....
|
'
|
0.5
"'l
!
i
|
1.0
Z
-0.05
-0.10
-0.15
-0.20
-0.25
9
0.0
|
=
=
I'"
0.5
'1
i
i
|
1.0
Fig. 4.9. Axial deflection vs. position, at various times, for a column subjected to a uniformly distributed load, for I-I diffusion boundary conditions. (a) h = 6 throughout; (b) h = 1 in the upper half and h = 6 in the lower half; (c) h = 6 in the upper half (x -- 0.5) and h = 1 in the lower half.
The situation just considered is when h and r/ are uniform along the length of the column (both equal 1). These are the physical parameters, which were not taken out of the computation when normalizing. If these parameters are not constant over the entire length of the column then Eq. (4.30) cannot be deduced from Eq. (4.29), and there is no reason to expect that the top of the column will remain stationary. As an example, suppose h is different (but constant) in the upper half of the column from in the lower half, while r / ( a n d E and K which were included in the normalization) remains constant along the entire column. It is possible to accomplish this by changing only the microgeometry and elastic
Ch. 4
52
Finite Element Formulation and Solutions for Quasi-Static Beams
properties of the solid skeleton, while the pore fluid is maintained uniform throughout. The influence of such a situation on the column's response for the I-I case is now examined. Take rt = 1 (as previously) throughout the column. Three cases will be considered: (a) A = 6 throughout; (b) A = 1 in the upper half while A = 6 in the lower half, and (c) A = 6 in the upper half while A -- 1 in the lower half. The vertical deflections, u, vs. the vertical location, x, are shown in Fig. 4.9 for all three cases, at several times. The results are surprising. In case (a) where A is uniform there is no qualitative difference from the previous example; again it is seen that the immediate axial deflection at the top of the column does not change at all with time. However, the relative portion of the column where the skeleton is in tension is greatly increased. That this should be so is shown as follows, u reaches its maximum value when U,x = 0. From Eq. (2.14a) we see that this requires N = r/Np
(4.31)
Now, for our loading, N is simply
N = -q(L-
x)
(4.32)
while at t = c~ we calculate from Eq. (2.43a) A L Np = - --q-------+---2 ag ,tAl
(4.33)
Inserting Eqs. (4.32) and (4.33) into (4.31), the position at which u is maximum is found to be given by 2 --
1 + (At//2) l+Ar/
L
(4.34)
So any increase in the product Ar/lowers the position on the column at which the maximum deflection is found. In the case considered here this position has been lowered from (3/4)L to (4/7)L. Now in case (b), where A in the upper part has the smaller value, the initial deflection at the top of the column continues to grow in time to its final value. But in case (c) where the upper A is the greater one, a very peculiar behavior is observed. After the initial (at t = 0) downward deflection of the top of the column, this region of the column reverses its direction and moves upward until reaching its final position!
Chapter 5 VIBRATIONS OF POROELASTIC BEAMS
Transverse vibrations of fluid saturated poroelastic beams are investigated in detail based on the formulations given in Chapter 2. Solutions are found for free and forced vibrations. The nature of the behavior patterns found is considered and comparisons are made with the types of behavior exhibited by damped elastic beams. For the sake of convenience, the governing differential equations in non-dimensional form are copied here
04W o~X4 -- 77 o~X2
+ q + 3,2
-- 0
(2.46b)
02Mp OMp 03W OX2 -- Ot - A Ox 2 0t -- 0
(2.48b)
Again, the superscript * is omitted, even though the variables considered are non-dimensional, q refers to the distributed normal load and the non-dimensional parameter 3' is defined in Eq. (2.35), and here is assumed to be constant in order to facilitate extracting closed form solutions (2.35)
y = Kx/p/EI
5.1. Initial Value Problems Consider a simply supported beam with permeable end surfaces. The boundary conditions for this case are given by w(0, t) = 0,
w(1, t) = 0
W,xxlx=o= W,xxL- = o Mp (0, t) -- 0,
(5.1)
Mp(1, t) -- 0
and the initial conditions are M p ( x , O) -- M ~
w ( x , O) -- w O(x),
wl~=0 =
v~
(5.2)
Ch. 5 Vibrations of Poroelastic Beams
54
Taking solutions for (2.46b) and (2.48b) of the form oo
oo
Mp(x, t) = E mn (t) sin(n Trx)
w(x, t) = E w,(t)sin(nTrx), n= 1
(5.3a, b)
n= 1
the boundary conditions (5.1) are immediately satisfied. The initial conditions on w(x,t) and Mp(x, t) are satisfied if wn(t) and mn(t) meet the following requirements: mn(0) -- 2 I 1M~p(X)sin(n'rrx)dx o
~o~(0) = 2 I 1 w~ o
dx
(5.4)
= where ~bn(0) refers to the derivative of % ( 0 at t = 0. Next, by expanding the load q(x, t) into a Fourier series (x)
q(x, t) = ~. bn(t)sin(n~rx)
(5.5)
n=l
where
bn(t) = 2;~oq(X,t)sin(nTrx)dx and substituting Eqs. (5.3) and (5.5) in (2.46b) and (2.48b) one obtains
T26bn(t) + (nTr)nogn(t) + rl(n'n')2mn(t) + bn(t )
= 0
rhn(t) + (nTr)Zmn(t) - A(nTr)Zdon(t)--0
(5.6a) (5.6b)
Eqs. (5.6) together with the initial conditions (5.4) constitute an initial value problem for Wn(t) and m~(t) for each n (n -- 1,2,3 .... ). This system can be reduced to a single differential equation involving only OOn(t). Solving Eq. (5.6a) for mn(t) and substituting this expression together with its time derivative into (5.6b) yields
q ~ n ( t ) + T2(nTr)Z ffgn(t) + (1
+
A~'l)(nTr)4(On(t) + (nTr)609n(t) + bn(t ) + (nTr)2bn(t) -~ 0 (5.7)
The additional initial condition required to solve (5.7) is obtained from (5.6a), i.e. 1
6in(0 ) -- - - - ~ [(n'n')4Ogn(0) +
rl(nTr)2mn(O) +
bn(0)]
(5.8)
Eq. (5.7) is solved by Laplace transformation. Denoting the Laplace transforms of w,,(t) and b,(t) by ff~n(S) and [~n(S), respectively,
Initial Value Problems
55
[T2s 3 -4- T2(nTr)2s 2 -4- (1 -4- A'q)(nTr)4s -4- (n'n')6l~n(S) --
--[~(_].)n(O) -4- T2 ( n Tr) 2 ton ( O ) ] s + S ffgn ( S ) -- ")/20) n ( O ) --
]/2(.0n(0)$2
T2(nTr)2tbn(O)
--(1 -4- ~T/)(nT"/')40)n(0) -- bn(O ) -4- ( n " I T ) 2 [ ) n ( S ) - - 0
(5.9)
from which
O)n(0)S2 -4- OIS -4- ~ 60n(S ) --
S[~n(S)/T 2 -- (nTl')2Dn(S)/'y 2
( S - ~I)(S- ~2)(S- ~3)
(5.10)
where
Arl 1 3 - -~(nTr)4tOn(0)-
c e - 6)n(0) + (nTr)2tOn(0),
~1 (nTr)2mn(O) -4- (n'n')26)n(0)
(5.11) and d)n(0) has been replaced by (5.8). sr so2 and so3 are the roots of the cubic equation 1
1 -4- At/(nTr)4 ~ -4-
+
+ ---7-
(nTr) 6 - 0
(5.12)
7
Letting ~ : - ~(nTr) 2, Eq. (5.12) can be cast into the form l+Ar/
1
Thus it is noticed that the nature of the roots of the cubic equation is independent of n. Now the nature of the solutions to,(t), i.e. the inverse of (5.10), depends on the nature of the roots of (5.13). The nature of the roots depends on A --
p3 27
+
0~
(5.14)
4
where P=
l+Ar/ y2
_ 1 ~
and
~_
2 27-
l+Ar/ 3y2
1 + 3t2
(5.15)
If A > 0 two of the roots are complex and one is real; oscillatory motion is possible (but may not necessarily occur, as will be seen in the examples). If A < 0 all three roots are real, and oscillatory motion is impossible. The case where A -- 0 is the critical one. The solutions tOn(t) for all of the various cases are given below. The following parameters are introduced for convenience A - ~-0/2
+ "v/A,
B-
~/-Q/2 - ~
where A and Q are given in Eqs. (5.14) and (5.15), respectively.
(5.16)
Ch. 5
56
Vibrations of Poroelastic Beams
C a s e 1. k < 0 Here/5 < 0. The cubic equation (5.12) has three distinct real roots
6
--
~:~,3 -
[j cos( 0, 2
--5-/-
(nTr)2' (5.17)
[
-2
--Scos
-5- - 5-
- 5
(n~)2
where
-Q COSO/o --
2x/-p3/27
Thus Eq. (5.10) can be written as
3 ki COn(S)- i ~ 1 S-- ~i
3 li bn(S) Z S -- ~i
(5.18)
i=1
where the constants are
ki =
13 .4_ O/~:i nt_ OJn(0)~2 3
and
~i + (nTr) 2 3
li -
H (~i-- ~) j=l (j#i)
,y2
(5.19)
H (~i-- ~j) j=l (j#i)
Several useful Laplace inverse transformations are given in Appendix C. By applying formula (C.la) to (5.18), the inverse form is obtained as 3
3
it
i=1
i=1
0
wn(t) -- ~ . ki e~t - ~ . li e~t
bn(r)e -~:i~ d r
(5.20)
C a s e 2. A -- 0 and both P # 0 and Q # 0 In this case there are three real roots, two being repeated ones ~1 -- (A + B - 1/3)(nvr) 2,
~:2 -- ~3 ---= - ( A + 1/3)(nTr) 2
(5.21a, b)
Then Eq. (5.10) can be written as
~n(S)- Z
i=1
,i
S - ~i -}-
(S-- ~2)2
where the constants are given by k1 =
/3 + ,~6 + ~ . ( o ) ~ ( 6 - ~2) 2
-- ~3n(s)
i=1 S - - g
-+
,3) (S-- ~2) 2
(5.22)
Initial Value Problems k2 =
k3 --
--~-
57
0~I -- 20)n(O)~l ~2 -I- OJn(O)~2
(~:1- ~2) 2 ]3 "4- 0~2 q'- O)n(O)~:2
~::
see-
~1 + (nTr)2
~2 + (nTr)2
ll ~- "Y2(~l- ~2) 2 '
12-
-ll'
13-
(5.23)
~t2(~2- ~1)
Using Eq. (C.1), the inverse form of Eq. (5.22) can now be determined as follows:
('On(t)- E2 kiewit + k3te~2t- E2 lie~ititobn(7-)e-~ir d 7 - - 13e~2t i, bn(7-)e-(;2r(t- 7-) d7i=1
i=1
0
(5.24) Case 3. A - 0 a n d / 5 _ 0 - 0 In this case there are three repeated roots
--(nTr) 2
6-6=6-
(5.25)
3 and Eq. (5.10) has the form
3 li 3 ki -- ff)n(S)E )i 6On(S)- E (S-- ~1) i ( S - ~1 i=1
(5.26)
i--2
where the constants are kl-
w,,(0),
12 -- 1/3'2,
k 2 - ce + 20)n(0)~ 1,
z3 - (6 + ,2 ~ ) / ~ (5.27) The inverse form of ff~n(s) is derived similarly to the previous case as
O)n(t) -- kl e~::t nt- k2te 6t + -~k3t2e 6t - 12e6t
1
_ -213e6t
X' bn(7-)e-6~(t
_
_
o
0
b~('.7-)e-6~(t- 7-) d7-
,./-)2d7-
(5.28)
Case 4. A > 0 There are two conjugate complex roots in this case
sc2,3 - ao ---/3oi where ao and 13o are real and given by
(5.29a)
Ch. 5
58 ce~
A +2 B + 13 ] (nTr)2'
-
Vibrations of Poroelastic Beams (5.29b, c)
Vc~A -2 B (n,n.) 2
/3~
The only real root ~1 is given by Eq. (5.2 l a). The denominator of Eq. (5.10) can then be rewritten as
( s - ~ ) ( s - ~2)(s- ~3)- ( s - ~)[(s- c~o)~ + ~3~]
(5.30)
and Eq. (5.10) becomes
kl -~1
COn(S) - -
-~
11
--
kz(sCeo) ( S - ~ o ) 2+~3~
k3
(s-
)2
(s-~o
12(S- CeO)
+
- ~1
"-~
+t~o~
t3 )t;.(s) + ( s - ~o)2 +/302
o~o)2 + ~
(5.31)
where the constants are given by k1
k2 --
k3 z
/3 + ~
+ O~n(0)~
( ~ - ~o) 2 + t~ --/3-
Ce~l + (ce~ + /32 -- 20~0~l)tOn(0 )
(r - ~o)2 +/3~ ('~o - ~,)t~ + (~o~ + t~o~ - ~or (~-
+ [(t~o~ - ~o~)r + (~o~ + t~o~)~o]o~(o) .o)2 +/32o
ll--
~1 + (n'rr) 2 ~t2[(~ 1 _ OLo)2 _q_ ~2] --
13 =
c~2o + 13o 2 - Ceo~:1 + (olo scl)(nzr) 2 ~[(~:1 - ao) 2 + 13o 2]
12
(5.32)
Taking the inverse transform and applying (C.2), the solution for tOn(t) is found
tOn(t)- kl e~It q- kee~~ - le[c~176
k3
t) + --g--e~~ Po
t) - lie ~t
~t
bn(~')e -~IT d~"
o
l--2-3[sin(/3ot)I n - cos(~ot)In]e ~~ ~o
+ sin(~~176
(5.33)
where Ic~ and I n are integrals, defined as follows:
Inc(t) --
0
bn(~')e-~~
d~',
I2(t) -
0
b~(~-)e-~~162
d~-
(5.34)
With %(t) in hand, ran(t) is then found from Eq. (5.6a). Thus solutions of the form (5.3) have been found for all cases. Note that the sign of A, and hence the nature of the behavior of the beam is dependent only on A 77 and y.
59
Initial Value Problems
Fig. 5.1. The T-A r/plane showing the region where no oscillatory motion is possible (shaded). The coordinates of the cusp of the no-oscillation area are shown.
In Fig. 5.1 the shaded area is the region where A < 0. Thus no oscillatory motion is possible for combinations of A r / a n d 3/there. In the language of damped elastic vibrations this is the over damped region. The boundary of the shaded region represents the case of A = 0. In the language of damped elastic vibrations this is the situation of critical damping. Outside the shaded region A > 0 and oscillatory motion is possible. In the language of damped elastic vibrations this is the region of light damping. Now recalling the definitions of the characteristic dissipation time, ~'D, the characteristic time of vibration of the drained beam, ~'s, and the characteristic time of vibration of the beam with fluid trapped, ~'T, we see that ~s _ ~/1 + An, ~'T
~s _ T, TD
~'T _ 3' ~'D ~/1 + An
(5.35a, b, c)
Thus the nature of the solution could have been presented as depending on any two ratios of characteristic times, ~'S/~'T, ~'S/~'D and ~'T/~'D. In practical systems it is unlikely to be situated in the shaded region of Fig. 5.1, and therefore our interest when considering examples will center on the outer region.
Free vibrations of a simply supported beam with a sinusoidal initial shape As the first example, consider a simply supported beam with the following initial conditions: w ( x , O ) = sin(vrx), w ( x , O ) = O , and Mp (x, O) -- O. Namely, the beam is displaced into a sinusoidal shape, held still until pore pressure is reduced to zero, and
Ch. 5
60
Vibrations of Poroelastic Beams
then released. We will also consider the situation when after displacement, the beam is released before the pore pressure has had time to decay at all, i.e. Mp(x, 0) is found from Eq. (2.49b). These two cases of initial pore pressure are denoted as I and II, respectively. The solution w(x, t) for 'light damping', the outer region in Fig. 5.1, is given by Eqs. (5.3a) and (5.33). For the initial conditions being considered, only one term of the series is required, and the solution becomes
w(x, t) -
kle ~'t + k2e'~~
t) + -~o e~~
t) sin(Trx)
(5.36)
where ki, ~1, Ceo and/30 are given by Eqs. (5.32), (5.21a), (5.29b) and (5.29c), respectively. The exponents ~1 and a 0 are negative;/30 is positive. ~:1, a0 and/30 depend only on A r/and y. There are thus two exponential decay processes occurring in the system; one results in a monotonic decay of part of the initial deflection, the other multiplies the time dependent sinusoidal oscillations. The relative importance and interaction of the two decay processes is dependent on the values of the k i, which in turn depend in a complicated fashion not only on the system parameters, but also on the initial conditions. Quite interesting patterns are possible. Consider first the simplest situation. If At/is very small (i.e. TS/~'Tis close to 1) or if y is very large (i.e. ~'S/TD is large) it is found that the first term in the square brackets is small compared to the terms involving sine and cosine. This is to be expected since At/ small implies that the presence of trapped fluid does not influence greatly the stiffness of the 1.0
I ,,
Condition I :Solid line Condition II: Dashed line
[ [
I
0.5
0.0
'
,
-0.5 Il
-I.0
0.0
2.0
4.0
6.0
8.0
t Fig. 5.2. Vibration pattern, w(x,t) at x = 0.5; Initial conditions w(x, 0) -- sin(Trx), w(x,0)- 0; AT/- 0.25 and y = 1.5.
Initial Value Problems
61
beam; there is little coupling between (2.46b) and (2.48b). Similarly, large Y implies that the period of the drained beam is very large compared to characteristic pressure dissipation time, so with a very small portion of a single period, pore pressure is lost and the beam vibrates essentially as a drained structure. But there will remain some damping, which is not at all insignificant. Typical deflections vs. time patterns in this circumstance are shown in Fig. 5.2 for conditions I and II of initial pore pressure; there is very little difference between the two deflection-time curves. Note that time has been non-dimensionalized with respect to ~'D,the dissipation time. The behavior is similar to that obtained for damped oscillations of an elastic beam. For these conditions the frequency of the system is identified as fi0/27r; using Eqs. (5.16) and (5.29), it is given explicitly as
f--
~
4
~+
x/~+~
~
Consider next cases for which the contribution of the first term in the square brackets is not negligible. While for the previous case the frequency of vibration was simply flo/27r, when the first term in the square brackets is not negligible no frequency can be defined. But it is convenient to define a pseudo frequency by
fpseudo --/3o127r
(5.38)
This is the frequency at which the system would vibrate if the first term in the square 5.0
4.0-
~.~
3.0
0.40--'--
+ 2.0....
0.60
i 1.0
" . . . .
0.0
i
1.0
. . . .
,
. . . .
2.0
y 141+ x,7
I
.
.
.
.
3.0
(:
Fig. 5.3. Normalizedpseudo frequency map in the ~'T/~'D-~'S/~'Tplane.
4.0
62
Ch. 5
Vibrations of Poroelastic Beams
1.0 k..
0.5
0.0 =
o Condition I o Condition II
-0.5
-1.0
,
,
0.5
0.0
|
|
,
t
1
i
1.0
|
,
|
1.5
Fig. 5.4. Vibration pattern, conditions as for Fig. 5.2; Condition I refers to initial pore pressure equal zero, and Condition II refers to initial pore pressure as developed immediately after rapid displacement of the beam. At1 -15 and y = 1 (ZS/ZT -- 4 and ZT/ZD = 0.25).
brackets were negligible. In Fig. 5.3 a map of constant values (solid lines) of the combination O is shown, where 0
2 y ~r ~/1 + a v fp~ua~
7r To
seudo
(5.39)
is plotted in the plane of y/~/i + Art vs. x/1 + Art (i.e. ~'T/ZD vs. ZS/ZT). Where the curves are nearly horizontal the behavior is as in Fig. 5.2, and the pseudo frequency is very nearly the frequency of the drained beam (O is asymptotic to 1/,,/1 + Art). The time scale of the beam vibration is sufficiently long with respect to that of the dissipation process so that the coupling through Art is of secondary influence in determining frequency. On the other hand, when the curves are nearly vertical the pseudo frequency is very nearly that of the beam with trapped fluid (O is asymptotic to ~/1 + Art/y). But now the initial conditions very strongly influence the relative importance of the two decay processes. Consider Fig. 5.4 where Art = 15 and y = 1 ('/'S/TT = 4 and 7"T/T D = 0.25). For the initial condition I, the monotonic decay process dominates and the sinusoidal process just adds a small tipple which rapidly disappears. For the initial condition II, it is clear that it is the sinusoidal decay which dominates. The practical implication is that for the first initial condition (I) at the time t -- 1 initial deflection has been reduced by only onehalf, whereas for the second initial condition (II) the deflection has essentially been
63
F o r c e d H a r m o n i c Vibrations
reduced to zero. This illustrates the crucial influence of initial conditions on the subsequent decay of vibrations. Finally, we point out that when Q - 0 the two exponents are equal, and ~:1 - a 0 - -~'2/3, which yield identical decay rate for the two processes. This occurs when 2y 2 - 9A~/+ 18 - 0
(5.40)
and is shown in Fig. 5.3 by the dashed curve. To the left of this curve the sinusoidal decay process is faster than the monotonic process (see Fig. 5.4); to the right of the curve the monotonic decay process is the faster one.
5.2. Forced Harmonic Vibrations
Consider now a beam subjected to the harmonic load (5.41)
q(x, t) = ft(x)exp(iwt)
where i - x/"-Z-1. Then the forced part of the solution has the form w(x, t) - ~ ( x ) e x p ( i w t ) ,
(5.42)
Mp(x, t) - Mp(x)exp(ioot)
Substituting these forms into (2.46b) and (2.48b) it is found that the amplitudes ~(x) and Mp(x) are governed by the following system of ordinary differential equations: d4~ d2/~p dx 4 - T] dx 2 -1- q - 3/2(.021,~ -- 0,
dZ/l~/p dx 2
-
iw/f/p
-
d2ff: iA~o-~-f - 0
(5.43)
Eliminating/f/p yields d6w
dx 6 - i w ( 1
d4w 2 d2~ d2q - imP/-- 0 + A~/)-~-~- - y z c o ~ + i~/2033w +
(5.44)
For a simply supported beam with both ends permeable the boundary conditions are given by Eq. (5.1). For this case the assumed solution is of the form co
-- ~ Wnsin(nTrx) n=l
(5.45)
By expanding the load amplitude into a Fourier sine series oo
g / - ~ . cnsin(nTrx)
(5.46)
n=l
where Cn - - 2 ~ i ~(x)sin(n~rx) dx
so that the coefficients Wn can be determined from Eq. (5.44) as follows:
(5.47)
64
Ch. 5
Vibrations
of Poroelastic
i~o] Ar/)(n'n) 4 + T2o92(nTr)2 + i'y2o93
Beams
Cn[(nTr) 2 + Wn - -
--
(nTr) 6 --
ioJ(1 +
(5.48)
and Mp(x) can be found from Eq. (5.43) in the form (3O
(5.49)
iVlp - - ~ . m n s i n ( n ' n ' x ) n=l
where Aoo(nTr)2 mn =
0 9 - i(nTr) 2
(5.50)
wn
Such a series solution would have been more difficult to obtain had we considered more complex boundary conditions, e.g. boundary conditions that are not homogeneous. In such situations, a general solution to Eq. (5.44) would be required. The general solutions of the homogeneous differential equation corresponding to Eq. (5.44) would have the form exp(r Here Ck is the root of the characteristic equation 6
- io~(~ + A n ) ~ 4 -
~0~2~
+ i~o
3 -
0
(5.51)
This equation is changed into the cubic equation (5.13) by letting
10o
10-' Q)
It.,
I
0_2
~
~ xn=o.o o xn=o.,
\\
lff" lff'
0.0
6.0
12.0
lB.0
24.0
Y Fig. 5.5. Deflection amplitude vs. frequency of harmonic load; first two resonance areas shown; y = 0.75" At/= 0 (elastic), 0.1 and 0.4.
Closure
65
2
i0)
~-
sZ.
(5.52)
Thus roots r (k = 1,2 ..... 6) can be found for all cases. When A # 0, all roots are distinct, and the general solution of Eq. (5.44) has the form
6 ~ ( x ) - ~. Bkexp(~kx) + l'~par(X) k=l
(5.53)
where B~ are constants which can be determined according to the boundary conditions, and Wpar(X) is a particular solution of the non-homogeneous equation (5.44). The following function can be taken as a particular solution if ?:/(x) is a polynomial of third order or less
WPar(x) -
,~(x)
3/20)2
(5.54)
For a more general loading, Eq. (5.45) can serve as a particular solution.
A simply supported beam subjected to a uniformly distributed harmonic load Consider a simply supported beam subjected to a uniformly distributed harmonic load (magnitude is 1, dimensionless), where the forcing frequency i s f = 0)/27r. The rs/rD (=3') ratio is 0.75. Fig. 5.5 displays the first two resonance areas, the first of which can be predicted by Eq. (5.37). The curve for A t / = 0 corresponds to the (drained) elastic case. For f - - 0 the solution is that of the quasi-static problem. Note that both the first and second resonant frequencies increase as the influence of the presence of fluid is increased ( A t / > 0); the shift is more prominent at the second resonant frequency. On the other hand it is clear that the amplitude response is significantly reduced as At/increases. We see that the amplitude is smaller than the static elastic value if the frequency of the loading is far from the resonance areas. In fact at very high frequencies essentially no fluid movement can take place at all and the beam behaves as an elastic beam whose Young's modulus is
(1 + An)E. In Fig. 5.6, T has been increased to 3; the behavior pattern is similar except that now the frequency shift is nearly imperceptible.
5.3. Closure A detailed solution has been carried out for the simply supported beam with permeable end surfaces. For free vibrations the nature of the deflection decay is strongly affected by the initial pore pressure. Whereas for the damped elastic beam (sub-critical) there is always some oscillatory motion, in our case this oscillatory motion is often completely wiped out by the existence of a monotonic decay term. It is seen that axial fluid diffusion in vibrating beams can easily provide a very strong damping mechanism. The mechanism might possibly be influential in reducing damage to plant elements when these are subjected to suddenly applied and short term loading. It is also possible that such a mechanism could be introduced into 'smart' elements in structures; by 'valving' the
Ch. 5
66
Vibrations of Poroelastic Beams
16'
10-2
tR
o X~=0.1 o X~=0.4
' ~
a),~=l.0 .
10 -3
10 -4 0.0
1.5
3.0
4.5
6.0
/ Fig. 5.6. Deflectionamplitude vs. frequencyof harmonic load; first two resonanceareas shown; y = 3" A~7-- 0.1, 0.4, 1.0 and 3.0.
diffusion boundary condition at the element end(s) one could rapidly change the response characteristics of the system, and thus provide an efficient control arrangement. Finally we point out that ), is inversely proportional to the viscosity of the contained fluid (see Eqs. (2.35) and (2.20)). Viscosity of most fluids is very sensitive to temperature; for instance a change in water temperature from 10~ to 40~ reduces viscosity by one half. Then considering Fig. 5.3 it is immediately apparent that such a temperature change in water saturated material can completely change the regime of vibration response of a structure. It is noted that structural damping is often related to and modeled as resulting from the viscoelasticity of the material in which case two time scales are involved, one which is related purely to the material, the second which also includes structural parameters. In the present model three time scales are involved. Additionally, in the present model a new type of initial condition must be included which can appreciably influence behavior patterns. For all of these reasons behavior patterns are possible which are not seen in viscoelastic structures.
Chapter 6 LARGE DEFLECTION ANALYSIS OF POROELASTIC BEAMS
In the previous chapters, the deflection was considered to be small and thus linear theory is sufficient since the constitutive law adopted is linear. Yet for some situations, it may be necessary to employ a large deflection theory in order to correctly describe the behavior of the porous components. The deformation can still be small and elastic; the large deflection is possible because of the slenderness of the components. Therefore, the components are modeled in this chapter as geometrically non-linear (but constitutively linear). Biot's constitutive law and Darcy's law are then still applicable, while new geometrical relations and equilibrium equations must be introduced.
6.1. Governing Equations As in all previous analyses, the beam considered is taken to be transversely isotropic in the cross-sectional plane and the microgeometry of the bulk material is such that the fluid flow is possible in the axial direction only. The basic assumptions for the small deflection theory are still justified for our problem, i.e. cross-sections remain plane after deformation and the shear strain and the transverse normal stress are negligible. Two more assumptions, which are implied in the small deflection problems, should be restated here: (1) the beam has a principal plane and bending occurs within the plane (i.e. the x-z plane, shown in Fig. 6.1), and (2) the deformation is small so that Biot's constitutive law is justified. A segment of the beam axis with an original length &x and deformed length As is shown in Fig. 6.1. s is always measured along the beam axis during the deforming process; 0 is the rotation angle of the cross-section, which may be large; u is the horizontal displacement and w is the vertical deflection of the beam axis. The area and moment of inertia (with respect to y) of the cross-section are A a n d / , respectively. N, M and Q are the resultants of the normal stress and shear stress at the cross-sections produced by the distributed axial load qs(x) and normal load qn(x). The directions of the two loads are taken to be unchanged after deformation, but their magnitudes are generally functions of x and t. (It is important to remember that x in this chapter always refers to a material point along the beam.) The geometry of the beam is considered first. Using e0 to represent the strain of the axis, we have
68
Ch. 6
Large Deflection Analysis of Poroelastic Beams
q.
a l..---
---.-I b
N M
'
I -
- ~
N+AN
Fig. 6.1. Free body diagram of beam segment.
As-(1
+ e0)Ax
(6.1)
Since the higher order of infinitesimal is omitted, Au + A x - As cos0 and Aw - As sin0 hold according to the geometrical relationships. Thus, using Eq. (6.1) the following transformations between the two sets of variables: u, w and e0, 0 are obtained: 3u 3x
--(1 + e 0 ) c o s 0 - 1,
Ow 3x
- - ( 1 + e0)sin0
(6.2)
The geometrical equation, i.e. the relation between the strain and displacement is e:
30 e0--Z-3x
where ~ is the normal strain of the solid matrix in the direction of the beam axis. Since the strains are small, the following relation is used instead of the above expression in order to simplify the formulations: 30 e = e0 - z - 3s
(6.3)
The difference thus produced is z(30/3S)eo which is a higher order term as compared with both e0 and z(30/3s). The constitutive relations for the beam are as introduced in Chapter 2. 7 = Ee-
~pf,
~"= r/e +/3pf
(6.4a, b)
The normal stress ~- at the cross-section leads to the integrated resultants N and M (see
Governing Equations
69
Eqs. (2.2)) as shown in Fig. 6.1, which can be given as follows by Eqs. (6.4a) and (6.3)
N-
EAeo + rlNp,
M-
- E l O0 + riMp 0s
(6.5a, b)
where Np and Mp are again defined as
-fpfdA,
P--
Mp=-~pfzdA
A
(6.6)
A
and the y-axis is taken to be the centroidal axis of the cross-section so that IA z d A = O. The equilibrium of the deformed segment as shown in Fig. 6.1 requires E
Mb' = 0:
(M + zkM) - M - QAs + o(As) = 0
~. Fz = 0:Nsin0 + Qcos0 +
qnZ]kx
-
(N + z~)sin(0 + A0) - (Q + AQ)cos(0 + A0) = 0
E Fx = 0: - Ncos0 + Qsin0 + qsZLr+ (N + zSaV)cos(0+ A0) - (Q + AQ)sin(0 + A0) -- 0 in which o(As) represents the second or higher order of the infinitesimal As. The first equation shown above is simplified as follows when o(As) is omitted:
OM Os
-- O
(6.7)
Further, multiplying the second equation by cos0 and the third by sin0 and then adding together, yields Q + (qncos0 + qssin0)zkr - (N + Z~)sin(A0) - (Q + AQ)cos(A0) -- 0 If the curvature O0/Os is not infinite, namely a moderately large deflection situation is considered, then A 0---, 0 when As ---, 0. For this case then, sin(A 0) -- A 0 and cos(A 0) -- 1. Moreover, ANAO is negligible since AN is of the same order as A0. Thus, the above expression leads to the following differential equation for equilibrium:
OQ oO - qnCOS0 + q s s i n 0 - N - Ox Ox
(6.8)
Similarly, another differential equation for equilibrium can be obtained
ON O0 -- % s i n 0 - qscos0 + Q ~ 0x 0x
(6.9)
Eqs. (6.7), (6.8) and (6.9) are all the differential equations required for equilibrium of the beam. Since Q has no relationship with the diffusion, it can be eliminated from (6.8) and (6.9) by using Eq. (6.7). Noting also Eq. (6.1), the two differential equations required for the problem in hand are qnCOS0+ qssinO
02M --
ON 3s
-
l+eo
o~S2 qnsin0--
qscosO
OM O0 +
l+eo
O0 Os
N--
3s Os
(6.10)
Ch. 6
70
Large Deflection Analysis of Poroelastic Beams
which, after (6.5) is introduced, can be written in new forms
-El
03 0 02Mp ~0 qncosO + qssinO --~ + 7q Os2 + (EAeo + r/Np) Os -1 + e0 (6.11)
020 E z -gj
OMp ) O0 - n - - ~s-
~
3Np + EA Oeo _ qnsinO- qscos0 + n
~s
-~s
-
1 + ~o
Now the fluid flow physics must be considered in order to complete the system of differential equations for the problem. This is governed by Darcy's law, which (cf. Eq. (2.18)) takes the following particular form when only the axial diffusion is considered: ~=
kll O2pf
(6.12)
]d~f o~S2
where kll is the permeability in the axial direction and ~f is the viscosity of the pore fluid. Substituting Eq. (6.4b) and then Eq. (6.3) into the above, the form below is obtained: 02pf
00
(6 13)
K ~s~- = pf + A E ~ 0 - AEz-~-s
and again K -- kll AE/(/zf r/). If differential equations expressed by the pore pressure resultant, Np, and pore pressure moment resultant, Mp, instead of the local pore pressure, are sought, the operators
fA [ ]dA'
fA [ ]zdA
should be applied to Eq. (6.13) respectively, yielding
K
o?2Np -- Np - AEA g:o, 3S2
K
o?2Mp 00 -- Mp + A E I ~ o~S2 o~S
(6.14a, b)
Eqs. (6.5), (6.10) and (6.14) are all the governing equations necessary for the six unknowns" e0, 0, Np, Mp, N and M. Alternatively, only Eqs. (6.11) and (6.14) will be involved if only four unknowns" e0, 0, Np and Mp are considered - when the boundary conditions permit us to do so. Note that as opposed to the situation for the small deflection problem, here the axial variables and the bending variables are inextricably coupled. Finally, the boundary conditions and the initial conditions for the problem are considered. The geometrical boundary conditions are Ule= / L
Wle-- ~,
0le = 0
(6.15)
where le represents that a variable takes on a specific value at a boundary, and a bar refers to a given function. No boundary conditions are required for e0 since none of its derivatives with respect to x or s are included in the differential equations. The mechanical boundary conditions are the same as those for the poroelastic beam with small deflections. The load conditions at boundaries are Nle-- iV,
Mle=/9/,
Qle = 0
(6.16)
E q u a t i o n s in N o n - D i m e n s i o n a l
71
Form
The diffusion for permeable boundary conditions are
Up]e- ~p,
Mp]e-
M--p
(6.17)
and for impermeable boundary conditions they are
ONe Io~Se--0,
0MP loS e - - 0
(6.18)
The initial conditions for e o, 0, Np and Mp are dependent. The relationship is determined by the requirement that ~']t=o--0, which means that there is no instantaneous diffusion. Using Eqs. (6.4b) and (6.3), this becomes 00 pf-
AEz-- AEeo, Os
(6.19)
t--0
and upon substituting into (6.6), the equivalent expressions in terms of the global quantities are derived 0O 0s'
Np -- A E A e o ,
Mp-- - AEI~
(6.20a, b)
t--O
6.2. Equations in Non-Dimensional Form when t~0 - 0 A particular but very likely situation is considered in the following. This is, when the axial strain is negligibly small, as is to be expected for a slender beam. Letting e0 -- 0, then, from Eq. (6.1), As = Ax. The differential equations in dimensionless form are obtained after the quantities are normalized as
x
, = _x L'
t* _
,
MpL
Mp--
E1
'
Kt L2,
, _ u
,
qn L3
qn--
E1
u L'
'
, _ w ,
qsL 3
qs--
E1
w L'
N* _
NL2 El'
M* = ML El'
(6.21) where L is the length of the beam. In accordance with Eqs. (6.2), (6.5b), (6.10) and (6.14b), the relevant equations are as follows" 0u* Ox*
= cos0 - 1,
0w* Ox*
= sin0
(6.22a, b)
which are the geometrical relations; M*-
00 0x*
+ r/Mp
is the constitutive law in terms of the global quantities;
(6.23)
72
Ch. 6 ,
, .
O2M* -- qncos0 + qsslnO - N* 00 Ox.2
Ox*'
Large Deflection Analysis o f Poroelastic Beams ,
ON* _ q~sin0- qsCOS0 Ox*
0114" 00 Or,* Ox*
(6.24) are the equilibrium equations; and 32Mp
00
0X.2 --/~'/; -nt-/~0X*
(6.25)
derives from Darcy's law. A dot now refers to a derivative with respect to the dimensionless time t*.
The dimensionless forms for Eqs. (6.5a) and (6.14a) are not included here since they are not relevant after e0 is omitted. N* must be determined instead by the equilibrium equations, as given by Eqs. (6.24). The forms of the boundary conditions (6.15)-(6.18) remain unchanged after the normalization. The initial condition, according to Eqs. (6.20b) and (6.5b), can be written as 00 M*-- - ( 1 + At/) 0x*'
,
t --0
(6.26)
or in another form ,
A
M p - - A1 -+~
M*
'
t* -- 0
(6.27)
Either of these will be helpful in computing initial values. The system of partial differential equations above is complete. It should be noted that certain unknowns may have to be solved simultaneously. For example, although the deflection w* appears only in (6.22b) and it can possibly be calculated separately after 0 is obtained, in general cases w* must be solved simultaneously due to coupling of the boundary conditions. For instance, there are two boundary conditions for w* but no boundary conditions for 0 for a simply supported beam. Elimination of N* and M* from the simultaneous equations is also possible but it will increase the order of the differential equations, which usually results in precision loss in numerical computation and also results in inconvenience for introducing boundary conditions. Therefore, the five equations (6.22b) and (6.23)-(6.25) will be solved simultaneously for five dependent variables, N*, M*, Mp, 0 and w*. Nevertheless, when computing the initial values for the variables, only Eqs. (6.22b), (6.24) and (6.26) need to be solved simultaneously for N*, M*, 0 and w. Then Mp* can be calculated by Eq. (6.27). On the other hand, u* can always be calculated separately by Eq. (6.22a) after 0 has been solved since its boundary condition is not coupled with others in our situation.
6.3. N u m e r i c a l F o r m u l a t i o n
Analytical solutions for the system of non-linear partial differential equations are not expected to be obtainable. Numerical solutions for these equations are derived in the following by employing the finite difference method for the spatial coordinate. The equations are employed in their non-dimensional forms; the superscript * will be omitted for convenience.
N u m e r i c a l Formulation
73
The numerical algorithm is presented now. The beam is discretized by nodes x i (i = 1,2, 3 ..... m). These nodes are not necessarily uniformly distributed and the domain is not necessarily from x = 0 to x = 1, but it is convenient to start from x - 0. Accordingly, the subscript i is used to denote that a variable is valued at x = xi. Discrete times are denoted by tj (j = 0, 1,2, 3 ..... n and to = 0); also the time intervals are not necessarily equal. A superscript j then implies that a variable is valued at t - - t j . The difference formulas derived by the Lagrange interpolation are adopted. The 3-node formulas for the first order derivative at the node i, for example for M, are given as
( 3M)Ji-- CoM~ i j + C1M~+ i j 1 ._[_ C2Mi+ i j 2
(6.28a)
( 3114 )i_
(6.28b)
/(3114 ,,1)i -~x
-
ij
C ilM~
i
+ CDIldj + C1M~i+ 1
j
C - 2M~_ 2 +
Ci
J -at- C~M~ 1M~-I
(6.28c)
where the coefficients C i, which depend on the coordinates of the node i and its two neighboring nodes, are given in Table D. 1 of Appendix D. Eqs. (6.28a,b,c) are the forward, central and backward difference formulas, respectively. The forward and backward formulas are used only at the end nodes; otherwise the central formula is adopted. In this way, the matrix obtained for the algebraic equations is banded and the bandwidth is five (pentadiagonal; tridiagonal if the first and the last equations are excluded). Alternatively, for any node i which is at least two nodes away from the end nodes (i.e. when 3 ~< i ~< m - 2), a 5-node central difference formula can be applied 9
2
=
C~M~+~
(6.29)
k=-2
where the coefficients C i are given in Table D.2 of Appendix D. This does not change the bandwidth but may improve the accuracy of the computation. In practice, one needs first to calculate the initial values for all variables by the initial condition (6.26). Eqs. (6.22b) and (6.24) are necessarily included in order to determine N, M, 0 and w at t - 0. Necessary reforms to these equations are to be made for the numerical computation. Replacing O0/Ox in (6.24) by M, using (6.26) and letting Q - OM/Ox, the ordinary differential equations for time t - 0 are dw dx dO dx dM dx
= sin0
(6.30a) M
-
l+h~/ -- Q
(6.30b)
(6.30c)
Ch. 6
74
Large Deflection Analysis of Poroelastic Beams
dQ MN - qncosO + qssinO + dx l+Art
(6.30d)
dN MQ - qnsinO- qscosOclx l+Art
(6.30e)
Together with the corresponding boundary conditions, the initial values can be determined. The shear force Q is involved in the numerical computation in order that all equations in (6.30) have identical forms and thus the same procedure in computation can be followed. This also helps with manipulating a boundary condition with a given shear force. The system of equations for the finite difference method can be obtained after (6.28), or alternatively (6.28) together with (6.29), are applied to each equation in (6.30). For example, the following apply for (6.30c) if only the 3-node formulas are used: i
i
Ciomi + c1mi+l + c2mi+2 -- Qi, i
C i l M i - 1 -1 CoMi + C1Mi+l -- Qi, i
c i 2 m i _ 2 -k- C i l M i _ l + C'om i -- Qi,
for i - - 1 for 2 --< i --< m -
1
(6.3 l c)
for i - - m
where all variables are valued at to and thus the superscript j ( = 0) is omitted. We will continue to do this when no confusion will be caused. Eqs. (6.3 l a,b,d,e) are omitted here since they have similar forms to (6.31c); only the right-hand sides are different. Thus 5m non-linear algebraic equations are involved, which are linearized, assuming that the right-hand sides are known. Since iteration is unavoidable in solving for the nodal values, each set of the algebraic equations (e.g. (6.31 c) alone) might as well be considered as an independent linear system. Every system of equations has the same coefficient matrix (m X m with bandwidth 5) before the boundary conditions are involved. The solution procedure is as follows: 1. Give initial values to Oi, Mi and Ni (an initial guess) in order to start an iteration; 2. solve the linear equations (6.31d) for Qi, (6.3 lc) for Mi, (6.31b) for Oi and (6.31a) for wi successively (always using the newest results to compute the right hand sides of the equations); 3. finally solve (6.31e) for Ni; 4. repeat steps 2 and 3 until the desired accuracy has been reached. An advantage of this scheme worth noting is that only in solving for Qi do we require the initial guess or the results from last round of iteration. This certainly helps the convergence of the algorithm. In an iterative procedure, the initial values for starting an iteration should be carefully chosen in order to guarantee a convergent result. In our situation, when the loads are not too high, the 'small deflection solution' (formally obtained from Eqs. (6.30) by letting 0--Ow/Ox, s i n 0 - - 0 , c o s 0 = 1 and deleting the coupling items MN/(1 + Art) and MQ/(1 + Art)) should be a good choice. However, this results in divergence when the loads are sufficiently high since the 'small deflection solution' deviates too much from the
75
Numerical Formulation
sought after solution. Nevertheless, the loading can alternately be performed step by step in the computation. In the first loading step, very small loading values are taken so that the small deflection solution can be justifiably adopted. Alternatively we can adopt a zero solution when the first step loading is taken to be zero, although the cost is higher. In each following loading step, the scheme stated above is applied with the loading values being increased by a proper increment and the results from the last step being used as the initial guess. The results are expected to converge to the solution for the required load after the loading values are finally increased to these values. Note that for a cantilever beam, there exists one boundary condition for each of the five unknown variables (w, 0, M, Q, N). Hence no difficulty occurs when they are solved separately. Otherwise, for instance, if there is no boundary condition for Q (when there are two for w, as encountered in a simply supported beam), the values of Qi cannot be determined in each step. In other words, w, 0, M and Q should be to some extent solved dependently for general cases (N has an independent boundary condition in all cases considered). Therefore, step 2 should be modified. One can always first suppose wl, 01, M1 and Q1 to be zero and solve for Qi, Mi, O i and W i successively. Wl, 01, M1 and Q1 could be determined by four simultaneous algebraic equations given by four boundary conditions and then Qi, mi, Oi and wi can be modified to satisfy the boundary conditions. This can be done in a straightforward way for a linear system (see Appendix E), but the relations for Wl, 01, M1 and Qa are not so simple in our non-linear situation. Considering the variations of (6.30) can solve the problem. When the variations of w, 0, M, Q and N are taken as unknowns, the corresponding differential equations and thus their difference equations (the latter are omitted here) are linear. For t -- 0 these are d(~ dx
= (cos0) k-16k0
d(6 k O)
6kM
dx
I+A'o
._ d(6kM)
dx d(6 kQ) dx
(6.32a)
(6.32b)
= 6kQ
(6.32c)
-- (q~cosO - q,,sinO) k-161'0 + (sinO) k-1 6kqs + (cosO) k-1 8kqn M k-16kN + N k-16kM
+
(6.32d)
I+A~/
d(6kN) -- (qssinO + qncosO) k-1 61'0 dx Mk-1 o~Q
(cosO)k-16kqs +
(sinO) k-1 o~q,,
+ Qk-1 o~M (6.32e)
l+AT/ where a variation, for example, is recorded as 6kw -- w k
-
Wk-1 , which might be produced
Ch. 6
76
Large Deflection Analysis of Poroelastic Beams
by a change of loading functions, boundary values or whatever. In a practical procedure (6.32) is considered as discretized, where k refers to an imaginary loading step. In each step the changes of the loading are finite so that the given loads can be applied in a finite number of steps. This is similar to what was done previously, except here variations are investigated. Consider now the time change. Rewriting Eq. (6.23) and substituting into (6.24) and (6.25) to replace Off&, the following system of differential equations for t > 0 is found: 0w
= sin0
(6.33a)
O0 3x - r/Mp - M
(6.33b)
OM -- Q Ox
(6.33c)
Ox
OQ Ox
-
-
-
-
qnCOSO + q s s i n O - N(rlM p - M)
(6.33d)
q , , s i n 0 - qscosO + Q(rlM p - M)
(6.33e)
-- ( l nt- /~'O)Mp -- ~ ' /
(6.33f)
ON Ox
o3X2
Integrating Eq. (6.33f) with respect to time over [tj-1, tj] and using the trapezoid formula to compute the integral of the left-hand side, as well as writing the remaining equations of (6.33) in incremental forms with respect to the time t, yields 09(AjW)
__ COS0 j - 1Aj 0
(6.34a)
3x
0(N0) -Ox
o(AJM) 0x
o(AJQ) Ox
rlAJMp - AJM
-- ~JQ
(6.34b)
(6.34c)
_ (qscosO j-1 _ qnsinOJ-1)AJ o - (riM p - M)J-I AJN - NJ-I (~qAJMp - AiM) (6.34d)
o(AJN) Ox
-- (qssinO j-1 + qncosOJ-1)AJo + (~TMp - M ~ - I A J Q + Q/-I(TIAJMp - A J M ) (6.34e)
77
N u m e r i c a l P r o c e d u r e f o r the F i n i t e D i f f e r e n c e M e t h o d
(1 + Ar/)(Mp)/ - --~
0x~
-- A A J M + (1 + A~7)(Mp)/-1 +
2 (6.34f)
where j --> 1, and the time increment at the time step j is defined as AJt -- tj - tj-1. The increments of the functions are recorded as AJw -- w/ - w/- 1, and so on. It was assumed that the loads are not functions of time. If the loads are time-dependent, additional items should be added to the fourth and fifth equations of (6.34). Thus, the solutions for longer times can be obtained successively, i.e. no simultaneity is required for all times. For the spatial coordinate x, when j is given, all equations of (6.34) can be manipulated in the same way as was done previously except for the last one, for which a second order difference for Mp should be introduced at time tj. A 5-node central difference formula given as
(
o92Mp ~
~. =
k=- 2
i j
(6.35)
C'kMi+ ~
is adopted when 3 ~ i -< m - 2. The coefficients are given by Eq. (D. 1) in Appendix D. When i : 2 or i = m - 1, a 3-node central difference formula is employed. The coefficients are shown in Table D.1 of Appendix D. When i = 1 or i = m, the difference equations for (6.340 are replaced by the diffusion boundary conditions in discrete forms. They are ( M p)/-= j 0
(6.36)
for permeable boundaries, and/or +
+
0
(6.37)
for an impermeable boundary at the le•hand end of a beam where the 3-node forward formula is used. If the right-hand end is impermeable, a similar condition to (6.37) can be obtained by a backward formula. The system of difference equations then obtained for (6.34f) has the same form as that for the remaining equations of (6.34), although the elements of the coefficient matrix (which is also banded with bandwidth being five) take different values. Therefore, Eqs. (6.34) can be solved in a similar way to Eqs. (6.32). The full procedure in solving the non-linear poroelastic problem is outlined in the following section.
6.4. Numerical Procedure for the Finite Difference Method The numerical procedure for solving the non-linear poroelastic problem is presented here. There would be no difficulty in extracting solutions for generally time-dependent loads and boundary conditions; however, for the sake of simplicity, it is assumed that the loads are imposed at t = 0 and that they are n o t dependent on time thereafter (the loads may of course be functions of x).
78
Ch. 6
Large Deflection Analysis o f Poroelastic Beams
(I) If any of the given boundary values (displacement and/or external load) are not zero, we can imaginarily load the non-zero values in steps and use k to refer to a loading step. At this stage the distributed loads are not to be loaded, i.e. q,, - qs - 6kq,, = ~ -- 0 (for t = 0). 1. Let k - 0 and take Wk, Ok, M k, Qk and N k to be identically zero (note that all boundary values are zero at this moment)" 2. let k increase its value by 1 and load a set of increments for the given boundary values; 3. determine the nodal values of o~w, o"k 0, 6kM, 64'Q, and o"kNproduced by the increments of the boundary values. First solve successively for o~Q, 6kM, o~ 0 and 6kw, respectively by the corresponding difference equations of (6.32d), (6.32c), (6.32b) and (6.32a). This is performed on the supposition that they are zero at the node xl and then modified by Eq. (E.3) with the constants being determined by the given boundary increments (when k > 1 initial guesses for ffM and 6kN are necessary, and can be taken from the last step, i.e. taken from those for k - 1). Then solve for 6"kN by the corresponding difference equations of (6.32e) using the given boundary increment; 4. repeat step 3 until the desired accuracy has been achieved; 5. calculate w k - w k- 1 nt- ~kw, ok _ ok-1 nt- ~k o, M k -- M k-1 + 6~M, Qk __ Q k - 1 + 8~Q and N k - N k- 1 + 6k N ; 6. if the boundary values have been loaded to the given values, go on to Stage II), otherwise return to step 2 above. (II) If the distributed loads (qn and/or qs) are not zero, they can imaginarily be applied in a stepwise manner, and use k to refer to a loading step; now all given boundary values are taken to be zero (for t = 0). 1. Let k = 0, (q,,)k = 0 and (q,)k = 0; Wk, Ok, M k, Qk and N k a r e taken to be the final results of Stage I, if these exist, or taken to be zero otherwise; 2. let k increase its value by 1 and give b'kq~ and 6"kqs; 3. give an initial guess for 6~0, 6"kM, and 6"kN, which are necessary for solving foro~Q: if k = 1 let them be zero, otherwise let them be equal respectively to those for k - 1; 4. determine the nodal values of 64'w, 6"k0, ~ M , o"kQ, and 6"kNproduced by the increments of the loads. First solve successively for o~Q, ~ M , ~ 0, and ~w, respectively by the corresponding difference equations of (6.32d), (6.32c), (6.32b) and (6.32a). This is performed on the supposition that they are zero at the node xl, and then modify them by Eq. (E.3) with the constants being determined by the correct boundary conditions (i.e. the value is taken to be zero at a prescribed boundary). Then solve for 64'N by the corresponding difference equations of (6.32e) using the correct boundary condition; 5. repeat step 4 until the desired accuracy has been achieved; 6. calculate w k = W k-1 + i~kw, Ok -- 0 k-1 nt- ~ko, M k = M k-1 + 6kM, Q k = Qk-1 + 6kQ N k = Nk-1 + 6kN, (qn) k = (%)~:-'+6~qn and (qs) k= (qs)k-l+ O~qs; 7. if (q~)k= q~ and (qs) k= qs go to Stage III, otherwise return to step 2 above. (III) Calculate Mp at t - 0 by (6.27), where M is now available. (IV) Calculate all nodal values for t > 0. qn and qs take on the given values (which are independent of time). All given boundary values are taken to be zero.
Examples and Discussion
79 9 j
J
1. Let j -- 0, and w/, 0/, Mj, Qr N and ( M . p ) are taken to be the final results for t = 0; 2. let j increase its value by 1 and give AJt; 3. give an initial guess for Aj 0, AJM and N N which are produced by the increment of time: if j = 1 let them be zero, otherwise let them be equal respectively to those for j - 1; 4. solve the corresponding difference equations of (6.34f) for M p using the given diffusion boundary conditions as denoted by (6.36) and/or ( 3 7 ) , etc. and let AJ(Mp) = (Mp)] -- (Mp.)j-1; 5. determine Nw, N 0, NM, AJQ and AJN. First solve successively for AJQ, AJM, Aj 0 and AJw, respectively by the corresponding difference equations of (6.34d), (6.34c), (6.34b) and (6.34a) on the supposition that they are zero at the node xl, and then modify them by Eq. (E.4) after the constants are determined by the correct boundary conditions. Then solve for AJN by the corresponding difference equations of (6.34e) using the correct boundary condition; 6. repeat steps 4 and 5 until the desired accuracy has been achieved; 7. calculate w/ = w]- 1 + Ajw, ~ = ~-1 + Aj O, M j = M j- 1 + AjM, ~ = ~ - 1 + Aj Q and N j = N j-1 + AJN; 8. if the results for more times are required, go to step 2 otherwise terminate the computation. \
]
6.5. E x a m p l e s a n d D i s c u s s i o n
In the following, various cases are analyzed using the above numerical procedure. The calculations are convergent for these cases when the loading values can be so high that the maximum rotation of a beam reaches approximately 0.7 rad (40~ When the loading is low, the numerical results obtained from the large deflection formulations are found to converge to the corresponding small deflection solutions, for which the closed form solutions are available (Chapter 3). Once again, it is noted that only two material parameters, A and r/, are involved since all variables are normalized.
A cantilever beam subjected to a uniformly distributed load Consider first the deflection of a cantilever. The conditions for Fig. 6.2a,b are the same except for diffusion boundaries. The load is suddenly applied, constant thereafter, and uniformly distributed. We see that the deflection can be both quantitatively and qualitatively changed if the flow process is changed by the diffusion boundaries. At t = 0 the deflections are the same; these are the instantaneous elastic responses to the load application. At t -- 0.85 the final shape of the beam has essentially been achieved in each case. One observes different 'creep' rates for the two cases. Note that for t -- 0.08 the deflection for the permeable case is quite close to the final deflection, while with the beam ends sealed the deflection is closer to that at t = 0 than it is to the final deflection. Furthermore, the rotation of the beam axis for the 'open' beam (the ends are permeable) is monotonic with respect to the position due to quick dissipation of the pore pressure. This is not the case for the sealed beam (i.e. both ends are impermeable).
Ch. 6 Large Deflection Analysis of Poroelastic Beams
80 0.0
o
a)
-0.3
o t=0.08
\
~ t=0.85 0.6
.0
.
.
.
.
I
0.5
.
.
.
. 1.0
0.0
0.3
0.6 0.0
0.5
1.0
Horizontal position Fig. 6.2. Shape of a cantilever subjected to a normal uniformly distributed load qn ( = 6) at times t = 0, 0.08 and 0.85 for different diffusion boundaries: (a) permeable at both ends; (b) impermeable at both ends (A = r / = 1).
Fig. 6.3 shows the variation of the beam rotation (0) at some positions along the sealed beam. The pore pressure does not finally vanish in this case. The long time pore pressure Mp is a constant, and it is known from Eq. (6.23) that it produces a constant curvature with the opposite sign to the one produced by the applied load. Consider next the pore pressure distribution for the sealed beam (Fig. 6.4b). At t = 1.00 the pore pressure is essentially constant and equal to 0.440, the value for infinite time. Eq. (2.52b) determines this long time pore pressure, which is valid for both the small and the large deflection theories (when e0 is taken as zero). If the small deflection theory were applied to the case corresponding to Fig. 6.4b, we would find Mp(x, oo) = 1/2. On the other hand, for both ends permeable the initial (t = 0) pore pressure distribution, which is identical to that for the impermeable case, undergoes rapid changes at very short times near the fixed end (see Fig. 6.4a) and of course finally decays to zero. Fig. 6.5 shows results for different loading values and 'mixed' permeability conditions, i.e. impermeable at the wall and permeable at the free end. Fig. 6.5a is the immediate response (t - 0) while Fig. 6.5b is for time t = 0.26. Noting that the curves for t -- 0 refer to elastic solutions when the fluid is trapped, we see for this specific case that when the
Examples and Discussion
81
-0.42
4
-0.47
nz=0.60 0:=:=0.80 Az=0.95 +x=l.O0
i
-0.52
-0.57
'
0.00
'
'
'
I
'
0.25
'
'
'
I
.
0.50
.
.
.
i
.
0.75
.
.
.
1.00
t Fig. 6.3. Rotation of beam axis vs. time for a cantilever. Conditions are as for Fig. 6.2b.
1.5
1.5 ~)
1.o-1 q
\
mr=0
ot--o.o8 ~t=0.z8
\
I
I
0.5
1,0
0.5
0.0
0.0 0.0
0.5 X
1.0
~r 0.0
0.5
1.0
27
Fig. 6.4. Pore pressure distributions of a cantilever with different diffusion boundaries, (a) and (b). Conditions are as for Fig. 6.2a,b, respectively.
Ch. 6
82
Large Deflection Analysis of Poroelastic Beams
0.0
0.0 I " ' 1 ~
mq=l oq=4 aq=6
\ ~
-0.2
-0.2
o
I
-0.4
0.4
d
-0.6
0.6
0.0
0.5
1.0
0.0
0.5
27
1.0
27
Fig. 6.5. Rotation of beam axis vs. position at times (a) t = 0 and (b) t = 0.26 for a cantilever subjected to a load qn = q (-- 1, 4 and 6, respectively) and impermeable at x = 0 and permeable at x = 1 (A = r / = 1).
l o a d i n g is s m a l l , the fluid flow d o e s n o t q u a l i t a t i v e l y c h a n g e the r o t a t i o n patterns. H o w e v e r , for a h i g h l o a d i n g , the p a t t e r n s will be c h a n g e d s o o n after the flow b e g i n s ; n o t e that h e r e t -
0.26 is a s m a l l t i m e c o m p a r e d to t -
c o m p l e t e (at that t i m e , the m a x i m u m r o t a t i o n for q 0.0 t
-0.2 -
A t the in-
0.0
\\"\,\. ~ , , ,
0.2
aq=6
-0.4 -
-0.6
3 w h e n the d i f f u s i o n is n e a r l y
6 is a b o u t 0.8 rad, or 45~
0.4
t
0.0
0.5 27
1.0
081
0.0
~
'
~
'
I
0.5
'
1.0
27
Fig. 6.6. Rotation of beam axis vs. position at times (a) t -- 0.10 and (b) t -- 0.26 for a cantilever. Dashed curves refer to those for t -- 0. Conditions are as for Fig. 6.5 except here the both ends are impermeable.
Examples and Discussion
83
b e t w e e n time, the r o t a t i o n is not m o n o t o n i c e v e n t h o u g h that at l o n g t i m e s m o n o t o n i c i t y m u s t n e c e s s a r i l y exist. O n the o t h e r hand, the i n c r e a s e o f l o a d i n g o n l y q u a n t i t a t i v e l y i n c r e a s e s the r o t a t i o n o f the b e a m axis in an elastic situation. T h e situation is m o r e c o m p l e x in the p o r o e l a s t i c case; w e see that the c u r v a t u r e (=O0/Ox) c h a n g e s its sign a l o n g x w h e n the l o a d i n g is sufficiently high. This is m o r e significant in the case o f the s e a l e d b e a m s h o w n in Fig. 6.6. H e r e the d a s h e d c u r v e s are for t = 0 for the v a r i o u s l o a d i n g
0.0
-0.2
a)
0 ....4
-0.4
o t=0.42
~"N~
,,t=2.40 + t=3.20
-0.6
'
l
0.0
\~~. '
0.2
l
'
0.4
l
0.6
'
"'"
0.8
0.0
o =
-0.2-
b) Q)
[] t=O o t=0.20
-0.4
-0.6
t=l.O0 + t=l.20
'
0.0
I
0.2
'
I
0.4
I
0.6
0.8
Horizontal position Fig. 6.7. Shape of a cantilever subjected to a lateral displacement at the free end, which is suddenly applied and remains unchanged thereafter (w(1, t) = A = -0.6). The fixed end (x = 0) is impermeable, A -- 6 and r / = 1. (a) Permeable at x = 1, the curve for t = 3.20 is overlapped by the one for t = 2.40; (b) impermeable at x --- 1, the curve for t = 1.00 and 1.20 essentially overlap.
Ch. 6
84
Large Deflection Analysis of Poroelastic Beams
values, and are the same as shown in Fig. 6.5a since they are elastic responses. It is interesting that the rotation at the free end changes very little from the initial elastic response, while the rotation at the middle positions changes significantly.
A cantilever beam subjected to an i n s t a n t a n e o u s deflection Consider now a cantilever, which is given an instantaneous vertical deflection by applying a lateral displacement A at the free end. Now restrain the free end (x = 1) from further vertical movement, i.e. let w(1, t) = A. We observe the shape of the beam as a function of time. Fig. 6.7 shows the cases for different diffusion boundaries. In the case of a beam sealed at both ends (Fig. 6.7b), the vertical deflection grows monotonically except at very near to the 'free' end. If the beam permits diffusion at the 'free' end (Fig. 6.7a), the middle part of the beam reaches its maximum deflection at approximately t - - 0 . 4 2 . The deflection here then decays and the beam approaches its final shape at t = 2.40. This final shape, however, is not the initial shape as would be the case for small deflection. To further clarify the situation we focus on the change of the vertical displacement w at the center of the beam, for situations when A takes different values; the beam is permeable at the free end. Fig. 6.8 shows that if the deflection is very small, the beam will finally
-0.0156
-0.062 t
-0.0178
-0.072
-0.13
-0.20
-0.15
-0.24 0
1
2
t
3
4
0
1
2
3
4
t
Fig. 6.8. Lateral displacement vs. time at x - 0.5 of a beam as in Fig. 6.7a with A taking different values.
Examples and Discussion
85
0.0 or=0.03
/
or:0.20
/ / / ~
0"0 I o x = O . O
//I
a~~"~_
J
-0.2"
-0.2
-O.4
-0.4
-0.6 0.5
1
0
z
1
2
t
Fig. 6.9. Pore pressure for a simply-supportedbeam subjected to a normal uniformly distributed load qn (--- 10) with boundaries impermeableat x -- 0 and permeable at x - 1 for A - 77= 1: (a) pore pressure distributions at times t -- 0.03, 0.20 and 1.00; (b) pore pressure decay at positions x - 0.0, 0.3 and 0.7. return to its initial elastic position. In a large deflection situation, the beam cannot return to the elastic position after diffusion is finished. The larger the deflection, the farther (relatively) the final position is from the elastic position.
A simply supported beam subjected to a uniformly distributed load Consider now the pore pressure for a simply supported beam (one end is free to move horizontally) subjected to a uniformly distributed load. One end of the beam is impermeable and the other is permeable. For such diffusion boundaries the diffusion progresses slower than it would with the same diffusion boundaries at both ends. Pore pressures are shown in Fig. 6.9. For this beam, the fluid flow is not negligible until about t = 5, although the pore pressure has been reduced to less than 20% of its peak value by t = 2. The initial state is such that M and thus Mp are symmetric with respect to x = 0.5 (necessarily zero at both ends). One observes that the pore pressure at the sealed end (x = 0) increases very quickly soon after the diffusion begins and the symmetry is broken. After it reaches the peak value at about t = 0.20, it decays and finally vanishes.
A clamped beam subjected to a uniformly distributed load Figs. 6.10 and 6.11 show the results for a beam fixed and impermeable at the both ends (one end is free to move horizontally), subjected to a uniform load. In Fig. 6.10, exactly antisymmetric patterns for the rotation are observed as expected. Fig. 6.11 illustrates the way the pore pressure vanishes in such a beam. (Note that the pore pressure finally vanishes even though the beam is impermeable at both ends; cf. Section 3.2.)
Ch. 6
86 0.70
Large Deflection Analysis of Poroelastic Beams
-
( I
0.35
U
0.00
-0.35
m t=0 o t=0.04 a t=0.26
-0.70
|
0.0
0.2
0.4
0.6
0.8
|
|
!
1.0
Fig. 6.10. Rotation of beam axis vs. position for a beam clamped and impermeable at both ends A = r / = 1).
(qn
--
100,
4.0 t=O o t=O.01 A t=O.08 + t=0.26 o
,
2.0
0.0
-2.0
~
0.0
0.1
0.2
I ~
0.3
0.4
0.5
Fig. 6.11. Pore pressure distributions of a beam clamped and impermeable at both ends (q, = 100, A = 77 = 1). Half is shown due to symmetry.
Examples and Discussion
87
0.2
0
0.0
~
-0.2
0.4
-3 :[ ~
0.6
,
0
1
t
2
-4 I 0
o Small deflection theory 0 Large deflection theory
. . . .
,
1
. . . . 2
t
Fig. 6.12. Pore pressure decay at positions (a) x = 0.27 and (b) x -- 0.60 for a beam which is fixed and impermeable at x = 0 and simply supported and permeable at x = 1. (qn -- 25), A = 4, 7/= 0.25.
A beam clamped at one end and simply supported at the other, subjected to a uniformly distributed load Finally, consider a beam clamped at one end and simply supported at the other, subjected to a uniformly distributed load. The differences between the solutions obtained respectively by the small and large deflection theories, when the loading is very high (i.e. the deflection produced is very large) are shown in Fig. 6.12. A and ~/have been assigned values for which the differences are easy to observe. Here one clearly sees the necessity of employing a large deflection theory. The pore pressure moment, which finally decays in the beam, can suddenly increase at certain positions for a period of time. This is seen dramatically in Fig. 6.12a for a position on the beam (x = 0.27) where the pore pressure is generally relatively low. The sign of the pressure even changes two times (cf. Fig. 4.3). A much sharper change is observed in the large deflection situation than for the small deflection approximation. For locations on the beam where pore pressure is relatively high (x = 0.60) we see from Fig. 6.12b that at short times there are still significant differences between small and large deflection theories. In summary, it has been shown that the numerical technique developed here makes it possible to determine deflections and pore pressures for beam-like elements made of a poroelastic material exhibiting axial fluid diffusion when subjected to general loading and having any boundary and diffusion conditions.
This Page Intentionally Left Blank
Chapter 7 STABILITY OF POROELASTIC COLUMNS
In this chapter, we consider three stability problems of poroelastic columns: quasi-static buckling, post-buckling and dynamic stability. For the buckling problem, the time-dependent behaviors of the critical loads and deflections are considered for various diffusion and geometrical boundaries. The time-dependent column response for the case of a constant load is also given. For the post-buckling problem, the time-dependent behavior of the columns, governed by three coupled equations, is obtained by using the large deflection theory. These equations are transformed into a single one, enabling the analytical derivation of the initial and the final responses. It is shown that, unlike the quasi-static response obtained by using the small deflection theory, the long time response derived here is bounded. The imperfection sensitivity of these columns is also investigated. For the dynamic stability problem, the stability conditions and boundaries are found, together with the critical (minimum) loading amplitude for which instability may occur.
7.1. B u c k l i n g o f C o l u m n s
Fluid-saturated poroelastic beams subjected to axial loads P are shown in Fig. 7.1. w denotes the transverse deflection of the beam axis if buckling occurs. Again, only nondimensional variables are considered. Quantities are normalized as in Eqs. (2.44). In addition, the point load is normalized as follows (7.1)
P* = PLZ/EI
The superscript * will be omitted hereafter for simplicity. The partial differential equations governing the problem are 0x2
32Mp o~X2
(2.45b)
- ~'/Mp + M = 0
3Mp -
Ot
33w - A
o~X2 0t
-- 0
(2.48b)
Among the three unknowns, w, Mp and M, the last can be determined in terms of the first by equilibrium of the beam in an obvious manner for the situation of statically determinate problems. For a simply supported beam, M = Pw.
90
Ch. 7
Stability of Poroelastic Columns
w P
x
(a)
(b)
Fig. 7.1. Beams subjected to axial loads: (a) a simply supported beam; (b) a cantilever; (c) a beam clamped at x -- 0 and hinged at x -- L.
The geometrical boundary conditions for w are the same as those in the elastic case. The diffusion boundary conditions imply conditions on w. For a free end, for instance, if no transverse loads are applied at the end, then considering Eq. (2.45b), the permeable diffusion boundary condition is given by W, xx = 0 and the impermeable diffusion boundary condition is given by W,xxx -- 0. After Mp is eliminated from the two governing equations, (2.45b) and (2.48b), one obtains 04w o~X4
o4M +
o~X2
33w = (1 + An)
o~X2 3t
3/14 +
3t
The required initial condition when a load is suddenly applied at t - - 0 obtained from Eq. (2.49b), by inserting (2.27b). Then we have 02w 3x 2
(7.2) is readily
m -- -
~
1 + h'q'
t:
0
(7.3)
7.2. L i m i t s o f C r i t i c a l L o a d
The buckling of the poroelastic beam is generally expected to be a time-dependent process due to the viscosity of the pore fluid. Three ranges of loading must be considered: for t - 0, for t = oo and for any t in between. The lower and upper limits of the middle range will now be determined.
Limits of Critical Load
91
Using PcE to represent the dimensionless critical load for the poroelastic beam with no pore pressure, i.e. the drained elastic solution, it is concluded from Eq. (7.3) that the critical load for the case in which the fluid is trapped is (1 + Ar/)PcE. This is the maximum critical load for the poroelastic beam because a load which is greater than this value will result in purely elastic buckling before any diffusion has time to take place (Biot, 1964). Thus, pmax E cr = (1 + Ar/)Pcr
(7.4)
Further, it is noted that the critical load for the poroelastic beam, Pcr, should not be less than PcE. If the beam is not sealed at both ends, then when P - PcE is applied and the pore fluid can flow out of the beam (due to even the tiniest imperfection), albeit slowly, the pore pressure would decrease. After an infinite time, the beam will reach its balance in an elastic state (i.e. there will be no fluid flow and no pore pressure at all). Thus the minimum possible critical load for a beam with at least one end permeable is pmin __ P~cr
(7.5)
Consider now the lower limit for a sealed beam, i.e. the beam is impermeable at the both ends. After being integrated over [0,1] with respect to x, and with the impermeable diffusion boundary condition (i.e. Mp,x--0) applied to both ends of the beam, Eq. (2.48b) can be written as follows:
0 10( w) Mp +
d
-0
(7.6)
where it has been assumed that the operations of derivative with respect to t and the integration can be interchanged. Thus the integral in Eq. (7.6) is not a function of t but is rather a constant. Further, that this constant should be zero, is implied by Eqs. (2.45b) and (7.3) at t = 0 when the fluid is trapped. Actually, the conclusion that this constant is zero does not depend on the initial conditions. It is a specific form of Eq. (2.41b), after M is replaced by (2.14b). Hence Mp+A~x 2
dx=0
(7.7)
This is true for all times. As a special case consider t = oo, when Mp(x, oo) should be a constant with respect to position since no diffusion occurs then. Then from Eq. (7.7), -
A
[ow
-
ow
]
(7.8)
As mentioned above, if Mp -- 0 then the lower limit is given by Eq. (7.5). Therefore, Eq. (7.5) can also apply to a sealed beam (i.e. both ends impermeable), as long as the requirement below is satisfied 0w 0w -~x (0, t) - -~x (1' t)
(7.9)
This is the case for the beams shown in Fig. 7.2a,b. For cases in which Eq. (7.9) is not
Ch. 7
92
Stability of Poroelastic Columns
(a)
(b) Fig. 7.2. B e a m s with rotation being zero at both ends.
justified Mp depends on w, which is not a priori determinable in a buckling situation. Thus the lower limit cannot be determined in an obvious manner, but clearly --crPmin~ PerE. 7.3. T i m e - D e p e n d e n c e of Critical Load and Deflections
The time-dependent nature of the critical load will now be investigated. We first define what is meant by a time-dependent critical load. Suppose that a load P0 (pcmrin --~ P0 -< p-cr m a x ]J has been applied to the beam at t __ 0 and the deflection is w(x, O) __ ~(x) at this moment; we now determine the time dependent change of the critical load Pcr(t) when the deflection is required to remain unchanged with time (i.e. w(x, t) = r t > 0). The point is that if at any time P(t) > Pcr(t) the deflection will increase, leading ultimately to buckling. Then, by Eq. (7.2), we arrive at d4~
oZm
t
dx 4
--
o~X2
om
(7.10)
o~t
M can be obtained by the equilibrium of the beams. For the beams shown in Fig. 7.1, M can be expressed as follows:
M--
P~
Fig. 7.1a
P ( v ~ - 6)
Fig. 7.1b
P~-R(1-x)
Fig. 7.1c
(7.11)
where 6 is a constant and the reaction R is a function of time. Thus Eq. (7.10) can be written as d4~ d2~, dx 4 + P - ~ --(~-
dP ~b) d-7
(7.12)
where (h, in accordance with the different cases, is defined as
~b-
0
Fig. 7.1a
6
Fig. 7.1b
Ro(1 - x)
Fig. 7.1c
('7.13)
Time-Dependence of Critical Load and Deflections
93
For the beam shown in Fig. 7. lc, the following relationship has been used:
R(t) = RoP(t)
(7.14)
where a dot refers to a derivative with respect to t and R0 is a constant. If the relation were not true, the fight-hand side of Eq. (7.12) would be ~(x)P(t) - (1 - x)[~(t). Then a particular solution of Eq. (7.12) for v~(x) would be (1 - x)[~(t)/P(t), which would be a function of t. This is in contradiction to the definition of v~(x). So Eq. (7.14) is justified. Actually Eq. (7.12) can be adapted to all types of beams, with q5 being a constant or a linear function of x. Now, solving Eq. (7.12) for P(t) yields
P(t)-
l~(4)(X) +
P0 +
~"(x)
exp
~"(x)
t
(7 15)
~(x) - 4,
where a prime refers to a derivative with respect to x. Since the fight-hand side of the above expression is not a function of x, one concludes that ~(4)(X )
~"(x)
~tt(X)
-- - C l ,
v~(x)- q5
-c2
(7.16)
where Cl and c2 are constants. Further, the two constants should be equal so that the solution can exist (~b is at most linear). Also it is clear that the constants are positive. Then c l = c2 - - c 2 can be determined by solving the boundary problems. However, its value can be determined by noting that p(oo) - c 2 and thus c 2 _ lcrPmin- W e note that c 2 can take on higher values also, which correspond to higher mode shapes. By taking c 2 to have the unique value pmin --cr we then limit v~(x). But for our purposes this is satisfactory, since the higher c 2 imply much greater elastic buckling loads and our purpose here is only to determine load paths guaranteed not to cause buckling (growth of deflection). Hence the critical load is given by Per(t) = --cr pmin + (P0 -- Pcr min)
e- P e rmin., x t) p
(
(
7
.
1
7
)
We can also determine the buckling modes by Eq. (7.16) and the corresponding geometric boundary conditions, giving as follows for the different cases shown in Fig. 7.1:
v~(x) =
8 sin(cx)
for Fig. 7.1 a, c = n 7r
811 - cos(cx)]
for Fig. 7.1b, c - (2n - 1 ) rr/2
Ro[-COS(CX) + sin(cx)/c + 1 - x]
for Fig. 7.1c, c = tan(c)
(7.18)
where n -- 1, 2, 3 ..... and only n = 1 is physically viable if there is no constraint between the ends of a beam. v~(x) is found to satisfy the permeable diffusion boundary conditions at both ends for the case shown in Fig. 7.1a, and the impermeable diffusion boundary condition at x = 0 and permeable diffusion boundary condition at x-= 1 for the cases shown in Figs. 7.1b and 7.1c. Thus, these diffusion boundary conditions are required in order that Eq. (7.17) is justified for the beams shown in Fig. 7.1. Generally speaking, the requirement is that the pore pressure vanishes at long times, or that pmin __ pcE. In the limiting situation, when P0 takes the value pmax _ (1 + Ar/)PcEr, the beam buckles instantaneously. Then Eq. (7.17) takes its special form (just below pmax)
94
Ch. 7
Stability of Poroelastic Columns
Pcr(/) = pcEr[1 -+- /~'17 exp(-PcErt)]
(7.19)
On the other hand, if P0 takes the value Pc~r, it will take an infinite time before the beam buckles. At this point it is necessary to reconsider Eq. (7.17) and to clearly understand its implications. First, recall that a critical situation is defined as that when an initial existing deviation of the column axis from straightness - imposed by any means whatsoever- will remain constant. Then Eq. (7.17) prescribes a loading path for a load whose initial value is P0 so that a critical situation is achieved. Any deviation, which brings the loading path above that given by Eq. (7.17), will ultimately lead to buckling. Now suppose the loading path up to some time t was such that it did not result in a critical situation. Then, what continued loading for time t + r (r > 0) will result in a critical situation? This is answered by considering Eq. (7.12) and noting that its solution was not in any way dependent on the past history (prior to r = 0+) of the column. Thus the following expression is found to be fully equivalent to Eq. (7.17): Pcr(t + r) = pmin + [ P ( t ) - pmin]exp(-pcmrinr)
(7.20)
In other words, Eq. (7.20) is identical in form to Eq. (7.17) with P0 replaced by P(t), the 'current' value of the load at time t which does not cause buckling. Again, any deviation, even for the shortest time, which brings the loading path above that given by Eq. (7.20), can lead to buckling. In other words to avoid buckling the slope must be more negative than that of the expression in Eq. (7.20). Taking the derivative of Eq. (7.20) with respect to r and setting r = 0+, the requirement dt <-
_cmri.[P(t)
-
-
pmin
]
(7.21)
guarantees that the column will not buckle at time t + (This inequality ceases to be of interest when P(t) falls below pmin since below this value buckling is not possible.) As an example, consider in Fig. 7.3 those loading paths for which the initial load value P0 is between pmin and pmax. Load path 'a' is that given by Eq. (7.17); the situation is always critical. Loading path 'b' (the dashed curve) will not cause buckling until point 'A'; at 'A' the column buckles. This is so since for times less than that at point 'A' inequality (7.21) is satisfied; at 'A' it is no longer satisfied. This is clear visually simply by comparing path 'b' with path 'a'. At every point on 'b' before 'A' the slope is steeper than that for the equivalent point on path 'a' with the current value of P. Obviously, the straight-line reduction of P from an initial P0, as shown in curve 'c', will not cause buckling. The load level will 'escape' to below pmin before any deflection, existing but reduced from the initial value, or imposed during the process, can possibly grow. Consider now the case in which a load P is given and held constant, pcmrin < P < ~pmax cr and the deflection will be time varying. For a beam with arbitrary geometric conditions, M can always be expressed by m(x, t) = Pw(x, t) - m(x, t)
where m is at most a linear function of x. This is an extension of Eq. (7.11).
(7.22)
Time-Dependence of Critical Load and Deflections
95
p
max cr
Po
P.(t)
<
0
\
A
P.min
time Fig. 7.3. Different loading paths.
The differential equation for this situation can be obtained by Eq. (7.2), using the constancy of P, to give
04w o~X4
+ P
OZw ~
-- (1 + At/)
03w
Ow
+ P~ o~X2 8t 8t
Om 8t
(7.23)
By the method of separation of variables, a solution for Eq. (7.23) can be obtained as follows:
w(x,t)- [blcos(wx) + b2sin(wx) + fit(x____)pa]exp(cet)
(7.24)
where fit(x) is given by rh(x, t) = fit(x)exp(at), which is justified because w is assumed to be the product of an exponential function of t and a function of x. Noting that to2 is the critical load for the corresponding elastic beam, a can be identified by the differential equation, Eq. (7.23), as (p
E
E
- Pcr)Pcr c e - (1 + Ar/)PcEr - P
(7.25)
Therefore the amplitude increases exponentially with time since a > 0. It should be noted that Eq. (7.24) is true only while the deflections are small enough for the linearized theory to be valid. Using dimensional quantities, the exponent of Eq. (7.24) becomes
atoc
(e-pmin)( g pmax_p
~-t
where the constant of proportionality depends on the geometric boundary conditions. Here we clearly see that the growth rate of the deflection is proportional to the ratio of the
Ch. ',1 Stability of Poroelastic Columns
96
variance of the load from the minimum buckling load to the variance from the maximum buckling load. We further note that for a given load level between the minimum and the maximum, growth rate is inversely proportional t o L2", such a situation is counter intuitive for elastic buckling, but in the present situation greater L increases diffusion times.
7.4. Post-Buckling: Formulation In the previous sections we discussed the buckling of poroelastic columns subjected to axial loading. The analysis was performed by using the small deflection theory, so that the problem in general was linear. However, in order to display the actual load-deflection relations it is necessary to employ large deflection theory, which makes the problem nonlinear. This kind of analysis is customarily called post-buckling. Post-buckling analyses of elastic columns have shown that the load carrying capacity is slightly increased beyond the buckling load. Consider the simply supported poroelastic column, as shown in Fig. 7.4. When slender columns are subjected to axial loading, the resulting deflection may be large, and thus the analysis should be according to the large deflection theory. Yet, the deformation can still be small and elastic. Hence, the columns can be considered as geometrically non-linear and constitutively linear. Biot's constitutive law and Darcy's law then still apply.
IP m
)
w(x,O
"~- w
~////////, Fig. 7.4. Column configuration.
97
Post-Buckling: Formulation
By using the large deflection theory presented in Chapter 6, and when the axial strain is negligibly small, as is to be expected for a slender column, the governing equations necessary for the post buckling analysis are Ow 3x
- sin0
M +
3O 3x
(6.22b)
- r/Mp - - 0
(6.23)
c92/ld~" *"P -- /~lrp -- A ~~---~ -- 0 3x 2 3x
(6.25)
These equations are in terms of non-dimensional variables, normalized as in Eqs. (6.21) and (7.1). The asterisks are omitted for the sake of convenience. 0 is the angle of rotation of the column cross-section, M is the bending moment, which for a simply supported column is given by M = P w , and P is the end load. Eq. (6.22b) is the geometrical relation, Eq. (6.23) is the constitutive law and Eq. (6.25) results from Darcy's law. The initial condition is determined by the requirement that there is no instantaneous diffusion, and results in the relationship 30
M - - ( 1 + Ar/)~x,
t= 0
(6.26)
Hereafter, a comma will denote spatial differentiation. It is noted that the critical buckling load of a simply supported drained column, in the non-dimensional form, is PeEr - "rr2 pmin~
Differentiating Eq. (6.22b) with respect to x yields O,x-
(7.26)
w, xx
cos0
Let c o s 0 - x/1 - s i n 2 0 - r
-
(W,x)
2 ~
1
1
-
-~(W,x)2
(7.27)
and 1
1 --~ 1 + -(W,x) 2 cos0 2
(7.28)
by which 1
O x -- W, xx + -~ W, xx(W,x)
2
(7.29)
It is noted that the above approximations imply an error of less than 0.2% for 0 -< 20 ~ By introducing Eq. (7.29) into Eqs. (6.23) and (6.25), one obtains 1 2 W,x x + -~ W, xx(W,x ) -- "liMp -+- P w -- 0
(7.30)
98
Ch. 7 Mpa x - Mp - A
+
Stability of Poroelastic Columns
W, xx(W,x) 2 - - 0
(7.31)
and by eliminating Mp the following single non-linear equation for the time-dependent response of the column is derived: 1 W, xxxx -[- ~ W, xxxx(W,x) 2 "-[- 3W, xW, xxW, xxx + (W, xx) 3 + Pw
o[
1
= (1 + A71)--~ W, xx + -~ W, xx(W,x)
2] + Pw
(7.32)
Using the separation of variables method, and in order to satisfy the boundary conditions of a simply supported column, the solution function is given by w(x, t) = f(t)sinTr x
(7.33)
This function does not satisfy Eq. (7.32) due to the coupling between the lower mode (sinTrx) and the higher mode (sin37rx). However, this coupling is rather weak and its effect can be neglected. Thus, and by making use of the trigonometric identities for sin3~b and cos 2 ~b, Eq. (7.32) is obtained in the form
4f+g1 rr6f 3
_
p@
[
= (1 + Ar/) - ~ r 2}
-
g
rr4f 2}
]
+ P}
(7.34)
1.2
e=O
1.0 0.8 0.6
0.4 0.2 J
0.0@ 0.0
0.1
0.2
fo
0.3
0.4
0.5
-,- cr , e = 0 (a perfect Fig. 7.5. The variation of the initial central deflection,f0, with the non-dimensionalload, Plpmax column) and e = 0.005, 0.01.
Post-Buckling: Results and Discussion
99
or
j~= [ -P + rr2 + -81 7.i.4f2],.~f [ 3 77.4f2] (1 + ArD r r 2 + g
P-
(7.35)
In order to solve Eq. (7.35) one needs to consider the initial condition. Introducing Eqs. (7.29) and (7.33) into (6.26) yields P-
[ 14 2]
(1 + A~/) -tr2 + g
,
t--0
(7.36)
from which
f(O)-
2x/~ A[ P 7r ~ (1 + An)rfl
1
(7.37)
Another form of Eq. (7.36) is obtained by noting that (1 + A~)rr 2 = pmax P 1 -- 1 + --7r2f 2, pmax 8 --cr
t-- 0
(7.38)
The expression given in Eq. (7.38) is in the form of the well-known elastic post-buckling curve of perfect columns, and is shown in Fig. 7.5 by the curve labeled e = 0, f0 --f(0).
7.5. Post-Buckling: Results and Discussion Eq. (7.38) reveals that at t -- 0 the load applied to the column can be greater than the maximum buckling load derived from the small deflection theory. Namely, P -> --crPmax-The 'penalty' is that the deflection will increase in a parabolic form. The question one should consider now is what is happening as time passes. The response, as we can see from Eq. (7.35), depends on two variables - the load P, and the magnitude of the poroelastic effect Ar/. In the following, we obtain the numerical solution of Eq. (7.35) by using the high order Runge-Kutta method. In Fig. 7.6 we check the influence of the first parameter; the load is changed from 1.1 to 1.15 and 1.2 times the maximum critical load while the magnitude of the poroelastic effect is kept unchanged ( A ~ - 0.1). Hence, in this case pcmrax = (1 + A~)w 2 -- 1.1 rr2. In order to see how AT/influences the response, the load in Fig. 7.7 is kept constant P = 1.32zfi, while the magnitude of the poroelastic effect is c h a n g e d - A~ = 0.01, 0.05 and 0.1 so that the cases of pmax _ 1.01 rr2, 1.0577"2 and 1.1 ~.2 are considered (in all cases the applied load is higher than the maximum critical load). It is first observed from these figures that in all cases the responses are increasing with time but approaching an asymptote at long time. That is, unlike the prediction obtained by using the small deflection theory, here the response is bounded. The long time response, the asymptote, can be found by noting that at this time 3~ = 0. Thus, from Eq. (7.35), we get
Ch. 7
100
Stability of Poroelastic Columns
0.55 -
0.50
p=1.2 9=1.15
0.45
9=1.1 0.40
0.35
0.30
0.25
I
'
0.0
I
0.2
'
I
0.4
'
I
0.6
"
I
0.8
'
I
1.0
Fig. 7.6.f vs. time, for p = 1.1, 1.15 and 1.2; At/= 0.1" p-- p]pmax.
0.52
0.50
0.48
7
Zxl=O.05
0.46 _~
--
o
0.44 0.42
0.40 0.0
'
i
0.2
'
I
0.4
'
I
0.6
Fig. 7.7.f vs. time, for At/-- 0.01, 0.05 and 0.1.
i
0.8
'
I
1.0
Post-Buckling: Results and Discussion P-
7r2(1 + g
t-
101
oo
(7.39)
or f(oo)---- 2 ~ J ~ j L
-I
(7.40)
7/" ~ T W
It can be seen from Fig. 7.6 that higher loading implies bigger response; the time dependent behavior of the curves is similar. The responses in Fig. 7.7 are starting from different points but approaching the same asymptotic value since the loading is the same for all these cases and the final response does not depend on At/. Another important difference from Fig. 7.6 is that here the time-dependent behavior is not the same. This is so since it depends on At/; higher value implies a slower approach to the asymptote. Knowing the exact initial and final deflections, we can ask about the contribution of the parameters discussed above on these results. The ratio between the final deflection and the initial one is f(oo) _ foo -f(0) f0
--
1 +An
-
P
-
P
=
P
1 +At/
--/Jcr-max
_ --X
(7.41)
p -- 1
p/pmax 9Fig. 7.8 shows the variation of f0, foo and X with p. The critical load wherep ---or is kept constant, pmax_ 1.1 w; (AT/- 0.1), while the loading is increasing from P = 1.117r2 to P - 1.32"n"2. For very small p, the curve of fo is non-linear, unlike the rest of this curve and that of foo, which are almost linear. As a result, the ratio between these _
5
_
~
4-
_
.
~
I
1.00
'
!
1.05
'
!
1.10
'
!
1.15
'
Fig. 7.8. The variation off0,f~ and X with p; At/- 0.1.
I
1.20
Ch. 7
102
Stability of Poroelastic Columns
1.3J 1.2 1.1 1.0
0.9 0.80.7-
i
0.6-" .
0.5f - . o _ ~
0.4 I
1.00
'
I
1.02
'
I
1.04
'
I
1.06
'
I
1.08
'
I
1.10
l+Lrl Fig. 7.9. The variation off0,foo and X with (1 + AT/);P = 1.32"n"2.
curves is sharply increasing when p is decreasing (X can reach very high values when p is close to one). Fig. 7.9 presents the variation of f0, f~o and X with (1 + At/). Here the applied load is kept constant, P - 1.32 77"2, while the magnitude of the poroelastic effect is increasing from A~7 = 0.001 to 0.1 (pmax _ 1.0017r2to 1.17r2). While f0 is decreasing when At/is increasing, the value of f~o remains unchanged since it does not depend on At/. This implies that X is strongly increasing with At/. It is noted that the case of A t / = 0 denotes the elastic case, for which the post buckling response is not time-dependent. For this case f0 = foo and X - 1.
7.6. Imperfection Sensitivity In the above formulation, it was assumed that the column is perfectly straight, that the loading is concentric and that the material from which the column is made is perfectly homogeneous. However, since nothing is really perfect, minor imperfections should be introduced into the analysis. In the case of elastic elements, such an analysis shows better agreement with the results obtained experimentally. In the following, a geometrical imperfection is introduced, assuming that that the centroidal axis is initially bent, say with 00 -- Oo(x). Since the bending strains are caused by the change in the curvature and not by the total curvature, Eq. (6.23) is now
103
Imperfection Sensitivity
(7.42)
M * + (Ox - Oo,x) - rtMp -- 0
Here 0 is the total angle of rotation, including the initial one and the angle developed due to the loading. However, since 00 is small (7.43)
Oo,x = Wo,xx
where w0 is the initial deformation. The corresponding form of Eq. (7.30) is hence 1
W, xx + -~ W, xx(W,x)
2
- Wo,xx - ~lMp + P w -- 0
(7.44)
Eq. (7.31) remains unchanged since the initial deformation is not time-dependent. For our case of a simply supported column, the initial deformation is assumed to be in the form w0 = e sinvrx
(7.45)
where e is an amplitude, considered to be small. Following the same steps as in the analysis of a perfect column, the column motion is now governed by the equation
f_[-
1]
4
P + Tr2 + _ 4 f 2
P-
vr2f _ vr e
(7.46)
3 r4f: ]
(1 + Ar/) vr2 + -~
and the initial condition is Pw-
- ( 1 + ,kr/)(0,x- Oo,x),
t= 0
(7.47)
yielding
[
e
P - - (1 + At/) "rr2 - 7rZf + g
,
t= 0
(7.48)
Note that heref is the total deflection, including the initial one. Its value at t -- 0 is the root of f2_8[ -~
P ] (1 + ,~/)7r2 - 1 -
8 e 7r2 f ,
t-0
(7.49)
which is applicable for any value of P (and not only greater than the maximum buckling load, as in the perfect case). Plots of (7.49) for several values of the initial imperfection are shown in Fig. 7.5. As can be seen, the major difference from the perfect case is at low loadings and small deflections. The time-dependent response of a column with a very small initial imperfection and a loading greater than the maximum buckling load is very similar to that obtained for a perfect column. Fig. 7.10 presents the results obtained for the case where p = 0.95 (the applied load is lower than the maximum critical one), A~/= 0.1, and e = 0.005, 0.0005, 0.00005 and 0.000005. It can be seen that all the curves start at different initial values (see Eq. (7.49)), they approach different asymptotes, and that the time needed to approach the asymptotes is longer for smaller imperfection. For very small imperfections there is a
Ch. 7 Stability of Poroelastic Columns
104 0.25
0.20
~~
e=O.O05
e=O.O005
0.15 0.10
.
0.05
0.00
I
0.0
'
I
0.5
'
I
1.0
'
I
1.5
'
I
2.0
Fig. 7.10.f vs. time, for e -- 0.005, 0.0005, 0.00005 and 000005; p = 0.95, At/= 0.1.
critical time for which the response remains almost unchanged. The smaller is the imperfection the longer is the critical time. At long time j ~ - 0, and the final response obtained from Eq. (7.46) is the root of f2
8[P --~-~--1
]
8 e
-- z f l f '
t = oo
(7.50)
The initial imperfection influences also the level of X, the ratio of the final response to the initial response. Fig. 7.11 a shows the variation of X with (1 + h~) for the same conditions as Fig. 7.9 but with e - 0.05, 0.005 and 0.0005, 0.005 and 0.0005. Since P > ~pmax for all cr cases, X basically behaves like in the perfect case. Yet, X is increasing with the decreasing of e. In Fig. 7.11 b the applied load is 1.067r2. As long as it is greater than --crpmaxthe behavior is as in Fig. 7.11 a. When it is lower than the maximum critical load, and the imperfection is very small, X is increasing very rapidly. The reason is that for this case the initial response tends to zero (as in the perfect case) while the long time response is finite. This increasing is more pronounced when the imperfection is smaller (even though the final response is reducing with an increase of the imperfection, see Fig. 7.10), and with a decrease of p (=
max
P]Pcr ).
7.7. Dynamic Stability of Poroelastic Columns Consider now the problem of the transverse vibration of a simply supported poroelastic
Dynamic Stability of Poroelastic Columns
105
1.30 = 1.25
a
e=O.O005 eiO,O05
1.20
.
~
~
1.15 1.10 1.05 1.00 i
1.00
'
I
1.02
'
!
'
1.04
I
1.06
'
I
1.08
'
!
1.10
I+~TI 1815 e=O.O005 12 e=O.O01
"-
I
1.00
'
I
1.02
'
.
I
1.04
'
I
1.06
'
i
1.08
'
!
1.10
1+~1 Fig. 7.11. The variation of xwith (1 + At/) for e = 0.05, 0.005 and 0.0005; (a) P = 1.32zr2; (b) P = 1.06zr2.
column with a uniform cross-section, subjected to a periodic longitudinal loading. The constitutive equations, in the form of global variables for the quasi-static buckling problem
Ch. 7
106
Stability of Poroelastic Columns
are
o~w
E1 ~
O2Mp K
o~X2
(2.14b)
- riM p + M -- 0
OMp -
Ot
O3W - AEI
OZxOt
-- 0
(2.22b)
where for a simply supported column M -- P w, and P is the end load. In our case the load is time-dependent, given by
P(t) = Po cos0t
(7.51)
In order to arrive at the equation for the transverse vibration of the column it is necessary to differentiate Eq. (2.14b) twice with respect to x and to add the inertia force acting on the column. Doing so one obtains
0%
o~Mp
o~w
E l y - ~- -- T] oqX2 -+- P o c o s O t ~
o~w
+ p--~
--0
(7.52)
where p is the mass per unit length of the column. Using the separation of variables method, and in order to satisfy the boundary conditions of a simply supported column, the solution functions are given by the following Fourier sine series: 130
w(x, t )
E 3 ~ sin j=l
jTrx L
oo
Mp(x, t) -- ~ . mj sin j=l
(7.53a)
J qT"X
(7.53b)
L
where L is the length of the column. In the following, the main stability region is investigated. To this end only the j = 1 term in Eqs. (7.53a,b) is considered. Let ce -- 7r/L, and substituting Eqs. (7.53) into (7.52) and (2.22b), the equations governing the motion of the column are now in the form
Elce4f -+- ~lce2m -
Poce2cos(0t)f
+ pf = O,
-Kce2m - rh + AElce2f -- 0
(7.54a, b)
where m = ml. By solving Eq. (7.54a) for m and putting this expression together with its time derivative into Eq. (7.54b) a single differential equation involving f only is obtained P f + [(1 + Ar/)E/ce2 - Pocos(Ot)~ + P10sin(Ot)f
OL2
-g[p? Let
+ (EIol 4 - Pooz2cos(Ot))f ]
(7.55)
Stability Boundaries and Critical Load Amplitude ~~2 __ ~Og4, E1
~2 = (1 + AT])o)2,
p
107
PE : (1 + A ?q)Eloe2, (7.56)
Po 2p E ,
T-
6 : Koe2
by which the equation of motion is in the form jr + s
- 23/cosOt)j~ + 20ysin(Ot)f]- - 6 [ f + w2(1 - 23,(1 + A~l)cos(Ot)f] (7.57)
where 3/and 6 are small and PE is the static buckling load of the fluid-saturated column. However, since the left-hand side of Eq. (7.57) is the time-derivative of the Mathieu equation, it is possible to proceed by investigating the stability of the following equation instead: f+
~2(1--23/cosOt)f:
--6(je + w2~f(t)dt -
o9223/(1
+ Arl)~cos(Ot)f(t)dt)
(7.58)
7.8. Stability Boundaries and Critical Load Amplitude In this section the stability properties of the solution of Eq. (7.58) are investigated. To this end the multiple scale method is employed in order to obtain asymptotic solutions (Bender and Orszag, 1978). This method is based on the fact that physical processes that determine the behavior of the structure take place on distinct time scales. In the system investigated here, three different time scales control its development. The first time scale serves as a reference to the others and hence is considered throughout this work to be of order 1. The second time scale, associated with the loading amplitude P1, is given by 1/% and is assumed to be much bigger than 1. The third time scale is associated with the diffusion characteristics and is given by 1/& As was shown in Chapter 5, the diffusion time of a beam of length L is equal to and is considered in the following to be much longer than the period time of the drained beam. The basic conditions for instability are derived by considering the original form of the Mathieu equation
L2]K,
f + ~22(1 -- 23/cosOt)f = 0
(7.59)
Assuming a regular perturbation expansion for f(t)
f(t) =f0(t) + yfl(t) + O(y 2)
(7.60)
where 3, is small, substituting it into Eq. (7.59) and comparing powers of 3/gives the equations
fo + 02fo = O,
f l + 02fl = -02(23/cosOt)f
(7.61)
It is first noted that when O -- 0 the solution to the lowest order equation is unbounded in time, known also as secular - grows linearly with time. Then, the solution for this equation is
108
Ch. 7
Stability of Poroelastic Columns
f(t) = Fexp(il2t) + c.c.
(7.62)
where c.c. stands for complex conjugate. Introducing this solution into the first-order equation yields ):l + J~fl - -F0[exp(i(J2 - O)t) + exp(i(O + 0)t)] + c.c.
(7.63)
In general, secular terms appear whenever the inhomogeneous term is itself a solution of the associated homogeneous constant-coefficient differential equation. In the above case, thus, secular terms appear only if O + 0
+s
(7.64)
which can only happen if (since g / ~ 0) 0 O -- 2
(7.65)
Weturnnowtofindingtheinstabilityboundariesinthe(J~2, 0, 7, 6) space in the limit of 7 and 3 being small. A regular perturbation expansion in 7 and 6 results in solutions that contain secular terms. In order to eliminate these secular terms, the multiple scales analysis is employed by introducing two new independent variables ~- 7 t and ( -- 3 t. Within the multiscale analysis t, ~-, and ( are considered as three independent variables. As a result, f is expanded in the following form: f(t, ~', ~) =f0(t, 7, ~) + yg(t, T,~ + 6h(t, 7 , 0 + O(3}, 82, y~)
(7.66)
and the second time derivative o f f as dt 2
Ot2
\
3td7
c)t2 ]
~ OtO;~
c)t2 ]
+ O(3,2,
73)
(7.67)
Further, since the regions of interest are near ~Q= 0/2, ~Qis expanded as s
= ~ + 7-(g + O(y~) 4
(7.68)
where, to the first order of 7, ~Ql measures how close half the loading frequency 0/2 is to the natural frequency /2. Eqs. (7.66), (7.67) and (7.68) are inserted into Eq. (7.58) and coefficients of equal powers of 7 and ~ are collected. The lowest order terms yield the following equation for f): c~t2
+ ~f0 = 0
(7.69)
and its solution is given by ]~)(t, 7,~) = A(~-,~ e x p ( i O t) + c.c. Terms of first order in y and 3 yield the following equations, respectively:
(7.70)
109
Stability Boundaries and Critical Load Amplitude
[,o A A+A*]expl,OI
Ot2
Ot2 +
4
0---~
-4- h
--
-2i
2 ~
-
-~ t
+ NST + c.c.
4(1 + a~/) ~-~Z - i ~ Z
exp i ~ t
(7.71)
+ NST + c.c. (7.72)
The terms on the right-hand side of Eqs. (7.71) and (7.72) that multiply exp(i0/2) are the secular ones since they give rise to unbounded solutions. The rest are non-secular terms (NST). In order to eliminate the secular terms, it is required that the coefficients multiplying exp(i0/2) vanish. Let (7.73)
a(r, ~ = T ( r ) Z ( ~
and by introducing this into Eqs. (7.71) and (7.72) 0 o-o r~ -i/2~zT + i ~02T* -- 0
(7.74)
az 1 An + - ~ Z asr 2 (1 + An)
(7.75)
= 0
These have solutions r(r)-
r, exp
-~ ~6 - ~
+ T2exp
-~
- ~
(7.76)
and Z(~) - Z0exp - ~ 1 + At/ respectively, where T1, T2 and Z0 are constants. Inserting Eqs. (7.76) and (7.77) into Eq. (7.70) yields the following form of the zero-order solution: fo(t, r, ~ - Foexp -
exp _+
- ~1
exp i-~t
(7.78)
where F o is a constant, obtained from T~, T2 and Z o. Then, by rewriting r and ~"in terms of t, we see that instability occurs when Y -0
- ~l > --
-2 (1 + A t / )
(7.79)
Hence, using Eq. (7.68), it is concluded that for small y and 6 the stability boundaries are given by
110
Ch. 7
Stability of Poroelastic Columns
0 2f~
stability region
1.0
I I I I I I I I I
I Y~
y
Fig. 7.12. Stability boundaries in a qualitative form.
(7.80) or
2~Q = 1 + -
-2--y
l+)~rt
~-
(7.81)
The critical (minimum) value of the loading parameter, for which instability occurs, is 2 Art 0(1 +Art)
Tcr - - - -
c~2K
(7.82)
A qualitative Strutt diagram of the above results is given in Fig. 7.12. It is noted that in the elastic (drained) case A = rt = 0, and thus ]/cr is equal to zero, as is expected since Eqs. (2.22b) and (7.52) are not coupled, and thus the column vibrates without a damping mechanism. The physical meaning is that the stability regions in the poroelastic case are expanded with respect to those of the elastic counterpart, mainly due to the shift of Ycr, which is identically zero in the elastic case. This phenomenon is similar to what is found in the case of viscoelastic structures (Touati and Cederbaum, 1994; Stevens, 1966).
Chapter 8 ANALYSIS
OF POROELASTIC
PLATES
In this chapter a theory is developed for fluid-saturated poroelastic plates made of a material for which diffusion is possible in in-plane directions only. Both transverse and inplane loading are considered. The plates considered are isotropic in the plate plane and obey the Kirchhoff hypotheses. Biot's constitutive law is adopted and Darcy's law is used to describe the fluid flow in pores. The basic equations are so derived that they could easily be extended to include orthotropic poroelastic plates. An analytical solution procedure is developed and examples are given.
8.1. Basic Equations for Thin Plates In this section, the necessary equations are established. First the three-dimensional constitutive equations are to be reformed to adapt them to the plate. The constitutive equations for the transversely isotropic poroelastic material, according to Biot (1962), are as follows: rll -- 2Bae]l + B2(~~ q- ~~
q- B3~~ q- B6ff
r22 = 2B1 42 + B2 (8] 1 + 42) + B3 e33s + B6 sr
7"33 --B48~3 -+- B3(8]1 -+- 42) -+- B7~
s
(8.1)
~'12 -- 2B1812 %3 = 2Bse~3
r13 = 2Bse]3 Pf -- B6(~~ + 42) q- B7e~3 + B8 where the material is taken to be isotropic in the x 1 - - X 2 plane. B m (m = 1-8) are the material constants introduced by Biot. rij are the total stresses of the bulk material (ij = 1,2,3), i.e. the stresses averaged over both the solid and the fluid phases, and e,~ are the strains of the solid skeleton, pf is the pore pressure and ~" is the fluid content increment, which can be calculated by
~ - ch(e~i-
eifi)
(8.2)
112
Ch. 8
Analysis o f Poroelastic Plates
where q5 is the porosity and eiif is the volume dilatation of the fluid. A repeated index (e.g. i or j) refers to a free index and the summation convention is adopted, except when specifically noted otherwise. Eliminating ~" from the first three equations of (8.1) by using the last equation of (8.1), the constitutive equations have the form
'7"22
--
7"33 pf-
F[sr-
/~12
Bll
B13
g~2
B13
B13
B33
43
-
C~l Pf
(8.3)
a3
t~l(6"]l nt- 8',~2) -- t~38'~3)]
and s ~q2 = 2G e12,
~'23 -- 2G3e~3,
~'31 -- 2 G 3 e ~ 1
(8.4)
where the material constants introduced here can be expressed in terms of Blot' s constants through /)ll
--
2B1 +
1)13 -- B3 Bll
G =
--
2
B 2 --
B6B7 B8
B12
~,
B8
/~12 -- B2
/~33 -- B4
-
B--8
-
B8
(8.5) ,
G3 =B5
B6 OL1
B8
B7 ce 3 - -
B8
F = Bs
Only eight independent material constants are involved since G can be considered as a derived parameter. We consider a Kirchhoff plate with an arbitrary boundary. The assumptions of Chapter 2 for the classical beam theory are valid here too. The Xl - x2 plane is taken to be the midplane of the plate, as shown in Fig. 8.1. Since for a state of plane-stress ~33 = 0 we have from the last equation of (8.3)
~176 =
o~3p f -- /~13(8~1 --I- g~2) /~33
(8.6)
Using Eq. (8.6), e~3 can be eliminated from the first and second equations of (8.3), and hence the constitutive equations for the plate can be simplified as (a, 13 and y, 6 = 1, 2) ~'~13 = D~3ya eva - rlPft~a3,
st"-/3pf + r/e~,~
(8.7a, b)
where the stress-strain relations for ~23 and ~31 are omitted because the transverse shear strains are zero in classical plate theory. Here and henceforth the superscript s has been
Basic Equations for Thin Plates
113
111 X2
Xt
I X3
Fig. 8.1. A rectangular plate with dimension L 1 • L 2 • h (h << LI ,L2) and subjected to distributed normal load.
omitted from the skeletal strains. 6~3 is the Kronecker delta, i.e. 6~t~ = 1 if 0 / = 13 and 6~13 = 0 otherwise. Here the stiffness coefficients for the transversely isotropic plate are D1111 - -
D2222 - - /~1
Dl122- vb
(8.8)
D1212 --/5(1 - v)/2 Dl112 = D1222 = 0 and --
1
F
nt-
0/~
/~33 '
B13
T~ - - 0/1 -- ~ 0/3 B33
(8.9)
where /5--
B l l -- /~23
/)33 and B12 - / ~ 2 3 / B 3 3 Bll -/~23//~33
Those elements of the fourth-order tensor D~e8 which are not given in Eqs. (8.8) are defined by the following symmetry relations: D~t~,~ = D~t~8e = D ~ , ~ = D~,8~
(8.10)
Thus far a total of four independent parameters,/5, v,/3 and r/, are involved; these are determined by seven of the eight independent material constants given in Eqs. (8.5). We
114
Ch. 8
Analysis of Poroelastic Plates
now consider the physical meaning of the parameters. From Eq. (8.7a) it can be seen that the Ds~3vs, which involve b and v, are the stiffness coefficients of the drained plate (or when the pore pressure is zero), and from Eq. (8.7b) that 77 is the ratio of the pore volume change to the sum of the two normal strains (area strain) within the plane for the drained plate. Thus Ds~3v~ and ~7 depend only on the elastic properties and the microgeometry of the solid skeleton. On the other hand,/3 depends also on the fluid compressibility, and can be interpreted as the fluid content increment when the pore pressure equals one and the area strain in the plane is zero. Furthermore, by considering a tensile test for the drained plate, i.e. Tll 5~ 0, 7"22- 0, it is found out that ~o22=--b'~~ and 7"11- D ( 1 - /,'2)811. Hence v is the Poisson's ratio and E - / 3 ( 1 - u2) is the drained Young's modulus as discussed in the beam case. The geometrical relations for the plate are considered next. These are
Ss~-
1
-~(Us,~ + u~,,~)- x3w, s~
(8.11)
where us is the displacement at the mid-plane of the plate in the x~ direction and w is the perpendicular deflection of the mid-plane, us,/3 refers to the partial derivative of us with respect to xt~, and so on. These relations are the same as those for a thin elastic plate. Taking notice of the symmetry of the tensor D~3va, Eq. (8.7a) can then be rewritten using Eq. (8.1 1) as %/3 = D,~va(uv, a - x3w, v s ) - r/Pf6s~
(8.12)
Further, in order to express the above relations in global quantities only, we introduce the stress resultants and the stress moment resultants (shown in Fig. 8.2) defined by
Ns~
_ f h/2 rs~ dx3 , J - h/2
M~ =
f h/2 r~x3 dx3 J - h/2
(8.13)
Inserting Eq. (8.12) into the above, yields the constitutive relations in the form
Ns~ = Ds~vahuv, a + rlSs~Np,
M ~ = -Ds~valw, va + rlS,~Mp
(8.14a, b)
where
p-- m
lh/2 pf dx3, - h/2
Mp = -
re/2 pfx3 dx3 .I- h/2
(8.15)
are pore pressure resultants and pore pressure moment resultants, respectively, and I -- h3/12. Note that the subscript p, as the subscript f previously introduced, does not refer to an index. The equilibrium of an element of the plate (Fig. 8.2) is now considered. The equilibrium equations should be the same as those for an elastic plate in terms of the resultants and the load, i.e. they are N,#3,~3 = 0,
Mst3,t3= Qs,
Qs, s + q - p # = 0
(8.16)
where p is the mass per unit area of the plate, which can be position dependent. The shearing forces Q~ (also shown in Fig. 8.2) are the resultants of 7.3~ over the thickness of the plate, and q is the distributed normal load, which is generally dependent on the
115
B a s i c E q u a t i o n s f o r Thin P l a t e s
JI
lq
dxl
-I il 12
Q2
Fig. 8.2. An infinitesimal free body from a thin plate, showing all the internal resultants on the visible sides. Moment vectors are shown with double arrows. coordinates xl and x2 and the time variable. These equations can also be given in terms of the displacements, the pore pressure resultants and the pore pressure moment resultants by using Eqs. (8.14) Dst3~hu~,t3 ~ + rINp, s -- O,
D s t 3 ~ l w , st3~ - *lMp,as - q + P ~ -- 0
(8.17a, b)
In summary, two sets of partial differential equations consisting of nine equations have been established: six are given in Eqs. (8.14) and three in Eqs. (8.17). They reduce to the equations for an elastic plate when r / - 0. It is noted that in these nine equations eleven variables, Nst~, Np, Mst~, Mp, us and w, are involved. In order to find the two additional equations required, the other aspect of the physics of the plate problem, i.e. the diffusion within the plate plane has to be considered. The fluid flow is in accordance with Darcy's law, t~(l) m
--
~f_~) --
kmpf,m,
m-
1,2,3; no summation on m
(8.18)
/xf where U_mand U_m "f are the displacements in the Xm direction for the solid and fluid phases, respectively. They are, generally speaking, functions of the coordinates Xl, x2, x3 and the time t./xf is the fluid viscosity and k m is the permeability in the Xm direction. This form of Darcy's law is obtained directly from the generalized Darcy' s law (Biot, 1962) when only elements on the diagonal of the permeability matrix are non-zero; this is justified for orthotropic materials (which includes the material considered here). Further, in our case kl - k2 due to the material being isotropic in the Xl-X2 plane and k3 - 0 since the present f f material does not permit flow in the x 3 direction. Noting that F_,ii D Ui,i and F_,ii -- Ui, i, the combination of (8.2) and (8.18) yields ~_
kl ~ P f , ss /xf
(8.19)
in which ~"can be replaced by (8.7b), and further the strains in (8.7b) can be calculated by (8.11); Darcy's law now takes the form
116
Ch. 8
g P f , ow~ -
Analysis o f Poroelastic Plates
- x3w,~=)
Pf + As
(8.20)
where the newly introduced constants are given by kl
r/ a -- D/3
K -- ~ f / ~ ,
(8.21)
Thus five independent physical parameters, 15, v, a, r / a n d K are required for defining the material properties of the problem./3 will be considered as a derived parameter hereafter. In order to express the results in terms of the pore pressure resultants, Eq. (8.20) is first integrated over the thickness of the plate, and is then multiplied on both sides by x3 and integrated over the thickness, to yield KNp,ao ~ - -
] V p -Jr-
aDha~,~ - o,
KMp,o, a
- - l~'Ip - -
~.L)Iw,,~o, -- 0
(8.22a, b)
Thus the two sought after additional partial differential equations have been found, and therefore the system of differential equations is complete for the 11 variables. In the remainder of this section, the boundary conditions and the initial conditions for the problem are discussed. The geometrical boundary conditions and the load boundary conditions are the same as those for an elastic plate. They are Uo~ - - -~oe,
W -- W,
W,r
(8.23)
-- W,r
and Nr - Nr,
Mn -- Mn,
Hn -- Qn + Ms,s - Hn
(8.24)
where the subscript r takes the value of n or s which denote the normal and tangent of the boundary respectively. Neither n nor s is an index. An overbar refers to a given function. As for the diffusion boundary conditions, the pore pressure or its derivatives with respect to the normal of the boundary are given. In terms of the resultants, for a permeable boundary they are Np -- Np,
Mp -- M e
(8.25)
and for an impermeable boundary N p , n - - O,
(8.26)
Me, n -- 0
Here the quantities in the directions of the normal and tangent to the boundaries can be calculated by the corresponding variables in the coordinate directions and the direction cosines of the boundaries. Let n and s be unit vectors normal and tangent to the boundary, with components n,~ and s~; i.e. n~ - cos(x,~, n),
s~ - cos(x~, s)
(8.27)
where x~ is a unit vector in direction a. s l - - n 2 and s2 - n l. N r --
N~n~r~,
M r --
m~n~r~,
Qn
-
Q~n~
where r - n, s are coordinate transformations (e.g. Washizu, 1982), and
(8.28)
Basic Equations f o r Thin Plates W,r = w , ~ r ~
117
(8.29)
Mp, n = M p , ~ n ~
Np, n --- N p , a n a ,
which are purely mathematical relations for derivatives. Finally, the initial conditions are considered. Physically the initial values for different variables should be compatible. If the loads are suddenly applied at t = 0 - , the relationship between the pore pressure and the displacements at t = 0 is determined by requiring ~"-- 0, which means that there is no instantaneous diffusion immediately after the application of the loads. Using Eqs. (8.7b) and (8.11), one obtains t-
f 3 p f + rl(u~, a -- x 3 w , a~) - - O,
(8.30)
0
Using Eq. (8.15) gives Np - ADhu~,~ - 0,
Mp + A[)Iw ~ -- 0,
t-- 0
(8.31a, b)
Two equivalents of the above can be obtained after Eqs. (8.14) and (8.17) are substituted respectively; they are Nail - w .-,inst ~ v s n. u v , ~,
M,~
__
- w. - -~, i nvs t~ w. , ~8,
t -- 0
(8.32)
and Dinst
~v~ uv,t3~ -- O,
.-,inst
, t)~/3vjw,,~3v ~ - q + pf~ -- O,
t -- 0
(8.33a, b)
where Dinst
(8.34)
~ 3 v 8 - D~t3v8 + h~lD6~t36v~
can be considered as the tensor of the instantaneous stiffness at t = 0 when the fluid is trapped, since Eqs. (8.32) and (8.33) have the same forms as the counterpart equations for an elastic plate. In terms of the Young's modulus and Poisson's ratio (note that E-/)(12 ) , as discussed earlier), the following are the instantaneous quantities, respectively: E(1 + v + 2At/) finst-- (1 + At/)(1 + v)'
v + At/ /"inst- 1 + An
(8.35)
Moreover, considering the definition of the fourth order tensor, Eq. (8.33b) can be simplified for the transversely isotropic plate as follows: DI(1 + an)w ~ t ~ - q + p# - 0,
t-
0
(8.36)
Thus all the necessary equations and conditions for solving the plate problem have been found. If we choose the eleven variables, N~/3, Np, M,#3, Mp, us and w, as unknowns, the governing differential equations are the constitutive relations (8.14), the equilibrium equations (8.17) and the motion equations (8.22). Together with the geometrical boundary conditions, (8.23), the load boundary conditions, (8.24), the diffusion boundary conditions, (8.25) or (8.26), and the initial conditions, (8.31) or (8.32), the problem is defined. Noting that the stretching and bending of the plate are not coupled, there are only six and five simultaneous equations respectively for the stretching and bending. Moreover, when the boundary conditions can be decoupled from N~/3 and M~I3, then if only five unknowns, Np,
118
Ch. 8
Analysis of Poroelastic Plates
Mp, us and w are taken, Eq. (8.14) will not be included in the simultaneous differential equations. The meaning of this is that there are only three and two simultaneous equations for the stretching and bending problems, respectively.
8.2. Bending Equations in Non-Dimensional Form The plate problem has been formulated for both in-plane deformation and bending. The two types of deformation are not coupled and therefore they can be considered separately. It is noted that the in-plane problem was considered for the case of plane strain, which is formally identical to the present plane stress problem. See, for instance, McNamee and Gibson (1960a,b) and Rajapakse (1993). In the next sections, closed form solutions of the bending situation are determined. For this purpose the relevant equations are now presented in non-dimensional forms. The quantities are normalized as follows: , _ x~ x~ -- L~' ,_ w
t*
w h'
Kt = L2,
M;~-
T--Kv
M~L~ DIh '
5 p DI, ,
Mp--
q MpL 2
DIh '
* - - qL4 DIh' L1
K-- L2
(8.37) where L~ is, generally speaking, the maximum dimension of the plate in the x~ direction. Then the governing equations, (8.14b), (8.17b) and (8.22b), take the following dimensionless forms respectively: Mll--(w, M~2-
ll
+ pK2w~22) -+- "oMp (8.38a, b, c)
-(/,,W?l 1 q-- K2W~22)--I- '0M;
M12---(1-
V)KW?I 2
V4W * -- T/V2M; -- q* + y2f0* - - 0
(8.39)
VZMp - Mp - AVZw* - 0
(8.40)
where the differential operator V2 is defined as ~72 -
02
02
2
+ K2 ~
(8.41)
x2:
and the higher order operators are defined as V4 - v z v 2, V6 - v z v 4, etc. The dimensionless forms of all boundary conditions remain unchanged. However the initial conditions, (8.31b) and (8.36), should be rewritten as follows: Mp +/~,V2w * - 0,
t* -- 0
(1 nt- /~'0)V4W * -- q* - y2fO*,
(8.42) t* = 0
(8.43)
Analytical Solutions for Quasi-Static Bending
119
8.3. Analytical Solutions for Quasi-Static Bending In the following, the quasi-static problem is investigated based on the formulation given in the previous sections. Accordingly, the inertia term is deleted from the relevant equations. All quantities discussed in this chapter are normalized as by Eq. (8.37). However, for convenience, the superscript * will be omitted. Eliminating Mp from Eqs. (8.39) and (8.40), gives the governing equation for w V 6 W - (1 + A'r/)V4fi;- V Z q -
0
(8.44)
Together with the initial condition (8.43) and the corresponding boundary conditions, it is now possible to obtain some analytical solutions from Eq. (8.44) for the deflection w. In order to simplify the solution procedure, the solution of the drained plate is introduced. It is denoted as w E - wire=0 and can be obtained from the elastic theory of thin plates. In other words, w E is the poroelastic solution when t approaches infinity if at least a finite segment of the plate boundary is permeable so that the fluid will finally be drained. We restrict ourselves here to this case. The solution is restricted to the plates subjected to time-independent loads, q - q(xl,x2) after t - 0, i.e. suddenly applied and then remaining constant in time. Taking the solution in the form w-
w E + Aw
(8.45)
Aw is to be determined now. The relevant governing equation can be obtained from Eq. (8.44). Noting that V4wE - q, vi,E - 0 and 0 - 0, we have V6(Aw)-
(1 -q- AT/)V4(Aki;)
(8.46)
The initial condition (8.43) is seen to be equivalent to AT/
Aw]t=o--
- w l+Ar/
E
(8.47)
by noting that WE
wit=~
1 + At/
For convenience, the boundary conditions for Aw are written according to the types of the constraints. For a simply supported and permeable boundary, when no moment applies at the boundary, the boundary conditions for Aw are mw -- 0,
Aw, 11 -- 0,
mw, 1111 - 0
(8.48a, b, c)
if the boundary is parallel to the x2-axis. Among these Eq. (8.48a) is a geometrical boundary condition. Eq. (8.48b) is true because of W, ll - - 0 which is given by (8.38a) due to Mll -- 0 (no moment), w,22 -- 0 (any derivative with respect to x2 is zero) and Mp = 0 (a permeable boundary). Finally, observing Eq. (8.40), VZw - 0 gives 72Mp - 0, which in turn gives V4(Aw) -- 0 by Eq. (8.39). Thus Eq. (8.48c) is derived, considering that the partial derivatives with respect to x2 are zero.
120
Ch. 8
Analysis o f Poroelastic Plates
Eqs. (8.48) can be extended to a boundary with arbitrary direction, i.e. for a simply supported and permeable boundary, as long as no moment applies at the boundary. Here Aw - 0,
02Aw On2
- 0,
04Aw On4
- 0
(8.49)
For a clamped boundary parallel to the xz-axis, if it is impermeable, the boundary conditions for Aw are A w - O,
OA w Ox , - O,
o~ A w oo A cv Ox~ - a ~ Ox---~-,
(8.50)
where the last equation is obtained from (8.39) and (8.40), noting that Mp,1 -- 0, wE _ 0 and that the derivatives of Aw with respect to x2 are zero. If the direction of the boundary is arbitrary, we write A w - O,
OA w On - O,
o~ A w On5
03 A Cv
- Ar/~
On3
(8.51)
Now consider the solution of (8.46). Let it be of the form (8.52)
A w -- f ( x l , x 2 ) e x p ( - c 2 t )
where c is a constant. The exponent is written as above because it must be negative so that Aw approaches zero for long time t. The unknown function can then be found from the following differential equation: V6f + c2(1 + Ar/)V4f = 0
(8.53)
Finally, Eqs. (8,39) and (8.40) can determine Mp as follows:
~/],~/p- V4(Aw)- AT]~721,i:
(8.54)
Alternatively Mp can be obtained in an integral form "qMp = -A~VZw +
I
t V 4{Aw(x 1,x2, "/')}dr
0
where the values of Mp and V2w at t - 0 (8.42).
(8.55)
have been eliminated by using the condition
A rectangular plate with all boundaries simply supported and permeable The boundary conditions (8.49) for Aw apply to all the boundaries. They are apparently satisfied, noting that the solutions of Eq. (8.53) have the form fmn -- Amnsin(mTrxl )sin(nerx2)
(8.56)
2 is determined from Eq. (8.53) as where m and n are natural numbers. Thus Cmn
2 Cmn =
(m2 at- KZn2)q7"2 l+A~7
(8.57)
Analytical Solutions for Quasi-Static Bending
121
Since the governing equation (8.46) is homogenous, a possible solution for Aw takes the form oo
AW -
oo
2
Z
~ fmnexp(-cmnt)
(8.58)
m--1 n=l
The coefficients A m n solution is given by oo
oo
w E -- ~
ZAEmnsin(mTrxl)sin(nTrx2)
m=l n=l
AEn
-
can be found by the initial condition (8.47), in which the elastic
4
-
7r4(m 2 + K2n2) 2
Xmfl qsin(m rrx)sin(n rry) dx dy 0
(8.59a, b)
0
and the coefficients can be given in explicit form after the integration is carried out for a particular loading. Those coefficients, which are required in the numerical computations considered in this chapter, are given as follows. For a uniformly distributed loading (i.e. q - constant), 16q+ K2n2)2, m,n odd a E n _ f rr6mn( m2 0,
otherwise
For a point load P applied at x 1 - AE,, =
4P
77-4(m 2 + KZn2) 2
-~1
and x 2 - -
-~2,
sin(mrrxl)sin(nrrY2)
Finally, the deflection can be written as co
co
(8.60)
w -- ~" ~. AZmnCbmn(t)sin(mrrx,)sin(nrrx2) m=l n=l
where dPmn(t)
--
At/ exp( 1 - 1 + h~ -
(m2 -+-g:2n2)772 ) 1 + An
t
(8.61)
and AEn is given by Eq. (8.59b). The pore pressure moment resultant can be extracted from Eq. (8.55) Mp -
-
(
)
~. ~ l BmneXp - (m2 + K2n2)~ t sin(m rrxl)sin(n rrx2) m=l = 1 + At/
(8.62)
where nmn - - ~
t~
l+ar/
7r2(m 2 q-
K2n 2)amn E
and AE,, is again given by Eq. (8.59b).
(8.63)
Ch. 8
122
Analysis of Poroelastic Plates
Before some examples are presented and discussed, some comments concerning the numerical results for all cases considered in this chapter are given. For the sake of convenience, x and y are used to refer to the coordinates instead of Xl and x2, and the expression pore pressure is used to refer to Mp since the pore pressure and Mp have the same patterns. All quantities involved in the figures have been normalized. In particular, the pore pressure, loads and time are normalized by L1 so that a larger K refers to a narrower plate (a smaller dimension in the y-direction when the dimension in the xdirection is fixed). In Fig. 8.3, some typical features of the time-dependent problem are shown for a suddenly applied uniform load on a square plate; A = 7 7 - 1 is taken for this situation since changing these parameters does not qualitatively change the figures. Fig. 8.4 compares pore pressure distributions for square and rectangular (K -- 2) plates subjected to a central point load. Note the shape difference of the distributions in the width and
%
% *
~
o
*
o
o .
.
0.0
.
.
.
I
.
.
0.5
.
.
Y
'
1.0
'
'
I
'
'
'
I
0.0
0.2
0.4
0.0
0.2
0.4
X o o
o
la,
's i/
0.0
o
o t=0.z0
0.5
1.0
t Fig. 8.3. Deflection and pore pressure vs. position and time for a square plate subjected to a uniformly distributed load q = 1. y = 0.5 is taken for w-x and Mp-x curves, and x - 0.5 and y = 0.5 is taken for w-t and Mp-t curves.
s
9
t~ r~
t~
t~
9
t~
II
9
.~.
t~
r~
0"
r~
t~
9
O'Q
e~
o_
~l .o_
0
r
[
.
0.00
.
'
0.00
I
1
t\
~1~
0.00
.
.
~.
'
.
~
~
'
'
~
I
0.05
~,
0.05
I
0.11
.
.
.
.
.
.
~
.
.
I
0.10
0.10
!
r
.
.
0
.
0
.
.
.
.
PP.'='.'='
+
.
o o o o
0.22
o
I
I
I
0.15
i
0.15
0.33
~.lvlS-!svn 0 1of suo!lnloS lW~.ldlvuV
Mp
Ch. 8 Analysis of Poroelastic Plates
124 0
cO-O.
GO--
0
C:~
C~
,,
_
ot=O.
.
o t=O.O02
_
,, t = O . 0 2 0
.
+
t=0.060
r
r
,s
i 0.00
r
0.25
0.50
z
0.75 1.00
0.00
0.25
0.50
0.75 1.00
y
Fig. 8.5. Pore pressure distributions for a square plate subjected to a point load P = 1 at x = 0.75 and y -- 0.5. The positions are taken to be across the loading point.
length directions. Fig. 8.5 considers the square plate but with the point load located at the three-quarter point on a symmetry axis. The pore pressure distributions are shown for traverses through the point of loading. Note that for the x-traverse the m a x i m u m pore pressure is not found at the point of loading except at time t -- 0.
Cylindrical bending of rectangular plates of infinite length in one direction Cylindrical bending occurs for a rectangular plate with, for example, L 2 - - ~ and loading which is independent of x2. In this case, all variables are functions only of Xl and t, and the plate problem degenerates to a plane-strain beam problem. By Eq. (8.53), the ordinary differential equation is
d6S
dx 6 + c2(1 + At/) dx~-~1 -- 0
(8.64)
from which a general solution for f can be found. Thus a particular solution can be obtained for a problem with the boundary conditions. Two examples with different boundaries are given. Consider first a plate which is simply supported and permeable at Xl = 0 and xl = 1 and is not subjected to moments at the edges. Taking
Analytical Solutions for Quasi-Static Bending
125
fm -- bmsin(m~-xl)
(8.65)
and 2
c =
(mYr) 2
(8.66)
l+A~7
where m is any natural number, and thus by (8.52)
zXw- ~. bmsin(mTrxl)exp
- 1 + At/t
(8.67)
m=l
the boundary conditions (8.48) are satisfied. Further condition (8.47) as follows:
b m can
2A~7 ~iwE(x)sin(m,n.x)d x b m = - i A-------~ +
be determined by the initial
(8.68)
The corresponding pore pressure moment resultant is
Mp
= - --1
y. bm(m~)2sin(m~.xl)exp
-
T/m=l
1 + A~ t
(8.69)
The second example is for a plate, clamped and impermeable at Xl = 0 and Xl = 1. We will assume for the moment that Aw(x, co) = 0 is true as is required for the technique to be valid. We demonstrate below that this is so. Following the steps above, after the boundary conditions (8.50) are applied, we get ~o
A w - ~. bm[1-
cos(2mwxl)]exp
((2m,rr)2) - 1 + At/t
(8.70)
m=l
and the initial condition (8.47) requires oo
AT~
~. bm[1 -
E
cos(2m~'xi)] - - 1 + A----~w
(8.71)
m=l
Noting that Aw(1 - x l , t ) = Aw(xl,t), i.e. the function is symmetric with respect to Xl -- 1/2, Eq. (8.70) is valid only for symmetric problems. Thus WE is also symmetric with respect to Xl -- 1/2. In order to determine the coefficients bm, derivatives are taken on the both sides of Eq. (8.71) as follows: oo /'" m=l
2m ~bmsin(2m'rrxl )
AT/
dw E
= - 1 + a----~ d x 1
(8.72)
Then the left-hand side of Eq. (8.72) must be a Fourier sine series of the right-hand side with period 1. Thus the coefficients are determined
bm = -
(1 + A~)m~
~1
sin(2m'rrXl)dXl
and the pore pressure moment resultant is
(8.73)
Ch. 8
126
Analysis of Poroelastic Plates
0 0
r
Q
O3 0 Q
0
r
o Simply-supp. P=I o Simply-supp. q=l A Built-in P=I
.
0 Q
Q Q Q
~&..~--'&
/
&
03
&
~~+-------+
'
'
'
0.0
I
'
'
'
0.2
r
!
0.4
,
,
,
----,
0.0
~
,
0.2
t
9
0.4
t
Fig. 8.6. Deflection and pore pressure vs. time at the center of the span of a plate for different loadings and boundaries in the cylindrical bending cases.
0 0
0
ox=0.
o
\ A
\
"o
X{-l}
o==0.15 x(-l)
\
,, ==0.35
Q Q
. . . .
0.000
I
. . . .
0.025
I
. . . .
0.050
I
. . . .
0.075
I
0.100
t Fig. 8.7. Pore pressure decay at different places in a cylindrical bending case for a plate with built-in ends and subjected to an uniformly distributed load q = 1. The curves for x = 0 and x = 0.15 are plotted with the sign reversed.
Analytical Solutionsfor Quasi-StaticBending 1 ~
Mp - ~ Z bm(Zmrr)2c~
127
((2mrr) -
2 )
1 + Arl t
(8.74)
m=l
Figs. 8.6-8.8 show the results for cylindrical bending, in which two types of boundaries are involved: simply-supported and permeable at the two ends, or clamped and impermeable at the two ends (built-in ends). The latter case is solvable by the present technique even though all edges are impermeable because special conditions are fulfilled such that the elastic solution with elastic constants taken as those under drained condition is the solution for long times. The built-in ends, together with their impermeability, guarantees that the long time pressure goes to zero. The proof of this is completely analogous to that given in Section 4.4. Two sets of loads are considered: a uniformly distributed load over the whole plate (q = 1) or a uniformly distributed line load along x = 0.5 (P = 1). Fig. 8.6 shows the time dependence of central deflection and pore pressure; note particularly that for the clamped and impermeable end conditions 'steady state' is achieved in a much shorter time than for the other condition. From Fig. 8.7 (built-in, impermeable), it is seen that the pore pressure decay is not necessarily monotonic. Now, consider Fig. 8.8 with both ends built-in. For the case of P = 1, the pore pressure necessarily remains antisymmetric with respect to x = 0.25 after such an initial pore pressure distribution is created, considering that diffusion is produced by the pore pressure gradients. Therefore the pore pressure at x = 0.25 remains zero. For the case of q = 1, the
Q
0
~,
P=I
o
g o
q=l
~ ~ o _
/ q
~ ~ o
ot=o.
o t=o.ol ~t=0.04
~
':
+t=o.1
. . . .
0.00
0.25
27
0.50
0.00
0.85
i
0.50
27
Fig. 8.8. Pore pressure distributions for different times in a cylindrical bending case for a plate with built-in ends. Only the left half figures are shown due to symmetry.
Ch. 8 Analysis of Poroelastic Plates
128
initial pore pressure is higher at the left end of the figure (i.e. compared to plate center) so that the zero point of Mp moves from left to right in order to meet the requirement that ~ l M p d x - 0 (see Section 4.4). It is seen that the distribution is approaching an antisymmetric pattern since the pore pressure will finally vanish. This can also be clearly seen from Fig. 8.7.
8.4. Transverse Vibrations of Simply Supported Plates For the present situation inertia is included and Eq. (8.44) is replaced by V6W -- (1 + ATI)V41~- VZ(q - 3/2/0) - (q - T2ig)
(8.75)
For simplicity in finding solutions, y is taken to be a constant hereafter. The following boundary conditions of w for a simply supported permeable boundary subjected to no boundary moments are found in a manner similar to that by which Eqs. (8.49) were derived, 2w w -- 0,
ogn2
04w -- 0,
0n 4
-- q
(8.76a, b, c)
where n again refers to the normal of the boundary. In order to solve Eq. (8.75) initial values of w, w and/0 are also needed. Given initial pore pressure moment, the initial condition on the second derivative can be obtained by --V4W -t- "qV2Mp + q
/0 --
(8.77)
which is derived from Eq. (8.39). Thus there are only three independent initial conditions. Alternatively, if a suddenly applied load produces the initial deflection, the following relation is derived from Eq. (8.43): /0=
q - (1 + Ar/)Vaw
,
t=0
(8.78)
Now consider a rectangular plate with the boundary conditions (8.76) existing at all four edges. In order to fulfill the boundary conditions (8.76a,b), the solution is taken as a double Fourier sine series oo
oo
w -- E E dpij(t)sin(iTrxl)sin(J'~rx2)
(8.79)
i=1 j--1
Eq. (8.76c) is also satisfied if q equals zero at the boundaries. In case q is not zero at the boundaries, the values in a small area near the boundaries can mathematically be changed, which physically will not alter the solution. Next, the load is expanded into the same type of series (x~
(3o
q -- E E Bij(t)sin(iTrxl)sin(j~x2) i=1 j=l
(8.80)
Transverse Vibrations of Simply Supported Plates
129
where the coefficients are functions of time
Bij -- 4 ~i ~i q(x, y, t)sin(iTrx)sin(j'try)dx dy Thus the problem now is to determine
(8.81)
chijusing
the following initial conditions:
t~ij(0)- 4 I10 ~low(X,y,O)sin(iTrx)sin(j'ay)dxdy, t ~ i j ( 0 ) - 4 I10
~ifv(x,y,O)sin(iTrx)sinQ"rry)dxdy,
(8.82)
~ ij(O) -- 4 f ; ylo fO(x,y, O)sin(iTrx)sin(j'try)dx dy and the ordinary differential equations
,y2 d3 dPij -k- ]/2aij d2 dpij -k- (1 dt 3
dt 2
+ Ar/)ce2 ddpij
3
dBij
dt + aijdpij- dt + aijBij
(8.83)
which are derived from Eq. (8.75), and where
Olij- (i 2 + K2j2)Tr2
(8.84)
The homogenous equation of (8.83) has characteristic solutions in the form where ~:g (k = 1, 2, 3) are the roots of the following cubic equation l+Ar/
r
~
1
~+7-~
exp(c~ij~kt)
(8.85)
Thus there are no particular difficulties in determining the solutions of Eq. (8.83) for any material constants and loading conditions. The same situations for beams were fully investigated in Chapter 5.
Free vibrations In this section, Eq. (8.83) is solved generally only for free vibrations, i.e. when the loading and thus B o are zero. Further, only the cases in which the oscillatory motion is possible are considered, which requires the existence of the complex roots. For convenience, two constants are introduced as follows:
0-27-
3~
+,/2,
A-
3,7
9 + 2
(8.86)
In other words A > 0 is required so that there will be one real root and two conjugate complex roots as given below: ~1 - -
--1
- 2c~,
where i - x/S- 1 and
~2 -- c~ + i~,
~:3 -- c~ - i ~
(8.87)
Ch. 8 Analysis of Poroelastic Plates
130
c~-~1 (~/g//2 +
1
'V~ - ( - g / / 2 + V/A) - -~
(8.88) are real quantities. Therefore the general solution in real form can be written as "" ff)ij __ exp(aij6t) [ C(exp(aij~ot) + C'~~jCOS(OgijJ3t)
+ C~sin(a/jB t)]
(8.89)
where ~:o = ~ 1 - 6 = - 1 - 3 6
(8.90)
Given the initial values as in Eq. (8.82), the constants C~ are found below: Cf = ((~2 _jr_~2)0/2(])ij(0)_
C~ -- r
c~ -
-
2~l.Olij~ij(O) + ~ij(O)
C~
(8.91)
d~o(o) _ a +ij(o)_+ ~oc[ /3~ij /3
Two types of initial pore pressure are investigated: (I) no initial pore pressure; and (II) the initial pore pressure is prescribed by Eq. (8.42), i.e. when after displacement the plate is released before the pore pressure has had time to decay at all. The initial deflection considered is w -- sin(Trx)sin(,rry) and the initial speed w is taken to be zero. 1
1
c=-5
tc=l
~0
--I
0
II |
|
1
t
2
0.0
0.1
|
!
|
|
0.2
t
Fig. 8.9. Decay of the deflection amplitude of a rectangular plate for At/= 0.25 and 3' = 1.5 in free vibrations for different initial pore pressure.
Transverse Vibrations of Simply Supported Plates
131
Some examples of free vibrations are shown in Figs. 8.9 and 8.10. As expected, the frequencies increase when K or At/increase (or y decreases which is not shown). In the cases shown in Fig. 8.9, the differences between the curves for initial conditions Types I and II are not large if other conditions are the same. However, Fig. 8.10 shows that the differences can be significant (cf. Fig. 5.4). It is clear that the natural frequencies and At/ make the difference: the higher the natural frequency, the bigger the differences are between the curves for the two types of initial pore pressure if At/ is sufficiently large. The phenomenon is explained here from the point of view of energy loss, which depends both on the actual fluid speed and on the comprehensive poroelastic characteristics represented by A~/. When the natural frequency is low, the fluid flows slowly so that little energy is lost in a cycle due to viscosity of the pore fluid. Therefore the behavior of the system approaches that of an elastic system and the existence of the initial pore pressure is not important. The same is true for a very small A~/. In the opposite situation, the energy dissipation is significant when the fluid flows quickly so that the initial pore pressure plays an important role in the process. When no initial pore pressure exists (Type I), there will be no initial impetus for fluid movement (i.e. there are no pressure gradients). The fluid flow, which is then produced by the deflection change, is not able to achieve a high velocity in a short time due to high damping. Hence the plate can only move slowly at first. In other words, the potential energy, which is produced by the initial deflection, is not able to be convened predominantly into kinetic energy but takes the path of being dissipated finally before the fluid reaches its high velocity. On the other hand, when there exist large pore pressure gradients at the beginning, the fluid flow can attain high velocity quickly and thus the plate vibrates. The energy is converting quickly among the following: the potential energy of the plate, the potential energy of the fluid produced by the pore pressure gradients and the kinetic energy of the plate. The energy loss occurs much quicker than for Type I due to the rapid fluid flow. 1
1
~r1=0.75
0
-I
a-------a
] I 0
. . . .
tion I ~176176 II , . . . . , 1 2
t
0
-I 0
, 0.5
, I
t
Fig. 8.10. Decay of the deflection amplitudeof a square plate for Y= 1 in free vibrations for different initial pore pressure.
Ch. 8
132
Analysis of Poroelastic Plates
Forced vibrations The long time solutions for harmonic loading problems of the same plate, q(x],x2, t) ~(Xl ,x2)exp(icot) can also be extracted. For this purpose, the coefficients in Eqs. (8.79) and (8.80) can be written in the forms
dPij(t) - wijexp(icot),
Bij(t) : bijexp(icot)
(8.92)
Then Eq. (8.83) becomes an algebraic equation for the given i and j and thus wij is determined
(c~ij + iw)bij wij :
(8.93)
a/~ + i(1 + A~/)c~2co- yzc~ijco2 - i y 2 c o 3
The first two resonance regions are shown in Fig. 8.11, where co is the circular frequency of the loading. For co = 0, the long time solutions for the corresponding quasi-static problems ( : elastic static solutions) is derived. A~ : 0 corresponds to the elastic cases. It can be seen that: (a) the resonance areas are shifted to the fight; (b) these shifts are more distinguishable for higher frequency; and (c) the amplitude response is reduced when A ~ increases. Fig. 8.12 shows the situation for plates with different ratio of length and width. Since L1 is kept constant, larger K refers to a narrower plate, for which the resonance areas are at higher frequencies and the amplitude responses are smaller, with respects to the counterpart results of the wider plate. The same behavior is observed also in Fig. 8.13, showing the influence of K, A~/ and y for a given frequency. Note that higher
10~
v ~,rl=O. o ),~=0.1
10-'
9" -
.
=lOo~,,i
10 -3
10 -4
.
0.0
.
.
.
'
I
6.0
.
.
.
.
.
I
.
2.o
.
.
.
.
I
.
.
.
.
18.o
Fig. 8.11. Resonance areas for a square plate for y = 0.75.
'
i
z4.o
Transverse Vibrations of Simply Supported Plates
133
lo o
-
o R:=I
;k~=O
x R:=I
X~=O
o~=l.
~,~/=0.
"0
",_.~
4
/
10.3 0.0
'
'
I
3.0
'
'
f -CO/2~r
Fig. 8.12. First resonance areas for plates with K(=
l~176 I
I
6.0
L1/L2)=
'
'
I
9.0
1.0 and 1.5 for y = 0.75.
o,:l.o x,7=o.o o R;=I.O ~,~=0.3
1 0 -~
#,
~=1.5
x~=0.0
I
+ ~:=1.5 X~=0.3
1 0 -2
1 0 -3
1 0 -4
, 0.0
2.0
, 4.0
6.0
Fig. 8.13. Resonance areas with respect to y for f = 2.5.
8.0
Ch. 8 Analysis of Poroelastic Plates
134
values of y denote higher permeability, when the other characteristics of the plate remain unchanged. Finally, as discussed in the beam case, the time scales of the present system are considered. Since the geometrical parameter K is involved, these time scales must be introduced as depending on the bending mode, denoted by positive integers m and n. The characteristic diffusion time is found to be
7/nDD n = (m 2 + K2n2)Tr2K
(8.94)
which refers to the case when after an initially imposed deflection the plate is restrained from further deformation after the initial pore pressure is produced. If the plate is free to deform when a sinusoidal load is suddenly applied the characteristic diffusion times are given by 7/flFn = (1 + /~)7/~D n
(8.95)
The characteristic times for the drained elastic plate are defined as the inverse of the natural (circular) frequency of a simply supported rectangular plate (elastic)
7~sn-- (m 2 + K2n2)Tr2
/~I
(8.96)
Accordingly, the characteristic times for the poroelastic plate when the fluid are trapped is introduced as follows" '~Tn =
~S n
(8.97)
+an This is obtained by comparing Eq. (8.36) with the parallel equation for the elastic plate. Note that as in the beam case, here too only three independent time scales exist. Namely, the time dependent features in terms of characteristic times are the same as in the beam case, though the ratio of length and width of the plate is necessarily involved.
Chapter 9 CLOSURE
In the previous chapters we have considered poroelastic structures for which fluid diffusion is possible only in the axial (or in-plane) directions. After formulating the model we presented solution procedures which made it possible to determine the response of various structures under different loading conditions. Many examples have been presented. Throughout, we have seen response patterns which are unique to this type of element; they are unforeseen, indeed surprising, in terms of experience with structures whose constituting material evinces the most common type of time-dependent behavior, viz. viscoelasticity. What makes such a system unique is that to a certain extent the right hand does know what the left hand is doing. Because of the axial fluid flow, information is being passed along the length of an element; thus the time dependent behavior of such structures shows anomalies which cannot appear in viscoelastic structures. This reality, together with the fact that these time-dependent behaviors are qualitatively changed when permeability end conditions are altered, makes such structures promising for use in a 'smart structure' environment; these end conditions should be readily controllable at any time based on prior response of the structure. Indeed the permeability conditions could be imposed, or altered, at any position along the beam; this broadens the range of control possibilities. (Such an example is given in Li et al. (1999a); there, an arbitrary value of Mp is imposed at a given point along the beam.) In addition, it is not necessary that the crosssection, or even the properties of the pore fluid, be uniform along the length of the beam. It also should be recalled that the viscosity of many fluids is very sensitive to environmental variables, most especially temperature, and thus characteristic times of the structure could be keyed to the environment. Human technology has not by choice utilized poroelastic materials up to now. Both geological structures and biological materials displaying poroelastic behavior have been imposed upon us; their behaviors have indeed been analyzed. It should be possible to design with poroelastic materials of the particular class considered here to achieve controllable structures. Now we underscore another interesting feature of such a poroelastic material when used in a structure. Under suddenly applied loads the stress experienced by the skeletal material may be significantly reduced, vis-a-vis the stress due to slowly applied l o a d s - especially in a structure with at least one permeable end condition. This is so since immediately after the load application a pore pressure is developed (see Eq. 2.13), and this pore pressure bears a portion of the load. Combining Eqs. (2.13) and (2.1) we have that the initial partial stress acting in the skeletal material at any point is given by
136
Ch. 9
crx --
(
1-
)
A 1 -+- - ~ q5 ~'~
Closure
(9.1)
We recall that 4, is the pore volume fraction. However, at long times o'x = %,. Thus a load applied for a very short time with respect to the characteristic time of the structure will cause greatly reduced stress in the skeletal material. The above-mentioned technological possibilities are brought home by considering that all of the analyses of structures in the previous chapters (except Chapter 8) are valid for beam-like elements which have geometrically discrete cross-sections, i.e. the crosssection is not treated as a single continuum. One can construct a beam cross-section consisting of two tubes, fluid filled, in which are permeable partitions (Fig. 9.1). Even the simple example of a cantilever beam with free-end valves subjected to tip loading as shown in Fig. 9.2 immediately stimulates speculations as to possible control schemes. We show the total analogy as follows. Consider the beam element shown in Fig. 9.3. For simplicity we limit ourselves to bending only (no axial load) and to a section symmetric about the bending axis; it is not necessary that the tube cross-section be round. The bending moment at the section is M; the fluid pressure in the lower cavity is pf. Relying on classical beam theory we can now immediately write 1 r
-- -- S l l M -+- S12Pf,
2
dVs-f -- S 2 1 M q- S22Pf dx
(9.2a, b)
Here r is the radius of curvature of the beam axis and dVs_f/dx is the change per unit length of the difference between the cavity volume and the fluid volume for the lower half. It is clear that $11 - 1~El, where E is the Young' s modulus of the s o l i d m a t e r i a l (taken here to
Fig. 9.1. Segment of discrete model of poroelastic beam.
Closure
137
Fig. 9.2. Cantilever beam.
Fig. 9.3. Beam slice.
138
Ch. 9
Closure
be isotropic) and I is the moment of inertia of the cross-section (excluding the cavities). It can be shown that the S matrix is symmetric, i.e. $12 -- $21. $12 and $22 can be calculated. $12 is dependent only on the cross-section shape and on the elastic properties of the solid material; $22 could be written as $22 -- S~2 nt-
2A
(9.3)
B
S~2 is also dependent only on the cross-section shape and on the elastic properties of the solid material; B is the bulk stiffness of the fluid and A is the cross-sectional area of the tube. For instance for the cross-section shown in Fig. 9.1 we find 1 _ 27rhRE[R 2 + 2b 2], Sll
"n'R3 ( $22 = - - ~
5-
4/,'-
S12 = ~
2 -
1
1 + (1/2)(R/b) 2
] (9.4a, b, c)
(1-21-') 2 ) "rrR2 1 + 2(b/R) 2 + 2 - - - ~
where it has been assumed that the solid material is isotropic (at least in the tube wall surface) and v is the Poisson's ratio of the material; we have taken h << R, h << b. For simplicity it has been assumed that the web connecting the two tubes does not contribute appreciably to the stiffness in flexure. Eqs. (9.2) could of course have been written in terms of a pore pressure moment, Mp, but in the present instance using the pressure in the initial formulation of the global constitutive equations is more natural. At this point we note that if the fluid is 'trapped', i.e. the situation immediately after a suddenly applied load, then using dVs_f/dx = 0, Eqs. (9.2) give 1 _S,1[1+ 7-
822 ]-1 1 1 8 1 1 S 2 2 _ $22 M =-- E1 (1 + ADr/D) M'
trapped
(9.5)
For convenience in showing the analogy we have introduced h D and ~)D given by h D - 2bA
S12Sll SllS22 - $22
77D . . . . . 1 S12 2 b A $11
(9.6a, b)
2b here is the distance between the centroids of the two tube cross-sections. The factor 2bA is included in the definitions above in order to make both h D and r/D non-dimensional. (1 + hDr/o) is the complete analogy of the factor (1 + At/)in the continuum formulation; it tells us to what extent the completely trapped fluid can stiffen the beam. We now consider Eq. (9.2a). Using the equilibrium equation (2.3b), and since for small deflections 1
\
.\
I
1
0%
r
o~X2
we have
04W __ D E10x-g-
~Mp --0 Ox2 + q, + P--zT.~ dr"
(9.7)
Closure
139
which we have written in terms of the pore pressure moment, Mp = -2bApf, in order to show more clearly the analogy between this equation and (2.16b). Next we formulate a global Darcy relationship for the lower tube as Cf o~pf
Q. . . . tzf Ox
(9.8)
Q is volume flow rate, and Cf is the flow resistance constant of the membrane geometry and spacing (or of any other flow resisting arrangement which might be used, for instance, the tubes might be filled with an open-pore polymeric foam, the foam geometry would determine Cf)./tZf is again the viscosity of the fluid. Now combining Eqs. (9.2b) and (9.8) and using (9.2a), we have KD ~Mp _ Mp - ADEI o~X2
dw o~XT
- 0
(9.9)
where, analogous to Eq. (2.20), KD _
2Cf ~kD ~
E1
(9.10)
Ixf qqD (2bA)2 Eqs. (9.7) and (9.9) are the exact analogies of (2.16b) and (2.22b), which are the governing equations for bending of poroelastic beams. And clearly all the boundary conditions could be cast into analogous forms to those presented in Chapter 2. Thus we see that all of the solutions and phenomena displayed in this book for beamlike elements are equally applicable for discrete cross-sections. The nature and extent of the responses we have seen are qualitatively dependent on A and ~/(A D and r/~ here). E1 and K (or K o) only manifest themselves quantitatively; they can be normalized out of the behavior patterns (load normalization, time normalization). If A and ~/(A ~ and r/~ here) are very low compared to 1, the poroelastic effect is small. For the very simple crosssection shown in Fig. 9.2 it is interesting to ask what values could be achieved. Using Eq. (9.4) together with (9.6), we have D r/ -- 1 -- 2v,
AD _ 1 -- 2v -- [4(1 -- ~ ) + (2hE/BR)][1 + (1/2)(R/b) 2]
(9.11a, b)
For Poisson's ratios equal to zero, r/D is 1 and the maximum value of AD is achieved for b >> R and is given by AD =
1 4 + 2(hE/RB)
(9.12)
So even for a very incompressible fluid, or relatively very thin tube wall,/~D could not be greater than 0.25. The product /~D'oD then could approach 0.25. This represents a not insignificant effect. As Poisson' s ratios increase towards 0.5, the limit for isotropic materials, the product decreases. Eq. (9.12) shows the great influence that the relative stiffness of the fluid with respect to the solid and the relative area of the fluid in the cross section have on determining the magnitude of the poroelastic effect; this is as expected. If the tube wall is not isotropic (say is orthotropic) far higher values could easily be designed into the system by properly adjusting the relative longitudinal and circumferential elastic properties.
140
Ch. 9
Closure
Table 9.1. Values of AD and /]D for the simple model /~
77D
0
1
0.25
0.5
0.5
0
0.75
- 0.5
1
-1
h D (b >> R)
hD
(b = R)
1
1
4 + 2(hE/RB)
6 + 3(hE/RB)
0.5
0.5
3.75 + 2(hE/RB)
5.625 + 3(hE/RB)
0
0 -0.5
-0.5
1.75 + 2(hE/RB) -1
2.625 + 3(hE/RB) -1
2(hE/RB)
3(hE/RB)
We now illustrate a very unique case demonstrating another peculiarity first noted in Section 2.1 for the continuum case. Consider a filament wound tube with + 45 ~ winding (say glass or graphite fibers in a polymeric matrix). We can use Eqs. (9.11) without having to generalize the S values given in Eqs. (9.4), for the isotropic case. This is so since such a material, in the tube's longitudinal and circumferential directions, has the same properties; thus Eqs. (9.4) and hence Eqs. (9.11) are valid as shown. Also it is known that such a material (in the tube surface, which is the operative situation vis-a-vis the elastic constants) has longitudinal and circumferential Poisson's ratios which can approach 1. If v is greater than 0.5 then both AD and ~/D are negative. This was pointed out in Section 2.1 to be physically possible for the continuum case. Examination of Eqs. (9.11) immediately shows the great sensitivity of the extent of the poroelastic effect to both the material and geometric parameters. As v approaches 1 very large values of the product AD~/D can be attained. Table 9.1 shows the values of AD and r/D for various Poisson's ratios and geometries within the very simple model considered. Finally we point out again that tube crosssections need not be circular; the cross-sectional shape would have a dominant impact on the values of the two parameters.
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Appendix A PROOF OF THE VARIATIONAL PRINCIPLES
Noting Eqs. (2.14), (4.3) and (2.28) and the fact that a variation of an initial quantity equals zero, the variation of U is as follows: 6U-
I
[ (2EAu'x + riNP)* 6(Ux) ' + riit'x * 6Np
AEA rI Np(x,O)6Np
+ (2EIw,xx - riMp), 6(ft,xx) - rift,~x * 6Mp + AEI rI Mp(x, 0)6Mp ]dx
(A.1)
It has been assumed that the order of a variation and a derivative can be interchanged. After integration by parts of the first and fourth terms once and twice respectively, (A.1) becomes 68-
l[ L
-(2EAu,xx + riNp,x)* t~ti + ritix * 6Np + - ~
+ (2EIw,xxxx -
riMp,xx)* ~ft - rlft:,xx * 6Mp +
Np(x, O),~Np
ri Mp(x, O)(~Mp]dx AEI
+ [(~A.~ + ~,~. ~]~+[.~,Wxx-.~. ~Wx~]~ -[(2EIw,xxx-
riMp,x)* t~ft]S
(A.2)
Since the variations of the loads and the given boundary quantities are zero, the variation of V is given as 6V - 2 f [qn * 6ft - qs * 6u]dx - [2N * 6ti] b + [2/17/* 6(ftx)] b ./ L
[
Up *
,
-
- - ~ Mp * 6(Mp,x) a
Consider now the variation of the functional (4.6)
(A.3)
Appendix A
146
t~TrmP--- AEIT] f [ _A l (KNpxx ' _ lVp + AEAit,x) , t3Np + (KMpxx ' _ Mp _ AEifv 'xx) , I
+ ~ Np * 6(KNp,xx - Np -~ AEAu,x) + Mp * 6(KMp,xx - Mp - ~EIw,xx)]dx (A.4) After integrating by parts or rewritten by (4.3) when necessary, it becomes 671"mP-- AEI rl I
[ --A I (2KNpxx ' - 21~Ip + AEAitx) ' * 6Np - AEINp'x * 6it - -~ Np(x, O)6Np
+(2XMp,xx - 2Mp - AEIwxx) * 6Mp - AEIMp,xx * 6w - Mp(x, 0)6Mp ]dx
[
][
nt-[T~Xp* ~/i]~-[- - ~ X p * 6(Xp,x)a- ~-~
a
Adding (A.2), (A.3) and (A.5) together, the variation of the functional 7r = U + V + is obtained as follows:
6"n'-- 2 f [-(EAu,xx .I L k
+ ~Np,x +
qs)* 6u + (EIw,xxxx - ~Mp,xx +
+ AEArl (KNp~x _ IWp q- AEAft,x) * 6Np + - ~ (KMp,xx --
7rmp
qn)*W
Mp -
hEIw,xx)* 6Mp ]dx
+[2(N - / V ) * &i]ab- [2(M - 19/)* 6(W,x)lab + [2(Ma - ~))* 6wlab + [ ~r/g
(Np --/Vp) * t~(Np,x) ]i + [ -~K(Mp-Mp)*a(Mpx)]ba ~ , (A.6)
If a displacement is given at a boundary, the variation of the displacement at the boundary is zero, thus the corresponding item vanishes. Noting that every item in (A.6) is independent and every non-zero variation is arbitrary, it is seen that 67r -- 0 gives the equilibrium equations (2.16), the motion equations (2.22), the mechanical boundary conditions (2.24) and the diffusion boundary conditions (2.25) and/or (2.26). In other words, the stationary condition of 7r is identical to those governing equations and all the boundary conditions except for the displacement boundary conditions (2.23) which are preconditions. Thus it has been proved that 7r corresponds to a
Appendix A
147
variational principle which includes two types of variables, the displacements and the pore pressure resultants, and hence refers to a generalized variational principle. Similarly, for the second functional 7r*, the following can be obtained: 6"n'* -- Jf I_[-(EAuxx + rINp'x + qs)* 6u + (Elwxxxx - rIMp,xx + qn)* 8w L
+
'
rl * (KNpxx - IVp -k hEAit,x ) * 6Np + rl * AEA ' AEI
(KMp,xx-- l~p
-- l~Ell,V,xx) * ~Mp ]dx
+ [(N -/~') * 6u]~-[(M-/17/), 8(W,x)]~+ [(M,x - {~)* 6W]ba
+
-
2AEA * (Np -/Vp) * ~(Np,x)
2AEA
* Np,~ * 6Np
-
2AEI
+ 2AEI 9(Mp - Mp) 9 6(Mp,x) * Mp,x * 8Mp
(A.7)
by which we see that the stationary condition of zr* gives the same Euler-Lagrange equations.
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Appendix
A FINITE ELEMENT
FOR POROELASTIC
B
BEAMS
A 3-node element is given here as an example of the beam finite elements, which are formulated earlier. The nodes are numbered as 1, 2 and 3. At each node, all the four unknown variables are adopted as nodal parameters. The nodal unknowns of the element are recorded as
{ T w } - [Wl, 01, W2, 02, W3' 03] T
(B. 1)
{/~M }
_
[ (Mp)l, (Mp)2, (Mp)3
]~
where Oi is the derivative of w with respect to x at the node i ( -- 1,2,3). For convenience, we define a polynomial as
Pi(x) -- 1 7 (x - xj) j#i
Then the interpolation matrices are given as
[Nu] = [SN] = [SM]-- [Pl(x)/Pl(Xl),
P2(x)/P2(x2),
Pa(x)/P3(x3)]
(S.2)
which is based on the Lagrange interpolation and
[Nw] = [al, bl, a2, b2, a3, b3]
(B.3)
where a i and bi, according to the Hermite interpolation, are as follows:
a i --
[
2 1
--
Pi(xi)
Pi(xi)
,
bi -
(x -
xi)
Pi(xi)
in which the derivative of the polynomial at the node i is
Pli(xi) -- 3 x i - (x 1 nt- x 2 nt- x 3) It should be noticed that the interpolation matrices [N,], [SN] and [SM] are independent quantities although they may have the same forms when the same nodes are adopted for u, Up and Mp.
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Appendix C SEVERAL USEFUL LAPLACE INVERSE TRANSFORMATIONS
In order to find the inverse transform of Eq. (5.10) for all cases so that the Ogn(t) in (5.7) are determined, some formulas for Laplace inverse transformations are required. The following two groups of formulas are derived for this purpose, using ~ to refer to the Laplace operator,
~,-1{ s ~,_1I 9o-1{
~L~,~(t)]l _ eat ~tof(~.)e_a~ d~., 1
( S - a) 2
~[f(t)]}-eatftof(~')e-a~(t-~-)
1 ~ [ f ( t ) ] } -(s _ a)3
eat f('r)e-ar(t -~-f'o
d~-,
(C.la, b, c)
T)2dT
and
~-1{
(s - s--ce a-~2 ~_/32 ~[b(t)]
}
-
e c~t[cos(flt)Ic(t) + sin(fit,Is(t)] (C.2a, b)
(s - a) 2 +/32 5~
-
[sin(flt)Ic(t) - cos(fit)Is(t)]
where
Ic(t) --
it b(z)e-~'cos(flz)dz, o
I~(t) -
;t b(z)e-~'sin(flz)dr o
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Appendix D COEFFICIENTS IN DIFFERENCE FORMULAS
The coefficients for all difference formulas required in the finite difference method for our problem are presented in this appendix. All of them are derived by Lagrange interpolation and for the computation of derivatives at the node i. The coefficients for the 3node difference formulas are given in Table D. 1, in which the first to third rows present the coefficients of Eqs. (6.28a,b,c), respectively. For second differences, only the central formula is employed in our work; the coefficients for the 3-node formula are given in the fourth row of Table D.1. T a b l e D.1. Coefficients for 3-node formulas ( h i - - x i -- X i - 1 ) F o r w a r d formula
Central formula
-- 2 h i + 1 - hi+2 + hi+z)
C~ =
Co = h i + l ( h i + l
Ci
i C~ =
9 hi Ct--2 -- h i _ l ( h i _ 1 4- hi)
2rid order (central)
C/
C~ =
-hi+l
(hi+l + hi+2)hi+ 2
hi+lhi+2
_ -hi+l 1 -- hi(h i + hi +1)
Backward formula
hi+ 1 -k- hi+ 2
Ci
2
+ hi+l hihi+ 1
_ - h i - 1 - hi 1 -hi_lhi 9
1 ---- h i ( h i + hi+ 1)
-hi
-2
i hi C1 = (hi + hi+l)hi+l i h i - 1 + 2hi C~ = hi(hi_ ~ --~-hi) i 2 C1 -- (hi + hi+ 1)hi+l
Cl~ -- hihi+ 1
The 5-node formulas are employed in the work for the node i which is at least two nodes away from an end node so that the central formula can be applied. The coefficients for the first order difference as defined in Eq. (6.29) are presented in Table D.2. Table D.2. Coefficients for 5-node central f o r m u l a (k -- i - 2 to i + 2)
c'_~
c~_,
I-I kr
(Xi -- Xk) k#i
H (Xi-2 -- Xk) k#i-2
I-I kr162
c~ (Xi -- Xk)
I-I (Xi-1 -- Xk) k#i- 1
~,
c~
c~ 1
kZ'~r Xi -- Xk
I-I kr162
(Xi -- Xk)
I--I (Xi+I -- Xk)
k#i+ 1
I-I k#i,kr
H
k#i+2
(Xi -- Xk)
(Xi+2 -- Xk)
The second order difference formula is written in the form of Eq. (6.35). The coefficients for the central formula are as follows:
Appendix D
154
CL 2 =
2[(Xi -- X i _ l ) ( X
i --
Xi+l) +
(X i -- Xi_l)(X i --
kr
CL1 =
2[(Xi -- X i _ 2 ) ( X
i --
Xi+l) +
I'-I (Xi-2 - 2
-- Xi+l)(X i --
Xi+2) ]
(X i - - X i + l ) ( X i - -
Xi+2) ]
-- xk)
(X i -- Xi_2)(X i --
1-1
Xi+2) -+- (X i
Xi+2) +
(xi-l-xk)
kr
1
e 0l
2[(Xi -- X i _ 2 ) ( X
i -- Xi_l)
-+ ( X i - - X i + l ) ( X i - -
Xi+2) qt_ ( 2 X i
_ Xi_2 _ Xi_l)(2X
i _ Xi+l
_
Xi+2)]
[ 1 (xi - x O kr
C~= 2[(xi
-
xi_2)(x
i -
Xi_l)
+ (x i -
xi_2)(x
H kr
i -
xi+2) +
(x i -
Xi_l)(X i -
xi+2) ]
(xi+ 1 -- Xk) + 1
c~ = 2 [ ( x / - xi-2)(xi - xi-~) + (xi - xi-2)(xi - xi+~) + (xi - xi-~)(xi - xi+ ~)] l-'I kr
(Xi+2 --
xk)
+ 2
(D.1)
Appendix E D E T E R M I N A T I O N OF B O U N D A R Y VALUES AT xl FOR THE FINITE D I F F E R E N C E M E T H O D
The fact that the boundary conditions for w, 0, M and Q produce coupling in the variables requires that the four unknowns be solved dependently. In the solution scheme presented in this book, we avoid solving them simultaneously due to considerations of convergence. Instead, the numerical solutions are pursued first on the supposition that all of the four variables, no matter whether they are previously given or not, are zero at x = X1. All of the four nodal parameters at Xl (i.e. wl, 01, M1 and Q1, or their variations, or their increments) can then be determined by four given boundary conditions and the supposed solutions. The real solutions are obtained by modifying the supposed solutions with the four nodal parameters. The procedures are presented in this appendix. We derive necessary equations for time t = 0 only. However, as will be seen later, they can be easily adapted to all times. Consider first the corresponding linear case, since its solution is adopted for the first loading step in an iterative approach when very small loading values (include the given boundary values) are taken. The governing equations for the four unknowns at t = 0, in accordance with Eqs. (6.30), are as follows: dw dx - 0 '
dO dx - -
M I+A~'
dM dx - Q'
dQ dx - q n
(E.1)
Supposing that wl -- 0, 01 -- 0, M1 = 0 and Q1 = 0, we get a supposed solution for (E. 1) recorded as Wsup, 0sup, msup and Qsup. The relationship between the real solution and the supposed solution can be analytically determined by (E.1). Integrating each equation of (E.1) from the last to the first successively yields Q0.
c1
c3
-~- Qsup,
M-
c 2 -Jr- C l X -Jr-
2c2x + Clx2 .2(1 + At/) . . + 0sup'
w
Msup, c4
+ c3x
3c2x2 + Clx3 6(1 + At/) + Wsup
(E.2)
from which four linear algebraic equations can be obtained in any case by the four given boundary conditions and thus the constants c l, c2, c3 and c4, whose values in fact are the real values of Q1, M1, 01 and Wl, respectively, will be determined. Hence, (E.2) gives the
156
Appendix E
real solution. In practice, we solve for the four constants using the supposed numerical solution. In a non-linear situation, when the boundary conditions cannot be decoupled, Eqs. (6.30) are replaced by their variational form (6.32). Similarly, the real solution of the variations is given by o~O = c 1 + (o~Q)sup 6kM
= c 2 4- r x 4- (o~'kM)sup
(E.3)
2c2 x 4- c1 x2
0 = c3 ~'kW -- C4 4-
2(1 + A~) + ( o~ 0)sup
0
C3 --
2(1 + At/)
(COs0)k-ldx 4- (~'kW)sup
The algebraic equations for the constants given by the four boundary conditions are also linear though integral relationships exist among the coefficients, which make numerical calculations necessary. Since ~Q, o~M, o~0 and o~w are obtained by iterations for any given k, the above procedure must be applied to every iteration. Apparently, the same can be done for time t > 0. Eqs. (E.2) and (E.3) are justified for the situation after (1 + At/) is removed from the two equations and o~ in (E.3) is replaced by AJ in accordance with Eqs. (6.34). For instance, the latter should be as follows: AJQ = cl + (AJQ)sup NM
- - c 2 4- c1 x 4-
zxJ0 _ c3 _
AJw--
c4 4-
(AJM)sup
2c2 x 4- c1 x2
2
o
c3 -
(E.4)
4- (N0)su p
2
(COS0r
dx 4- (~Jw)sup
SUBJECT INDEX anisotropic, 3 antisymmetric, 24, 25, 45, 85, 127 beam
cantilever, 28-32, 46-48, 75, 79, 84, 87, 90 clamped, 47-48, 85 drained, 14, 15, 16,20,21,26,59, 61, 62,91,107, 110 simply supported, 15, 16, 21-27, 41, 45, 54, 59, 65, 72, 85, 89 sealed, 16, 18, 46, 50, 79, 80, 83, 91 statically determinate, 14, 20, 89 biomechanics, 1, 2, 7 Biot, 1, 3-6, 10, 13, 33, 67, 91, 96, 111, 115 bone, 2 boundary conditions, 2, 4, 5, 13-14, 18, 26, 34, 4251, 146, 155 buckling, 3, 6, 89-96 bulk, 2, 4, 12, 67, 111 cartilage, 2 characteristic times, 15-16, 59, 61, 134, 136 column, 6, 49-52 consolidation, 1 constitutive law, 5, 6, 11, 17, 111, 138 control, 7, 66, 135, 136 convergence, 41 convolution, 5, 33-34, 40 coupling, 1 creep, 47, 79 cylindrical bending, 124-128 damping critical, 5, 55, 59, 65 light, 5, 6, 59, 60 over, 5, 59 structural, 66 Darcy's law, 4-6, 12, 33, 67, 70, 96, 97, 111, 115, 139 deflection large, 5, 6, 67-87, 96 small, 4-6, 10, 67, 70, 71, 74, 75, 79, 80, 84, 87, 89, 99 diffusion boundary condition, 4-6, 7, 14, 24, 27, 35, 40, 4345, 66, 87, 89, 116, 117
thermal, 1 diffusion time, 134 discrete, 34, 39-40, 136-139 discretization, 75-77 dissipation time 59, 61 drained, 6, 14, 15, 39 end clamped, 47,48 free, 43-44, 47-48 impermeable, 14, 16, 22, 41, 44, 47-49 permeable, 14, 21, 47, 53, 63, 65 equilibrium, 1, 4, 5, 9, 12, 18, 39, 67, 69, 72, 114, 117, 138, 146 Euler-Lagrange equations, 5, 15, 34, 36 finite difference method, 5, 72-79, 155 finite element method, 33-41 fluid compressibility, 11, 12, 114 Fourier series, 19, 54, 63, 125, 128 frequency, 15, 61-62, 65, 108, 131-134 functional, 5, 33-37, 41 geomechanics, 1 Galerkin method, 33 imperfection sensitivity, 6, 89, 102-104 impermeable, 14, 16, 21, 22, 23, 77, 116 inertia, 4, 9, 10, 35, 106, 119, 138 infinite time, 34, 41, 80, 94 initial conditions, 4, 13, 14, 16, 18, 21, 29, 53, 60, 62, 70, 117 instantaneous response, 14, 25, 79, 84 interpolation, 5, 37, 39, 41, 73, 149, 153 jacketed, 12 Kirchoff hypothesis, 6, 111 Kirchoff plate 3, 112 kinematic, 10 Lagrange multipliers, 5, 33-36 Laplace transform, 5, 54, 56, 151 loads critical, 6, 90-95, 104, 107-110 suddenly applied, 12, 14-16, 65, 90
Subject index
158 uniformly distributed, 79, 85, 87 Mandel-Cryer effect, 7, 29, 42-44, 48 mass, 9, 10 Mathieu equation, 107 microgeometry, 3, 9, 114 modulus bulk, 2 drained, 114 Young's, 12, 14, 19, 20, 21, 65, 114, 117 multiple scale method, 107-108 non-dimensional, 9, 16, 17-18, 19, 53, 61, 71, 72, 97, 118, 138 non-linear, 5, 67, 72, 74, 75, 77, 96, 98, 101,156 orthopedic, 2 orthotropic, 6, 111, 115 overshooting, 26, 29 permeable, 14, 16, 20, 22, 53, 63, 65 plate, 2-4, 6, 42, 111-134 rectangular, 120-134 square, 122, 124, 131-132 Poisson's ratio, 114, 117, 138-140 pore pressure, 59-61, 65 pore volume, 9, 11, 12 porosity 9, 11 post-buckling, 6, 96-104 potential energy, 35, 131 quasi-static, 1, 4, 5, 6, 19, 34, 65, 89, 105, 119, 132 relaxation, 47 resonance, 5, 7, 64-66 roots
complex, 55, 108, 129 conjugate, 57, 108, 129 real, 55, 56 Runge-Kutta method, 99 saturated, 1, 2, 6, 13, 66, 89, 107 secular terms, 108-109 separation of variable method, 21 skeletal material, 3, 5, 11, 50, 135 smart structures, 7, 65, 135 soft tissue, 2 stability boundaries 6, 89, 107-110 condition, 6, 89 dynamic, 6, 89, 104-110 steady state, 127 stem, 3 Strutt diagram, 110 superposition, 39 symmetric, 9 thermoelastic analogy, 1 transversely isotropic, 3, 4, 9, 10, 11, 67, 111, 113 trapped, 12, 15, 16, 62, 117, 138 variational principle, 5, 34-37 vibrations, 53-66, 128-133 forced, 5, 63-65, 132-134 free, 5, 15, 59-63, 65, 129-131 viscoelastic, 6-7, 24, 26, 46, 47, 66, 110, 135 viscosity, 7, 13, 66, 70, 90 volumetric change, 1 water, 3, 66 wave propagation, 1
