Advances in Constitutive Relations Applied in Computer Codes (CISM International Centre for Mechanical Sciences)

~ SpringerWienNewYork

CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien

The Secretary General Bernhard Schrefler - Padua

Executive Editor Paolo Serafini - Udine

The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 515

ADVANCES IN CONSTITUTIVE RELATIONS APPLIED IN COMPUTER CODES

EDITED BY JANUSZ R. KLEPACZKO UNIVERSITÈ DE METZ, METZ CEDEX, FRANCE TOMASZ ŁODYGOWSKI POZNAN UNIVERSITY OF TECHNOLOGY, POZNAN, POLAND

This volume contains 158 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2009 by CISM, Udine Printed in Italy SPIN 12754533

All contributions have been typeset by the authors.

ISBN 978-3-211-99708-6 SpringerWienNewYork

PREFACE Many important industrial applications incline toward better understanding of the constitutive properties of matter. Nowadays, the development of measurement possibilities, even in nanoscale, allows for multiscale formulations that drive to the more sophisticated models used in continuum mechanics. These phenomenological models are particularly important and useful for solutions of very concrete initial boundary value problems. Our interests are focused mainly on detailed descriptions of material behavior that depend not only on simple stress-strain relationships but also includes the strong influence of loading type, which introduces temperature, strain rate dependence, fracture, etc. Understanding these physics phenomena is of fundamental importance for successful and responsible computations. In particular, using the popular commercial programs requires deep understanding of constitutive formulations and their restrictions. These lectures are addressed to industrial users who are responsible for making crucial decisions in design, as well as, to young scientists who work on new models that describe the behavior of materials which also account the new influences and reflect the complexity of the material behavior. At the end, let me express my gratitude to the lecturers of the CISM course No. 328 on “Advances in Constitutive Relations Applied in Computer Codes”, held in Udine in July 2007, who finally prepared the included materials. Unfortunately, during the preparation and collecting papers for this book, our friend and colleague Prof. Janusz R. Klepaczko passed away. This is a very big loss for the society of mechanics. Prof. J.R.Klepaczko was the initiator of this course and the editor of this material. We dedicate this book to His memory. Tomasz Lodygowski




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CONTENTS

Preface Tomasz Lodygowski Preface Janusz R. Klepaczko Constitutive Relations in Dynamic Plasticity, Pure Metals and Alloys by Janusz R. Klepaczko .................................................... 1 Modeling of Rubberlike Materials by A. Luis Dorfmann .................................................... 147 Imperfect Bonding with Nonpenetration Condition by George Mejak ........................................................... 203 Selected Constitutive Relations in Practical Computations by Tomasz Łodygowski .................................................. 261

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Modeling of Rubberlike Materials A. Luis Dorfmann Department of Civil and Environmental Engineering Tufts University, Medford, MA 02155, USA E-mail: [email protected], Home Page: http://ase.tufts.edu/msml/ Abstract In this chapter we provide an overview of the basic equations governing the mechanical response of rubberlike materials capable of finite deformations. We use the term rubberlike materials to refer to highly deformable, nonlinear elastic continua, which respond differently in their reference state and their deformed configuration. An elastic material, by definition, will always return to its original state after the application and removal of forces. Nonlinear elastic materials, in particular, exhibit a change of material behavior during deformation, but assume the original shape after removal of all applied forces. Therefore, we do not include the effects of time and rate dependence, nor do we assume that the applied forces are of such magnitude to induce damage to the material. We first introduce the mathematics of deformation and different stress tensors for the analysis of rubberlike materials. Subsequently, we summarize the general constitutive theory and specialize its use to isotropic compressible and incompressible materials. A large number of constitutive functions are available in the literature to describe the nonlinear elastic response. We provide an overview of the most frequently used formulations. The last section of this chapter covers the solution of some boundary value problems. In particular, we consider incompressible, isotropic and elastic materials to illustrate the mechanical response when different strain energy functions are used. We apply the theories to circular, cylindrical thick-walled tubes subjected first to a pure azimuthal shear deformation, followed by combined axial extension and radial inflation.

1 Introduction Modeling of rubberlike materials requires the use of constitutive equations, which in the simplest form account for nonlinear elasticity. Pioneering contributions have been made by R.S. Rivlin, W. Noll, C. Trusedell, A.J.M. Spencer, M.L. Beatty, B.D. Coleman, A.C. Pipkin and many others and are described by Truesdell and Noll (1965) and by Ogden (1997) in his classic monograph. The challenge in mathematical modeling is to select or develop an appropriate formulation and to

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experimentally determine associated model parameters. Examples of highly deformable materials include, rubber in its natural state or with added reinforcing fillers, and a broad range of synthetic elastomers. Constitutive laws have been used extensively to model elastomers such car tires, seals and flexible joints, medical appliances, etc. The basic theory to describe nonlinear elastic materials has been augmented to account for pseudo-elasticity by Ogden and Roxburgh (1999). Pseudo-elasticity describes a modeling effort where the material behavior during loading is described by an elastic constitutive law and during unloading by a different elastic formulation. The theory of nonlinear elasticity is restricted to isotropy in the mechanical response, which has been modified by Spencer (1971) and Zheng and Spencer (1993) to account for anisotropy. Holzapfel et al. (2000) modified the basic theory to describe arterial wall mechanics including the influence of collagen fibers, its orientations and the effect of residual stresses on the macroscopic response. The time and rate dependence has been addressed, among others, by Reese and Govindjee (1998) and Holzapfel and Gasser (2001). Highly deformable elastic materials are also found abundantly in living systems such as arterial wall, skin, tendon, muscle and cornea. The mathematical framework describing rubber elasticity has been transformational in the field of soft-tissue biomechanics. There are a number of books available on the mechanical response of soft biological systems, for example, Fung (1993) and Humphrey (2001). A brief history of continuum biomechanics, an overview of the theoretical framework and some open questions are summarized in the recent review article by Humphrey (2003). Applications are in all major biological systems such as the heart (Hunter et al., 1998), arteries (Holzapfel et al., 2000), (Holzapfel et al., 2002), (Humphrey, 2001), (Humphrey, 2003), skin (Tong and Fung, 1976), brain (Franceschini et al., 2006) and muscles (Dorfmann et al., 2007), (Dorfmann et al., 2008), (Bose and Dorfmann, 2009). Recently, we have extended the theory of nonlinear elasticity to include the interaction of multi-functional materials with electric and magnetic fields under finite deformations. These materials, which consist of an elastomeric matrix infused with uniformly dispersed microscale ferrous particles, rapidly change their mechanical properties by the application of an external field. We have developed a theory of nonlinear electro-mechanical coupling and developed a constitutive framework at the quasi-static level, which accounts for the finite deformation in the presence of an electric field (Dorfmann and Ogden, 2005), (Bustamante et al., 2009). Similarly, we have recently produced pioneering work that provides a transparent theoretical framework for the analysis of isotropic magneto-sensitive elastomers subjected to large deformations (Dorfmann and Ogden, 2003a), (Dorfmann and Ogden, 2004b), (Dorfmann and Ogden, 2004c) and Bustamante et al. (2008). To appreciate the typical behavior of rubberlike materials, we show experimen-

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tal stress-strain data of three different natural rubber compounds, the first with 1 part per hundred of rubber (phr) of carbon-black (CB) filler, the second with 20 phr and the third with 60 phr. A filler content of 1 phr does not affect the mechanical behavior significantly compared with that of an unfilled rubber and can therefore be considered as essentially unfilled. A specimen from each compound is subjected to six cycles of periodic loading-unloading up to a pre-selected extension with stretch   . The pre-conditioning is performed to monitor the progression of stress softening, to evaluate the accumulation of residual strain and to determine the ultimate stress-deformation response for stretches up to   . The stress-strain data are shown as nominal stress versus stretch . Figure 1 shows the stress-stretch response for the natural rubber compound with 1 phr of CB filler. Figures 2 and 3 show corresponding results for compounds with 20 and 60 phr of filler, respectively. The data show a large difference in the mechanical response depending on the amount of filler content. There is a significant reduction in the stress for a given strain during each successive loading. This reduction, known as the Mullins effect, is largest on the first and second loading-unloading cycles and becomes negligible after 5-6 cycles, see Dorfmann and Ogden (2004a) for a detailed discussion on stress softening in natural rubber. In addition, the area enclosed between the loading curve and the relevant unloading curve is a measure of the energy dissipated in the material. The data show that the magnitude of non-recoverable energy depends on the amount of filler content. Energy dissipation during loading-partial unloading and reloading is analyzed in detail by Dorfmann and Ogden (2003b). After the first loading-unloading cycle a residual strain is generated in all compounds, the magnitude increases with the amount of filler content. The mechanical response of rubberlike materials, as briefly outlined in the above paragraph, differs significantly from biological tissues. Figure 4 shows experimental data of tensile loading-unloading cycles of bovine coronary artery tissue. The specimen is first loaded from the reference state to    and then unloaded. When the stress is removed, the material returns to its reference state and no residual strain is generated. The material is subjected to two additional loading-unloading cycles. During the first and second reloading, the material retraces the virgin loading curve and therefore no stress soften occurred. Also, all loading and unloading curves coincide and no energy is dissipated. The mechanical behavior of this material, contrary to the filled rubber compound, is therefore nonlinear elastic. Importantly, this chapter focuses on the nonlinear elastic response of rubberlike materials. We do not account of energy dissipation, stress softening, accumulation of residual strains nor do we include time and rate dependence. The modeling of inelastic effects in highly deformable materials is not addressed by the theory summarized below.

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Nominal Stress (MPa)

1.5

1

0.5

0 1

1.5

2 L

2.5

3

Figure 1. Pre-conditioning of a particle-reinforced dumbbell specimen with 1 phr of carbon black with maximum stretch   . 3

Nominal Stress (MPa)

2.5

2

1.5

1

0.5

0 1

1.5

2 L

2.5

3

Figure 2. Pre-conditioning of a particle-reinforced dumbbell specimen with 20 phr of carbon black with maximum stretch   .

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8 7

Nominal Stress (MPa)

6 5 4 3 2 1 0 1

1.5

2 L

2.5

3

Figure 3. Pre-conditioning of a particle-reinforced dumbbell specimen with 60 phr of carbon black with maximum stretch   . 450 400

Nominal Stress (KPa)

350 300 250 200 150 100 50 0 1

1.1

1.2

1.3 1.4 Stretch L

1.5

1.6

1.7

Figure 4. Characteristic stress-strain data of cyclic tensile loading-unloading of bovine coronary artery tissue with maximum stretch    .

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2 Kinematics 2.1 Configurations Consider a body, denoted by , as a set of points referred to as particles (or material points), which have a one-to-one correspondence with points of a region in a three-dimensional Euclidean point space. We denote such a region a configuration of the body. In the reference state the body occupies the three-dimensional region  , in the current configuration (independent of time) the region is denominated by , see Figure 5. To describe the mapping we introduce appropriate coordinate systems in the reference and in the current configuration. For simplicity we use rectangular Cartesian coordinate systems, however depending on the geometric shape of the body a cylindrical or spherical coordinate system may be more appropriate. In the reference configuration the Cartesian coordinate system is defined by the origin  and by the unit basis vectors      . The position of a generic material particle  in the reference configuration is then specified by the position vector . In the current configuration we define similarly a Cartesian coordinate system with origin 3 and unit basis vectors      . The position of the same material particle  in the current configuration is denoted by the position vector  with respect to the unit vectors      . In Figure 5 the origin of two coordinate systems and the basis vectors for the reference and current configuration coincide, but in general it is convenient to differentiate between them. For example, if a circular cylinder tube is unfolded to a rectangular sheet, a cylindrical coordinate system for the undeformed configuration and a rectangular Cartesian coordinate system for the deformed configuration may prove convenient. Applied mechanical loads deform and move the body so that the generic material particle, defined by the position vector , occupies the new position  

 in the deformed configuration . The vector field
, which is a one-to-one, orientation-preserving mapping with suitable regularity properties, describes the deformation of the body. The deformation
of the body will generally change its shape, position and orientation. A body able to change its shape is called deformable. The components of the position vectors  and  are denoted by ; and % , where Roman indices are associated with the current configuration  and Greek indices with the reference configuration  . The triple
%  %  %  or %
    identify the different points of the body and are known as material coordinates. Similarly, the coordinates
;  ;  ;  or ;
.     identify the material points in the current configuration and are known as the spatial coordinates of the point . The mapping  

 from the reference configuration  to the current

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Figure 5. The deformation of a body
from the reference configuration  to the current configuration . The inverse mapping is denoted by
 .

configuration  is given in component form by ; ; ;




%  %  % 

 


%  %  %  
%  %  % 

(1)

or ;  
%  %  % 

.   

(2)

We assume the mapping
to be invertible. The inverse mapping, which is unique, can be expressed as  

 with  ,  being the current configuration. The components of the inverse mapping are %  
;  ;  ; 

   

(3)

There are a number of books available that describe the mathematics of deformation, for example, the classic monograph by Ogden (1997) and the textbook by Holzapfel (2000).

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2.2 The deformation gradient The deformation gradient tensor 
 is defined as the material gradient of deformation by 
  Grad  Grad

 (4) where Grad being the gradient operator with respect to . Using Cartesian coordinate systems, the components of  are  

;  %

(5)

with ;  
. Since the mapping  

 is invertible, the inverse of the deformation gradient exists and is given by   grad   grad



(6)

where grad denotes the gradient operator with respect to  and as a consequence    . In component form the inverse of the deformation gradient is given by
   

%  ;

(7)

It follows that an infinitesimal material line element d at the point  transforms linearly under the deformation
into a line element d at . This important relation between d and d is given by d 

  d

  Grad
d

(8)

which alternatively can be written as d   d

(9)

d;   d% 

(10)

and in component form Note that the current configuration  of the material point  can equivalently be expressed in terms of the displacement field  as 
    


(11)

Using the Grad operator on both side of equation (11), the deformation gradient  becomes     Grad  (12) where  is the identity tensor.

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2.3 Examples Rigid body translation:

The rigid body translation of a body is given by     

(13)

where  identifies a typical material particle in the reference configuration  . The body is subjected to rigid body translation so that the point  occupies the new position . Recall that the explicit form of the vector describing the location of the point in the reference configuration and current configurations is   %   %   %  

(14)

  ;   ;   ;  

(15)

and The displacement vector, which is independent of , is denominated by . That is, every material point is displaced in an identical manner, with the same magnitude and the same direction. Rigid body rotation: Rigid body rotation is described by the equation    
 

(16)

where  is a constant vector,              is a proper orthogonal tensor and  denotes the tensor product. If the rotation is about the origin, then   and   . Note that the rotation tensor  is called a twopoint tensor since one set of unit vectors
  involves the current configuration, the second set
  is defined in the reference configuration. Simple shear: Consider the motion given by   

(17)

       

(18)

where Equation (18) may be written alternatively as                   

(19)

The deformation (17) is known as simple shear in the  plane with amount of shear . It can be seen easily that line elements parallel to the  direction remain parallel and the perpendicular distance between any pair of such line elements

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does not change. The components of the deformation gradient tensor , which we denote by , are            (20)    Uniaxial extension: Consider a circular cylindrical body extended uniformly along its axis so that the circular symmetry is maintained. Let  and ' by the reference and current length of the cylinder. The change in length in the axial direction is given by   ' and the lateral contraction, due to symmetry, by    . The deformation is then described by   

(21)

                

(22)

where It can be shown that for isochoric deformation     , therefore for uniaxial extension the deformation can be described in terms of the axial stretch only. The components of  are then given by 

   

  

 

 

(23)



where we introduced the notation    to identify the stretch in the axial direction. Pure dilation: Suppose a cube of material has edges of lengths  and ' in the reference and current configuration. Then, by symmetry         ' and  has matrix representation 

   

  

    

(24)

Example: In rectangular Cartesian coordinates, the motion is given by ; ; ;

 %  % 7  % % 7 % 7  %  % 7 % 7

(25)

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where 7 is a variable. Using equation (5), the components of the deformation gradient  are       % 7  7 7   (26)   7 7 It can be seen immediately that for 7   no deformation occurs,    and d  d. Let the value of the variable 7 be equal to 1, then the components of the deformation gradient become 

  %   

  

     

(27)

2.4 Polar decomposition theorem The polar decomposition theorem applies to any real and invertible second order tensor. It states that there exists a unique decomposition into a product consisting of a proper orthogonal tensor and a symmetric tensor. Applying this theorem to the deformation gradient  we obtain the important decomposition     

(28)

where  and  are positive definite and symmetric and are known respectively as the right and left stretch tensor. The polar decomposition theorem therefore implies that the local deformation  can be decomposed into a pure stretch  at  followed by a rigid body rotation , or by the same rigid body rotation followed by a pure stretch  at . The rotational tensor  is proper orthogonal and characterizes the local rigid body rotation of a material element. Recall that proper orthogonality implies that      . Equation (9) shows that any material element d at  is transformed linearly by the deformation gradient  into the vector d. Using equation (28) we have d   d  d

(29) (30)

Equation (29) shows that d describes a pure stretch in which a material volume element stretches in three mutually perpendicular directions (the eigenvectors of ), but does not rotate. The effect of  is then simply to rotate the stretched volume element into the final position. Similarly, the deformation can be decomposed using (30) into an initial rigid body rotation  followed by a pure stretch resulting in the same final configuration.

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2.5 Spectral decomposition The spectral decomposition theorem provides a method to represent any second order symmetric tensor in terms of its eigenvalues and the corresponding eigenvectors forming an orthonormal basis. The right and left stretch tensors  and  do have the same eigenvalues  , which have the physical interpretation of principal stretches. The spectral decomposition of the tensor  has the form 

 

 
  
 

(31)



where the eigenvectors 
 of  determine the directions in the undeformed body for which the stretch  receives extreme values. It is a volume element in the reference configuration whose edges are oriented along the unit vectors 
 . After deformation no change in angles occurred, but each edge undergoes a stretch  . Similarly,  can be represented in the spectral form by 

 

 
  
 

(32)



where each eigenvector 
 of  obeys the transformation 
  
 

.    

(33)

Equation (33) shows that  rotates the principal axes of  at the material point  into the principal axes of  at the spatial point  so that 

 


  
 

(34)



It follows that the spectral decomposition and the determinant of the deformation gradient  can be expressed as 

 

 
  
 

        

(35)



which shows that  involves two distinct configurations and is therefore a twopoint tensor. According to equations (29) and (30) the deformation of a continuum body in the vicinity of a given particle  can therefor be regarded as a two-step process: I. Impose the stretches  along the directions 
 in the reference configuration  . This represents the Lagrangian deformation
d in  since  is a Lagrangian measure of deformation.

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II. Perform a rigid rotation which maps the intermediate configuration from the previous step to the current configuration . This is accomplished by the application of the two point rotation tensor  given by equation (34). Alternatively, the deformation is performed in reverse order: I. Perform first a rigid body rotation which maps d from the reference configuration  to a new position in the current configuration . This corresponds to d (remember that  is a two-point tensor). II. In the deformed configuration impose the stretches  along the directions 
 . This is achieved by the application of  to the intermediate configuration at the end of the previous step. The result is going to be in current configuration  because  is an Eulerian measure of deformation. 2.6 Right and left Cauchy-Green tensors The polar decomposition of the deformation gradient      and the orthogonality condition T   allow to introduce additional measures of deformation. These are   T   T       T  T   

(36) (37)

where  and  are called, respectively, the right and left Cauchy-Green deformation tensors, the superscript T signifies the transpose of a second order tensor. Equations (36) and (37) show that  and  describe deformations without being influenced by a pure rotation. If there is no deformation we have      and     . It is straight forward to show that  and  are related by  

      

 

        

(38)

In addition, if follows that tr   tr
T   tr
T   tr 

(39)

which holds for arbitrary powers tr   tr  

(40)

This shows that the invariants of  and  coincide. The set of invariance most commonly used in the literature are 
 
 


 
  tr   
tr  tr
    
    
       

(41)

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Equivalently, these invariants can also be given in terms of principal stretches      by 
 
 


 
         
            
 

  

(42)



 

The physical meaning of the right and left Cauchy-Green deformation tensors  and  is now given. Consider a material line element d at  with original length d!, which transforms into the spatial element d with corresponding length d6. The transformations between d and d are given by d   d

d  d

(43)

We find d6 d!



 

d
d  d
d  d
T d  d
 d d
d  





d


T

d  d






(44) 

d  d


d

Recall that   T    and the spectral decomposition of  given by equation (31). The right Cauchy-Green deformation tensor can then be written as   

      
  
   
  
  



   



   
  
 

 

 

 
  
 

(45)



Following similar steps, the left Cauchy-Green tensor has the spectral form 

 

 
  
 

(46)



Equation (45) shows that the principal axes of  are identical with those of the stretch tensor , i.e. the tensors  and  are coaxial. The eigenvalues of  are identical with squares of the eigenvalues of . Therefore, the invariants of  and  are given by 
 
 


 
         
            
      

(47)

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2.7 Green-Lagrange strain tensor We have seen that the deformation gradient  is a fundamental quantity to describe the local deformation of a material element at . On the other hand, there exist a large number of different and equivalent measures to characterize the local strain. From mechanics of materials we know that in uniaxial deformation the strain is simple given as the ratio of the change of length to the original length of a material element. It is therefore reasonable to expect that any measure of strain to vanish if no deformation occurs. A suitable measure of strain is given by the Green-Lagrange strain tensor    T (48)
   
  
     where the  is a normalization factor. Note that if there is no deformation    and   , as expected. To provide a physical interpretation consider a material element d with original length d!, which transforms during deformation into d having final length d6. The difference of the final length squared minus the initial length squared is 

d6 d!   d
d d
d  d

  d   d
d

(49)

Given an undeformed material element parallel to the unit vector  and denoted by d  d!  . The material element deforms into d  d6 , where  is a unit vector. We then get d6 d!   (50) d!  equivalent interpretations can be given for  and  . A similar procedure, using two elements originally normal to each other, show that      describe the change in angle. Equation (12) shows that the displacement vector  can be used to express the deformation gradient, which for convenience we recall as     Grad. Using this equation in (48), the Green strain tensor can be written in the form    Grad 
Grad T 
Grad T
Grad   (51)  
T      In the linear approximation, where all higher order terms are negligible, equation (51) simplifies to   Grad 
Grad T   (52)   which defines the infinitesimal strain tensor . The components of  are   8 8 +    (53)  ; ;  

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2.8 Almansi strain tensor The Almansi strain tensor is defined in the spatial representation and can be obtained easily from the definition of the Green tensor  d  d   d
 d   d
T  d

(54)

The Almansi strain tensor can now be written as


   T        T   T T        From the definitions of the Almansi and Green strain tensors, we find that d
 d  d
 d

(55)

2.9 Deformation of area and volume elements The equations that describe the change in volume and the change in an area element in going from the reference configuration to the current configuration are of fundamental importance in nonlinear solid mechanics. These correlations are used in transforming vector and tensor quantities from a spatial description to a material description (pull back operation) or vice versa (push forward operation). Volume element: Consider a parallelepiped in the reference configuration  formed by line elements d, d
and d

at , see Figure 6. The volume is obtained by the triple scalar product, which may equivalently be written as a determinant. We have d#  d

d
 d

  
d

d


d



(56)

After deformation, the corresponding material element in the current configuration occupies the volume d9

 d

d
 d

  
d d
 
d d
*

    
d

d




d

 (57)





d 

where equation (9) has been used. Alternatively, equation (57) can be written as d9   d#

(58)

where the determinate of the deformation gradient has been denominate by , i.e.    . If the volume does not change locally during deformation then          

(59)

at  and the deformation is said to be isochoric at . If equation (59) is satisfied for all  in  then the deformation of the body is isochoric.

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Figure 6. Deformation of a volume element d# from the reference configuration  to the current configuration .

Area element: Consider an infinitesimal surface element d
in  that deforms into the infinitesimal surface element d in . Let  be a point on d
and  the corresponding point on d. Also, let d be a material line elements with image d under the deformation . Using equation (58), we write d9  d
d   d

d

(60)

with d   d( and d
  d denoting vector elements defined in the current end reference configurations. Equation (60), using the linear transformation of a material line element given by (9), may be written in the equivalent form
T d  d

d  

(61)

This equations shows that the infinitesimal areas d and d
in the current and reference configuration are related by d   T d
  d(   T  d

(62)

which is known as Nanson’s formula and applies to area elements of arbitrary shape. 2.10 Isochoric and volumetric deformations Consider a mapping
   , which describes the deformation of a deformable body with deformation gradient . Let d# be the volume element at the

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Figure 7. Deformation of a surface element d from the reference configuration  to the current configuration .

point  in the reference configuration  . The volume d9 at the point  

 in the deformed configuration  is then given by d9   d#

         

(63)

The determinant of  can also be expressed in terms of the third strain invariant as 



   
  
  
  


(64)

Importantly, during deformation, we require conservation of mass of an arbitrary volume element such that  d#   d9

(65)

which using equation (58) simplifies to   

(66)

where  and  denote the mass densities in the reference and current configuration, respectively. An isochoric (volume preserving) deformation is therefore given by       or    

(67)

A materials is defined as incompressible when every possible deformation is isochoric.

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2.11 Compressible materials There are different starting points to develop constitutive equations for compressible materials. Often, it is convenient to split the deformation gradient into a dilational (volume-changing) and isochoric (volume-preserving) contribution in the form (68)  
       where    describes the volume- changing part and  the volume-reserving part of the deformation. It follows that    

(69)

Recall the polar decomposition of the deformation gradient   . Inserting into equation (68) defines the modified stretch tensor        
    

(70)

The eigenvalues of the modified stretch tensor  can be obtained by writing  in spectral form as shown by equation (31). We have        

 

 
  
 



 

 
  
 

(71)



where the modified principal stretches are defined as      

(72)

which satisfy the incompressibility condition     

(73)

The right Cauchy-Green deformation tensor  can now be split into an isochoric and a dilatational component as   T      T  
   

(74)

where the volume preserving deformation is given by  with     and the volume-changing contribution is specified by   .

3 Stress tensors 3.1 Cauchy’s stress vector Consider a body in the deformed configuration  subject to external forces such that equilibrium is satisfied. We assume the existence of internal forces that

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Figure 8. The traction vectors and  acting respectively on an area element d in the reference configuration  and on an area element d( in the current configuration . The surface boundary in the reference configuration is denoted by  , in the current configuration by .

account for the interactions between neighboring parts of the body. These internal forces are best visualized by cutting the body by an arbitrary surface, see Figure 8. The two regions of the deformed body so created are no longer in equilibrium. Cauchy’s stress principle states that equilibrium of each region is possible by assuming the existence of internal forces acting across the surface. These forces are represented in the current configuration by a vector field , which is known as the Cauchy stress vector having physical dimensions of force per unit surface area. The vector  acts on a deformed surface element d( and depends on the orientation of d(, which is specified by the unit normal vector . 3.2 Cauchy’s stress theorem Consider a virtual surface dividing the body into two separate regions, see Figure 8. In the current configuration, an arbitrary point on that surface is denoted by , the area element by  d( and the corresponding orientation by the unit normal vector . Cauchy’s theorem states that for each orientation  the stress vector  is given by (75)   T  where the second-order tensor is defined in the current configuration and known as the Cauchy stress tensor. Equation (75) represents a tensor operation whereby the unit vector  normal to the surface area d( is transformed by into the Cauchy stress vector  acting on d(. For any orientation , the Cauchy stress tensor is unique and symmetric.

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There are alternative ways to describe the state of stress. In particular, we are interested in formulations defined in the reference configuration. Define the contact force on a unit reference area d with unit outward normal  such that  d(  d

(76)

which shows that  and have the same orientation, vector  acting in the current configuration, vector in the reference configuration. Using equation (75) and Nanson’s formula (62) gives T  d(    d

(77)

which defines the (non symmetric) nominal stress tensor by   

(78)

The traction vector in the reference configuration is than simply given by

 T 

(79)

In computational mechanics the first Piola-Kirchhoff stress tensor is often used in place of the nominal stress . It is simply the transpose of and is obtained from the Cauchy stress tensor by  T  T 

(80)

and is in general not symmetric. We introduce the symmetric Biot stress tensor defined in the reference configuration and denoted by
 . It is given in terms of the nominal stress and rotation tensor  as  (81)


 
  T T   An important symmetric stress measure, again defined in the reference configuration, is the second Piola-Kirchhoff stress tensor


    T   T

(82)

which is work conjugate to the Green-Lagrange strain measure defined in (48).

4 Equation of equilibrium Let  denote the body forces defined per unit mass and acting throughout the body. Examples of body forces are gravitational or electro-magnetic forces. Then, equilibrium in the current configuration requires that the sum of all body forces and and the sum of surface traction vanish. This can be written as     d9   d(   (83) 



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where  is the boundary surface of the body, see Figure 8. Using equation (75) the equilibrium equation can be written in terms of as     d9  T  d(   (84) 



Using d9   d# and Nanson’s formula in equation (84), we obtain the equilibrium requirement in the reference configuration     d#  T d   (85) 



where  is the boundary surface of the body in the reference state and  the corresponding density. Using the divergence theorem, equations (84) and (85) yield the equilibrium equations in local form div     Div     

(86) (87)

where div and Div denote the divergence operators in  and  respectively.

5 Constitutive equations 5.1 Hyperelastic materials The second law of thermodynamic, applied to perfectly elastic materials, shows that there exists a strain energy function, denoted $  $
. The function $ is defined on the space of deformation gradients such that the nominal stress and Cauchy stress are given respectively by $ $       (88)   These materials are defined as hyperelastic. Therefore, the stress response of a hyperelastic material is given by the derivative of the scalar-valued energy function $ , being defined per unit volume in the reference configuration and representing the work done per unit volume at  in changing the deformation gradient from  to . We take $ and the stress to vanish in the reference configuration, so that 

$
   (89)  which implies no residual stresses. We assume further that an infinite amount of energy is needed in order to expand a hyperelastic solid to infinity or to compress it to a point with vanishing small volume. $
  

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For incompressible elastic materials the strain energy function is modified to $  $
 4
 , where 4 is an indeterminate Lagrange multiplier associated with the incompressibility constraint    . The stress-deformation relations (88) are then modified to $ $  4   4     (90)   Alternatively, the strain energy function $ can be formulated in terms of the right Cauchy-Green deformation tensor  as $  $
. It is therefore necessary to find the derivative of $ with respect to . It is easy to show that 

$
 T $
      and an alternative expressions of the Cauchy stress is then    

$
 T   

(91)

(92)

5.2 Material objectivity Consider a mapping
   , which describes the deformation of a body with deformation gradient . Let d be an infinitesimal element in the current configuration , which transformed according to equation (9). If an orthogonal tensor is applied to the current configuration, the body occupies the new config The infinitesimal element transforms according to uration .  d   d  d  d

(93)

  . where we have shown that the deformation gradient  transforms as  Recall that the right and left Cauchy-Green deformation tensors are given respectively by   T  and   T . Applying the property of proper orthogonal tensors T  , it follows immediately that    

   T  

(94)

which shows that the  is an objective tensor field and is not effected by a rigid body motion applied to the current configuration. Similarly, the elastic stored energy is required to be independent of superimposed rigid deformations and the principle of objectivity requires that   $
  $
 $


(95)

for all rotations and for all deformations . A strain energy function satisfying this requirement is said to be objective. Using the polar decomposition    and the choice  T in the objective strain energy function gives $
  $


(96)

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This statement is of fundamental importance and implies that the strain energy of a hyperelastic material depends on  only through the stretch tensor . $
 automatically satisfy the objectivity requirement. It follows that the Biot stress tensor
 , which for shortness we indicated by

, is given by $ (97)

  for an unconstrained material and



$ 4  

   

(98)

for an incompressible material. 5.3 Isotropic elastic materials A very important class of hyperelastic materials exists when the mechanical behavior is isotropic. As a matter of fact, most rubberlike solids can be considered isotropic and many constitutive equations exist to describe the behavior of this class of materials. Let  be a material point in the reference configuration  and consider the motion described by the deformation gradient . We have seen in equation (88) that the Cauchy stress is a function of the deformation only. This can be expressed as 
 (99) Consider a different reference configuration 
and apply the deformation 
in order to have the same final configuration, i.e. the same stress . We write 


 

 


(100)

where is the deformation gradient describing the motion from the reference configuration  to 
and where we have used the important relation   
. In general we have 
 (101) however, for isotropic materials and for  , where is an orthogonal tensor we have 
and 
 
  
T  


(102)

Isotropy means that the response of a material specimen is independent of its orientation in the reference configuration and that the stress function
 depends on  only through the left stretch tensor .

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Given the deformed configuration  of an isotropic material, which is rotated into  by an orthogonal tensor . A constitutive material law is said to be objective if it satisfy the condition  
 T  
 T  


(103)

for arbitrary orthogonal tensors . This is equivalent of saying that
 is an isotropic tensor function, which according to the representation theorem (shown in Subsection 5.4) can be represented as a quadratic polynomial in . In the above equation, the left Cauchy-Green tensor is objective, i.e. it satisfies the condition    T .  5.4 The representation theorem The isotropic tensor function
 can be written as 
          

(104)

where the three scalar coefficients 
.     are functions of the invariants of the deformation tensor  and thus itself invariant. Equation (104) is known as the Rivlin-Erickson representation theorem and shows that for an isotropic elastic materials and and  are coaxial, i.e. they have the same eigenvectors. Equation (104) represents the most general form of a constitutive equation for isotropic elastic materials. For the undeformed configuration for which   , the state of stress is given by       (105) 
   where  
.     are the material functions in the reference configuration. If the material is stress free in the reference configuration, then it is defined as a natural state for the material. For a stress free state to exist, the material functions must satisfy the condition       (106)   The representation theorem may be written alternatively in terms of    as 
          

(107)

where the coefficients 
.     depend on the three invariants of the left stretch tensor , which are defined by equation (47). 5.5 Constitutive equations in terms of strain invariants Constitutive relations of the form 
 or 
 are isotropic if the strain energy function $ is given as a symmetric function of the invariants of 

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(equivalently ). This requires that $  $

 
 
  $

 
 


(108)

The explicit expression of the nominal stress tensor , defined by equation (88) , can be obtained easily by using the chain rule of differentiation. Starting from equation (108), we write 

$  $  $  $           

(109)

where the explicit expressions of the derivatives of the invariants with respect to the deformation gradient  are required. The differentiations are straight forward and give   T  

   T T T  

     

(110)

Inserting into (109) yields the following expressions of the nominal stress 

$ $  $ T  
 T T T        

(111)

The Cauchy stress shown in equation (88) is obtained from the nominal stress by using the transformation     . It has the form    $ $ $ $            (112)     Comparing with equation (104), we find the expressions of the response functions 
.     in terms of derivatives of the strain energy function $ . These are   $ $ $ $                       (113) To conclude this section, it can easily be shown that 

$ T $ $        

(114)

which allows to use an isotropic hyperelastic strain energy function in terms of the left Cauchy-Green tensor . The Cauchy stress tensor is then given by    

$  

(115)

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5.6 Constitutive equations in terms of principal stretches For an isotropic elastic solid W depends on  only through the principal stretches      , as shown by equation (96). We write this dependence as $  $
       $
       $
     

(116)

which implies that $ it is a symmetric function of the principal stretches. A consequence of isotropy is that Biot stress is coaxial with the stretch tensor , i.e. and  have the same eigenvectors. This motivates to write and  in the spectral forms 

 

 
  
 





 

7 
  
 

(117)



The direct calculation of the nominal stress , as shown by equation (88) , requires a lengthy calculation. However, we obtain the nominal stress indirectly by first calculating the Biot stress and then using the connection  T . Thus, we have   $ $
  or    
  (118)

     where the principal values of the Biot stress are given by 7 

$  

(119)

The nominal stress tensor is then given by  T 

  $ 




  
 

(120)

which shows that is a two-point tensor having one index in the reference configuration, the other in the current configuration. Using equation (78) to transform from nominal stress to Cauchy stress yields     

 

  



 



$
   
 

 
  
 

(121)



where

$   are the principal Cauchy stresses defined in the current configuration.     

(122)

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5.7 Incompressible isotropic hyperelasticity A significant simplification can be obtained in the solution of boundary value problems if the material can be treated as incompressible and therefore subject to the constraint          
  
  
  
  

(123)

This implies that the deformation can be described in terms of  and  only, the third invariant   . Equivalently, only two of the three principal stretches remain independent. It follows therefore, that the strain energy function $ is given as a function of the two invariants    or equivalently two principal stretches. To obtain the correct expression for the stress tensor, we write the strain energy function as 4 (124) $  $
   
   where 4 is an indeterminate Lagrange multiplier. The nominal stress is now given by equation (90) , and in terms of the strain invariants becomes 

$ T $  
 T T T  4    

(125)

The expression of the Cauchy stress is given by (90) . It can be obtained directly form the nominal stress using the transformation   as   $ $ $        4  (126)    When the strain energy is expressed in terms of principal stretches, the strain energy formulation for incompressible materials modifies to $  $
      4
 

(127)

subject to the incompressibility constraint     . The principal components of the Biot stress and Cauchy stress are then given by 7 

$  4  

  

$ 4 

(128)

Alternatively, the incompressibility condition      allows at most two of the three principal stretches to be independent. The strain energy can therefore  be written as symmetric function of  and  with      . This allows to eliminate the unknown pressure 4 from equation (128) and leads to the principal stress differences    

$  

   

$  

(129)

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Since the material is incompressible, it is always possible to superimpose an arbitrary hydrostatic stress, which allows to set    in the above equations   

$  

  

$  

(130)

The principal Biot stresses are then given by the very simple equations 7 

$  

7 

$  

(131)

5.8 Compressible hyperelasticity We have seen in Subsection 2.11 that for compressible materials the total deformation can be decomposed into two parts. For example, equation (74) shows that the right Cauchy-Green deformation tensor  can be modified to obtain an isochoric and a volume-changing contribution. This motivates to define a strain energy function $ composed of two additive terms, one as a function of the determinant of the deformation gradient , the other as a function of the modified right Cauchy-Green deformation tensor . We therefore write $
  $vol
  $iso


(132)

where $vol
 and $iso
 describe the volumetric and isochoric components of the strain energy. For $vol
 we require, in particular, that the conditions $vol
  


$vol
  

(133)

and

$vol  0

for

  

$vol
   as

      (134)

are satisfied. Therefore, an infinite amount of energy is required to increase the volume to infinity or compress it to zero. We have also introduced the bulk modulus in the reference configuration 0 . This suggests a similar decomposition of the total stress tensor. For example, the additive components of the nominal stress tensor are obtained by differentiation of $ with respect to  

$
 T   vol  iso  

(135)

where vol represents the volumetric contribution and iso the isochoric contribution.

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Similar arguments apply to formulations defined in the current configuration. The left Cauchy-Green deformation tensor can be decomposed similarly in isochoric and volume-changing contributions. Again, using the split     , we have T

  T      
  

 
   (136)

   

such that the strain energy function can be separated into two contributions $
  $vol
  $iso


(137)

For the special case of isotropic hyperelastic materials, we need to define the modified strain invariants and the modified stretches. These are given by     
   


    
   


   

(138)

 tr   tr       
tr  tr
  
tr  tr
           

(139)

which can be obtained from  or    

or alternatively from the invariants  and  by       

      

   

(140)

The expressions of the modified stretches will not be repeated here. They have been derived previously and are given by equation (72).

6 Examples of strain energy functions There are a large number of strain energy functions available in the literature to describe the nonlinear elastic response of rubberlike solids subject to large deformations. In this section we provide a brief overview. The strain energy function $ of an isotropic elastic solid can be expressed either as a symmetric function of the principal stretches      or as a function of three independent invariants, for example,      . The strain energy is defined per unit volume in the reference configuration  and represents the work done (on the unit volume) in changing the deformation gradient from  to . One convenient way to formulate a strain energy function in terms of the invariants      is to expand $ in the Taylor series $
      

 


 
 
  



where  are material coefficients to be determined experimentally.

(141)

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6.1 Incompressible hyperelastic materials For an incompressible material   , which removes the strain energy dependency from the third invariant. Therefore, for an isotropic incompressible material we have   $
     
 
   (142) 

An important example to describe the behavior of incompressible materials is the linear expansion known as the Mooney-Rivlin energy formulation. It can be expressed in terms of invariants or equivalently as a function of the principal stretches. The Mooney-Rivlin strain energy function has the form $

 
   
     
       
       (143)

where    are constants and     . When    this reduces to the simplest model, the so-called neo-Hookean strain energy formulation. It is given by   (144) $  
   
        where  has been replaced by ,  being the shear modulus of the material in the undeformed reference configuration. The neo-Hookean strain energy function may also be derived from the statistical theory of rubber elasticity, which is based on the concept of Gaussian network distribution. A detailed discussion on the elasticity of a molecular network is given by Treloar (1975). Using the statistical theory, Treloar (1975) found that the shear modulus  is equivalent to    0" , where the value  represents the number of freely jointed segments in a molecular chain, each of length 1. The extended total chain length is then  1, 0 is Boltzmann’s constant and " is the temperature. Another important strain energy formulation is due to Varga (1966) given by $  
     

(145)

which is used extensively in basic stress-strain analysis,  is a material coefficient to be determined by fitting experimental data. Knowles (1977) proposed a strain energy formulation of the type     $   
    (146)  2 where and 2 are positive dimensionless material parameters and  is the shear modulus. When 2   we obtain the neo-Hookean model (144), for 2   we

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recover the important strain energy formulation initially proposed by Fung (1967) to model biological materials. It has the form $ 

 
    

(147)

which accounts for an exponential stiffening at increasing deformation observed in many biological tissues. 6.2 Compressible hyperelastic materials The Mooney-Rivlin strain energy formulation in (143) is modified to account for compressibility by adding an addition term. The compressible Mooney-Rivlin model has the form $  
   
   $
 

(148)

with $ describing the pressure-volume response of the material. Similarly, Blatz and Ko (1962) modified the neo-Hookean model by introducing additional terms to account for the compressibility    
,  
   $  
    
       
 ,  (149)  
 where  is the shear modulus,
a material parameter describing the compressibility and , a value between 1 and 2. An obvious step to generalize the strain energy functions described above is by introducing nonlinearities in the material coefficients. In this context, we like to mention the model by Haughton (1987) in form $  $
   
   
 

(150)

where the coefficient  has been replaced by an unknown function $ to be determined experimentally. Carroll (1988) introduced the nonlinearity with respect to the second strain invariants such that the strain energy function has the from $  
   $
   
 

(151)

with the unknown function $ to be again determined experimentally. Most of the strain energy functions shown above are of the form $  :
   :
   :
 

(152)

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179

which was introduced by Valanis and Landel (1967). Next, we consider the well known Ogden model. It is is formulated in terms of powers of the stretches as $ 

    
         

(153)

where  and  are material constant, the latter not necessarily being integers, (Ogden, 1972a), in general   . Consistency with the linear theory requires that the parameters in equation (153) satisfy 

   

(154)



where  is the shear modulus in the reference configuration. Ogden (1972b) modified the above theory to compressible materials, it is expressed in terms of the modified principal stretches  and has an additional energy term to describe the purely elastic volumetric response $ 

     
       

  

 

(155)

where  is the volume ratio and
 , (Ogden, 1972b). Several alternative expressions exit for the additional volumetric strain energy formulation but they will not be given here. 6.3 Other I models If limited test data are available for model calibration, it is strongly recommended to select a strain energy formulation that depends on one invariant only, a so called  model. A classical example is the neo-Hookean model given by (144). It is well known, that the neo-Hookean model is not able to capture the stiffening effect at larger strains observed in many filled rubber compounds, see stress-strain data shown in Figure 3. Gent (1996) generalized the neo-Hookean model by introducing the additional parameter  , which represents the limiting chain extensibility and accounts for the material stiffening at larger strain. The model has the from      $      (156)   and it can be seen that for    the Gent model reduces to the neo-Hookean material (144).

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A very popular example, in addition to the Gent model, is the strain energy formulation due to Arruda and Bocye (1993). The Arruda-Boyce model is based on a micro-mechanical representation of the rubbery solid in the form of a eight chain model and uses statistical theories to physically link the 2 material parameters to the polymeric network. It is given by  $ 

   
  
  
            

(157)

where  is the shear modulus and  is the number of statistical segments in a chain between permanent crosslinks, each of length '. Note, that the energy formulation only depends on  . For compressible materials, the strain invariant  is replaced by the modified invariant       and an additional energy terms is added to describe the change in volume .

7 Boundary value problems We now apply the equations described in the preceding sections to two representative boundary-value problems in order to illustrate the theory. We consider an incompressible isotropic material and apply the theory to problems with circular cylindrical geometry. We first consider pure azimuthal shear of a thick-walled tube, followed by combined extension and inflation. Various energy functions are used to show the difference in the mechanical response. 7.1 Pure azimuthal shear Geometry: Consider a circular cylindrical tube of a hyperelastic, isotropic and incompressible material, which is defined in the reference configuration by 

 

   

  &  

(158)

where
  & are the coordinates relative to a cylindrical polar basis 
    &. A pure azimuthal shear is defined as an isochoric plane strain deformation with no change in the radial direction. We therefore describe the deformation by 5      -
 <  & (159) where
5  < are cylindrical polar coordinates in the deformed configuration associated with the cylindrical polar basis 
.  5  <, and -
  -
5 is a function to be determined. The boundary condition on the inner and outer surfaces of the tube are -
(   -
)   (160)

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where , which may be positive or negative, is the angle of rotation of the outer boundary 5  )   relative to 5  (  . We take   to correspond to the positive (anticlockwise) sense of shear (the sense of increasing ). Referred to cylindrical polar coordinates the deformation gradient tensor  has the form (161)         where         is a proper orthogonal rotation tensor in cylindrical coordinates. The corresponding right and left Cauchy-Green tensors, defined respectively by equations (36) and (37), are   T     
                 T    
              

(162) (163)

where   5-

5, the prime on - indicates differentiation with respect to 5  . Note that  is the amount of shear and represents locally a simple shear in planes normal to  . For the pure azimuthal shear deformation, the principal invariants of  are         

  

(164)

Constitutive law: Consider an incompressible hyperelastic material with strainenergy function $  $
 . For such materials, the Cauchy stress tensor given by equation (126) simplifies to  4  $ 

(165)

where 4 is a Lagrange multiplier associated with the incompressibility constraint and the abbreviation $  d$ d . For the deformation here considered and using (165), we find the components of being equal to    

  

4  $  4  $
     4  $ 

(166) (167) (168)



$ 

(169)

while the remaining shear components are   

  

(170)

Alternatively, from (166), (167) and (169) we find that the stress component  can also be written as       .

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A. Luis Dorfmann

The invariants in (164) depend only on . We may therefore introduce a new function, denoted :, such that :
  $
 

(171)

Differentiation of (171) with respect to  gives the simple formula     :

d:  d

(172)

Equilibrium equations: For the considered plane strain deformation, in the absence of body forces, only two of the equilibrium equations (86) are not satisfied trivially. This is the equations in the radial direction  d  
     d5 5

(173)

and in the azimuthal direction  d     d5 5

(174)

d 
5     d5

(175)

The latter can be rewritten as

which after integration gives   : 

)   5

(176)

wherein we have introduced the notation  
). This equation determines  and by integration the function -
5. The equilibrium equation in the radial direction, using (166) and (167), becomes d  -
:   (177) d5 which determines  and the Lagrangian multiplier 4 after -
5 has been determined from (176). For a full discussion of the azimuthal shear problem we refer to, for example, Ogden (1997), Jiang and Ogden (1998), Horgan and Saccomandi (2001).

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7.2 Illustration Rubberlike materials:  has the form

Consider the strain energy formulation, which in term of  
  $    0 

(178)

where  is the constant shear modulus for infinitesimal deformations and 0  . In terms of  equation (178) can equivalently be written as  
       (179) : 0  Using equation (176) we get 
      

 )  5

(180)

This equation can in principle be solved for  and hence for the function -
5. For simplicity we assume that 0   so that an explicit solution can be found. Equation (179) reduces to the classical neo-Hookean model :

    

(181)



 )  5

(182)

and from equation (176) we get

After integration and using the boundary condition -
(  , we obtain the well known result   

 )  -
5   (183)  ( 5 which is shown in Figure 9. The relation between the angle of rotation of the outer surface 5  ) and the shear stress  is therefore linear  

 )  

 -
)   (184)  ( ) and shown graphically in Figure 10.

184

A. Luis Dorfmann 0.15

Function g(r)

0.12

0.09

0.06

0.03

0 1

1.2

1.4

Radius r

1.6

1.8

2

Figure 9. The function -
5 versus radius 5. The inner radius (  , the outer radius )  , the applied shear stress    and the stiffness   . 0.15

Angle Y

0.12

0.09

0.06

0.03

0 1

2

3

4

5 6 Applied Shear T

7

8

9

10

Q

Figure 10. The angle of rotation versus shear stress  on the outer surface 5  ). The inner radius (  , the outer radius )  , the applied shear stress

   and the stiffness   .

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185

It has been shown by Horgan and Saccomandi (2001) that the remaining normal stress components have the form   )

     (185)  5  

 )     (186)  5 

  

(187)

These results show that the radial and axial components are compressive, while the hoop stress is tensile. Gent model: We consider the strain energy function (156), which is due to Alan Gent (1996). The additional parameter  represents the limiting chain extensibility and accounts for the material stiffening at larger strains. For pure azimuthal shear, with       , the strain energy in terms of  has the form     :      (188)   Equation (176) now becomes )      5    

(189)

which can be solved explicitly, see Horgan and Saccomandi (2001). Figure 11 shows the function -
5 for values of the parameter         . Note that for increasing values of  we approximate the solution from the neoHookean model, see Figure 9. Similarly, Figure 12 shows the amount of rotation of the outer surface as a function of the applied shear stress  . For smaller values of  , the material becomes stiffer, as expected.

186

A. Luis Dorfmann 0.15

Function g(r)

0.12

0.09

0.06

0.03

0 1

1.2

1.4

Radius r

1.6

1.8

2

Figure 11. The function -
5 versus radius 5 for the Gent model with         . The inner radius (  , the outer radius )  , the applied shear stress    and the stiffness   . 0.15

Angle Y

0.12

0.09

0.06

0.03

0 1

2

3

4

5 6 Applied Shear T

7

8

9

10

Q

Figure 12. The rotation versus shear stress  on the outer radius for the Gent model with         . The inner radius (  , the outer radius )  , the applied shear stress    and the stiffness   .

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187

7.3 Inflation and extension of a circular tube Geometry: Consider a circular cylindrical tube whose reference geometry is defined by           &   (190) where    are positive constants and
  & cylindrical polar coordinates associated with basis vectors
     . The deformed configuration is specified in terms of cylindrical polar coordinates (5  <) with basis vectors
     . Consider the combined deformation consisting of uniform axial extension and radial expansion (inflation). This implies that the deformed tube maintains its circular symmetry and therefore all variables depend only on the coordinates 5 and <. The location of an arbitrary point  in the deformed configuration is specified by the position vector (191)   5  <  The geometric shape of the deformed tube is defined in terms of the coordinates 5  < such that (  5  )        <  1 (192) For simplicity, we again assume an incompressible material and describe the deformation by  5  ,
  (   




 





  

<   &

(193)

where  is the constant axial stretch. The deformation gradient for inflation and extension of a circular tube is given by 

 ,

   

5

        

(194)

                where   and  are the principal stretches associated respectively with the radial, azimuthal and axial directions. Explicitly, these are      

   

5



   

(195)

where we introduced the notation  and used the incompressibility constraint      to define  . Rewrite equation (193) as 



 
( 5 

(196)

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A. Luis Dorfmann

and after a few intermediate steps we find    






    

 
    

(197)

where we introduced the notations  

(  

 

)  

)  ,


(198)

The expressions of the principal invariants of the left Cauchy-Green tensor , as defined in (41), are            

         

  

(199)

Stress components: The Cauchy stress tensor for incompressible materials has been defined by equation (90) , which in component form becomes   

$ 4  

(200)

with . / 
5  < and 
  &. Let the strain energy function of an isotropic and incompressible material be given in terms of two invariants $  $
    with   . It follows, that the nonzero components of the Cauchy stress tensor are     4     $  $
         4   $  $
        4   $  $
     

(201)

where the abbreviation $
.    signifies differentiation with respect to the strain invariant, i.e. $  $ . The stress components in the deformed configuration need to satisfy the equilibrium equation (86). In the absence of body forces and using cylindrical coordinates, the equilibrium equations in 5  and < directions are           
     5 5  < 5           
      5 5  < 5               5 5  < 5

(202)

Modeling of Rubberlike Materials

189

Due to symmetry requirements of the deformation, only the equilibrium equations in the radial direction is not trivially satisfied. It reduces to d   
     d5 5

(203)

From equation (199) we note that the invariants  and  only depend on the two independent quantities   . It is therefore convenient to define an alternative strain energy function : in the form :
    $
   

(204)

which allows to write the stress differences in the simple form    

:  

   

:  

(205)

and the equilibrium equation (203) can now be written in the alternative form 5

: d    d5 

(206)

Let the tube be subjected to a pressure  applied to the inner surface 5  ( with no mechanical loads applied to the outside surface 5  ). The two boundary conditions have the form   

on

5  (

  

on

5  )

(207)

Integration of equation (206) and the use of the boundary conditions (207) leads to   : d5     (208)  5

which, in principle, for a given : allows to determine . From equation (196) we have      
5 (   and therefore the stretch  is   5  5   
5 (   

(209)

(210)

To change the variable of integration in equation (208) from 5 to  we need the derivative of (210) with respect to 5. This is 5

d 
     
   d5

(211)

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A. Luis Dorfmann

We then have the alternative formulation   : d  
    


(212)

where  depends on  through equation (197). Equation (212) therefore provides an expression for  as a function of  (equivalently of the deformed radius) when  is fixed. The resultant axial force  on any cross-section of the tube is calculate from 

  



  

 5 d5  




  5 d5 

 5 d5 

(213)

The second integral on the right hand side can be integrated by parts to give 





(   5 d5   





  5



5 d5 

(214)

where we used the equilibrium equation (203) and the boundary conditions on the inner and outer surfaces 
(   and 
)  . The resultant axial force can then be written as    
    5 d5  (  (215)

The use of equation (205) allows to rewrite the force  in terms of the strain energy function  



 

 

    

 5 d5  ( 

(216)

Using equation (211), together with (197) we find that 5 d5 


      d
  

(217)

which allows to change the variable of integration in equation (216), namely  

 


  



 




   


     

  d  (218)

where we switched the limits of integration to eliminate the negative sign in equation (217).

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191

7.4 Illustration Biological tissues: The rapid stiffening of biological tissues, when large loads are applied, can best be described by a constitutive model based on an exponential strain energy formulation. For the illustration below we use the constitutive law suggested by Fung, see equation (147). The shear modulus of the material  and the constant need to be determined from experimental data. For the incompressible material considered, the explicit form of the first strain invariant is given by equation (199) . To evaluate equation (212) we need the derivative of the strain energy with respect to . This is given by :  
    
  

(219)

In the numerical evaluation of equation (212) we consider a circular tube with an exterior diameter of 25.4 mm and an initial wall thickness of 1.2 mm and representative values of    kPa and  . Figure 13 shows the internal pressure versus  for increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top in the figure. The axial stretch    and is constant. Figure 14 shows the corresponding results for an axial stretch of    . These figures show that for larger values of  more pressure is required to induce the same amount of radial expansion. The axial force  of a circular cylindrical tube is given by equation (218). Figures 15 and 16 show the increase of  as a function of the circumferential stretch  . Both figures show an initial step in the applied axial force corresponding to the extension of    and    , respectively. These figures show that the normal force  increases exponentially to keep the axial stretch constant during inflation.

192

A. Luis Dorfmann 2 1.8 1.6

Pressure (KPa)

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.98

1

1.02

1.04

1.06

1.08

1.1

L

b

Figure 13. Plot of the internal pressure  versus  of a circular cylindrical tube. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top. The axial stretch    and is constant. 2 1.8 1.6

Pressure (KPa)

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.98

1

1.02

1.04 Lb

1.06

1.08

1.1

Figure 14. Plot of internal pressure versus  of a circular cylindrical tube. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top. The axial stretch    and is constant.

Modeling of Rubberlike Materials

193

1.8 1.6

Axial Force (N)

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.98

1

1.02

1.04

1.06

1.08

1.1

L

b

Figure 15. The resultant axial force  versus  of a circular cylindrical tube. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top. The axial stretch    and is constant. 1.8 1.6

Axial Force (N)

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.98

1

1.02

1.04

1.06

1.08

1.1

Lb

Figure 16. The axial force  of a circular tube versus  of a circular cylindrical tube. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top. The axial stretch is    and constant.

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A. Luis Dorfmann

Rubberlike materials Neo-Hookean model: A very important strain energy formulation to model nonlinear elastic deformations is the classical neo-Hookean law (144), which we repeat here for convenience  (220) : 
   and where the expression for  is given by equation (199) . To determine the pressure-deformation response defined by equation (212) we require the derivative of the strain energy with respect to  : (221)  
      Consider a tube having the same geometric dimensions as specified previously in this subsection and use a representative value of      MPa. Figure 17 shows the internal pressure  versus the radial stretch at the outer surface  for increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top in the figure. The axial stretch is constant and equal to    . Note that the value of  corresponding to zero internal pressure is smaller than one, i.e. during the initial stretch the tube contracts in the lateral direction. These graphs show that the neo-Hookean model is not able to account for the material stiffening at large deformations. Evaluating equation (218) gives the resultant axial force  as a function of the radial stretch  . These results are shown in Figure 18, where the initial step represents the axial force needed to stretch the tube by an amount    , before inflation begins. During inflation, the axial force changes to keep the elongation constant. The graphs are for increasing wall thicknesses and again show no stiffening for large values of  . Gent model: Gent (1996) modified the neo-Hookean formulation to account of the experimentally observed stiffening of rubberlike solids at large deformations, see Figure 3. The Gent model accounts of the material stiffening by including the limiting polymeric chain extensibility through the model parameter  . Recall that the Gent model has the form      :      (222)   where  is the shear stiffness in the reference state and  is the first strain invariant. The Gent model reduces to the neo-Hookean formulation by considering an infinite chain extensibility, i.e. by letting   . The value of  needs to be determined from experimental data, for illustration purposes we here consider   .

Modeling of Rubberlike Materials

195

0.2

Pressure (MPa)

0.15

0.1

0.05

0

1

2

3

4

5

L

b

Figure 17. Plot of the internal pressure versus  of a neo-Hookean circular cylindrical tube. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top. The axial stretch    and is constant. 400 350

Axial Force (N)

300 250 200 150 100 50 0

1

2

3

4

5

L

b

Figure 18. Resultant axial load  versus  of a circular cylindrical tube made of a neo-Hookean material. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm. The axial stretch    and is constant.

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A. Luis Dorfmann

To evaluate the pressure-inflation response (212) and to determine the resultant axial force  given by (218), the derivatives of (222) with respect to  and  are needed. These are, respectively,

and

 :      
   
 

(223)

 :      
   
 

(224)

The graphs in Figure 19 show the pressure-inflation response of a cylindrical tube having the same geometric dimensions, the same value of the shear stiffness and the same amount of axial stretch as consider for the neo-Hookean model. The difference in evaluating the pressure-inflation response using the neo-Hookean formulation, shown in Figure 17, and the Gent model in Figure 19 is the stiffening at large values of  . Note, that for an inflation up to about   , the two models give essentially the same material response. Graphs are for increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, the tube with the the thinnest wall obviously being the most flexible one. Figure 20 shows the resultant normal force acting on any cross-sectional area of the cylindrical tube versus the radial expansion  , which is required to keep the axial elongation constant during inflation. Comparing these graphs with the results shown in Figure 18 we immediately note that the response for small inflations is equal, for large deformations the Gent model captures the material stiffening typical of rubberlike solids. Ogden model: To simulate the material response during inflation and extension of a circular tube we now use the elastic strain energy function due to Ogden (1972a). Recall that the Ogden model is given by $
      

    
          

(225)

where  and  are material constants that must satisfy the requirement (154). The nonlinear iterative method known as the Levenberg-Marquardt algorithm is used frequently to determine the valued of the constants  and  , for a given set of experimental data (Twizell and Ogden, 1983). For the numerical results shown in this section, we use the values listed in Table 1.

Modeling of Rubberlike Materials

197

0.2

Pressure (MPa)

0.15

0.1

0.05

0

1

2

3

4

5

Lb

Figure 19. The internal pressure versus radial stretch for a Gent material. The axial stretch    and is constant. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, reading from the bottom to the top. 400 350

Axial Force (N)

300 250 200 150 100 50 0

1

2

3

4

5

Lb

Figure 20. The resultant axial force versus the radial stretch  using the Gent model. The force is necessary to keep the axial stretch    constant during inflation. Graphs correspond to wall thicknesses of 1.2, 1.4, 1.6 and 1.8 mm.

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A. Luis Dorfmann

Table 1. Summary of model parameters used to simulate inflation and extension of a circular tube made of an incompressible rubberlike material. The values of      are given in MPa. Material Model Parameter, Ogden     1.26

 1.3

 0.024E1

 5.0

 0.2E1

 2.0

For the biaxial deformation here considered the strain-energy function $
      becomes 
            :
      

(226)

and the derivative of : with respect to  is 
:             

(227)

The pressure  versus circumferential stretch  using Ogden’s model is shown in Figure 21. Results are given for increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm, again reading from the bottom to top. Data in Figure 21 are obtained for a constant axial stretch of    and shear modulus      MPa. Note the non-uniqueness in the relationship between applied pressure and radial expansion, which was not captured using the neo-Hookean model or the Gent model. Initially the pressure increases rapidly with the diameter. After the pressure has reached a maximum value an instability occurs with an associated loss of symmetry in the configuration of the inflated tube, i.e. bulges may develop. The corresponding total axial force  , calculated from equation (218), is shown in Figure 22 as a function of the radial stretch  . It represents the axial force that is necessary to keep the total length of the tube constant during inflation. The initial vertical jump represents the necessary force for an axial elongation of    before inflation begins. A very detailed analysis on the loss of symmetry during extension and inflation of thin and thick walled cylindrical tubes is given by Haughton and Ogden (1979b) and Haughton and Ogden (1979a).

Modeling of Rubberlike Materials

199

0.2

Pressure (MPa)

0.15

0.1

0.05

0

1

2

3

4

5

Lb

Figure 21. Pressure versus circumferential stretch  using Ogden’s material model. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm and constant axial stretch    . 400 350

Axial Force (N)

300 250 200 150 100 50 0

1

2

3

4

5

Lb

Figure 22. Total axial force  versus circumferential stretch  using Ogden’s material model. Graphs correspond to increasing wall thickness of 1.2, 1.4, 1.6 and 1.8 mm and constant axial stretch    .

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Fung, Y. C. (1993). Biomechanics: Mechanical Properties of Living Tissues. Springer. Fung, Y. C. B. (1967). Elasticity of soft tissues in simple elongation. American Journal of Physiology, 213:1532–1544. Gent, A. N. (1996). A new constitutive relation for rubber. Rubber Chemistry and Technology, 69:59–61. Haughton, D. M. (1987). Inflation and bifurcation of thick-walled compressible elastic spherical-shells. IMA Journal of Applied Mathematics, 39:259–272. Haughton, D. M. and Ogden, R. W. (1979a). Bifurcation of inflated circularcylinders of elastic-material under axial loading–I. Membrane theory for thinwalled tubes. Journal of the Mechanics and Physics of Solids, 27:179–212. Haughton, D. M. and Ogden, R. W. (1979b). Bifurcation of inflated circularcylinders of elastic-material under axial loading–II. Exact theory for thickwalled tubes. Journal of the Mechanics and Physics of Solids, 27:489–512. Holzapfel, G. A. (2000). Nonlinear Solid Mechanics. John Wiley & Sons. Holzapfel, G. A. and Gasser, T. C. (2001). A viscoelastic model for fiberreinforced composites at finite strains: Continuum basis, computational aspects and applications. Computer Methods in Applied Mechanics and Engineering, 190:4379–4403. Holzapfel, G. A., Gasser, T. C., and Ogden, R. W. (2000). A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity, 61:1–48. Holzapfel, G. A., Gasser, T. C., and Stadler, M. (2002). A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis. European Journal of Mechanics A-Solids, 2:441–463. Horgan, C. O. and Saccomandi, G. (2001). Pure azimuthal shear of isotropic, incompressible hyperelastic materials with limiting chain extensibility. International Journal of Non-Linear Mechanics, 36:465–475. Humphrey, J. D. (2001). Cardiovascular Solid Mechanics. Springer. Humphrey, J. D. (2003). Review paper: Continuum biomechanics of soft biological tissues. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 459:3–46. Hunter, P. J., McCulloch, A. D., and ter Keurs, H. E. D. J. (1998). Modelling the mechanical properties of cardiac muscle. Progress in Biophysics & Molecular Biology, 69:289–331. Jiang, X. and Ogden, R. W. (1998). On azimuthal shear of a circular cylindrical tube of compressible elastic material. Quarterly Journal of Mechanics and Applied Mathematics, 51:143–158. Knowles, J. K. (1977). The finite anti-plane shear field near tip of a crack for a class of incompressible elastic solids. International Journal of Fracture, 13:611– 639.

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Ogden, R. W. (1972a). Large deformation isotropic elasticity - correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences, 326:565–584. Ogden, R. W. (1972b). Large deformation isotropic elasticity - on the correlation of theory and experiment for compressible rubberlike solids. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences, 328:567–583. Ogden, R. W. (1997). Non-Linear Elastic Deformations. Dover Publications. Ogden, R. W. and Roxburgh, D. G. (1999). A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 455:2861–2877. Reese, S. and Govindjee, S. (1998). A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures, 35:3455–3482. Spencer, A. J. M. (1971). Theory of invariants. In Eringen, A. C., editor, Continuum Physics, Vol. 1. Academic Press, New York. Tong, P. and Fung, Y. C. (1976). Stress-strain relationship for skin. Journal of Biomechanics, 9:649–657. Treloar, L. R. G. (1975). The Physics of Rubber Elasticity. Oxford University Press. Truesdell, C. and Noll, W. (1965). The Non-Linear Field Theories of Mechanics. Springer. Twizell, E. H. and Ogden, R. W. (1983). Non-linear optimization of the material constants in Ogden stress-deformation function for incompressible isotropic elastic-materials. Journal of the Australian Mathematical Society B-Applied Mathematics, 24:424–434. Varga, O. H. (1966). Stress-strain behavior of elastic materials, selected problems of large deformations. Wiley - Intersience. Zheng, Q. S. and Spencer, A. J. M. (1993). Tensors which characterize anisotropies. International Journal of Engineering Science, 31:679–693.

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Imperfect Bonding with Nonpenetration Condition

211

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212

George Mejak

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Imperfect Bonding with Nonpenetration Condition

213

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214

George Mejak

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Imperfect Bonding with Nonpenetration Condition 1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

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215

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216

George Mejak

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0 0

5

10

15

20

25

30

0

5

10

15

20

25

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Imperfect Bonding with Nonpenetration Condition

217

2.5

2.0

1.5

1.0

0.5

0.0 0

50

100

150

200

(&2/$  4EIA ARKHQPEKJ KB @EOLH=?AIAJPO BKN PDA JKJ EILQHOERA ATPAJOEKJ KB PDA NECDP AJ@ KB PDA >=N

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50

100

150

200

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218

George Mejak

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2.0

1.5

1.0

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0.0 0

50

100

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(&2/$  4EIA ARKHQPEKJ KB @EOLH=?AIAJPO BKN PDA EILQHOERA ATPAJOEKJ KB PDA NECDP AJ@ KB PDA >=N

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0.04

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50

100

150

200

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Imperfect Bonding with Nonpenetration Condition

219

JQIANE?=H AT=ILHAO PDA IAPDK@ EO ?KJOEOPAJP OP=>HA =J@ PDQO ?KJRANCAJP !P PDA >=OA KB PDA IAPDK@ EO R=NE=PEKJ=H BKNIQH=PEKJ KB PDA EJEPE=H >KQJ@ =NU R=HQA LNK>HAI 4DQO EJ LNEJ?ELHA = R=NEAPU KB LNK>HAIO SDE?D =@IEP R=NE=PEKJ=H BKNIQH=PEKJ ?=J >A OKHRA@ >U PDA O=IA =LLNK=?D 4DEO EJ?HQ@AO =IKJC PDAI HEJA=N PDANIKREO?KAH=OPE?EPU DULANAH=OPE?EPU JEPA AH=OPE?EPU =J@ H=NCA PDAKNU KB AH=OPKLH=OPE?EPU /@AJ =J@ 2A@@U = !HOK = H=NCA ?H=OO KB LNK>HAIO SEPD EILANBA?P EJPANB=?E=H >KJ@EJC BKN SDE?D =J =LLNK LNE=PA >KJ@EJC AJANCU BQJ?PEKJ=H ATEOP ?=J >A OKHRA@ 2=@E=H KO?EHH=PEKJ KB = ?KILKOEPA OLDANA SEPD EILANBA?P >KJ@EJC ?=J >A OKHRA@ QOEJC AOOAJPE=HHU PDA O=IA =LLNK=?D =O EJ PDA =>KRA 'AJAN=HEV=PEKJ PK PSK =J@ PDNAA @E IAJOEKJ=H LNK>HAIO EO EJ LNEJ?ELHA OPN=ECDPBKNS=N@ >QP EP I=U >A BNKI PDA EILHAIAJP=PEKJ =J@ ?KILQP=PEKJ=H LKEJP KB REAS MQEPA ?D=HHAJCEJC

 

      ,1/-#2"1(-,

)J PDA LNAREKQO OA?PEKJ = LKSANBQH E@A= S=O AJ?KQJPANA@ .=IAHU NA@Q? PEKJ KB PDA LNK>HAI PK PDA NA@Q?A@ LNK>HAI SDE?D EO LKOA@ KJ = OI=HHAN OAP KB @ACNAAO KB BNAA@KI 4DA KNECEJ=H OAP KB PDA @ACNAAO KB BNAA@KI 2      S=O NOP NA@Q?A@ PK PDA OAP KB @ACNAAO KB BNAA@KI 2     =J@ PDAJ PK (    )J PDEO OA?PEKJ PDEO E@A= EO BQNPDAN AH=>KN=PA@ !?PQ=HHU EP NEOAO PK = RANU A?EAJP ?KILQP=PEKJ=H IAPDK@ SDE?D SA ?=HH = PSK O?=HA JEPA AHAIAJP IAPDK@ /B ?KQNOA PDA E@A= KB PDA NA@Q?PEKJ EO JKP JAS )P EO SAHH GJKSJ =J@ EO =HOK GJKSJ =O PDA 3?DQN ?KILHEIAJP OAA BKN AT=ILHA 3==@

 "QP EP =LLA=NO PD=P OK B=N EP S=O JKP AH=>KN=PA@ PK PDA HARAH SDE?D EO DANA LNAOAJPA@ /NECEJ=HHU PDA IAPDK@ S=O @ARAHKLA@ PK OPQ@U PDANI=H =J@ IA?D=JE?=H LNKLANPEAO KB ?KILKOEPA I=PANE=HO SEPD IE?NK OPNQ?PQNA OAA -AF=G  3Q?D LNK>HAIO D=RA PSK HAJCPD O?=HAO = I=?NK O?=HA SDE?D EO ?KIL=N=>HA PK PDA HAJCPD O?=HA KB PDA K>FA?P =J@ = IE?NK O?=HA SDE?D EO PDA HAJCPD O?=HA KB PDA IE?NK CAKIAPNU ! CHK>=H @EO?NAPEV=PEKJ SDE?D ?=LPQNAO IE?NK O?=HA EJ BQHH @AP=EHO EO PKK ATLAJOERA EJ PANIO KB IAIKNU =J@ ?KILQP=PEKJ=H PEIA ! OP=J@=N@ =LLNK=?D EO PK QOA = PSK O?=HA ?KILQP=PEKJ 4DA I=EJ MQAOPEKJ KB OQ?D =LLNK=?D EO DKS PK AT?D=JCA NAOQHPO >APSAAJ PDA IE?NK =J@ I=?NK O?=HA ?KILQP=PEKJ )J L=NPE?QH=N DKS PK LNKFA?P PDA I=?NK O?=HA ?KILQP=PEKJ KJPK PDA IE?NK O?=HA =J@ DKS PK =RAN=CA PDA IE?NK O?=HA ?KILQP=PEKJ PK PDA I=?NK O?=HA ,EPAN=PQNA =>KQP PDA IQHPE O?=HA EO PK@=U =>QJ@=JP OAA BKN AT=ILHA 0EANK =J@ /SAJ  =J@ 3=@KSOGE  &KN = NAREAS KB PSK =J@ IQHPE O?=HA ?KILQP=PEKJ SA NABAN PK %JCMQEOP  =J@ PDA NABANAJ?AO PDANAEJ 7A NAI=NG PD=P PDA J=IA PSK O?=HA JEPA AHAIAJP IAPDK@ EO =HNA=@U EJ

220

George Mejak

QOA BKN ?ANP=EJ IAPDK@O SDE?D @A=H SEPD PSK O?=HA ?KILQP=PEKJO OAA BKN AT=ILHA #DAJ =J@ #QE  .ARANPDAHAOO SA QOA PDEO J=IA >A?=QOA EP EO EJPNEJOE?=HHU ?KJJA?PA@ SEPD PDA >=OE? E@A= KB PDA JEPA AHAIAJP IAPDK@ =J@ @KAO JKP =O OKIA KPDAN IAPDK@O SDE?D OD=NA PDA J=IA =OOQIA =JUPDEJC OLA?E=H =>KQP PDA LNK>HAI OQ?D =O LANEK@E?EPU DKIKCAJEV=PEKJ KN =OUIL PKPE? ATL=JOEKJ KB PDA OKHQPEKJ 4DA IAPDK@ SDE?D SA LNAOAJPO SKNGO BKN =HH GEJ@ KB >KQJ@=NU R=HQA LNK>HAIO BKN SDE?D = OP=J@=N@ JEPA AHAIAJP IAPDK@ =LLHEAO )P ?KI>EJAO EJPK PDA JAS IAPDK@ PSK OP=J@=N@ JEPA AH AIAJP ?KILQP=PEKJO SEPD @EANAJP IAOD OEVAO KRAN PDA O=IA @KI=EJ )J PDA ?KJPATP KB PSK O?=HA ?KILQP=PEKJ SA ?=J OLA=G =>KQP IE?NK =J@ I=?NK O?=HA JEPA AHAIAJP IAPDK@ (KSARAN BKN PDA IAPDK@ EP EO JKP JA?AOO=NU PD=P PDA IAOD OEVAO @EAN >U =J KN@AN KB I=CJEPQ@A 4DA IAOD OEVAO ?=J >A ARAJ PDA O=IA =J@ =O SEHH >A ODKSJ PDA JAS IAPDK@ EO EJ PDEO ?=OA BNKI PDA ?KILQP=PEKJ=H LKEJP KB REAS OECJE?=JPHU >APPAN PD=J PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ !P PDA DA=NP KB PDA IAPDK@ EO L=NPEPEKJ KB PDA @KI=EJ EJPK ?AHHO 4K A=?D ?AHH = ?AHH >KQJ@=NU R=HQA LNK>HAI SEPD = OAP KB >KQJ@=NU ?KJ@EPEKJO EO =OOK?E=PA@ 4DAOA LNK>HAIO =NA OKHRA@ >U PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ 4DAEN OKHQPEKJO =NA ?=HHA@ ?AHH >=OEO BQJ?PEKJO !P PDA I=?NK O?=HA PDA OKHQPEKJ KB PDA LNK>HAI EO ATLNAOOA@ =O = HEJA=N ?KI>EJ=PEKJ KB PDA ?AHH >=OEO BQJ?PEKJO 4DAEN ?KA?EAJPO =NA @APANIEJA@ >U PDA 0APNKR'=HANGEJ OP=PAIAJP SEPD PDA SAECDPEJC BQJ?PEKJO BNKI PDA I=?NK JEPA AHAIAJP @EO ?NAPEV=PEKJ )J L=NPE?QH=N ?=OA SDANA PDA IAOD OEVAO =NA O=IA = OAP KB I=?NK @ACNAAO KB BNAA@KI EO FQOP PDA NA@Q?A@ OAP KB PDA @ACNAAO KB BNAA@KI KB PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ 4DA OA?PEKJ EO KNC=JEVA@ =O BKHHKSO )J PDA OA?KJ@ OQ>OA?PEKJ @AO?NELPEKJ KB PDA IAPDK@ EO CERAJ SEPDEJ PDA BN=IASKNG KB PSK O?=HA ?KILQP=PEKJ &KN = IK@AH AHHELPE?=H >KQJ@=NU R=HQA LNK>HAI L=NPEPEKJ KB PDA LNK>HAI EJPK ?AHH LNK>HAIO EO EJPNK@Q?A@ 4SK JEPA AHAIAJP @EO?NAPEV=PEKJO =NA QOA@ PDA IE?NK =P PDA ?AHHO HARAH =J@ PDA I=?NK =P PDA CHK>=H HARAH 4DA ?AHH >KQJ@=NU R=HQA LNK>HAIO =NA =OOK?E=PA@ =J@ OKHRA@ >U PDA IE?NK JEPA AHAIAJP IAPDK@ SDE?D EO FQOP = OP=J@=N@ JEPA AHAIAJP IAPDK@ =P PDA IE?NK @EO?NAPEV=PEKJ HARAH 5OEJC PDA 0APNKR'=HANGEJ IAPDK@ PDA OUOPAI KB AMQ=PEKJO KB PDA I=?NK JEPA AHAIAJP IAPDK@ EO @ANERA@ )P EO ODKSJ PD=P = ?KJRANCAJ?A KN@AN KB PDA PSK O?=HA IAPDK@ EO PDA O=IA =O EO PDA KN@AN KB PDA I=?NK JEPA AHAIAJP @EO?NAPEV=PEKJ (KS PK =OOAI>HA PDA CHK>=H OUOPAI KB PDA I=?NK AHAIAJP @EO?NAPEV=PEKJ EO @EO?QOOA@ EJ JATP OQ>OA?PEKJ )P EO ODKSJ PD=P PDA OUOPAI EO LKOEPERA @AJEPA EB PDA LNK>HAI EO OPNKJCHU AHHELPE? #KIL=NEOKJ KB PDA PSK O?=HA IAPDK@ SEPD PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ EO @KJA EJ 3A?PEKJ  #KIL=NEOKJ EO EJ PANIO KB PDA ?KILQP=PEKJ=H PEIA =J@ PDA NAMQENA@ IAIKNU )P EO ODKSA@ PD=P PDA PSK O?=HA IAPDK@ EO O?EAJPE?=HHU

Imperfect Bonding with Nonpenetration Condition

221

IKNA AA?PERA .QIANE?=H AT=ILHAO BKN OKIA IK@AH LNK>HAIO =NA CERAJ EJ PDA ?KJ?HQ@EJC OQ>OA?PEKJ 

   

,AP  >A =J KLAJ >KQJ@A@ OQ>OAP KB  SEPD = LEA?ASEOA NACQH=N >KQJ@ =NU =J@ HAP QO ?KJOE@AN PDA BKHHKSEJC >KQJ@=NU R=HQA LNK>HAI EJ PDA R=NE= PEKJ=H BKNI BKN = CERAJ 0   J@ =    OQ?D PD=P + = >

0 >

BKN =HH

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(ANA +   EO = >EHEJA=N BKNI KJ      =J@   EO = O?=H=N LNK@Q?P KJ  (ANA = OP=J@=N@ JKP=PEKJ BKN 3K>KHAR OL=?AO   =J@    EO QOA@ &KN @AP=EHO =>KQP PDAI OAA BKN AT=ILHA /@AJ =J@ 2A@@U > KN "NAJJAN =J@ 3?KPP  !OOQIEJC PD=P +   EO >KQJ@A@ =J@ ?KAN?ERA KJ   EA "="  + = = BKN OKIA LKOEPERA =J@ =HH =    EP EO SAHH GJKSJ PD=P PDA LNK>HAI D=O = QJEMQA OKHQPEKJ 7A =NA EJPANAOPA@ PK J@ = JQIANE?=H OKHQPEKJ KB PDA LNK>HAI EJ PDA ?=OA SDANA PDA ?KA?EAJP KB PDA >EHEJA=N BKNI + =NA DECDHU KO?EHH=PEJC   BQJ?PEKJO )J PANIO KB IA?D=JE?O IE?NK IA?D=JE?=HCAKIAPNE?=H LNKLANPEAO KB PDA LNK>HAI =NA CERAJ >U PDA KO?EHH=PEKJ >AD=REKN KB PDA ?KA?EAJPO KB + =J@ LKOOE>HU =HOK KB 0 4K I=GA = OPAL BQNPDAN SA AJ@KS IE?NK CAKIAPNU SEPD = ?AHH OPNQ?PQNA )J L=NPE?QH=N EP EO =OOQIA@ PD=P      SDANA   EO = B=IEHU KB KLAJ L=ENSEOA @EOFKEJP >KQJ@A@ OQ>OAPO KB  )P EO ATLA?PEJC PD=P PDA L=NPEPEKJ EJPK PDA OAPO  D=O OKIAPDEJC EJ ?KIIKJ SEPD PDA IE?NK IA?D=JE?=H OPNQ?PQNA KB PDA LNK>HAI =O EO ATLH=EJA@ >AHKS 2AOPNE?PEKJ KB PDA BKNI +   PK        EO @AJKPA@ >U +   5          3EIEH=NHU 0 EO = NAOPNE?PEKJ KB 0 PK   4K PDA L=NPEPEKJ   SA =OOK?E=PA = IEJEI=H OAP KB NABANAJ?A @KI=EJO   3 5        OQ?D PD=P A=?D  EO CERAJ >U = >EFA?PERA =JA PN=JOBKNI=PEKJ &  5  6 BNKI  KJPK   "U O=UEJC PD=P PDA CAKIAPNU KB PDA LNK>HAI D=O = ?AHH OPNQ?PQNA EP EO IA=JP PD=P    4DA OAPO  =NA ?=HHA@ ?AHHO =J@  PDA NABANAJ?A ?AHHO &KN AT=ILHA EB   EO = QJEP OMQ=NA I=@A KB E@AJPE?=H OMQ=NA ?AHHO   PDAJ   =J@ A=?D & EO FQOP = PN=JOH=PEKJ ! I=LLEJC & LQHHO >=?G + PK PDA BKNI +   @AJA@ KJ          )J L=NPE?QH=N EB  + = > %=
%> .6   
  

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222

George Mejak

SDANA 

  6   5



     5 5      6 6

5OEJC PDA BKNIO + SA @AJA =J AMQER=HAJ?A NAH=PEKJ  KJ PDA OAP  >U 5  5  + +  3 5  3 5  4K PDEO AMQER=HAJ?A NAH=PEKJ EP EO =OOK?E=PA@ = B=?PKN OAP   4DA B=IEHU   PKCAPDAN SEPD PDA I=LLEJCO & =J@ PDA BKNIO +   5   @APANIEJA PDA ?AHH OPNQ?PQNA KB PDA LNK>HAI 7A ?=J O=U PD=P  EO = OECJ=PQNA KB PDA ?AHH OPNQ?PQNA 4DA IAPDK@ SA =NA LNAOAJPEJC =LLHEAO JKP KJHU BKN LANEK@E? ?AHH OPNQ?PQNA SDANA  !  !   >QP =HOK BKN JKJ LANEK@E? OPNQ?PQNA =J@ ARAJ BKN   &KN AT=ILHA ?KJPEJQEJC PDA LNAREKQO KJA EB A=?D  EO = ?KILKOEPA I=@A  KB PDA I=PNET  =J@ PDA EJ?HQOEKJ  SEPD I=PANE=H ?KA?EAJPO  =J@      4         NAOLA?PERAHU PDAJ   6 4 SDANA 6 4 5 EO = JQI>AN KB @EANAJP CAKIAPNE?=H =NN=JCAIAJPO KB PDA EJ?HQOEKJO  I=@A KB PDA O=IA I=PANE=H !BPAN PDEO LNAHEIEJ=NU @EO?QOOEKJ SA PQNJ PK PDA @EO?NAPEV=PEKJ KB PDA LNK>HAI ,AP  >A = JEPA AHAIAJP @EO?NAPEV=PEKJ KB  OQ>KN@EJ=PA PK PDA L=NPEPEKJ KB PDA @KI=EJ EJPK PDA ?AHHO   )J =JKPDAN SKN@O A=?D  EO = JEPA QJEKJ KB PDA AHAIAJPO BNKI   4DA ?KNNAOLKJ@EJC JEPA AHAIAJP OL=?A SEPD VANK >KQJ@=NU R=HQAO EO @AJKPA@ >U ( =J@ PDAEN >=OEO BQJ?PEKJO >U !  8      $  4DA OL=?A ( EO ?=HHA@ = I=?NK JEPA AHAIAJP OL=?A &QNPDAN >U (; EP EO @AJKPA@ = OQ>OAP KB ( KB PDKOA >=OEO BQJ?PEKJO BKN SDE?D =J EJPANEKN KB PDAEN OQLLKNP D=O = JKJ AILPU EJPANOA?PEKJ SEPD = QJEKJ ; KB PDA ?AHHO >KQJ@=NEAO     4DA JEPA AHAIAJP >=OEO KB ( EO @AJKPA@ ; ; ; >U !  8      $  &QJ?PEKJO ! =NA ?=HHA@ ?AHH >KQJ@=NU >=OEO BQJ?PEKJO ,AP  >A = JEPA AHAIAJP @EO?NAPEV=PEKJ KB  =J@ ( PDA ?KNNAOLKJ@EJC JEPA AHAIAJP OL=?A 4DA OL=?AO ( =NA ?=HHA@ PDA IE?NK JEPA AHAIAJP OL=?AO !JKPDAN =LLNKLNE=PA J=IA BKN ( EO PDA ?AHH JEPA AHAIAJP OL=?A )P EO NAMQENA@ PD=P PDA NAOPNE?PEKJ KB PDA I=?NK JEPA AHAIAJP OL=?A ( PK  =NA ?KJP=EJA@ EJ PDA IE?NK JEPA AHAIAJP OL=?AO (  )J =JKPDAN SKN@O IE?NK =J@ I=?NK JEPA AHAIAJP @EO?NAPEV=PEKJO ODKQH@ >A ?KIL=PE>HA SEPD PDA IE?NK JEPA AHAIAJP @EO?NAPEV=PEKJ AMQ=H KN JAN PD=J PDA I=?NK AHAIAJP @EO?NAPEV=PEKJ 3AA &ECQNA  SDANA = JEPA AHAIAJP @EO?NAPEV=PEKJ  2 KB PDA QJEP OMQ=NA SEPD BKQN OMQ=NA ?AHHO EO O?DAI=PE?=HHU ODKSJ ; !  (; = @EO?NAPA 4K A=?D   5   =J@ ARANU I=?NK >=OEO BQJ?PEKJ   ?AHH R=NE=PEKJ=H LNK>HAI EO =OOK?E=PA@ J@ !!  ( OQ?D PD=P  !!  + !!  

; ! !   

0  

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Imperfect Bonding with Nonpenetration Condition 18

19

223

20 21

17

15 16

14

10

11

12 13

9

7 8

6

5

1 2

3

4

(&2/$  -=?NK JEPA AHAIAJP @EO?NAPEV=PEKJ KB PDA QJEP OMQ=NA  9 : 9 : PKCAPDAN SEPD PDA IE?NK JEPA AHAIAJP @EO?NAPEV=PEKJ KB PDA ?AHH  9  :  9  :  2 &ECQNA ODKSO AJQIAN=PEKJ KB PDA ?AHHO >KQJ@=NU JK@AO KB PDA I=?NK JEPA AHAIAJP @EO?NAPEV=PEKJ

&QNPDAN BKN A=?D 5   =J =@@EPEKJ=H @EO?NAPA ?AHH R=NE=PEKJ=H LNK>HAI EO =OOK?E=PA@  J@ !  ( OQ?D PD=P + !  

0  

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(ANA ( EO = OQ>OAP KB ( SEPD PDA VANK >KQJ@=NU @=P= =HKJC   &KN   =J@ ! SA @AJA ?AHH >=OEO BQJ?PEKJO !! !!   ! CERAJ !! SDANA  =NA UAP QJOLA?EA@ L=N=IAPANO )P BKHHKSO BNKI PDA >KQJ@=NU ; ! KJ  =J@ BNKI PDA ?KJOPNQ?PEKJ KB R=HQAO  =J@  PD=P !!  !! PD=P

 + !!    0   BKN =HH   (  0=P?DEJC @AJEPEKJO KB !! KRAN PDA ?AHHO  SDE?D D=RA ; ! =J@ ATPAJ@EJC PDAI >U = JKJAILPU EJPANOA?PEKJO SEPD PDA OQLLKNP KB  VANK R=HQA KQPOE@A KB PDA OQLLKNP PDA CHK>=H >=OEO BQJ?PEKJ !! =NA K>P=EJA@ .KPA PD=P PDA ?AHH >=OEO BQJ?PEKJO =NA =?PQ=HHU ?KILQPA@ KJ PDA NABANAJ?A ?AHHO   5    )B PDA ?AHH OPNQ?PQNA KB PDA HK=@ 0 ?KEJ?E@AO SEPD PDA ?AHH OPNQ?PQNA KB PDA BKNI + PDAJ =HH >=OEO BQJ?PEKJO !! =NA K>P=EJA@ >U OKHREJC  $;   R=NE=PEKJ=H LNK>HAIO  =J@  &KN =J =N>EPN=NU

224

George Mejak

HK=@  $;   R=NE=PEKJ=H LNK>HAIO SEPD  @EANAJP HK=@O 0 D=RA PK >A ; !  8       $; OKHRA@ (ANA $; EO = JQI>AN KB JKJVANK NAOPNE?PEKJO KB  PK   (=REJC BKQJ@ ?AHH >=OEO BQJ?PEKJO !! = OKHQPEKJ = KB PDA LNK>HAI  EO =LLNKTEI=PA@ >U = HEJA=N ?KI>EJ=PEKJ ' 

 

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Imperfect Bonding with Nonpenetration Condition

225

EO K>P=EJA@ BKN QJGJKSJ ?KA?EAJPO ' !  (ANA  9 5  OQLL "    =J@ $ 9 8  OQLL !!  OQLL "    3KHQPEKJ KB  CERAO ' ! =J@ =BPAN PD=P  =NA K>P=EJA@ BNKI  4DEO CERAO PDA ?AHH >=OEO BQJ?PEKJO !! =J@ J=HHU = PSK O?=HA JEPA AHAIAJP OKHQPEKJ '  ' ! !! KB PDA LNK>HAI  (KS PK NASNEPA  EJ PANIO KB ?AHHO IE?NK JEPA AHAIAJP OPEJAOO I=PNE?AO SEHH >A ODKSJ EJ PDA JATP OQ>OA?PEKJ #KJRANCAJ?A KB PDA PSK O?=HA JEPA AHAIAJP IAPDK@ BKHHKSO BNKI 0APNKR '=HANGEJ OP=PAIAJP  )P BKHHKSO BNKI  =J@  PD=P + = '   ) 



BKN =HH

)  (  

&QNPDAN + = '   = '  

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SDANA  '  EO PDA LNKFA?PEKJ KB '  KJPK (  4DANABKNA QOEJC PDA >KQJ@A@JAOO =J@ ?KAN?EREPU KB + SA D=RA "= '  " 


"+" "= ) "  " '  '  "



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 "N=AOO  KN "NAJJAN =J@ 3?KPP  

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!OOAI>H=CA KB PDA OUOPAI  EO @KJA ?AHH >U ?AHH &KN = TA@ ?AHH  SA D=RA  !  ) ' ! 0  )  + !  ) 

 + !! BKN =HH )  *   (ANA *   EO = OAP KB NAOPNE?PEKJO KB PDA BQJ?PEKJO BNKI PDA I=?NK JEPA AHAIAJP OL=?A ( PK PDA ?AHH   )J  PDA OQI ; !     ,AP QO I=PEKJ KB 8 EO KRAN PDA OAP KB EJ@E?AO 8 BKN SDE?D OQLL  @AJKPA >U *   = OQ>OAP KB *   KB PDKOA BQJ?PEKJO SDE?D D=RA VANK ;   *   # *   PN=?A =HKJC PDA ?AHH >KQJ@=NU  =J@ >U * ; 4DQO *   ?KJOEOPO KB PDA EJPANEKN =J@ *   KB PDA >KQJ@=NU I=?NK JEPA AHAIAJP BQJ?PEKJO 3EJ?A *   (  AMQ=PEKJ  EO >U PDA  ?KJOPNQ?PEKJ  =J@  KB PDA ?AHH >=OEO BQJ?PEKJO !! =J@ ! O=PEOA@ ;   BKN =JU )  *   4DANABKNA SA NAOPNE?P KQNOAHRAO PK )  * ;        ; ; " =NA PDA JEPA AHAIAJP EJ L=NPE?QH=N PK )   SDANA    ;   -KNAKRAN ;   =J@ $; EO PDA @EIAJOEKJ KB * >=OEO BQJ?PEKJO KB * ; *   ( =J@ PDANABKNA A=?D " ?=J >A ATLNAOOA@ =O = HEJA=N ?KI>EJ= PEKJ KB PDA BQJ?PEKJO BNKI (  4DA OL=?A ( EO = @ENA?P OQI KB PDA EJPANEKN

226

George Mejak

=J@ PDA >KQJ@=NU JEPA AHAIAJP BQJ?PEKJO EA ( (  (;  3EJ?A

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0  ? + 

+ "!  ?

BKN =HH ?  (  4DA OKHQPEKJ "! D=O = BKNI "! "!  !
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 PD=P 7 PQLHA ! EO CERAJ =O PDA OKHQPEKJ KB PDA JEPA AHAIAJP HEJA=N OUOPAI ) !

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SDANA )  
EO PDA JEPA AHAIAJP OPEJAOO I=PNET SEPD ?KILKJAJPO 5  +     !   EO PDA HK=@ RA?PKN SEPD ?KILKJAJPO , 0    =J@ )    
EO PDA I=PNET SEPD ?KILKJAJPO 5   + ;    3EJ?A + EO OPNKJCHU AHHELPE? ) EO = JKJOEJCQH=N I=PNET 3EIEH=NHU !! !
 SDANA ! )  ! =J@ PDQO !! !!

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Imperfect Bonding with Nonpenetration Condition

227

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SDANA    
 EO PDA I=PNET SEPD ?KHQIJ RA?PKNO !  .KPA PD=P OEJ?A ; ! EO QJEMQAHU CERAJ >U PDA HEJA=N ?KI>EJ=PEKJ KB ;   EO EJFA?PERA A=?D  EA +AN    )B PDA BKNI + EO OUIIAPNE? PDAJ PDA ?AHH IE?NK AHAIAJP OPEJAOO I=PNET ) EO OUIIAPNE? =J@ >U  =HOK   -KNAKRAN EB ) EO LKOEPERA KN LKOEPERA @AJEPA OK EO =HOK   &KN EB
   PDAJ >U 
!
)  )  )  ) !   BKN !





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!  ! 
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0  !

0  ;
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228

George Mejak

=J@ KLAN=PEKJ=H ?KQJP EO ?KJOE@ANA@ 7A NAOPNE?P KQNOAHRAO PK = PSK @EIAJ OEKJ=H IK@AH LNK>HAI KRAN PDA OMQ=NA @KI=EJ &KN IKNA CAJAN=H ?=OA PDA ?KJ?HQOEKJO =NA OEIEH=N 4DA @KI=EJ  EO L=NPEPEKJA@ EJPK  " AMQ=H OMQ=NA ?AHHO   %=?D ?AHH EO QJEBKNIHU @EO?NAPEVA@ EJPK 6  6 
PNE=JCQH=N AHAIAJPO 4DA ?KN  NAOLKJ@EJC IAOD OEVA EO 2  =J@ EO EJ@ALAJ@AJP KB 5 4DA QJEKJ KB PDA ?AHH PNE=JCQH=PEKJO CERAO PDA OP=J@=N@ JEPA AHAIAJP @EO?NAPEV=PEKJ  KB  EJPK 7 6  JEPA AHAIAJPO #KILQP=PEKJ=H A?EAJ?U @ALAJ@O QLKJ PDA KN@AN KB PDA JEPA AHAIAJP =LLNKTEI=PEKJ 4K AHEIEJ=PA @EANAJ?AO @QA PK @EANAJP KN@ANO KB =LLNKTEI=PEKJO PDA I=?NK JEPA AHAIAJP @EO?NAPEV= PEKJ EO OAP PK >A ?KEJ?E@AJP SEPD PDA CHK>=H IE?NK @EO?NAPEV=PEKJ PDQO  2   4DQO PDA I=PNET  EJ  EO FQOP PDA E@AJPEPU I=PNET 5O =J@  EJC HEJA=N PNE=JCQH=N AHAIAJPO = OP=J@=N@ JEPA AHAIAJP @EO?NAPEV=PEKJ D=O

6"   @ACNAAO KB BNAA@KI SDEHA PDA I=?NK JEPA AHAIAJP @EO 7 ?NAPEV=PEKJ D=O " 6"   "    6 " "   ?AHH >KQJ@=NU @ACNAAO KB BNAA@KI OAA &ECQNA  BKN 6  "  =J@  2 !LLHUEJC = J=PQN=H HEJA >U HEJA AJQIAN=PEKJ KB PDA JK@AO PDA D=HB >=J@ SE@PD KB PDA OP=J@=N@ JEPA AHAIAJP OPEJAOO I=PNET EO 7$ 6"   #KIL=NEOKJ KB PDA IAPDK@O EJ PANIO KB KLAN=PEKJ=H ?KQJP EO @KJA >U ?KQJPEJC KLAN=PEKJO KB IQHPELHE?=PEKJ =J@ @EREOEKJ EJ PDA OKHQPEKJ KB PDA HEJA=N OUOPAI 4K OEILHEBU @EO?QOOEKJ ?KIL=NEOKJ EO @KJA BKN = @ENA?P >=J@ OKHRAN ! JQI>AN KB HKJC KLAN=PEKJO BKN = OUIIAPNE? @A?KILKOEPEKJ KB PDA OPEJAOO I=PNET OPKNA@ EJ PDA >=J@ OUIIAPNE? OPKN=CA O?DAI= EO 7!# 7$ 7   7$  (ANA EP EO =OOQIA@ PD=P 7  7$  ! BKNS=N@ =J@ >=?G OQ>OPEPQPEKJ NAMQENAO A=?D 7!# 7$ 7 7$  HKJC KLAN=PEKJO =J@ PDQO PDA PKP=H KLAN=PEKJ ?KQJP EO 7!# 7!#  7!# 7$ 7 7$   7$ 7$   #KJOE@ANEJC 7!#  7!# =J@ 7!# =O PDA BQJ?PEKJO KB 2 SA D=RA   2  2  2 2  2

  2   2 7!# 7!# 2  2   2  2 2

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Imperfect Bonding with Nonpenetration Condition

229

PDQO PDAEN ?KILQP=PEKJ=H ?KOP IQOP >A =@@A@ PK PDA ?KILQP=PEKJ=H ?KOP KB PDA PSK O?=HA IAPDK@ &KN A=?D ?AHH  PDA ?AHH JEPA AHAIAJP LNK>HAI BKN 6   HK=@O D=O PK >A OKHRA@ 4DA LNK>HAI D=O 7 6   @ACNAAO KB   BNAA@KI =J@ PDAEN KLAN=PEKJ ?KQJP EO 7!#  7!#    6  7!#   2AIAI>ANEJC PD=P  @AJKPAO = JQI>AN KB ?AHHO SEPD @EANAJP I=PANE=H LNKLANPEAO =J@ @AJKPEJC >U  = JQI>AN KB @EANAJP HK=@O 0  PDA PKP=H KLAN=PEKJ ?KQJP KB OKHREJC PDA I=?NK AHAIAJP OPEJAOO OUOPAI EO "!#

 "!#   7!#

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4DA A?EAJ?U N=PEK EO @AJA@ =O 

"!#  7!#



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

   "       "    "   "    "      " 

 2A?=HHEJC PD=P 6  SA ?KJ?HQ@A PD=P =O OKKJ =O PDA IAOD ?KJOEOPO KB IKNA PD=J  AHAIAJPO PDAJ PDA PSK O?=HA IAPDK@ BKN "   EO OQLANEKN  PK PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ 3I=HHAN EO 2 IKNA A?EAJP EO PDA PSK O?=HA IAPDK@ 2 I=U CK PK VANK SEPD "  =P PDA TA@ 6 KN SEPD 6  =P PDA TA@ "  )J PDA NOP ?=OA

HEI  2

 

6 

6



=J@ PDQO  2   BKN OI=HH 2 EB 6   .KPA PD=P PDA O=IA ?KJ?HQOEKJ  4DA ?=OA "  ?KN BKHHKSO BKNI  =BPAN ?KILQPEJC HEI   NAOLKJ@O PK PDA LNK>HAI SEPD RANU JA IE?NKOPNQ?PQNA &KN OQ?D LNK>HAIO QOQ=HHU   " =J@   "  )J PDEO OA?KJ@ ?=OA SA D=RA HEI  2



   "



230

George Mejak

=J@ EJ PDA SKNOP ?=OA SDANA 



HEI  2



" EP BKHHKSO PD=P   "



1.5 1.25 1 0.75 0.5 0.25 0.002 0.004 0.006 0.008

0.01

(&2/$  4DA A?EAJ?U N=PEK  RANOQO 2 BKN "



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  "

    "  "  "  "  

!C=EJ PDA PSK O?=HA IAPDK@ EO OQLANEKN EJ PANIO KB PDA IAIKNU PK PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ =O OKKJ =O PDA IAOD ?KJOEOPO IKNA PD=J    AHAIAJPO -KNAKRAN

6 

 HEI 2   6

=J@  HEI 2 

  "

)J 4=>HA  SA CERA JQIANE?=H AT=ILHAO BKN PDA 0KEOOKJ AMQ=PEKJ SEPD VANK $ENE?DHAP @=P= -APDK@KHKCU EO OHECDPHU @EANAJP =O EJ PDA =>KRA .=IAHU NAOQHPO =NA CERAJ BKN PDA MQ=@N=PE? PNE=JCQH=N AHAIAJP =J@ PDA OGUHEJA OKHRAN EO QOA@ EJOPA=@ KB PDA >=J@ OKHRAN 4DA OPKN=CA NAMQENAIAJP EO CERAJ EJ PANIO KB PDA HAJCPD KB PDA OPEJAOO I=PNET EJ PDA OGUHEJA OPKN=CA O?DAI= 7?5 EO PDA HAJCPD KB PDA ?AHH OPEJAOO I=PNET ) =J@ " )  KB PDA I=?NK O?=HA OPEJAOO I=PNET  7 EO PDA JQI>AN KB PDA IE?NK JEPA AHAIAJPO LAN = ?AHH -8= EO ?KILQP=PEKJ=H PEIA EJ OA?KJ@O BKN OKHREJC PDA ?AHH LNK>HAIO =J@ $ ' EO PEIA BKN OKHREJC PDA I=?NK O?=HA LNK>HAI )J KN@AN PK D=RA NAHE=>HA

Imperfect Bonding with Nonpenetration Condition

231

PEIEJC = N=PDAN OHKS 0# -(V '> 2!- S=O QOA@ .KPA PD=P EJ PANIO KB PDA OPKN=CA PDA IAOD L=N=IAPAN 2 KB PDA MQ=@N=PE? AHAIAJP EO AMQER=HAJP  PK PDA IAOD OEVA  2 BKN PDA HEJA=N AHAIAJP )J PDA AT=ILHA   3EJ?A  EO =N>EPN=NU PDEO ?KNNAOLKJ@O PK PDA ?=OA SDANA PDA BKNI + =J@ PDA HK=@ 0 =NA ?KJOP=JP 4DA NOP NKS EJ PDA P=>HA SEPD   ?KNNAOLKJ@O PK PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ 4K @N=S ?KJ?HQOEKJO =>KQP A?EAJ?U KB PDA PSK O?=HA IAPDK@ EJ PANIO KB PDA ?KILQP=PEKJ=H PEIA ?KILQP=PEKJ=H PEIAO $ ' =J@ -8= IQOP >A =@@A@ 4DQO PDA PSK O?=HA IAPDK@ BKN   EO PDA KLPEI=H KJA )P EO IKNA PD=P  PEIAO B=OPAN =O PDA OP=J@=N@ JEPA AHAIAJP =J@ NAMQENAO  PEIAO HAOO IAIKNU  !*$  #KILQP=PEKJ=H A?EAJ?U KB PDA PSK O?=HA JEPA AHAIAJP IAPDK@  BKN @EANAJP JQI>ANO KB ?AHHO + BKN PDA IAOD 2   +      

7          

JSG               

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#KJ?ANJEJC PDA QOA KB PDA EPAN=PERA OKHRAN SA NAI=NG PD=P BKN H=NCA 7 =J EPAN=PERA OKHRAN ?=J OQ>OP=JPE=HHU NA@Q?A PDA ?KILQP=PEKJ=H PEIA KB PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ !OOQIEJC PD=P BKN H=NCA AJKQCD 7 = JQI $ >AN KB PDA NAMQENA@ EPAN=PEKJO OPALO EO 7 PDA KLAN=PEKJ=H ?KQJP BKN PDA $ 7 7$ 7  4DA OP=J@=N@ JEPA AHAIAJP IAPDK@ EO =LLNKTEI=PAHU 7!# PSK O?=HA IAPDK@ NAMQENAO BKN A=?D ?AHH PDA OKHQPEKJO KB PDA HEJA=N OUOPAI BKN

6 @EANAJP NECDP D=J@ OE@AO 4DANABKNA EP EO IKNA A?EAJP PK OKHRA PDA ?AHH LNK>HAIO >U = @ENA?P IAPDK@ (KSARAN BKN H=NCA AJKQCD " PDA I=?NK O?=HA OUOPAI EO IKOP A?EAJPHU OKHRA@ >U =J EPAN=PERA OKHRAN /IEPPEJC @A P=EHO PSK O?=HA IAPDK@ EO =C=EJ OECJE?=JPHU IKNA A?EAJP )J L=NPE?QH=N =C=EJ     HEI  2  " >QP

6  HEI 2    6 4DA PSK O?=HA IAPDK@ EO =OUILPKPE?=HHU ARAJ IKNA A?EAJP EB PDA EPAN=PERA OKHRAN EO QOA@ ! LHKP KB PDA A?EAJ?U N=PEK BKN "  EO =HIKOP E@AJPE?=HHU PK PDA LHKP KB  2 EJ &ECQNA 

232

George Mejak

4K ?KJ?HQ@A PDA PSK O?=HA IAPDK@ EO OECJE?=JPHU IKNA A?EAJP PD=J PDA OP=J@=N@ JEPA AHAIAJP IAPDK@ 3I=HHAN EO PDA IAOD OEVA KN CNA=PAN EO PDA JQI>AN KB PDA ?AHHO IKNA A?EAJP EP EO 5OEJC L=N=HHAH ?KILQP=PEKJ=H A?EAJ?U KB PDA PSK O?=HA IAPDK@ >A?KIAO ARAJ IKNA LNKJKQJ?A@

(&2/$  -=PANE=H SEPD PDA ?AHH OPNQ?PQNA SEPD = N=J@KI @EOPNE>QPEKJ KB AHHELPE?=H EJ?HQOEKJO



2+$/(" * $5 +.*$0

4K EHHQOPN=PA LN=?PE?=H =LLHE?=PEKJ KB PDA PSK O?=HA JEPA AHAIAJP IAPDK@ SA ODKS DANA OKIA NAOQHPO BNKI LH=JA OPNAOO AH=OPE?EPU )J L=NPE?QH=N @ENA?P JQIANE?=H ?KILQP=PEKJ KB PDA AA?PERA I=PANE=H LNKLANPEAO KB PDA I=PANE=H SEPD = N=J@KI OPNQ?PQNA EO LNAOAJPA@ ,AP  >A = QJEP OMQ=NA KB PDA I=PANE=H )P EO =OOQIA@ PD=P  D=O = ?AHH OPNQ?PQNA >QEHP BNKI  " OMQ=NA ?AHHO   OAA &ECQNA  %=?D ?AHH EO = ?KILKOEPA I=@A KB PDA I=PNET   =J@ PDA AHHELPE?=H EJ?HQOEKJ   #    )P EO =OOQIA@ PD=P =HH EJ?HQOEKJO =NA ?KJCNQAJP =J@ =NA I=@A KB PDA O=IA EOKPNKLE? I=PANE=H 4DAEN I=FKN =J@  =J@ ,  !HH I=PNE?AO =NA =HOK I=@A KB IEJKN OAIE =TAO =NA + PDA E@AJPE?=H EOKPNKLE? I=PANE=H 4DAEN 8KQJC IK@QHE =J@ 0KEOOKJ N=PEKO =NA @AJKPA@ >U       =J@  =J@   NAOLA?PERAHU /NEAJP=PEKJ KB PDA EJ?HQOEKJ

Imperfect Bonding with Nonpenetration Condition

233

 EO @AJKPA@ >U  =J@ EO CERAJ =O PDA =JCHA >APSAAJ PDA I=FKN =TEO =J@ PDA @ @ENA?PEKJ KB PDA OAHA?PA@ OE@A KB PDA @KI=EJ 4SK R=HQAO KB  =NA  =HHKSA@   =J@   $EOPNE>QPEKJ KB PDAEN R=HQAO @APANIEJA PATPQNA KB PDA I=PANE=H 2AOQHPO =NA CERAJ BKN PDA BKHHKSEJC I=PANE=H L=N=IAPANO     =J@ 

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10

500

5 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

(&2/$  4=HHEAO KB PDA AA?PERA OPNAOO ?KILKJAJPO <&& CNAAJ <'' NA@ =J@ <&' >HQA ,ABP   NECDP  

1000

1000

800

800

600

600

400

400

200

200

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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234

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252

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Imperfect Bonding with Nonpenetration Condition

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0.00125

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George Mejak

Debonded area

254 70 60 50 40 30 20 10 0 0.2 0.4 0.6 0.8 1 Strain

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Imperfect Bonding with Nonpenetration Condition

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255

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256

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Stress

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Debonded area

Stress

Imperfect Bonding with Nonpenetration Condition

257

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Young modulus

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258

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262

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278

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