Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

which is another important stress relation characterizi~g the behavior of isotropic hyperelastic materials at finite strains. Note that, since v is the unique square root of b, aw /8b also com:mutes with v.

Constitutive equations -in terms of principal stretches. If the strain-energy

tion \JI is an invariant, we may regard \JI as a function of the principal stretches ,\a, the place of (6.27), we .may rep.resent Win the form

(6.44)

For the stress-free reference configuration the normalization condition (6.4) takes on

the form '11(1, 1, 1) = 0.

Consider the left stretch tensor v = b 112 describing the -defonned state of an isotropic hyperelastic material. .From the e-igenvalue problem (2. l l 6h we know that

,,\a denote the three principal stretches (the real eigenvalues) of v. Since the principal directions of v coincide with those of b (compare with eqs.. (2..116.h and (2. .l l7h)

they also coincide with the principal directions of the Cauchy stress tensor u (recall

"o.

with the volume ratio

(6.46)

according to (.5.9.lh.

In addition to (6.45), we introduce equivalent relations for the three principal PiolaKirchhoff stresses Pn and Sa, namely

(compare with the fo.llowing Example 6.2)., which may be expressed in terms of the Cauchy stresses (6.45) as

a = 1, 2, 3 .. (6.48)

Constitutive relations (6.45) and (6.47) show that principal stresses in an isotropic hyperelastic material depend only upon the principal stretches. They are simply obtained by differentiating the strain-energy function with res.pect to the corresponding principal stretches.

-~,,~,,-~-·· ----·'""''''"'''•:"''''"'''''"'"'"'"''"'"'J..•''''''''''•·'-"•"''Y•'•','•. -';,,,,,,••.,•"o.. •,,,, . , -- ..•"'~':"V'~'"'"""___ ,.,,,,•.,,,....':Y'''''"'••-,••'"'.''-'"':':''~-·'--'''''"'"''":''''''''...'•-'•'..'•'''-.','-''•"''':•'"''"'•'''•""'•'•''•'•'•'•"'''''•''•'''''••'•'•'''''""''','•:

where Ga and Pc.., Sa, a = 1, 2, 3, are the principal values of the Cauchy stress tensor er and the two Piola-Kirchhoff stress tensors P, .S according to the expressions (6.45) and (6..47)., respectively. The orthonormal vectors Na and Du = RNa, a = 1, 2, 3, denote t~e principal referential and spatial directions (axes qf stretch), respectively.

Thes.e constitutive equations describe isotropic response of byperelastic materials

and hold if and only if ;\1 '# ..\2 # .Ai1 # /\1.

Solution. with constitutive equation (6.50)2 which is expressed in tenns of second Pio.la-Kirchhoff stresses Sa, a = 1, 2, 3. We compute the derivative of the isotropic function w{C) with respect to the symmetric tensor C. By means of the chain rule and kinematic relation (2..123), we obtain for .the general case ~\1 # A2 # .1\~ =f )q,

(6.51.)

In (6.51 )2, .,,\~ are the eigenvalues (the squares of the principal stretches) and N0 the corresponding eigenvectors (principal .referential directions) of C (compare with the eigenvalue problem (2.115)). With (6.5 lh and the chain rule we find from (6.1.3)2 that

(.6.52)

which gives the desired .results (6.47h and (6.50h.

By the use of (6.52)2 , the relation according to ( 1.58.) and eq. (2..132) 1, i.e. FN0 = Aufi,u a = 1, 2, 3, the spectral fonn of .the first Piola-Kirchhoff stress tensor ·pmay be

with the scalar-valued scalar functions er = J- 1;\8'11 / [)/\ and with P

8,T! IDA, s ·=,\-18\J! /D,\.

6.3 Incompressible Hyperelastic Materials

Numerous polymeric mate.rials can sustain finite strains without noticeable volume changes. Such types of material may be regarded as incompressible so that only .isochoric motions are possible. For many cases, this is a common idealization and accepted assumption often invoked in continuum and computational mechanics. In this sect.ion we present the constitutive foundation of incompressible hyperelastic materials.

1-ncompressible hyperelastidty. Materia]s which keep the volume constant throughout a motion are characterized by the incompressibility .constraint

or hy some other equivalent expressions according to (2.1.77) (recall the expression (2.5 l) for Lhe volume ratio J). In general, a material which is subJected to an internal constraint, of which incompressibility is the most common, is referred to .as a .con- strained material.

In order to derive general constitutive equations for incompressible hyperelastic materials, we may postulate the strain-energy function

(6.57) serves as an indeterminate Lagrange .m.ultiplier, which can be identified .as a hydrostatic pressure. Note that the scalar v may only be determined from the equilibrium equations and the boundary conditions. It repres·ents a workless reaction to the kinematic constraint on the deformatio.n field.

in tenns of the spatial strain variable b. This is only valid for .incompressible .isotropic hyperelastic materials.

If we express \JI as .a function of the .three principal stretches Aa we write \JI = '11("\11 A2 , A3 ) - p(J - 1) in the place of (6.57), with the-indeterminate Lagrange mul- tiplier JJ. Using 8.JIoAa = J)_-;;1_, a = 1, 2, 3, which is relation (2 ..1.74) expressed in principal stretches, eqs. (6.45) and (6.47) are then replaced by

These stress relations -incorporate the unknown scalar p, which .must be determined from the equilibrium equations and the boundary conditions. The incompressibility constraint J = 1 takes on the form

(6.71)

.leaving two independent stretches as the deformation measures. Expressing the first and second Piola-Kirchhoff stresses in terms of the Cauchy stresses (6.69), we obtain, by analogy with (6.48), ~1 = A;1cru and 80 = /\;2cra, a = 1, 21 3.

EXAMPLE 6.3 Consider a thin sheet of .incompressible hyperelastic material which is embedded in a reference frame of (.right-handed) coordinate axes with a fixed set of orthononnal basis vectors ea, a = 1, .2, 3. Suppose that the .axes are aligned with the major faces of the sheet.

A deformation created by the stretch ratios ,.\ 1, /\2 along the directions -e1, e2 results in a (lw11u~ge11eous) biaxial deformation with the kinematic .relation (2.130). The assodated stress state is assumed to be plane throughout the sheet so that the Cauchy stress

·components a1:~, 0"231 a 33 are equal to zero which is in accordance with (3.59).

Show that the biaxial stress state of the homogeneous problem is of the form

(6.72)

(6.73)

(see RIVLIN [1948, eq. {6.5)]), with the principal .invariants 11 = ~W + ..-\~ + .1\)2,,\22 ,

I2 ·=,,\i A~ + /\12 + /\i··2 and l:i = 1.

(b)Consider a ho.mogeneous pure shear deformation with the kinematic relation ,,\ 1 = ,\, --\2 = 1, Ai = l/ A. (compare with eq. (2.131 )). Show that the nonzero Cauchy stress components are

(6.76)

(6. 77)

2.Consider a th.in sheet of incompressible hyperelastic material with the same setting as formulated in Example 6.3 but subjected to a homogeneous simple shear deformation which is caused by a motion in the form of (2.3) (compare also with

Exercis-e 2 -on p.. 93).

(a)Show that the associated stress state is completely defined by

(6. 78)

where Jt > 0, called the shear .modulus, is a measure of resistance to distortion and p is a scalar to be determined from the boundary conditions.

(b) Consider a plane stress state throughout the sheet in the sense that the face of the body nonnal to the direction e;1 is free of surface tractions, i.e. a 1;i = o2a = a3:~ =O. Show that the nonzero Cauchy .stress components ure

0-.12 = JJ.C •

6.4 Compressible Hyperelastic Materials

which can undergo changes of volume is said to be compressible. Foamed elastomers~ for example, .are able to sustain finite strains with volume changes. The only restriction on this class of materials is that the vo]ume ratio J must be positive.