Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

Finally, according to the inverse.of relation (6.79)2, property (l .256).and relation (6..82h, we obtain the fourth-order tensor

in which pT defines the transpose of the fourth-order tensor JP governed by the identity (1.1.57). We call Jr the projection tensor with respect to the reference configuration, therefore expressed thro~gh C. With the associated property (1.159), the relation for the projection tensor JP> reads, with reference to (6.83)a, as

where Edenotes the .fourth-order unit tensor, as defined in eq. ( l. l 6Q) 1, with representation ( l ~161).

Earli~r we agreed to study the purely mechanical theory. To characterize processes

where \ltv0 1(J) and Wiso(C) are given scalar-valued functions of J

sumed to be obje·ctive. They describe the so-called volumetric (or dilation.al) elastic response and the isochoric (or distortional) elastic response of the .material, re~pec­ tively.

Additionally, we .require that 'Wvol ·is a strictly convex function taking on its unique minimum at J = 1 (for formal definitions of strictly convex functions see, for example, OGDEN [1997~ Appendix l]). With reference to normalization condition (6.4) we daim that Wvor(.l) = 0 .and Wiso(C) = 0 ho.Id J = 1 and C =I, respectively.

We now determine constitutive equations for compressible hyp.erelastic materials. In ·order to particularize the second law of thermodynamics through the ClausiusPlanck inequality (4.154) to the specific strain energy (6.85) at hand we determine the derivative of '11 with respect to time t first. By means of the chain rule we obtain from (6..85)

(6.86)

Hence, we need to compute j and C, which, with eqs.(6.82}1 and (6.83),1, simply

with the multiplicative split of the left Cauchy-Green tensor b = F'FT in the form

(6.99)

(compare with eq. (6.79)2). The terms J213I and b = FFT represent the volumechanging (dilational) and volume-preservjng contributions to the deformation. We call b the modified "left Caucby...Green tensor, with detb = 1.

The derivative of the volume ratio J = (detb) 112 and the modified left Cauchy-

Green tensor b relative to bis given by (6.82)1 and (6.83h (with C replaced by b). We obtain

Following arguments analogous to those which led to eqs. (6.88)-(6.91), we obtain from (6.98) the spatia·1 version of constitutive equations which are expressed in terms of J and b. Given entirely in the spatial description and characterizing the isotropic behavior of -compressible hyperelastic materials, we have

where the stress contributions are defined by

(6.102)

(6.103)

Use has been made of eqs. (6.99) and (6.100) ·and properties (1.95) and (1.155).

The constitutive equation for the hydrostatic pressure pis given in (6.91 )1 and the fictitious Cauchy stress tensor u is defined to be

3

which .furnishes the physically correct deviatoric operator in the Eulerian description, i.e. dev (•) = (•) - (1/3)[(•) : I ]I, so that

For the characterization of the stress response in the spatial description in terms of the projection tensor=n:- see also, for example., the work of MIEHE [1994].

Note the sbnilar structure of the stress relations in the Lagrangian description (6. 88)-(6.91) to those presented in (6.101)-(6.104).

Compressible isotropic hyperelastic.ity in terms of invariants. We now introduce a strain-energy function for compressible isotropic hyperelastic m.ateria]s in terms of strain invariants. By analogy with the decoupled representation (6.85), or (6.98), we write

(6.107)

with the first two strain invariants f1 and 12 of the symmetric modified Cauchy..Qreen tensors. Since C and b have the same eigenvalues, we deduce that

(6. I 08)

The strain invariants Ia, a = 1, 2, 3, are referred to as the modified invariants and

With the kinematic assumption (6.79.h,-or (6..99), and properties (L92) and .(l.101), we conclude from (6.109)-(6.111) with reference to (5.89)-(5.91) that

Finally, we formulate the associated constitutive equation in terms of the volume ratio J and the modified invariants lt. 1 12 , which reads in the material description as

with the volumetric contribution S\'olto the second Piola-Kirchhoff stress, i.e. (6.89),

4.Consider- the strain-energy function \JI = \ITvur( .J) + \JI iso [f1.(b), l2 (b)] with the

.associated constitutive equation for compressible isotropic hyperelastic materials in the spatial description, i.e. u = u vol + O'jso·

(a)By analogy with the above Exercise 3(b)., obtain the constitutive equation for the isochoric (Cauchy) stress contribution ui~o = P : .u, where TI!\ is the fourth-order projection tensor, i.e. eq.(6.105), and u the fictitious Cauchy stress tensor, defined as

for the fictitious Cauchy stress tensor, which .is responsible for volumepreserv.ing deformations. The response coefficient ')'2 is given by (6.1.l

Hint.: Recall .identity (5.93).

6.5 Some Forms of Strain-energy Functions

From previous sections we have learnt that the stress response of hyperelastic .ma-

terials the given strain-energy function \JI. Numerous specific forms

of strain-energy functions to describe the elastic properties of incompressible as well compressible materials have been proposed in the literature and more or less efficient new specific forms are published on a daily basis.

The aim of this section is to specify some forms of strain-energy functions which are well tried within the constitutive theory of finite elasticity and frequently employed in the literature.. In particular, we present a se-lection ·Of representative examples of

'1t known from rubber elasticity describing isotropic hyperelastic materials within the

.isothermal regime (for a collection of constitutive models for rubber see the book edited by DORFMANN and MUHR [1999]). We start by presenting suitable strain-energy functions for incompressible materials and continue with some particular forms which

are able to describe compressibility.

Ogden model for incompressible -(rubber-Hke) mater.ials.. The only materials

undergoing finite strains relative to an equilibrium state are biomateria.ls such as bio...

logictll soft tissues and solid polymers such as rubber-like materials. On the latter we

2"38 6 Hype-relastic l\tlaterials

Mooney-Rivlin, neo~."Hookean, Va.rga model for incomp·ressible (rubber-like) materials.. As a special case we obtain from eq. (6..1.l 9) the Mooney-Riv.Jin .model~

the neo-Hookean model and the Varga model (see MOONEY [1940], RIVLIN [l948, 1949a, b], TRELOAR [1943a, b] and VARGA [1966], respectively).

For example, the very useful Mooney-Rivlin. model results from (6.119_) by setting 1\T = 2, n 1 =-= .2, n 2 = -2.. Using the strain invariants 1 1 , !'2 .as presented by (5.89h and (5.90h, with the constraint condition 1:1 == "'fA~;\~ = 1, we find from (6.119) that

.ployed in the description of the behavior of isotropic rubber...li.ke materials. Mooney derived -it on the basis of mathematical arguments employing considerations of symmetry.

The neo-Hookean model results from (6..l 19) by setting 1V = l, c1: 1 = 2. Using the first principal .invariant / 1, see eq. (5.89)2._, we find from (6.119) that

(6.128)

with the constant c 1 = -11 1/2 and the shear modulus p = fl·i according to (.6.120) 1• This strain-energy function involves .a single .parameter only and provides .a mathe-

maticaUy simple and reliable constitutive model for the nonlinear deformation behavior of isotropic rubber-like materials. It relies-on .phenomenological considerations and in- cludes typical effects known from nonlitiear elastidty within the small strain domain.

However, the important strain-energy function {6.128) may also be motivated from the statistic.al theory in which vulcanized rubber is regarded as a three-dimens.ional net- work of long-chain molecules that are connected at a few points.. A brief discussion is given in Section 7.2 on p. 318.

Constitutive relations for the Mooney-Rivlin and the neo-Hookean model follow from (6.65) by means of (6.127h and (6.128)2 • Derivatives of \JI with respect to the strain invariants 11 and 12 give the simple assoc.iated stress relations u = -pl+ 2c 1b·- ·2c2h-1 and u = -pl+ 2c1b, respectively.. Compare also the considerations on p. 203.

As the last special case of Ogden's model we introduce the model by \q1rga. It results from (6.119) by setting JV = 1, n 1 = 1, i.e.

(6.129)

with the constant .c1 =1-1. 1 and the shear modulus 11 = J.Li /2 according to (6. 120) 1•