Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

Table 6.2 Specified coefficients for the constitutive equations of some ma-

terials in the decoupled form.

2. Consider a s.pherical balloon of incompressible hyperelastic material (Au\2.A3 .

1), The material is characterized by a strain-energy function in terms of principal stretches according to

(6.150)

with material constants tl and a. For n = 2 we obtain the classical neo-Hookean model.

(a)Determine the inflation pressure Pi as a .function of the circumferential stretch A in the form

and a relative minimum.if -3/2 < a < 0 (see OGDEN [1972a])..

Consequently, balloons made of-materials which .are described by the strainenergy function (6.150) will not 'sn~p throu.gh' for a:~ 3. Typical examples for this type of (stab.le) material behavior are biomaterials such as an

.artery (for the mechanics of the arterial wall see the excellent survey text by HUMPHREY [1995]) or a -ventricle (see NEEDLEMAN et aL [ 1983]). Specific results on membrane biomechanics including illustrative examples from the literature .are reviewed in the article by HUMPHREY [1998].

3.Consider a thin sheet of incompressible hyperelastic material with the same setting as formulated in Example 6.3~ The homogeneous stress response of the material is assumed to be .isotropic and based on Ogden's model. Discuss the stress states (which are plane throughout the sheet) for the fol.lowing two modes of deformations (for the .associated kinematic relations, comp.are with Exercise 1 on .P· 226):

(a_) Consider a simple tension for which ,.\.1 = A (,\2 = /\3 ). Show that the only nonzero Cauchy stress er, in the direction of the applied stretch ,.,\, is

N

a= L/.Lp(Aa,. - ,,.\-0,,/2) ..

11=1

In addition, find the stress required to produce a final extended length of

.,\ = 2 for each of the Mooney-Rivi.in and the neo-Hookean -models.

(b) Consider a homogeneous pure shear deformation and show that the biaxial stress state (aa = 0) of the problem is of the form

(c)Compute the associated constitutive equations given in (a) and (b) for the Mooney-Rivlin, neo-Hookean and Varga models and compare the results with Ogden's model (plot the relation between the Cauchy stress and the associated stretch ratio for each material model).

4.The so-called Saint..Venant Kirchhoff model is characterized by the strain-

energy function

two constants of Lam·e. The Lame constant "Y is usually denoted in the literature by the symbol A. However, in order to avoid confusion with the stretch ratio ,\

we use a different symbol for it. The Saint-Venant Kirchhoff model is a classical

nonlinear model for compressible hyperelastic materials ·often used for metals. Note that the volume ratio J does not appear explicitly in this material model.

(a)From the given strain energy w(E) derive the second Piola-Kirchhoff stress S, which linearly depends on the Green-Lagrange strain E.

(b)Consider the one-dimensional case of the constitutive equation derived hi

(a). For a uniform deformation of a rod (with uniform -cross-section), i..e. :i: = AX'", .derive the relation between the nominal stress P and the associ-

ated stretch ratio A (which is .a cubic .equation in .,\) and plot the function

P = P(A).

Show that P(A) .is not monotonic in compression and derive the critical

fails (zero stiffness, the tangent of P( A) at Acrit is horizonta]). This failure is not influenced by the material constants / and JL.

In addition show that the· material model does not satisfy the growth condi- tion (6.6)2 (in fact for A -7 o+ the stress tends to zero which is physically unrealistic). CIARLET [.1988] showed, with the proof by RAOULT .(1986], that the Saint-Venant Kirchhoff mo~el does not satisfy the requirement of

.polyconvexity either.

Note that this material model is suitable for large displacements -i:mt it is not recommended to use it for large -compressive strains.

5. In the first term in eq. (6.151) replace trE by lnJ and / by the bulk modulus Ii > 0 in order to obtain a modifi-ed Salnt-Venant K..ircbhoff model of the form

where J = detF denotes the volume ratio and JL > 0 is identified as the shear modulus. The proposed material model (6.152) circumvents the serious draw-

ba9ks of the classical Saint-Venant .Kirchhoff model (see Exercise 4) when used for::l~arge compressive strains.

(a)From the strain energy (6.152) derive the .second .Piola-Kirchhoff stress S =

S{C) as a function of the right Cauchy-Gre-en tensor C. The result is similar

to a stress relation proposed by CURNIER II994, eq. ·(6..1I3)] which also

has the aim of avoiding the defects of the classical Saint-Venant Kirchhoff

model occurring at large co:mpressive strains.

(b)Consider a one-dimensional problem as described in Exercise 4(b) and ob- tain the nominal stress Pas a function of A. Discuss the function P-(.1\) for the two regions .,\ > 1, ;\ < 1 and show that the modified Saint-Venant

Kirchhoff model satisfies the growth condition in the sense that the stress tends to (minus) .infinity for A --t o+.

We say C possesses the minor symmetries. The symmetry condition (6.156) is independent of the existence -of a strain-energy function w and holds for all elastic

materials. Note that the minor symmetry of C follows from the symmetries of the right Cauchy-Green tensor C (or equivalently from the Green-Lagrange strain tensor E) and the second Piola-Kircbhoff stress tensor S.

If we .assume the existence of a scalar-valued energy function w(hyperelastici.ty), then S may be derived from \JI according to S = 28\ll(C)/8C (see (6..13h). Hence,

using (6.154), we arrive at the crucial relation

We say C possesses the major symmetries. Thus, tensor C has on·Iy 21 independent components at each strain state. The condition (6.158) is a.necessary and sufficient

.condition for a material to be hyperelastic. The symmetry condition CAncn

is often referred to as the definition of hyperelasticity. Hence, the major symmetry of C is basically equivalent to the existence -of .a strain-energy function. Note that the ma-

tensor is associated with the of the (tangent)

stiffness matrix arising in a finite -element discretization proc-edure.

The elasticity tensor in the spatial description or the spatial tensor of elasticities, denoted by c, is defined as the push-forward operation of C times a factor· of J-1

(see MARSDEN and HUGHES [1994, Section 3.4]), in other texts the definition -of c frequently excludes .the factor J-1• .It is the Pio/a transformation of C on each large

index so that

Ccdab· The fourth-order tensors C and c are crucial within the concept of linearization,

.as will become apparent in Chapter 8, particularly in Section 8.4.

The spatial representation of eq. (6.153) can be shown to be

Oldroyd stress rate (5.59) -of the contravariant .Kirchhoff stress tensor T, the rate of deformation tensor (2.146), and the spatial elasticity tensor, as defined in eq. (6.159),

254 6 Hyperelastic Materials

respectively. A material is said to be .hy.poelas-tic if the associated rate equations of the form (6.. 161) are not obtained from a (scalar-valued) energy function. For more on hypoelastic materials see the classical and detailed account by TRUESDELL and NOLL

.[ 1992, Sections 99-103].

Systematic treatments of the elasticity tensors have been given by, for example,

TRUESDELL .and TOUPIN [1960, Sections 246-249], CHADW"ICK and OGDEN [197la, b], HILL [198.1..J, TRUESDELL and NOLL [1992, Sections 45, 82] and 0.GDEN [1997,

Chapter 6].

Decoupled representation of the elastidty tensor. Based on the kinematic assump- tion (6.79h and the decoupled structure of the strain-energy function (6~85) we derive the associated elasticity tensor. We focus attention solely on the material .description of the elasticity tensor.

The elasticity tensor (6.154) may be written in the decoupled form

of the purely volumetric contribution Cvot and the purely isochoric .contribution Ciso·

By analogy with eq. (6.157) we express the two contributions Cvol and Ciso in terms

of the strain-energy function \JI. Before this exploitation we introduce the definition

of the fourth-order tensor ac-J Iac, for convenience (recall Example 1.11, p. 43, and

Starting with (6.163)i, a straightforward computation yields~ with definition (6.89h, property (1.256), the derivative of J and c-1 with respect to C, Le. eqs. (6.82) 1 and (6.164), and the product rule of differentiation,

For convenience, we have introduced the scalar function p, defined by

with the constitutive equation for p given in (6.91) 1• Note that the only values which must be specified for a given material are p and 1J.

The foil owing example shows a lengthy but r~presentative derivation of an elastic-

ity tensor.

EXAMPLE 6..8 Show the following explicit expression for the second contribution to the elasticity tensor, .i.e.. the isochoric part C;~m' as defined .in eq. (6. l63h,

Cisn = P : C : pT + ~Tr(r 213S)IP

The first term in this equation yields, through property (6.82.h and definition (6.90)..h

which gives the last expression of (6.168) we show.

Hence, in the following we analyze exclusively the second t~rm in (6.171). With the definition of the projection tensor (6.84), identities (l.160) 1, (l.152) and the chain

and the set {Na}, a = 1, 2, .3, of.orthonormal eigenvectors of the right Cauchy-Green

tensor C. They define principal referential directions at a point X, with the conditions

INuf = 1 and N.n ·Nb = <~ab·

The important fourth-order tensor Cina more general setting was given by OGDEN

[1997, Section 6.1..4]. Compare a]so the work of CHADWlCK and OGDEN [ 1971 a, b]

with some differences :in notation.

The proof of representation (6.180) is as follows:

P.rooJ In order to prove relation (6.180) we follow an approach which takes advan-

tage of isotropy. ·we know from Chapter 5 that for isotropic elastic materials the second Piola-Kirchhoff stress tensor S .is ·coaxial with the right Cauchy-Green tensor C, so that S has the same principal directions as C.

For notational convenience we use henceforth a rate formulation rather than an infinitesimal formulation. In particular, we now compute the material time derivat)ves ofthe stress and strain tensors S and C, and compare them with the rate form of relation (6..153), i.e. S= C: C/2, in order to obtain the elasticity tensorC, as defined in (6.154)

and specified .in (6. I·SO).

To begin with, consider a set of orthonormal basis vectors ea, a = 1, 2, 3, fixed in space. Consequently, the set {Na}, a = 1, 2, 3, of orthonormal eigenvectors may be

governed by the transformation law

components Qal1 == e,.t ·Nb ·=cosO (ea, Nb), representing the cosine of the angle between the fixed basis vectors ea and the ·orthonormal eigenvectors Nb (principal referential directions). Tensor ·Q is ·characterized by the orthogonality condition, i.e. QTQ =

QQT = J.

and by means of the skew tensor n, as introduced in (5..:15), and transformation (6.182), we may eli 1ninate the basis {ea} and find that

Note that the components of the skew tensor n = -nT with respect to the bas.is {c,,} are obtained from (6.183)2 in the form

(6. 184)