Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

with n,u1 = 0. By me~ns of identity (l .65h, we may deduce from (6. l 83h the repre-

Knowing Na~ we now can determine -the .material time derivative of the spectral representation of C. Converting eq. (2. l 26) to the rate form, we obtain, by means of eq. {6..183h .and the spectral decomposition of the right Cauchy-Green tensor, i.e.

where 2Aa.~a =.Na· CN0 = C.'0 a, a== 1, 2, 3 {compare with eq. (2.127)), denote normal components (diagonal elements) and nab(,\~ - A~) = Na · t:Nb = Cub,, a ¥ b, denote

shear components of C (off-diagonal elements) with respect to the basis {Na}·

a=l

in which the material time derivative of the principal second Piola-Kirchhoff stresses is defined to be

(6.188)

By expanding the numerator and denominator of the second term in (6.187) with

·On comparing the derived ·eqs. (6."I 89) and (6..186) with (6.153) in the rate form, i.e. S= C : C/2!f we find by .inspection that the elasticity tensor C emerges, as given by eq. (6.180). II

260 6 Hyperelastic Materials

For the case in which two or even all three ei.genvalues /\~ of C {and also of b) are equal, the associated two or three stresses Sa are also equal, by isotropy. HenceT the divided difference (Sb-Sa)/(.Al-A.~) in expression (6.1"80) represents an indeterm.inate form of type g. However, it can be shown that the divided difference is well-defined as _,\b approaches Aa. Namely, app.lying l'Hopital 's rule, we see simply that

(6.190)

(compare also with the work of CHADWICK and OGDEN [1971Q]). Consequently, the elasticity tensor, as defined in eq. (6. I.80), is valid for the three cases: ,.\ 1 f. .A2 -:/: A3 # /\i, A.1 = ~\2 '# ~\1 and ,.,\1 = ~\2 = .t\3.

Finally, in order to set up the spectral form of the elasticity tensor ·c in the spatial description we use the Piola transformation of C for principal values. According to

Cabcd = J- 1 AaAb.Ac,\d Cabcd' with the principal stretches /\1, .-\2, ,\3 and the volume ratio J = A1A2 A3 . A straightforward computation leads to

(6J.9l)

with the principal Cauchy stresses aa = J- 1>..·~Sa, a = 1, 2, 3 (see the inverse of eq. (6..48h), and the principal spatial directions n,z, a= 1, 2, 3, which are the orthonor- mal eigenvectors of v (and also ofb), with .Jiia I = 1 and Da ·nb = Oab· From the property {2.l 18) we know that the two..-point tensor R rotates the principal referential directions Na into the .Principal spatial directions Da.

If /\a = Ab we may conclude that aa = ab, by isotropy. Hence, the divided difference (ab/\~ - aaA.~)/(/\~ - /\~) in expression (6.19]) gives us *and must therefore be determined applying l 'Hopitat's rule. Differentiating the numerator and de.nominator by Ab and takin.g the limits Ab -7 /\a, the divided difference becomes

(6.192)

An alternative version of solution (6.192)2 in terms of the principal se.cond Piofa-Kirch- left as an exercise.

EXERCISES

I.For the description of isotropic hyperelastic materials at .finite strains consider the strain-energy function w = \f!(l.i, r2, /3) in the coupled form, with the principal invariants Io., a= 1, 2, 3.

(a)Use the stress relation (6.32.h, the chain rule and the derivatives of the in· variants with respect .to C, i.e..eqs. (6.30):i .and (6.31 ), in order to obtain the most general form -of the elasticity tensor C in terms of the three principal invariants

=811®I+02(.I ® c + c ®I)+ 03(1 ® c-1 + c-1 ®I)

+8.-iC ® c + Or,(C 0 c-l. + c-1® C)

cording to (6.164) and (1.160), respectively.

(b)Particularize the coefficients Ou, a = 1, ... , 8, (6.194), for the compress- ible Moon~y-R.iv.lin, neo-Hookean, Blatz and Ko models, Le. eqs. (6.147)- (6.149). For convenience, summarize the (nonzero) coefficients for the three .material models to fonn the entries of Table 6.3.

For notational simplicity we have introduced the abbreviation ~ = 2tt(1 -

f)/Ia with ~ > 0. Note that the coefficients 6a, a = 1, ... , 8, for the neo- Hookean ·model are simple those of the Blatz and Ko model obtained by

setting ~ = 0 and µf = 2c1.

Compare the .corresponding constitutive equations of the three material models of Exercise l(a) on p. 248, with the specified coefficients summa-

rized in Table 6.1.

Table 6.3 Specified coefficients for the clasticily te.nsors of some materials in the ·coupled form.

2. The strain-energy function '11(.J~11, 12) = Wvut ( J) + Wisa(lL, 12 ) is given in terms

of the volume ratio J and the two modified principal stretches 11, T • This type

2

of strain energy in tlecoupled representation is suitable for the characterization

of compressible .isotropic materials at finite strains.

(a)The associated decoupled elasticity tensor C is given by (6 ..162), with the

volumetric .and .isochoric parts (6.1·66)4 and (6.168), respectively. Particularize the fictitious elasticity tensor C in the material description, i.e.. eq. (6.169.).i, to the specific strain energy at hand.

Start with the constitutive equation for the fictitious second Piola-Kirchhoff

stress S, as defined in eq.·: (6J.15), and use the derivatives of with

respect to tensor C., i.e. (6.117), in order to obtain the most general form of C in terms of 11 and 12 in the form

with the fourth-order unit tensor Il defined by ( 1.160) and the coefficients J,H l1. = 1, • .. l4, by

(6.195)

(b)To be specific, take the decoupled form of the compressible Mooney-Rivlin model (6.146) and the compressible neo-.Hookean model (set c2 = 0). Additionally, take the .Yeoh mode.I and the Arruda and Boyce model of the

forms

Note that the associated constitutive equations, with the specified coeffi- c.ients as summarized .in. Table 6.2, are presente,d in Exercise I.(b) on p. 248.

3. Consider a compressible isotropic material charad.~rized by the strain-energy function in the decoupled form of \}I (,.\I' ,\2' A:i) = ....\JI vu) ( J) + \{I iso (Xi, ...\2' ,\:i)' with the volume ratio J = >iu\2.,\1 and the modified principal stretches ,.\1 = J-1./:i .,.\a, a = 1, 2, 3. The associated decoupled structure of the elasticity tensor C in the material description is given as C(,.\., .,.\2 , ,,\a) = Cvol + <Ciso' with the volumetric contribution Cvot specified by the expression (6.166)4 •

(a)Show that the spectral .form of the isochoric contribution <Ciso may be given by

(6.196)

where Sison = {8\J!iso/D,\a)/,\1' a = 1, 2~ 3, denote the principal values of the second Piola-Kirchhoff stress tensor Siso ~

Representation (6.198) was given by ·MIEHE and STEIN .(1992]. In their work the definition of the elasticity tensor c = x* (C) in the spatial description excludes the factor J- 1.

7.Consider the additive split of the Cauchy stress (6.101)-(6.104) in terms of J and b, based on the strain-energy function of the form (6.98). Show that the associated elasticity tensor <C in the spatial description may .be written in the decoupled form

of the purely volumetric contribution Cvol and the purely j:sochoric contribution

>Ciso• the latter being based on the spatial projection tensor p." ][ :..... ~I® I and on the definitions T = .lu, Tiso = J Ui50 , with O"iso = n) ; u, as given in (6.103) and {6.104).

In addition, we introduced the .definitions of the fourth-order fictitious elasticity tensor c in the spatial description and the trace tr{•) according to

Db8b

For an explicit derivation, ·see MIEHE [ 1994~ Appendix A].

6.7 Transversely Isotropic Materials

Numerous materials are composed of a ma.tr.ix material (or in the literature often calJed ground substance) and one or more families of fibers. This type of materiat

which we cal1 or fiber..reinforced com:posite, is heterogeneous

in the sense that it has different compositions throughout the body. We consider only composite materials in which the .fibers are continuously arranged in the matrix mate- rial. These types of composites have strong directional properties and their mechanical responses are regarded as anisotropic.

266 6 Hyperelastic Materials

The challenge in the design of fiber-reinforced composites is to -combine the matrix material and the fibers in such a way that the resulting material is most efficient for the desired application. For engineering applications composite mate.rials provide many advantages over monolithic materials such as high stiffness and strength, low weight and thermal expansion and corrosion resistance. However, the drawbacks in using composite materials seem lo be the high costs when compared with those of monolithic (more dassical) materials and_, from the practical point of view, limited knowledge of how to combine these types of material.

A material which is re.inforced by only one family of_ fibers has a single prefe.rred direction. The stiffness of this type of composite material .in the fiber direction is typically much greater than in the directions orlhogonal to the fibers. It is the simplest representation of material anisotropy, which we call .transve.rsely isotrop.ic with respect to this preferred direct.ion. The .material response along directions orthogonal to this preferred direction is isotropic. These composite .materials are employed in a variety of

.applications in industrial engineering and medicine. For manufacturing and fabrication processes and for typical features and properties of transversely isotropic materials the reader should consult, for example, the textbook by HERAKOV.ICH .[ 1998].

The aim of the following section is to investigate transversely isotropic materials capable of supporting finite elastic strains. As mentioned above, all .-fibers have a single preferred direction. However, the fibers are assumed to be continuous.ly distributed throughout the material. We derive appropriate -constitutive equations which are based solely on a continuum approach (excluding micromechanical considerations). Constitutive equations which mod.el transversely isotropic materials .in the small elastic strain .regime .are well established and may be found, for example, in the textbooks by

TSAI and HAHN [1980], DANIEL and .lSHAI [19.94], HERAKOVlCH [1998.] and JONES

body B which initially occupies atypical region 0 0 at a fixed reference time t

region is known as a fixed reference cm~figuration of that body B. A point in n0 may be characterized by the position vector X (with material coordinates .Xr.h .A = 1> 2, 3) related to a fixed set of axes. At a subsequent time .t > 0 the continuum body is in .a deformed configuration occupying a region ·n.The associated point .inn is characterized by the position vector .x (with spatial coordinates :c0 , a = 1., 2, 3) related to the same fixed set of axes. For more details about the relevant notation .recall Section 2.1.

We suppose that the only anisotropic property of the solid comes from the presence of the fibers. To start with, for a material which .is reinforced by only one family of fibers, the stress at a material point depends not only on the deformation gradient ·F but also on that -single preferred direction, which we call the fiber direction. The direction of a fiber at point X E H0 ..is defined by a unit vector field a0(X)T laol = 1, with material

coordinates .a0 A. The fiber under a deformation moves with the material points of the continuum body and arrives at the deformed configuration n. Hence, the new fiber direction at the associated point x E !1 is defined by a unit vector field a(x, t), Jal = 1,

with spatial coordinates aa. For subsequent use it is beneficial to review the section on material and spatial strain tensors introduced on pp. 76-81..

Allowing length changes of the fibers, we must determine the stretch ~\ of the fiber idong its direction a0 • It is defined as the ratio between the length of a fiber element in the deformed and reference configuration. By combining eq. (2.60) with (2. 71) we

whkh relates the fiber directions in the reference and the deformed configurations.

which we already have .introduced in relation (2.62). This ·means, that the fiber stretch depends on the fiber direction of the undeformed configuration, i.e. the unit vector field a0 , and the strain measure, i.e. the right Cauchy-Green tensor C.

We now assume the transversely isotropic material to be hyperelastic, characterized by a Helmholtz free-energy function wper unit reference v·~fli·nl-~. Because of the

directional dependence on the deformation., expressed by the unit vector .fie.Jd a0 , we require that the free energy depends explicitly on both the .right Cauchy-Green tensor c and the fiber direction ao in the reference configuration.

Since the sense of a0 is immaterial, '11 is taken as an even function of a 0 • Hence, by

From previous sections we know that the free energy must be independent of the coordinate system; hence \Jl(C, a0 0 a0 ) must be ·Objective. Since C and a 0 ® a 0 are defined with respect to the reference configuration (which is fixed), they are unaffected by a rigid-body motion superimposed on the current configuration. Consequently, the principle of material frame-indifference of the postulated free energy w(C 1 a 0 0 a0 ) is satisfied trivially.

"EXAM.PLE 6...9 The free energy W(C, a0 ® a0 ) must be unchanged if both the matrix material and the fibers in the reference configuration undergo a rotation around a

certain axis described by the proper orthogonal tensor Q.

Show that the requirement for transversely isotropic hy.perel.astic materials formally reads

(6.202)

which holds for all proper orthogonal tensors Q.

Solution. For the solution it is beneficial to review eqs. (6.19)-(6.25) of Section 6.2. A rotation of the reference configuration by tensor Q transforms a typical point X into position X* = QX. Consequently, fiber ..direction a0 transforms into the new fiber direction a0= Qa0 so that a0®a~ becomes Qa0 ® a0 QT. Now, after a subsequent motion of the rotated reference configuration, X* maps into position x. Thus, the deformation gradient F* and the strain measure C* = F*TF"" relative to the rotated reference

.configuration are F* = FQT (compare with (6.22)) and C* = F*TF* = QFTF·QT = QCQT, respectively.

We say that a hyperelastic material is transversely isotropic relative to a reference configuration if the identity \P (C, a0 ® a0 ) = '11 (C*, a0® a~) is satisfied for all proper orthogonal tensors Q. Hence, restriction (6.202) follows directly. Note that in view of (6.202), \JI may be seen as .a scalar-valued isotropic tensor function of the two tensor variables C and a0 ® a0 . II

According to (6.27), an isotropic hyperelastic material may be represented by the first three invariants 11, 12 , 13 of either C o.r .b, characterized in (5.89)-(5.91 ). These invariants can be used to fulfil requirement (6.25), i.e. '1t (C) = \II (QCQ'1) for all (Q, C). Followjng SPENCER [1971, 1984], two .additional (new) scalars, 11 and 15 , are necessary to form the imegrity bases of the tensors C and a0 ® a0 and to satisfy relatio.n (6.202). They a.re the so-called .pseudo-invariants -of C and a0 0 .a0 , which are given by

(6.203)

The two pseudo-invariants L1, / 5 arise directly from the anisotropy and contribute to the free energy. They describe the properties of the fiber family and its interaction with the other material constituents. Note that invariant / 4 is equal to the square of the stretch A in the fiber direction a0 (compare with eq. (6.200))..

For the definition of the integrity bases and the related theory of invariants see the lecture notes by SCHUR (1968], the articles by SPENCER .[1971] and ZHENG Il994, and references therein]. For applications .in continuum mechanics the reader is referred to the works by RIVLIN [.1970], BETTEN [1987a, Chapter D and 1987b], TRUESDELL and NOLL [1992] among others. A brief review of the theory of invariants may also ~e fou~d in SCHRODER [1996]. .