Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

For a transversely isotropic material, the free energy can finally be written in tenns of the five independent scalar invariants, and eq. (6.27}1, valid for isotropic mate.rial response, and may consequently be expanded according to

The free energy (6.204) provides a fundamental basis for .deriving the associated constitutive equations.

Constitutive equations 'interms of invariants. In order to derive the constitutive

equations we apply (6.13h. Then, by use of the chain rule, the second Piola-Kirchhoff stress tensor S is given as a function of the five scalar invariants, i.e.

in which 8!1 /8C and 812 /BC, 8I3 /8C are given by (6.30) and (6.3.l), respectively. The remaining derivatives follow from (6.203) and ·have the forms

which extends the constitutive equation (6.32) by ·the addition of the last two tenns. Using arguments similar to those used for the derivation of the spatial version of

the stress relation (6.34), namely a push-forward operation on the material stress tensor

Recall that the unit vector a(x, .t) denotes the fiber direction in the deformed ·con-

figuration while b is the (second-order) left Cauchy-Green tensor. Observe the

Note that the indeterminate terms -qa0 0 a 0 and -qa ~ a are .identified as ·fiber

.reaction stresses which respond Lo the inextensibiHty constraint L1 = 1.

EXERCISES

1.Starting from the pseudo-invariants L1 and 15 , i.e. eqs. (6..203), show their deriva- tive with respect to C, eqs. (6.206) and (6.207).

2.We characterize a transversely isotropic material by the decoupled free energy in the form

(6.214)

where \1'.vol and wi:m are the volumetric and isochor.ic contributions to the hy- perelast.ic response (recall Section 6.4). The modified invariants 11, 12 are given according to eqs. (6.109) .and (6.110), while f:s = detC = 1 (note that 11, !2, f, are the modified principal invariants of the modified tensor C = .1-2tic). The

remaining modified pseudo-invariants are ex.pressed by f 1 = J-'2/a I.1 and 15 ·=

J-4/:l h.

(~)Having in mind the free energy (6.214) and the derivatives (6.206) .and (6.207), show th.at the constitutive equation S = Jpc- 1 + J-2/:ip = S

specializes to

with the response coefficients

·- - ') awiso

!::. - J.J 0/5

for the fictitious second Piela-Kirchhoff stress S. Note that the coefficients 1 1 and 1 2 re.fleet the isotropic stress response, as given in eqs. (6. I.1.6).

(b)By recalling Section 6.6, a closed form expression for the elasticity tensor

C in the material description .is given by relation (6.1.62), with contrjbutions

Cvoh i.e. {6.166),b and C1so' i.e. (6.168).

272 6 Hyperelastic Materials

By use of the important property (1.256) and the constitutive equation for the fictitious second Piola-Kirchhoff stress (6.215)3 , show that the fictitious elasticity tensor·C "in the material description takes on the form

with the fourth-order unit tensor I defined by (1.160) and the coefficients

Ja,a= 5, ... ,12,by

Note that the coefficients 8a., a = 1,,. ... ., 4, were given previously in rela- tions (6.. 195) and reflect the isotropic contributions.

6.8Composite Materials with Two Families of Fibers

In the following we discuss appropriate constitutive equations for the .finite elastic response of fiber-reinforced composites in which the matrix -material is reinforced by two families of fibers. We assume that the fibers are continuously distributed throughout the material so that the continuum theory of fiber-reinforced composites is the constitutive theory of choice.

There are many different fibers and matrix materials now in use for composite materials. Examples of specific fibers for structural applications .are boron and glass,. The latter is an important engineerin.g fiber with high strength and low cost. Further examples are carbon and graphite (the difference is in the carbon content), the organic

.fiber aramid and the ceramic fibers silicon carbide and alumina among others. Many specific matrix materials are available for the use in composites; for example~

plastic polymers, tlzermoset polymers, metals (such as alum.inum, titanium and and ceramics.

Vast numbers of applications in industry are concerned with .composite materials~ such as the finite elastic response of belts and high pressure tubes, steel reinforced rubber used in tyres, and integrated circuits used in electronic computing devices. Typical medical applications are lightweight wheelchairs and implant devices such as hip joints (see also the textbook by HERAKOVICH [1998, Chapter l]). The five-volume encydopedia of composites edited by LEE .[ 1990, 1991] includes a detailed account of special types of fiber, matrix materials and composites .as engineering materials. Typical engineering properties, manufacturing and fabrication processes and details on how to use

composite materials for different applications are also prov.ided.

However, it is important to note that numerous organisms such .as the human body, animals and plants are heterogeneous systems of various composite biomaterials. The textbooks by FUNG [ 1990, 1993, 1997] are concerned with the biomechanics of vari-

ous biomaterials, soft tissues and organs of the human body. One important example of a fibre-reinforced b.iomaterial is the artery. The layers of the arterial wall are composed mainly of an isotropic matrix material (associated with the .elastin) and two families of ·fibers (associated with the collagen) w·hich are arranged in symmetrical spirals (for arterial histology see RHODIN [1980]). For mechanical properties and constitutive equations of arterial walls, see the reviews by, for example, H"'YASHI [1993], HUMPHREY

[1995] and the data book edited by AJH~ et al. (1996, /Chapter 2]. A simple finite element simulation of the .orthotropic biomechanical bepavior of the arterial wall is provided by HOLZAPFEL et al. [.1996d, 1996e] and HOLZAPFEL and WEI.ZSACKER

[199.8]. For a review of finite element models for arterial wall mechanics, see the article by SIMON et al. [1.993].

Free energy and constitutive ·equations. We may now consider a body built up

of a matrix material with two families of fibers each of which is unidirectional with preferred direction. The matrix material .is assumed to be hyperelastic. The preferential fiher directions in the reference. and the current con figuration are denoted by the unit vector fields a 0 ., g0 and a, g, respectively. By analogy with relation.(6.201) we may postulate the free energy

per unit reference volume. For notational simplicity we have introduced the abbrevia-

The free energy must be unchanged if the fiber-reinforced composite (i.e. a hy- perelustic (matrix) material with two families of fibers) in the reference configuration undergoes a rotation described by the proper -orthogonal tensor Q. Using arguments similar to those used for a single fiber family (see the previous Section 6.7), the re-

which holds for all tensors Q (recall Example 6.9).. Here1 \JI "is a scalar-valued isotropic tensor function of the three tensor variables C, Ao and G0 •

According to SPENCER [.1971, 1984], requirement (6.217) is satisfied if \lJ is a function of the set of invariants

The three invariants 11 , 12 , 1:.1 are identical to those from the isotropic theory pres.ented in eqs. (5,.89)-(5.9.1). The pseudo-invariants J..1, In are given by eq. (6.203) and char- acterize one family of fibers with direction a 0 • The pseudo-invariants 14 , ••• , .Iu are associated with the anisotropy generated by the two families of fibers. The dot product a0 • ·g0 is a geometrical constant determining the cosine of the angle between the two fiber directions in the reference configuration. Therefore, the invariant /~> does not depend on the deformation and is subsequently no longer considered. Note that I 1 and ft; are equal to the squares .of the stretch in the fiber directions a0 and g0 , respectively.

The constitutive equation for the second Piola-Kirchhoff stress S follows from the postulated free energy (6.216) by differentiation with respect to C. By means of the chain rule, S is given as a function of the remaining eight scalar .invariants in the form

(6.219)

in which DI1/8C, ... ~ 8I:d8C and 8I.1/8C., Dh/fJC are given by eqs. (6.30), (6.31) and (6.206), (6.207), respectively.

The remaining derivatives ·Of the invariants fo.llow from (6.2 l 8)n-(6.218)8 and have the forms

EXERCISES

l.Consider a locally orthotropic material with the free energy wexpressed as a function of the invariants presented by (6.227) using (6.228). Assume an incompressible isotropic matrix material and two families of inextensible fibers.

Show that the constitutive equation for the Cauchy stress tensor u is given by

where the first three terms .characterize reaction stresses.

2.We characterize a compressible composite with two families of fibers by the decoupled representation of the free energy

(6.229)

(comp.are also with Exercise 2 on p. 271), with the volumetric and isochoric parts '11vol and W180 , and the modified invariants 11, l2 .given by eqs. (6..109) and (6.1.10)

Use the free energy (6..229) to particularize the fictitious second Piola-Kirchhoff stress S which appears in .the constitutive equation for S = Jpc-1 + J-2/ 3 p: S,.

Show that

with the explicit expressions

6.9Constitutive Models w·ith Internal Variables

Many materials used in the fields of engineering and physics are inelastic. It turns out that the constitutive ·models introduced hitherto are not adequate to describe this class of ..materials, for which every admissible process is d:issipative. Within the remaining sections of this chapter we study inelastic materials and, based on the concept of internal variables, we derive constitutive models for viscoelastic materials and hyperelastic mate.rials with isotropic damage.

Conc·ept of inte.rna:J variables. The current thermodynamic state of thermoelastic

materials can be determined solely by the current values of the deformation gradient

F and the temperature 8. Variables such as F or 8 are measurable and collfrollable quantities and .are accessible to direct observation. In practice these type of variables are usually called external variables.

The current thermodynamic state of ·materials that involve dissipation can be determined by a finite number of so-called inte.rnal variables, or in the literature sometimes called bidden variables (hidden to the eyes of external .observers). These additional thermodynamic state variables, which we denote collectively by e, are supposed to describe aspects of the internal structure of materials associated with irreversible (dissipative) effects. Note that strain (stress) and temperature (entropy) depend on these internal variab]es. The evolution of internal variables replicates indirectly the history of the deformation, and hence they are often also termed history variables. Materials that involve dissipative effects we refer to as dissipative mate.rials· or materials with dissipation.

Hence, the concept of internal variables postulates that the current thermodynamic state at a point of a dissipative material is specified by the triple (F, -8, e) (the current thermodynamic state may .be imagined as ajictitious state of thermodynamic equilibrium). Then, the current thermodynamic state is represented in a finitedimensional state space and described by the current v(.llues (and not by their past history) of the deformation gradient, the temperature and the finite number of internal variables.

The nature of .internal variables may be physical, describing the physical structure of materials. In the course of phenomenological experiments one may be able to iden~ tify :internal variables; however, they are certainly n.ot controllable or observable.

We use the internal variables as phenomeno1ogical variables which are constructed mathematically~ They are mechanical ..(or themmI, or even chem.ical or electrical ...) state variables describing structural properties within a macroscopic .framework, such as the 'dashpot displacements' in viscoelastic models, damage., inelastic strains, dislocation densities, point-defects and so on. Hence, here we introduce both external and interna1 variables as macroscopic quantities without referring to the internal mi-