Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

(7.87)

for the general case .A1 # .,\2 :j:. A3 i= A1 (see the analogues of eqs. (6.140)i, (6.143)).

The constitutive equation for the hydrostatic pressure p, essential for relation (7 .86h, may be specified in terms .of the free energy (7;81 )1 as

where the term dQ(J)/dJ was particularized in eq. (6.141)1 or (6.141.)2 depending on whether the scalar-valued function (6.137h or (6.138) is used.

In addition, with free energy (7.81)2 (and (7.82)) we may compute the three princi- pal isochoric stress functions Sisoa, a = 1, 2, 3, in the form

a=l,2,3, (7 ..89)

which are needed for (7 .87)2 (compare with the derivation which led to eqs. (6.144)2 and (6.145)).

The derived set of expressions (7.86)-(7.89) completely defines the constitutive model for rubber-like materials, allowing thermoelastic deformations with strain chan- ges unrestricted in magnitude. It is a straightforward thermodynamic extension of Og- den's mode.I known from the isothermal regime, i.e. (6.140), (6.14.l) and (6.143)-

(6.145).

Alternative stress measures follow directly from eqs. (7.86) and (7.87) by means of suitable transformations. For exam.pie, the stress response expressed by the Cauchy

stress tensor u simply results from a push-forward (and Piola) transformation u =

J-1 x*(S~) = J- 1FSFT of s.

As a second step in the consistent linearization process, we compute the change in the stress tensor S, i.e. dS = C : dC/2 + Td8, with the definitions of the isothermal elasticity tensor C in the material description, i.e. (7.49), and the referential stress-

for the tensors C and T, w·here the first expression represents the familiar additive split

From (7.8·8)2 and (7 ..89h we find, using the relations for the bulk modulus (7 .83) and the shear moduli (7.84), which are temperature dependent quantities, that

;•

standard push-forward (and Piola) transformation.

Heat conduction. The considered thennoelastiC problem, for which (Vint = 0).. is governed essentially by Cauchy ,s .first equation of motion (see, for example, the

local fonns (4.53) or (4.63)) and by the balance of (mechanical and thermal) energy in entropy or temperature form (see ·eq.{7.45) 1 or (7.46)). Hence, in regard to the energy balance equation we need an additional constitutive equation for the heat ·flux vector governing heat transfer. One example which satisfies the heat conduction inequality .is

Duhainel's law of heat conduction (see the considerations on p. 1.70).. For the dass of

.thermally isotropic materials we may express the constitutive equation as

(i.e. Fourier's Jaw of heat conduction, which should be compared with eq. (4.148)), where ·Qis the Piola-Kirchhoff heat flux and k0 > 0 denotes the coefficient of thermal conductivity associated with the reference configuration. Note that this coefficient is, in general, not a constant. In fact, for vu Icanized elastomers, ko decreases linearly with increasing temperature (see .SI.RCA R and WELLS [ 1981]) according to

The solution of the coupled thermomechanical problem may be performed by adopting the staggered solution technique. Within a time step this technique leads to a decomposition of the coupled problem (compare with the statements on p. 331). As a result we must solve two smaller, in general., symmetric decoupled sub-problems on a staggered bas.is. For algorithmic aspects of the entropic theory of rubber thermoelasticity see, for example, M.IEHE [1995b] and HOLZAPFEL and SIMO [l996b] ..

ing .to eq. (6.137h (due to Ogden) and the energetic contribution e0 (J)/80 = 8 0 )1-1 ( J 1 - 1), with/ > 0, according to eq. (7.85) (due to Chadwick)~

Use (7 .88h, (7.9]) and the relation for the bulk modulus (7.83) in order to obtain

the constitutive equations

which completely determine the isothermal elasticity tensor Cval in the material description.

2. Consider the extension of the isothermal Ogden material to the non-isothermal domain (7 ..80)-(7.84) (including the quantity e0 (J) which represents an ener- getic (volumetric) contribution to the free energy). Recall the definition of the structural thermoelastic heating (or cooling) 1lc~ Le. eq. (7.44h, and show that

1le may be written in the decoupled structure of the form He = He vot + He iso, with the de-finitions

The response functions \JlvoJ o = Wvol (J, 80), Wisoo = '11-iso(/\1, .,,\2, /\3, 80) are given by (7.77) with (7.78).

The analogue of the decoupled structure of 1ie was derived by MIEHE fl 995b] .

.In his work the structural thermoelastic heating 1le is, however, based on the multiplicative split of the (spatial) left Cauchy-Green tensor b = FFT and is defined with the opposite sign.

7.8 Simple Tension of Entropic Elastic Materials

The aim of this section is to illustrate the ability and performance of Ogden's model for the non-isothermal do.main as outlined in the last section. We set up the basic equations required to describe the realistic physical stress-strain-temperature response of rubber- like materials. In particular, here we consider the simple tension of an entropic elastic

materiat a class of material defined in Section 7 .6. A re.presentative example conce.med with the adiabatic stretching of a rubber band will contribute to a deeper insight in the coupled thermomechanical phenomena.

Thermoelastic deformation. Before we start our studies with the simple tension of

entropic elastic mate.rials it is most beneficial to point out briefly some aspects of the therm-oelastk deformation of a continuum body.

Consider a fixed reference configuration of a body with the geometrical region 0 0 correspondin.g to a fixed reference time t = 0. The position of a typical point may

identified by the position vector X (with material coordinates {..Y1, .X2 ,

a fixed origin 0 (see Figure 7.5). The reference configuration is assumed to be

.free and possesses a homogeneous (uniform) reference temperature value

map of the reference configuration 0 0 to a current configuration (with the new

characterized by the spatial coordinates (:r 1, :1:2 , :r:3).

As a measure of the thermoelastic deformation we use the deformation gradient

The multiplicative decomposition (7.99) separates the total thermoelastic deformation into a purely mechanical contribution F ~·i, J~.1 and a purely thermal contribution Fe,

J8 , which represents the Duhamel-Neumann hypothesis for the nonlinear theory (see, for example, CARLSON [1972, p. 310]).

The two successive motions establish a new intermediate (i.magined)

tion with geometrical region Q 0 , as illustrated"in Figure 7.5. The new configuration is assumed to be isolated from the body so that a thennal stress-ji·ee deformation may occur. Hence, a relative temperature field .f) = 8 - 8 0 causes a (free) thermal expansion (or contraction) about the reference configuration 0 0 characterized by the associated variables Fe and J0 . The intermediate configuration with the region Q 8 is .given by the macroscopic motion Xe = which carries points X located at -00 to points

_............

F,J

x

Figure 7.5 Multiplicative decomposition of the thermoelastic deformation into a purely me.. clumical contribution Fr...i, JM = detFrv·r and a purely thermal contribution Fe, Je = detFe.

. .

Xe in the intermediate configuration He. A typical point Xe is characterized by the coordinates (..-Ye i, )(02, ..rYe3).

According to the multiplicative-split (7.99) we ·have defined, additionally, a macroscopic motion x = XM (Xe, t) at a constant (fixed) temperature e along with the stre~fs­ producing deformation gradient F:rvt and the volume change JM. A so-called mechani-

EXAMPLE 7.4 Consider a mechanically incompressible and thermoelastic material under a non-isothermal deformation process. The thermoelastic material is. assumed to be thermally :isotropic so that the deformation gradient F 9 may be given by an isotropic

346· 7 · Thermodynamics of Materials

where F(E>) is a scalar-valued scalar function determining the volume change relative to the reference configuration. In eq.(7.102h we have assumed a particularization of F(8), where o: = 0:(8).denotes the temperature dependent expansion coefficient.

Detennine an expression for the total volume change J of the ~aterial due to the non-isothermal deformation process. Linearize the result by assuming a constant value n0 = a(80) for the expansion coeffic.ient at a reference temperature 8 0 .

Solution. Since the considered thermoelastic material is mechanically incompressible (no volume change during an isothermal process) we introduce the constraint condition throu.gh J~.1 = L Thus, the total volume change J within a non-isothermal deformation process from region no ton is characterized by eq. (7.9.9h which degenerates7 using eqs. (7.101 h and (7 .102), to

It defines the total volume change J of a mechanically inc~mpressible and thermally.

.isotropic material within the infinitesimal strain theory. The approximate solution (7.105) is a well-known relation in linear continuum mechanics. It may be viewed as the volume of a unit cube .at temperature e with values .n0 and 8 0 which correspond to a reference state. •

In the following, attention will be ~onfined t~ the the~m.oelasti.c description of isotropic and entropic elastic materials (such as elastomers) at finite st.rains. The stress-de.formation-temperature response of a piece of rubber under simple tension, in particular, is examined. We assume that the material under consideration obeys the modified entropic theory of rubber thermoelasticity. .Furthermore, the material is assumed to be mechanically incompressible,

which .motivates the use of a multiplicative split of the thermoelastic deformation, ·as introduced above. Consequently, the total volume change is given in (7.103)i, i.e·..

J = Je(8) (detF1v1 = 1). ·

Now we consider a thin sheet of rubber stretched in one direction from the reference (undeformed) state to A.1 = .-\ (s.imple tension). Then, obeying condition J = J0 (8) ·:

A ,,\ A , we deduce by symmefry that ,,\ = ,\ = (J/A) 112 • Hence, the Helmholtz

1 2 3 2 1

free-energy function per unit reference volume

(7.106)

is given in terms of one independent mechanical variable and one thermal variable, i.e. the stretch ratio~\ and the temperature 8. For notational convenience we use the same Greek letter \JI for different free-energy functions.

Since the material is isotropic it is appropriate to use the thermodynamic extension of Ogden's ·model, .as discusse.d in Section 7.7 (see eqs. (7.80)-(7.84)).. Recall that, in general, one contribution to the free energy is due to volume changes and purely thermal causes (comp.are with relation (7.8.1 h). Since we study a mec·hanically incompressible material we need only to consider the purely thermal contribution. We use, without loss of generality, the standard form (7.73}, i.e. T (e) = Co [19 - eln (eI80)]' where the specific heat capacity c0 > 0 is a positive constant.

Hence, the free energy relative .to the reference configuration, which is stress-free and free of thermal expansion (or contraction), results from eqs. (7.80)-(7.82) and (7. 84), using the temperature change {) = e - 80 and the specified kinematic relations,

where the additional condition (7.79) must be enforced. Note that the volume change due to thermal expansion governed by .J = Je (8) is of considerable importance, which will be pointed out in more detail within the Example 7.5 below.

· ··.Next, we derive the associated thermal equations of state, namely the stress and eiuropy functions, as given in eqs. (7.24) and (7..25). For simple tension we may write

348 7 Thermodynamks of Materials

Relations (7..·109) and (7.110) specify the (non-zero) nominal stress P1 = P (also called the first Piola-Kirchhoff stress) P2 = P3 = 0, and the entropy 'I} whic.h are generated by the stretch ratio...-\ and the temperature 8.

In order to describe the coupled thermoinechanical problem of the piece of stretched rubber completely we must add a constitutive equation for the heat flux vector governing heat transfer. For the class of thermally .isotropic materials we may adopt Fourier's law of heat conduction and refer to eq. (7.97). Having in mind the specified kinematic relations, the inverse of the right Cauchy-Green tensor, c-1, as needed in eq. (7.97), may be given in the form of its matrix representation

(7.111)

where the diagonal elements are the eigenvalues of_ c-1 .

EXAMPLE 7.5 Consider the stretching of a mechanically incompressible piece of rubber, for example a rubber band, obeying the modified entropic theory of rubber thermoelasticity. The rubber band is elongated rapidly so that no ti-me remains for isothermal removal of heat. Hence, the homogeneous deformation process is viewed as an adiabatic process for which the heat flux on the boundary surface is zero and, .in addition, heat sources are zero (thermal energy cannot be generated or destroyed within the material). The non-isotherma] deformation process is assumed to be reversible.

.In the present example attention .is paid to the effects of structural thermoelastic heating (or cooling) and to the stress-strain-temperature response of the rubber band. In particular, show how the nominal stress P depends on the temperature change 19 = G - 8 0 at a fixed elon_gation, .i.e. a fixed stretch .-\,.and derive the temperature -evolution of the rubber band with stretch. Finally, discuss the results of this classical example of rubber thermoelasticity, which demonstrates one of the great differences betwe-en rubber and 'hard' solids, nam-ely the distinctive effects of temperature~

As known from Section 6.5 the Ogden model exce11ently rep.licates the finite extensibility dom_ain of rubber-like materials. Hence, as a basis for the con~_titutive model take the thermodynamic extension of Ogden's model with three pairs of constants (1V = 3) characterized by the Helmholtz free-energy function (7.107). The constants O:p, Jip(G0 ), p = 1, 2_, 3, at the reference temperature 8 0 are those given by OGDEN [1972a], listed in eq. (6.121) of this .text. The volume change due to thermal expansion