Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

Note that the symbols T .and t have already been used and must not be confused with the traction vectors.

We now express the symmetric spatial stress...temperature tensor t in terms of the latent heat v' as de-fined in eq. (7.43). Knowing the transformation (aw (F' e) I8F)T ::;: 2(8\J!(C, 8)/ BC)FT, which is in accord with eq. (6.11), we find from (7.51)2 that

or

Both tensor quantities T and t measure the change of the stress in a process in which

the temperature is raised by one unit keeping the deformation fixed.

~ Finally we ~-ompute the change .in the fu-~cti~n f~r...th~ entropy 'f} = 17-(C_, 8). ·with constitutive equation (7.24)2 and the equivalence \Jl(F, 8) = w(C, 8), we obtain

By .applying the de.finitions of the referential stress-temperature tensor T and the spe~. cific heat capacity at.constant deformation CF, the entropy change may be expressed as

.1nechanical and thermal variables and consider the existence of the internal-energy

.function per unh refe~e vohuri.e in the f~rm of -e = e(C, 17) (and equivalently e = e(F_, -rj)). ..

Theisen.tropic elasticity tensor in the material description or the referential ten- sor of isentropic elasticities, denoted by cise' .is derived from the internal energy e in the same way that the isothermal elasticity tensor is derived from the free energy W.

Hence, by analogy ·with the above, the change in Sis given by

which furnishes the desired expression .for .the isentropic elasticity tensor in the material description for which the .entropy .is held constant during the thermodynamic process.

This important relation expresses the isentropic elasticity tensor cisc in the material description as .a function of the free energy lp = w(C, 8) (compare with eqs. (7.49)2, (7.50h). Note thaUn terms· of the internal-energy function e = e(C, 17), relation (7..60)

Some.. numerical aspects.. The distinction between isothermal and isentropic -elas-

ticities plays a crucial role in the analysis of nonlinear numerical stability.

Coupled thermomechanical p(oblems in solid mechanics may be solved numerically within one .time step. leading· .to simultaneous (monolithic) solutions of all the fields involved in the problem which have the feature of a good stability characteristic. However, this approach leads to large non-symmetric systems which are inefficient to solve and are associated with high computational cost. This typ~ of fundamental numerical solution strategy goes back to NICKELL and SACKMAN. [1968] .and ODEN

[l 969, 1972].

Alternatively, the coupled system of nonUnear differential equations

using the classical staggered solution technique (or also known as the fractionalstep method or staggered method) {see, for example~ YANENKO I1971], MARCHUK

[1982, and references therein]). In this method, the key idea is to partition the monolithic system of equations into smaller (symmetric) sub-systems by making use of the physical meaning of the problem considered. Within the concept of a staggered solu-

tion techri.ique.ithe system can be solved sequentially with much lower computational cost. For each~sub-system ~~-~an apply existing algorithms and solution strategies.

The dassical (merely standard) staggered solution technique for a coupled thermomechanical problem in solid mechanics is. based on the solution of a mec·hanical

(isothermal) :pro~!~!Il at a.ft-red temperature of the system (elastodynamic phase), whiCh····

_.,,,.,...--.,, ... . ~'-··-...,--.-.~-~c-

involves the isothermal elasticity tensor (7.49), followed by the solution of .a heat con-

duction problem at a fixed configuration--- in the temperature form (7 .46),...,.This---Cliissf:..

.........,,

cal ·p~ition is referred to as th~ .isother.mal operator -spl~t, which was used within the context of .coupled thermomechanical problems (see, for example, ARGYRIS et al. [1979, .1981, 1982], MIEHE [1~8.8] and SIMO and MIEHE (1992]).. However, this type ·Of staggered soluti~n technique is assoc~~ted wit~ the crocia.I restriction of ·Conditional stability (see ARMERO and SIMO .[1992, 1993])~

It e·merges that an alternative .partition of a strongly coupled thermomechanical problem leads .to a so-called unconditionally stable (time-stepping) solution -tech- nique, characterized as indepe·ndent from the chosen time step. This technique allows

solutions in an efficient The analysis is based on the .solu-

332 7 Thermodynamics of Materials

tion of a l_!!~Ch!ln!.~12roblem .at a fi'ted entropy of the system {elastodynamic phase), which involves the isentrop.ic elasticity tensor (7 .56), followed by the solution of a heat--- conduction problem at a.fixed configuration (thermal phase) in the entropy,...,,.fOfm

(4.142).

This alternative methodology is referred to as the isentropic operator split when dissipative materials are "involved (damage, viscous or plastic effects may occur). Likewise for peifectly thermoelastic materials the split is referred to .as the adiabatic ope·r-

·ator ·split. Within this solution technique it is possible to show that a defined Lyapwzov functional for the coupfod system of evolution equations - regarded as the canoni-

cal free-energy .function for thermoelasticity, first introduced by DU.HEM [1911, Vol.2,

.........

pp. 220-23 .I], is decreasing along the ft.ows for each. of the two sub-problems involved. This approach was proposed by ARMERO and SIMO [1992] for linear thermoelasticity and by ARMERO .and SIMO [1993] for finite thermoplasticity. For a successful application to rubber the.rmoelasticity see the papers by MIEHE [1995b] and

HOLZAPFEL and SIMO [1996b].

It must be emphasized that this class of staggered solution technique can be applied not only to cou.pled therrnomechanical problems in solid mechanics but also .to coupled problems of, for example. fluid flow in a porous medium, magnetohydrodynamics in fluid mechanics or to stress-diffusion problems. All th.at must be·done is to replace the temperature 8 and the ·entropy ·11 by the associated field variables···of the COttpJed problem considered.

EXERCISES

1.Recall the spatial stress-temperature tensor (7.5 I). By means of eq. (3 .66) derive the alternative expression

Du{C,8)

where u denotes the symmetric Cauchy stress tensor. In addition, derive the spatia'l stress-entropy tensor which has the form tise = au{ c, 'T/) IOT/.

2.Suppose that a body admits the (spatial) left Cauchy-Green tensor b = FFT and the temperature e as .independent variables and consider a free-energy function in the form of w= w(b, 8). By analogy with (6.38) we may find the associated constitutive equation .in the form of T = 2b(Dw{b, 8)/8b), where r denotes the

symmetric Kirchhoff stress tensor (note that this type of constitutive equation

represents isotropic thermoelastic response on.ly).

with the definitions

of the (fourth-order) isothermal elasticity tensor c in the spatial description and the (second~order) spatial stress-temperature tensor t.

For explicit.derivations of eqs. (7.61) and (7.62) see the work of MIEHE [l995b]. Therein, the definitions of the tensor variables c and .t exdude the factor J.

7.6 Entropic Elastic Materials.

We consider so-cal.led entropic elastic materials, which have the property that the change in internal energy w~th deformation is. small or even zero (recall Section 7..l ).

The underlying concept of entropic elastkity· is used particularly for the .thermo-

mechan.ic.al description of rubber-like m..~~~r!H,h~.,,..~.uch.,.;as...,.~la~lOJn.~rn..,J~.ee, for exam-

r:~·8~~~;A~~~:~1[1~;~:~k~:~~7~1.,.B:~~:w~~i~~~~~~~;ifi~b~~LLER

From experimental observations it is known that the bulk modulus for rubbe.r-like materials considerably exceeds the shear modulus. For an ideal rubber the internal

energy e .is assumed to be a function of the temperature e alone, e = e(8), which

is a typi~-~i~h~r~~t~~i;tic nf i~comp~~~~ib1e materiafs· (see~ for ex.ample, TRELOAR

[1975, p. 34] and ANTHONY et al. [.1942] for more details). This assumption leads to the .purely (?.r. strictly) ..entro.pic theor! of.D~P.~~r .th~rmoela~t~~ity (for a theoretkal treatment see the work of CHADWICK [ 1.974]). Consequently, the change .in internal energy with de.formation .at a. given reference temperature 8 0 is constant. We assume that

Here and elsewhere the subs·cripi'{•}cr:charactetizes quantities at a reference tempera- ture 8 0 so that, for example, e0 (F) = e(F, Go).

Allerna~.i~~·~.~?....~.Jh.ermoelastjc material obeys the CQ.9dified entropic. theo~y if its

internal energy e .is expressible as the sum of e(8), already known from the purely

entropic theory, and the internal energy e0 ( J) = e( J, 8 0 ). The additional contribution en (J) to the internal energy depends on the ·deiorffia~a.o·n only through the volume ratio

J .at a given reference temperature 8 0 (for a theoretical treatment see the work of

CHADWICK [1974] and CHADWICK and CREASY [1984]). Consequently, we conclude that

334 7 Thermodynamics of Materials

Change in temperature, internal energy, entropy. For both the purely and the

modified entropic theories of rubber thermoelasticity we may conclude from eq. (7.42).1 that the specific heat capacity CF is a function of the temperature 8 only. Thus, we write

between a selected reference state with reference temperature 8 0 (the choice is quite arbitrary) and the current state with absolute tem_perature e._ Hence7 by means of (7.65) the change in internal energy for a process· is determined by integrating eq. (7.42)4 with

the

Recall from eqs. (7.64) and (7.63) that e0 (F) = e0 (J) for the ·modified entropic theory while e0 (F) is assumed to be zero for the purely en.tropic theory.

Similarly, by means of (7.65), the -entropy change si~ply results from (7.42)2 by

the integration

e=e "'

potential in the form of the uniquely defined free energy W may be expressed in terms of the internal -energy e0 , the entropy 1/o and an additional function T for the purely thermal contribution. Thus,

.···········................. e::::::eo·.""·····=······..···"'. .'"·'"'········•·.......···,,,,_,,,.......,,.~

Another common.ly used alternative form of the thermodynamic potential may be found by considering only state functions which are assumed to characterize the reference state at a given reference temperature 8 0 • These functions are interrefated by

Determine the change .in entropy for the material which ·changes its temperature by tJ = e - E>o.

2.Assume that a thermoelastic material obeys the pure·Iy -entropic theory {e0 (F) = 0). The change in the strain energy from the reference to the .current configuration

where the strain energy vanishes in the reference configuration according to ~greement (6.4). For convenience assume that the specific heat capacity is a constant c0 •

3. Consider a thermoelastic material with a _given .isothermal free energy '11 (F) and

0

a constant specific heat capacity c0 • Assume that the material obeys the purely entropic theory.

(a)Recall the physical -expression .(7.:24.h and show that the evolution (change) of the entropy ·17 is given by

(b)Consider an is.entropic process in which the -entropy possessed by the given thermoelastic material remains constant. Deduce that

. . e

co8 = Wo(F)- ,

8:- 0

which gives the same temperature evolution as in (7.75).. Interpret this result.

338 7 Thermodynamics of Materials

where the para.meter µo denotes the shear modulus in the reference configuration at 80.

The strain energy '11 vol ( J, 8 0 ) is associated with the volumetric elastic response and is, in general1 decomposed into .a bulk modulus Ii(8 0 ) at a fixed reference temperature

0 0 and a scalar-valued scalar function Q{J). One example for Q was introduced in eq. (6.137h. The strain energy Wiso{A1, A~h /\3 , 8 0 ), however, is associated with the isochoric elastic behavior in the space of principal directions of C, i.e. the ValanisLandel hypothesis {see VA LAN.IS and LAND EL [1967]). The parameters .Jl.p(H0 ) denote the (constant) shear moduli at 0 0 ·and lt11 are dimensionless constants, TJ = 1; .... , N.

Next, we derive a relatively simple but very efficient thennodynamic potential for rubber-like materials which is basically a thermodynamic extension of Ogden,s model. We employ the modified entropic theory with assumption (7.64). Applying the thermo- dynamic .potential .in the form of (7.72) and using the strain energy (7.76) together with relations (7.77) and (7 ..78), we obtain the non-isothermal free energy \JI for .isotropic

thermoelastic material response, i.e.

(7.80)

and on the condition (7.79). .In (7.81}1 the purely thermal contribution T(8) is given in eq. (7.70).

The first contribution \IIvot( J, 8) to the thermoefastic response defined by (7.8 l )i is due to volume changes and .purely thermal causes. The bulk modulus n.( e) in (7.81) 1 depends linearly on the absolute temperature 8 (see eq. (7 ..83)). Note that the energetic contribwion. e0 ( J) to the function \JI vor( J, 8) occurs only within a modified entropic theory. An empirical expression was proposed by CH~DWICK [ 1974] and has the form

This relationship is based upon experimental considerations., where ')' > 0 -is a positive non-dimensional parameter and the quantity n0 = n{00 ) denotes the so-called