Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

sian statistical theory becomes increasingly inadequate for the finite strain domain.

.In order to account for the finite extensibility of chains some significant refinements must be taken into account (SPERLING [1992, Section 9.10]). The more accurate non- Gaussian statistical theory is required (see, for example, TRELOAR [1975, Chapter 6] and MARK and ERMAN [1988, Chapter 13]). Within the non-Gaussian statistical theory the finite extensibility of cha.ins is considered in the form of correction terms leading to a more realistic form of the distribution function which is valid over the whole range of r-values up to the maximum or fully extended length.

One example of such a refined theory is based on the Langevin distribution function. The exact treatment of the freely jointed chain is considered by KUHN and GRUN l.1942], JAMES and GUTH [1943], and summarized by FLORY [1953, Appendix B of Chapter X]. In this type =af refined theory the Gaussian distribution function is included as a special case.

Other examples are phenomenologically motivated and based on mathematical arguments (see the material models introduced .in Section 6.5).

EXAMPLE 7.2 At a given temperature E> we consider a thermodynamic process in a closed system within some closed time interval t E "[O, T]~ in which the values 0 and

T denQte the initial (reference) and the .final time, respectively. Assume that the closed

I .

system' is thermally isolated and conservative. During the thermodynamic process a unit cube of ideal (incompressible) rubber deforms homogeneously to a parallelepiped with sides of length /\i., .,\~ and /\:i = {A 1..\2)- 1, i.e. the principal stretches. The cross-

.linked network of the rubber cube consists of N Gaussian chains per unit volume..

Based on the Gaussian statistica"J theory find the heat .g·enerate·d (or destroyed) .and the induced total entropy change of the .rubber biock due to the homogeneous .de-

where K denotes the 'Kelvin temperature' and °C the 'Celsius temperature'. The term

·1·8in/r2out is equal to 1.

.Solution. Since rubber is incompressible, the change in internaJ energy, i.e. £, during deformation is zero. Hence, the first law of thermodynamics (4.. 122) reads

The .expressions for the thermal power Q(t) = .f~0Rdll (the system is thermally isolated, i.e. thermal energy ·can not enter Of leave .the boundary (QN = 0), no heat transfer) and the stress power Pint = D /Dt J~0 \lldll (the system is conservative) are adopted from (4. l 18) and (4.1.16), respectively. After integration over time interval t E (0, T] we find from the last equation (7.20), by means of (7 .18), the particularized

first law of thermodynamics, i.e·.

(7.21)

We used the fact that according to assumption (6.4) the strain-energy function wfor the

unit cube of rubber vanishes in the reference configuration (normalization condition). The term ./~='{'Rdt in relation (7 .21) represents the heat per unit reference volume

E (0., TJ (thermal work). All work done which appears

as the strain energy is transformed to heat.

The total entropy change of the network ~·77 induced by the thermodynamic process

is, in accord with (7..17), given by

By substituting the given values into -(7.21) and (7.22) we .find using Boltzmann's constant k = 1.38 · 10-23Nm/K that J Rdt = -[3.0 ·1021 1.38 -10-23 293~15 (4 + 9 +

1/36 - 3)]/2 = ·-60.85N/m2 • The negative sign means that energy in the form of .heat :is destroyed within the solid body. The entropy change gives 6.11 = -60..85/293.15 =

-0.208N/m2K which shows dearly that entropy is decreasing as the rubber block is deformed. II

·EXERCISES

1.A drunken man starts to walk on a -flat field at a ·starting point 0. He makes one step per second each step of length 0.5 m. The path of the walk of ·Course me- anders· randomly (the man is drunke·ri), which means that there is no correlation between the directions of successive steps.

By applying relation (7.3) compute the average distance r from point 0 he has

.moved after three minutes.

2,. A rubber band of .initial cross-sectional area Ao is applied to a mass m,. At a certain temperature 8 the mass causes a 200% increase in length.

Compute the number of chains JV per unit volume for the assumption that the material is modeled as neo-Hookean and rfiin = (Boltzmann's constant

k = 1.38 · 10-23 Nm/K).

3.Two rubber bands. A and B, with identical material compositions and length l are tied together at their ends. Then the assembled band is stretched up to a total

322 7 Thermodynamics of Materials

which holds at every point of the ·continuum body .and for all times. As usuat the subscripts in (7.23) indicate variables that are being held constant during the partial differentiation of lJI. For convenience, in the following we will sometimes omit the subscripts. The coupled equation (7.23h is known as the Gibbs relation for elastic solids (Gibbs postulated the equation only for the case of a fluid).

By comparing terms, we may evaluate physical expressions imposed on require-

which are the general forms of constitutive equations for the first Piola-Ki.rchhoff stress P and the entropy ·q describing thermoelastic materials. Note that for the case of any

:= const) the free-energy .function '11 .is identified with the isothermal strain-energy function (compare with eq. (6.1)). Consequently, tll(F) =

w(F, 8) le==r.onst· By (7.24), the stress and the entropy are determined by the free energy

'1!, which has the status of a potential for the stress, the entropy and their respective conjugate thermodynamic varjables.

From physical expressions (7.24) we deduce the stress and entropy functions de-

.·.··'·_.··.

Eqs. (7.24) and (7.25), .also known as thermal equations of s.tate, are crucial in spec- ifying material behavior and are complete1y .determined once the free energy \JI =

stress S follo~s from .relation (3.65h., by means of (7.24)l and the analogue of (6.1 l ), as

where D'll(C, B)/DC is a symmetric tensor. From (7.24):h using the relation (6.9)t, we, obtain the alternative constitutive e.qualion for the entropy, Le.

In order to describe the thermodynamic state of a system by an alternative thermodynamic potential we require that the entropy function (7..25)2 is uniquely invertible with respect toe for each fixed F, so that we have loc.al.ly the condition 817/88 # 0.

7.3 Thermodynamic .Potentials

We assume th.at the inversion of eq. (7.25h is given by 8 = 8(F, 17) and we postu-

late an associated thermodynamic .potential e, which is the internal-energy.function per unit reference volume introduced on p. 157. Knowing that the .internal energy is refated to the free energy \P through the Legendre transformation (4. l 52), we have the (canonical) representation

e = e(F1 ''I) = \JI (F,. 8(F, 1"/)) + 8(F, 17)'IJ , (7.28)

.

also known as the caloric equation of state. It is .an equation that determines the inter-

. .

nal energy as a function of the deformatio·n gradient .F and a non-mechanical variable, i.e. the entropy TJ.

With potential (7 .28) we may deduce fundamental physical expressions from the entropy principle based on the Clausius...Planck inequality (4.141). For clll .admissible thermoelastic processes the second law of thermodynam.ics reduces to the identity

Dint = P : F - e+ 8iJ = 0. Hence., the Gibbs r~lation is obtained by determin.ing the total rate of change of e = t!{F:, 17) and by use of the chain rule~ Thus,

for the first Piola-Kirchhoff stress P .and the temperature 8. Note that for the case of an isentropic process (17 = const) there exists a state function e, whose partial derivative with respect to F gives the corresponding first Piola-Kirchho.ff stress P.

Comparing the physical ·expression (6.l)t i.e. P = D\JJ(F)/DF, with eqs. (7.24) 1 and (7.30) 1 we recognize that the strain energy \JI = \Ji(F) serves as the free energy l{I (F) = \JI (F, 8) le=mmst or as the internal energy e(F) = e(F, 11) l1i=const.~ depending on the process considered, isothermal or isentropic.

Knowing that e = e(F, 11), we deduce from (7.30) the .stress and temperllturejimc- tions which depend on ·F and 11. We obtain the thermal equations of state in .the form

P = P(.F, r1) , e = 8(F, ·11) • (7.3.1)

The two thermodynamic potentials '11 and e .introduced are commonly applied in solid meclumics. They are suitable for modeling so-called tbe.rmoel.astic materials (no 'memory effects' occur).

For the sake ·of completeness two additional potentials .are reviewed briefly.. These are the Gibbs free energy (or in the literature sometimes called the Gibbs function or che.mical potential), denoted by fl (or sometimes in the literature by G), .and the

.in the material description). For a more specific constitutive assertion see, for example, the phenomenological Duhamel's law of heat conduction (4.144) on p. 1.69 (given in terms of .material coordinates). A material for which the heat flux Q = o .and the heat source R = 0 vanish foi: any point and time is known as an adiabatic material.

EXERCISES

1.Using constitutive equations (7.24), (7.30) and (7.36), (7.37) obtain four relations combining .P, 17, F, 6 in the forms

These identities are known as the the·rmodynamic Maxwell (or reciprocal) re- lations and are very valuable in thermodynamjc analysis.

2. Each of the four (most common) thermodynamic potentials introduced, i.e. '1t, e, g, h, is related to any other by a Legendre transformation.· Show tbat the identity

e ·- \JI - h + g = 0 ·

is satisfied.

7.4 Calorimetry

Two centuries ago calorimetry became a branch of experimental physics. In the present day calorimetry deals with both the 1.neasurement of the amount of heat generated (or destroyed) within a give·n body duri_~g a change of state, and with its formulation with.in the theory of c·ontinuum thermodynamics..

temperature of a unit volume of the body keeping the defonnationfu:ed.

The specific heat capacity eF = CF (F' e) > 0 for .all (F' e) is, in general, defined

3.26 7 Thermodynamics of M-nted~ds

ible material l* depends on the deformation gradient Fas well as on the temperature -E>. The positiveness of CF may be related to the stability of the material (see, for .example,

SILHAVY [1997, Section 173]).

Hence, the specific heat capacity at constant deformation cF may also be expressed through the internal energy e whicht in general, depends on F and 8.

Secondly, we introduce the latent heat whkh is denoted by the symmetric tensor v . .It .is a ~patial field defined to be

(7.43)

·Of

Note that the latent heat v is proportional to the mixed second derivative of the free energy ·\JI.

Structural (or We define the general relation for the structural thermoelastic heating (or cooling) 1ie as the double contraction of the

latent heat, as given in (7.43) 1, and the symmetric part -of the spatial velocity .gradient I = :FF-1, i.e. the rate of deformation tensor d. Thus, with property (] ..95)

The scalar quantity 1ie represen ·st e thermoelastic coupling effect This so~called Gougl~-J~ule effect occurs, for ex~mple., during an adi_abatic stretching of a rubber band which typically changes its temperature., as pointed out in Section 7. I. In some p.robfoms the thermoelastic coupling effect is neglected due to the fact that this change -of temperature is. s-malL . ..

For a thermoelastic process (Vint. = 0) the rate of change of the ·entropy, as derived in (4. l42)~ -may be written by m·eans of (7.25h and (7.24h and the chain ru:le as

with DivQ and R denoting the material divergence .of the Piola-Kirchhoff heat flux Q and the heat source per unit time and per unit reference volume, respectively. Hence, from (7.45) we obtain finally the (coupled) energy balance equation in tempe·rature

.i\.j\

On comparison with the associated energy balance equation .in entropy form, that is eq. (4.142), we recognize that the structu.ral thermoelastic heating (or cooling) 1le appears on]y explicitly in the temperature form (7.46).

Consider the case of a process during which DivQ vanishes. We deduce from (7.46)

which we may regard ns defining the theory of calorimetly. For a historical study see

TRUESDELL [1980, Chapter 2C]~

Within the theory of finite thennoelasticity, in general, we observe three different types ofthermo~echanical coupling effects, nam·e1y

(i)the influence of .a change in temperature on the stress (thermal stress),

(ii)structural thermoelastic" heating (or cooling) - Gough-Joule effect, eq. (7A4).,

.and

(iii) geometric coupling (influence of a change in deformation on heat conduction)

(see, for example, eq& (4.144).).

EXERCISE

] . By me.ans of physical expres.~ion (7.36h, show an aHernative version of the .en- ergy balance equation in temperature form (7.46), i.e.

where Cp = c11(P, e) > 0 de.notes the specific heat capacity at constant stress (:P .. canst), defined to be cp(P, 0) = -8(82g(P, E>)/8888)h,·In words: ~!pis the energy .required to produce unit increase in the temperature of a unit volume of the body keeping the stress flxed. Alternatively to eq. (7 .44):i the term 1l(~1• represents the structural thermoelastic heatfog (or cooling) which is defined to be

11.cr == -e(a2!J(P, 8)/DP88): i>.

7.5 Isothermal, Isentropic Elasticity Tensors

The following presentation of the isothennal and isentropic elasticity tensors is based on the concept introduced in Section 6~6. It .is an extension of the purely mechanical framework to thermodynamics by one thermal variable, i.e. the temperature e or its conjugate quantity, the entropy 'fJ.

Isothermal elasticity tensor and stress-temperature t--:n~or. Suppose that a body

admits the right Cauchy-Green tensorC = FTF and the temp~raiiii·e 8~Jndepend.eot

~ec_!!anica1 and ..1hermaf:·variab1e·s anci. suppose the ·existence or the.fi.e1mh0Itz free- energy func.tion in the form of \JI = \Jl(C, 8) ·(and equivalently \JI= w{F, 8)).

Then, according to relation (7.2<5)a, we can find the second .Piola-Kirchhoff stress tensor Sofa point at a certain ti-met, which may be seen as a nonlinear tensor-valued tensor function of the two variables C and 8. By analogy with Section 6.6, we now compute the change .in S. According to-considerations (1.247) and (l.248), we obtain

which gives expressions for a·purely _mechanical part, C, and a (mixed} me-chanical- thermal part, T, in the material description. In the first term .in (7.48) we have duced the definition of the fourth-order tensor C, which is proportional to the second partial derivative of Wwith respect to C. By analogy with eqs. .(6.154) and (6.1.57) we write

or

evaluated at (C,·8),with the major symmetries C =<CT or CABCJJ = CcDA.B· Here we call C .the isother.mal elasticity tensor in the material description or the re.fer~

ential tensor of isothermal elast.ici.ties, which is defined by keeping the temp·erature jixecrdiii'ifljfffiei}~iicess. The isotherm~ elasticity tensor in the spatial description

c = J-1x*(9_j9{ the spatial tensor.of isothermal elasticities) is defined as a push-

Jorward-:={ana·--Piola) transformation of <C·on -each large index by analogy with relation

(6.159).

The second terin in eq. (7.48) is the referential___ stress-temperature tensor or the

__,.._ '""'""'----·~~-.-.,..,.-·--:-·-~--·-···· .

referential t~ermal -~oefficfon~ of stress, denoted by T, which is proportional to the

:mixed second ·p~ial derivative of W with respect to C and e. It is a symmetric second- order tensor defined as

The spatial counterpart, denoted by- t," results via a standard push-forward (and Piola) transformation t = .1-1x* (Ttt) of the (contravariant) referential stress-temperature