Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

which is known as the Claus.-ius-Duhem inequality here presented in the spatial and material descriptions, respectively.. The Clausius-Duhem :inequality is widely used in modern thermodynamic research initiated by TRUESDELL and TOUPIN [1960] and first clearly studied in the work of COLEMAN and NO.LL [1963.].

In order to derive the local :form, which we only present in the material description, we convert the surface integral in -eq. (4.135) to a volume integral according to the divergence theorem (1 ..297).. By the .product rule (.1.287) we obtain

where Grade denotes the material gradient of the smooth temperature field 8.

By substituting eq. (4.136)2 back into (4.135) and noting that the reference volume ll .is arbitrary and .independent of time, the local.form of the Clausius-Duhem inequality in the material description reads, in symbolic and index notation,

An alternative local version results, for example, from (4.137) by elimination of the heat source R by means of eq. (4..128)~ leading to

The last term in eq. (4.1 J8) determines entropy production by conduction of heat.

Clausius-Planck inequality and heat conduction. .Based on physical observations heat flows from the warmer to the co.Ider region of a body (free :from sources of heat), not vice versa. Hence, entropy production by conduction of heat must be non-negative, i.e. -(1/8)Q ·Grade > 0 (see the last term in (4.138)). The spatial version of this condition reads -(1/B}q ·grade > 0, where grade de·notes the spatial gradient of the smooth temperature field e.

Knowing that 8 is non-negative, we have

170 4 ·Balance Principles .

which .is known as the classic.al heat conduction inequality, here presented in terms of spatial and material coordinatesT respectively. The heat conduction inequaHty expresses that heat does ·flow against a temperature gradient It imposes an essential restriction on the heat flux vector.

Obviously the Cauchy heat flux -q depends on the spatial temperature gradient. With restriction (4.l 39) we may deduce that q(grad8) · grade < 0 holds identically if

grade = o. Therefore, the dot product takes -on a local maximum at grade = o and its derivative must vanish. Consequently, q = o is implied, which means that there is no heat flux without a temperature gradient.

According to restriction (4.1.40), the Clausius-Duhem .inequality (4.138) leads to .an alternative strongerform of the second law of thermodynamics, often referred to as the

i.e.

or

with the inte~al diss~pation or local production of -entropy 'Dint > 0, which is required to be non-negative at any particle of .a body for all times.

The non-negative internal dissipation V1nt consists of three terms: the work conjugate pair P : F, i.e. the rate of internal mechanical work (or stress-power) per unit. reference volume, then the rate uf internal energy, e, and the absolute temperature ·multiplied by the rate of entropy, -e.,;. The internal .d.iss.ipation is zero for reversible processes while ~he inequality holds for .irreversible processes.

By means of the Clausius-Planck inequality (4.141 )" the local form of the balance

of energy (4.128) can be rephrased in the convenient entropy form

_•\. ..4

in which the local evolutia.n (change) of the entropy 11 appears explicitly.

A suitable constitutive assertion which relates the Cauchy heat ftux q = q(x, t) in x = x(X, t) to the spatial temperature gradient grad8 = grad8(x, t) .is furnished by

This classical (phenomenological) law is ·motivated by -experimental observations and is known .as Duhame-J's law of :heat conduction, here presented in the spatial descrip-- tion.

The symmetric second-order tensor "' denotes the -spatial therma:J -conduct,ivi-ty tensor; its components Kab = fb·ba are e.ither-cons-tants or functions of deformation and temperature. Substituting Duhamel's law of heat conduction (4.143) into restriction (4..l 39) we obtain K-grad8 · _grad8 > 0. This implies that"" is a positive semi-definite tensor at each x.

·EXAMPLE 4.6 Derive Duhamel's law of heat conduction in terms of material coordinates,

U ..·"\J)

where F- 1 is the inverse of the deformation gradient F according to eq. (2.40) and

and the material temperature gradients. By using the absolute temperature 8 instead of

<I> in relation (2.47) we find that

(4.145)

Subsequently, by recalling the Piola transformation (4.121 _) for the heat ·flux vector we obtain from e-q. (4.143)

(4.146) which after recasting gives the desired result {4J 44). Sinc.e ~o .is a positive semi-

definite tensor the imposed restriction on the inequality (4. .I 40) is satisfied. •

lf K is an isotropic tensor we say that the material is thermally isotropic (no preferred direction for the heat conduction). For such a type of material the -conductivity

where c- 1 = F- 1F-T characterizes the .inverse of the right Cauchy...;Green tensor C. The scalars k > 0 and k 0 = Jk > 0 denote coefficients -of thermal conductivity

(constants or, in general, deformation and temperature-dependent) and .are naturally associated with the current and reference configurations of a body, respectively. The conditions k > 0 and k 0 > -0 imply that heat .is conducted in the direction of decreasing temperature.

The relations (4.147) and (4.148) .are well-known as Fourfor's law of heat con- duction which are basically constitutive relations.

If the heat flux qn = -q·n across the surface.an of a certain region n (QN = -Q·N on 8f20 ), and the he.at source r in that region n (R in r20) vanish for all points of a body at ·each time, then a thermodynamic process is said to be adiabatic (coming from the Greek word a&u5f3o:roc;·, which means impassable). This definition is based on the work of TRUESDELL and TOUPIN [1960, Section 258]. However, in engineering thermodynamics an adiabatic process is frequently defined as a process which cannot involve any heat transfer, qn = 0 (or CJN = 0). This means that the heat source r (or R) need not be zero.

.In our case an adiabatic process is based on the condition that thermal energy can neither cross (enter or leave) the boundary surface nor be generated or destroyed within the body, so that the thermal power Q is zero (compare with eq. (4.118)). Additionally, under the assumption of relations (4.133) the rate of entropy input Q is also zero (compare with eq. (4.130)). Consequently, the second law of thermodynamics .in the form of (4.131) reads r = S > 0, wbich means that the total entropy S cannot decrease in an adiabatic process. Nevertheless, the entropy TJ at all points and times is not necessarily non-decreasing.

For an adiabatic process the energy balance equation, for example, in the entropy form (4.142), degenerates to

(4.149)

.If, in addition, an adiabatic process is .reversible (no entropy is produced er = 0), entailing that S is zero), the balance equation (4J 49) reduces further to

(4.150)

which bas· some important applications in thermodynamics.

If the absolute temperature e .during a .thermodynamic process remains constant

(8 = 0)~ the process is said to be isothermal and if the entropy .S possessed by a body

,.

remains ·constant (S = 0), the .process is said to be isentropic.

For an isentropic process the second law of thermodynam.ics (4.131) takes on the

If an is·entropic process is reversible, the rate of entropy input Q (and the thermal power Q via assumption (4.133)) is zero since there is no entropy production r anymore. For this special case the .equal signs in (4.15.1) hold. Consequently, it is im-

portant to .note that adiabatic processes and isentropic processes are identical for the case in which both are reversible.

.In discussing deformations of elastic materjals (see Chapter 6_) it is convenient to work with the strain-energy function \JI introduced in eq. (4..116)2. However, for the case in which W is us-ed within the thennodynamic regime incorporating thermal vari- ables such as 8 or 'f/, then wis commonly referred to as the ·uelmholtz free-energy

function or referred to briefly as the free energy.

.Next, we express the Helmholtz free-energy .function .in terms of the internal energy e and the entropy rJ. Having in mind this aim, we apply the Legendre transformation, generally defining a procedure which replaces one or some variables with the conjugate variables_, particularly used in analytical mechanics (see, for example, ABRAHAM and

Note that all three quantities W, e, J} are introduced so that they refer to a unit volume of the reference configuration.

By using the material time derivative of the free energy, i.e. W= Dw /Dt, we may

write .the Clausius-Planck inequality (4.141) in the convenient form

(4.153)

often employed when the absolute temperature is used as an independent variable. Note that for the case of a purely mechanical theory, that is, if thermal effects are

"ignored (8 and r1 are om.itted), inequality (4..15J) degenerates to

and the free energy \JI coine:ides with the internal energy e (see the Legendre trans- formation (4.] 52)). For a reversible .process for which the internal dissipation Vint is zero (no -entropy production, r = 0), we conclude .from (4J54) that the rate of internal mechanical work (or stress-power) Wint = P : :F per unit reference volume (see eq. (4.113)) equals w.

EXERCISES

J.Recall relation (4.145) and Piola transformation (4.121) for the heat flux vector. Show the equivalence of relations (4.139) and (4.140) with manipulations according to identity ( 1.81 ).

2.Starting from (4.132), show that the local spatial form of the second .law of thermodynamics is

(compare with Exercise 2 on p. 166).

3.Derive the local forms of the Clausius-Duhem inequality and the Clausius-Planck inequality in the spatial description associated with (4.137), (4.138) and (4."141.), (4..142), respectively. Note the fact that the .internal energy and the entropy possessed by a body were introduced per unit volume rather than per unit mass (see relations (4.91.) and (4.129)).

Hint: Use Reynold's transport theorem in the form of (4.26).

4.Show that if the absolute temperature E> is merely a function of time, the ClausiusDuhem inequality and the Claus.ius-Planck ine·quality coincide, leading to the local material fo.rm

4.7 Master Ba·Jance Principle

We recognize from previous sections that conservation of mass, the momentum balance principles and the two fundamental laws of thermodynamics are all of the same mathematical structure. To generalize these relations .is the objective of the following section.

Master balance principle .in the global form. The present status of a set of particles occupying an arbitrary region n of a continuum body B with boundary surface an at time t may be characterized by a tensor-valued function I(t) = J~ .fdv (compare w.ith eq. (4.22)). In the following let f = f (x, t) be a smooth spatial tensor field .per unit current volume of order -n. In particular, f may characterize some physical scalar, vector or tensor quantity such as density, linear and angular momenlllm, total energy and so forth.

A change of these quantities may now be expressed as the following master bal-

The first term on the right-hand side is the integral of the so-ca] led surface density ¢> = .cp(x, ·t, .n), which is a tensor nf order n. The surface density is defined per unit current are.a and distributed over the boundary surface an. Note that c/J depends not only on position x and time t, but also on the orientation of the infinitesimal spatial

EXE.RCISES

1.Find the Cauchy flux '11., as derived in (4.16.1 }, for each of the balance principles (see Table 4.1.).

2.Define the Piola-Kirchhoff tlux q,R = <I>R(X, t) so that the corresponding surface density per unit reference area is .PRN, where N is the unit outward normal to the boundary surface 800 • Find the maste.r balance principle in the global and local material .fonns equivalent to eqs. (4.155) and (4.161), respectively.