Objective rate equations and memory properties in continuum physics

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Objective rate equations and memory properties in continuum physics A. Morroa ,∗, C. Giorgib a

b

DIBRIS, University of Genoa, 16145 Genoa, Italy DICATAM, University of Brescia, Section of Mathematics, 25133 Brescia, Italy

Received 24 April 2019; received in revised form 25 July 2019; accepted 27 July 2019 Available online xxxx

Abstract The paper deals with the modelling of material behaviours in continuum physics by means of rate equations. The research has a twofold purpose. First, to review the structure of objective time derivatives, namely invariant derivatives within the set of Euclidean transformations; known derivatives occurring in the literature are shown to be particular cases of the whole family of objective time derivatives. Second, to investigate the thermodynamic consistency of some models involving objective time derivatives. In particular, two topics are developed. One is the improvement of the constitutive equation of viscous fluids. The other topic is a possible rate equation for the stress. The thermodynamic consistency is shown in connection with the co-rotational derivative. c 2019 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights ⃝ reserved. Keywords: Rate equations; Objective derivatives; Thermodynamic consistency

1. Introduction Rate (or evolution) equations are commonly used for the modelling of material behaviours such as memory and/or delay effects. Such equations are usually expressed by relating the time derivative of proper fields to a set of state variables thus resulting in differential equations in time. In the mechanical literature, rheological models motivate rate equations for the stress tensor T in terms of the infinitesimal strain E or the stretching D in the form [7] ˙ + T = kE, τT

(1)

˙ + T = 2µ(D + ξ D), ˙ τT

(2)

the superposed dot denoting the total (or material) time derivative, these equations being referred to as Maxwell model and Jeffreys model. In nonequilibrium thermodynamics the evolution of appropriate fields is often governed by rate equations. Perhaps the best known rate equation in thermodynamics is the Maxwell–Cattaneo equation for the heat flux q [16], τ q˙ + q = −κg,

(3)

∗ Corresponding author.

E-mail address: [email protected] (A. Morro). https://doi.org/10.1016/j.matcom.2019.07.014 c 2019 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights 0378-4754/⃝ reserved.

Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.

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where g is the temperature gradient. Within the rheological modelling, the scalars k and 2µ are the modulus of the spring and the viscosity of the dashpot. In Eq. (1) k denotes the elasticity, in Eq. (2) µ and 2µξ denote the viscosity coefficients. Also κ in (3) is the heat conductivity. In all cases τ is a (positive) relaxation time. As any constitutive equation, rate equations are required to comply with the principles of continuum physics chiefly the objectivity and the second law of thermodynamics. In this sense we need to clarify the general structure of objective time derivatives and to establish the thermodynamic consistency of the whole set of constitutive equations. Accordingly, the purpose of this paper is twofold. First, to review the structure of objective time derivatives, namely invariant derivatives within the set of Euclidean transformations. Upon establishing the general structure, it follows that known derivatives occurring in the literature, are particular cases of the whole family of objective time derivatives. Second, to investigate the thermodynamic consistency of some models involving objective time derivatives. While the influence of the rate equation for the heat flux, on the constitutive equation for the stress, has been investigated in [10], here attention is addressed to two topics. One is the improvement of the constitutive equation of viscous fluids by adding the contribution of the derivative of D. While the literature has investigated the use of the Cotter–Rivlin derivative for incompressible fluids, here the analysis is developed with both the Cotter– Rivlin and the Oldroyd derivatives for compressible fluids. The other topic is a possible rate equation for the stress. The thermodynamic consistency is shown in connection with the co-rotational derivative. Notation and assumptions. We consider a body occupying the time dependent region Rt ⊂ E 3 . The motion is described by means of the function χ (X, t) providing the position vector x ∈ Rt in terms of the position vector X in a reference configuration R, and the time t, so that Rt = χ (R, t). The deformation is described by means of the deformation gradient F = ∂X χ or, in suffix notation, Fi K = ∂ X K χi . The symbol ∇ denotes the gradient in the current configuration Rt . Throughout v is the velocity, T is the Cauchy stress, L is the velocity gradient, q is the heat flux vector, r is the (external) heat supply, ρ is the mass density, and η is the entropy density. 2. Objectivity It is a fundamental principle of classical physics that material properties are independent of the frame of reference or observer ([17], §19). Yet we need to specify the class of frames under consideration. As we see in a moment the class is different according as we refer to the motion of the material or that of the observer. In the literature, objectivity and material frame-indifference are often regarded as synonyms. Earlier Noll [15] used the term principle of objectivity to express that processes related by a change of frame must be compatible with the same constitutive equation. Accordingly, the material properties of a body should not depend on the observer, no matter how he moves. More frequently nowadays this statement is referred to as principle of material frame-indifference [6,17]. Sometimes distinction is made among the two concepts (see, e.g., [13,14]). The balance equations are invariant relative to Galilean frames and then the material properties should be the same in all of the set of Galilean frames. Hence one might expect that, e.g., the stress T and heat flux q are frame independent within the set of Galilean frames but T and q might be frame dependent relative to the set of Euclidean (non-inertial) frames. This point is investigated in [11] within a kinetic theory approach and the conclusion is that stress and heat flux are frame dependent because the corresponding relations contain the spin tensor. Accordingly, it seems natural to assume that material properties are independent of the frame of reference within the set of Galilean frames and to view this assertion as the content of the principle of material frame-indifference. In this sense the material frame-indifference is related to the material properties arising from dynamic processes. Instead we confine objectivity to the constitutive equations viewed as the mathematical description of material behaviour. As the statement of the principle of objectivity we assume that the constitutive equations are form-invariant within the set of Euclidean frames. We now give a mathematical form to the content of objectivity. Let F, F ∗ be two frames of reference. The position vectors x and x∗ of a point, relative to F and F ∗ , are related by the Euclidean transformation (see [17], §19) x∗ = c(t) + Q(t)x,

(4)

where c(t) is an arbitrary vector-valued function of t ∈ R while Q(t) is a proper orthogonal tensor function, det Q = 1; for formal simplicity we let t ∗ = t. Under the Euclidean transformation F → F ∗ the vector u and the tensors M, K are transformed into u∗ = Qu,

M∗ = QMQT ,

K∗ = QKQT ,

(5)

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ˆ while scalars, say φ, remain unchanged, φ ∗ = φ. If M(φ, u, K) is the constitutive equation for T then the objectivity requires that ˆ ˆ ˆ ∗ , u∗ , K∗ ), M∗ = QM(φ, u, K)QT = M(φ, Qu, QKQT ) = M(φ

(6)

and the like for vector functions, for any proper orthogonal tensor function Q(t). If ψ is a scalar then the requirement is the invariance ˆ ∗ , u∗ , K∗ ) = ψ(φ, ˆ ψ ∗ = ψ(φ u, K) = ψ.

(7)

Sometimes vectors and tensors satisfying (5) are said to be objective (see, e.g., [12], §1.4). In this sense, in this paper vectors and tensors stand for objective vectors and objective tensors. 2.1. Non-objectivity of the standard time derivatives It is a crucial point, within the formulation of objective constitutive equations, that the standard time derivative is not objective. For any function f (X, t), where X ∈ R and t ∈ R, we let f˙ = ∂t f (X, t).

(8)

If, instead, f = f˜(x, t), x ∈ Rt , then f˙ = ∂t f˜ + v · ∇ f˜.

(9)

Accordingly, throughout a superposed dot denotes the material time derivative, that is the derivative with X ∈ R fixed. Also, a superposed dot denotes the time derivative of a function only of t, as is the case for Q. We denote by Ω the spin tensor associated with Q, ˙ T = −ΩT . Ω := QQ

(10)

˜ t). Hence u∗ (x, t) = Qu(x, ˜ t) and Let u be a vector field given by the spatial description u = u(x, ˙ + Q∂ u˜ + Q(v · ∇)u˜ ˜ + (v · ∇)Qu = Qu u˙∗ = ∂ (Qu) t

t

whence ˙ + Qu. ˙ u˙∗ = Qu

(11)

(12)

Likewise, for the partial time derivative ∂t we have ˙ + Q∂t u. ∂t u∗ = Qu

(13)

Hence, though u is a (objective) vector both u˙ and ∂t u are not (objective) vectors. The same assertion holds for any time derivative ∂t u + (w · ∇)u,

(14)

where w is any velocity, not necessarily the velocity of the point of the continuum. Hence we say that any time derivative (14) is not objective. The same assertion is found to hold if the derivative is applied to tensors. Since time derivatives are not objective then objective evolution equations require generalized time derivatives. For later convenience the next section reviews the approach of [9] and shows how a general class of objective time derivatives can be established. 3. A general class of objective time derivatives A derivation [1,8] ∂ of a vector algebra V, over R, is a rule ∂ : V → V such that, for every u, w ∈ V and α, β ∈ R, ∂(αu + βw) = α∂u + β∂w,

∂(u ⊗ w) = (∂u) ⊗ w + u ⊗ (∂w).

(15)

The conditions (15) are referred to as the linearity and the Leibnitz rule. Owing to linearity, a derivation ∂ such that ∂u has the meaning of a time derivative is considered in the form ∂u = u˙ − Λu

(16)

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where Λ : V → V is a function on R × R. Hence ˙ − Λw). ∂(u ⊗ w) = (u˙ − Λu) ⊗ w + u ⊗ (w

(17)

A derivative ∂u of any vector u is objective if, under the Euclidean transformation (4), it is (∂u)∗ = ∂u∗

(18)

namely Q∂u = ∂(Qu).

(19)

Let Λ∗ be the transform of Λ under the change of frame F → F ∗ given by (4). The following statement characterizes the set of operators Λ making (16) an objective derivative. Proposition 1. The derivative (16) of any vector u is objective if and only if Λ∗ = QΛQT + Ω.

(20)

Proof. Since ∂u = u˙ − Λu then the objectivity requirement (18) becomes ˙ − Λ∗ Qu = Q(u˙ − Λu). Qu

(21)

Hence we have ˙ − Λ∗ Qu = −QΛu. Qu

(22)

The arbitrariness of u implies that ˙ Λ∗ Q = QΛ + Q.

(23) T

Right multiplication by Q gives the conclusion. Conversely, if Λ∗ = QΛQT + Ω then ˙ − QΛu − ΩQu = Q(u˙ − Λu), u˙∗ − Λ∗ u∗ = Qu

(24)

that is ∂u∗ = (∂u)∗ . □ By a change of frame a dyadic product u ⊗ w and a tensor K obey the transformation law (u ⊗ w)∗ = Q(u ⊗ w)QT ,

K∗ = QKQT .

(25)

To determine the transformation law of the derivative of a tensor we now consider the derivative of a dyadic product u ⊗ w. The objectivity requirement is [∂(u ⊗ w)]∗ = ∂(u ⊗ w)∗ = ∂(u∗ ⊗ w∗ ).

(26)

We now verify that if (20) holds then ∂(u ⊗ w) is objective. Upon substitution we have ∂(u∗ ⊗ w∗ ) = ∂u∗ ⊗ w∗ + u∗ ⊗ ∂w∗ = (u˙∗ − Λ∗ u∗ ) ⊗ w∗ + u∗ ⊗ (w˙∗ − Λ∗ w∗ ) ˙ + Qu˙ − (QΛQT + Ω)Qu] ⊗ Qw + Qu ⊗ [Qw ˙ + Qw ˙ − (QΛQT + Ω)Qw]. = [Qu

(27)

˙ = ΩQ and Qw = wQT then it follows Since Q ∂(u∗ ⊗ w∗ ) = Q[∂u ⊗ w + u ⊗ ∂w]QT = Q(∂u ⊗ w)QT = [∂(u ⊗ w)]∗

(28)

and the objectivity of ∂(u ⊗ w) is proved. The derivative of a dyadic product yields ˙ − Λw) = u ⊗˙ w − Λ(u ⊗ w) − (u ⊗ w)ΛT . ∂(u ⊗ w) = (u˙ − Λu) ⊗ w + u ⊗ (w

(29)

Hence we define the derivative of a tensor K in the form ˙ − ΛK − KΛT . ∂K := K

(30)

Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.

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By definition, the derivative ∂K is objective if (∂K)∗ = ∂K∗ .

(31)

Again we show that if Λ satisfies (20) then the derivative ∂K is objective. By direct substitutions we find ∂(K∗ ) = K˙∗ − Λ∗ K∗ − K∗ Λ∗T ˙ = QKQT − (QΛQT + Ω)QKQT − QKQT (QΛQT + Ω)T ˙ − ΛK − KΛT )QT = (∂K)∗ = Q(K

(32)

and the conclusion follows. Based on the transformation property (20) we observe that the operator Λ is non-unique as is shown by the following ˜ satisfies (20) and B is any tensor, namely Proposition 2. If Λ ˜ ∗ = QΛQ ˜ T + Ω, Λ

B∗ = QBQT ,

(33)

˜ + B satisfies (20). then Λ = Λ Proof. A direct substitution gives ˜ ∗ + B∗ = QΛQ ˜ T + Ω + QBQT = Q(Λ ˜ + B)QT + Ω = QΛQT + Ω, Λ∗ = Λ

(34)

which proves the assertion. □ Proposition 2 characterizes classes of objective time derivatives. Given an operator Λ satisfying (20), the class of operators {Λ + B}, B belonging to the set of (second-order) tensors, generates objective time derivatives. The derivative of vectors and tensors need not be the same in that ∂u and ∂K may involve different operators Λ. What is more, the derivative of a tensor ∂K may involve two different operators Λ. To see this, let Λ1 and Λ2 satisfy (20) and consider the derivative ˙ − Λ1 K − KΛ2T . ∂K = K

(35)

˙ (∂K∗ ) = QKQT − (QΛ1 QT + Ω)QKQT − QKQT (QΛ2 QT + Ω)T ˙ − Λ1 K − KΛ2T )QT = (∂K)∗ = Q(K

(36)

Now,

and hence the derivative (35) is objective. It is worth remarking that if Λ1 and Λ2 satisfy (20) then Λ = Λ1 + Λ2 does not satisfy (20). This is so because (Λ1 + Λ2 )∗ = Λ∗1 + Λ∗2 = QΛ1 QT + Ω + QΛ1 QT + Ω = Q(Λ1 + Λ2 )QT + 2Ω.

(37)

4. Objective derivatives in continuum physics It is natural to ask whether operators satisfying (20) occur in (continuum) physics. In this connection we consider (4); upon time differentiation we have ˙ + Q˙x = c˙ + Qx ˙ + Qv, (38) v∗ = x˙∗ = c˙ + Qx v and v∗ being the velocities of the point relative to F and F ∗ . By (38) we have the function v(x∗ (x, t), t), ˙ v(x∗ (x, t), t) = c˙ (t) + Q(t)x + Q(t)v(x, t).

(39)

Since x∗ (x, t) is given by (4) and ∂x j xi∗ = Q i j

(40)

then it follows that ∂xi∗ v ∗p Q i j = Q˙ pj + Q ph ∂x j vh ,

(41)

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whence L∗ = QLQT + Ω.

(42)

Accordingly, the velocity gradient L obeys the transformation law (20). Indeed, since L = W + D, where W ∈ Skw, D ∈ Sym, then W∗ + D∗ = QWQT + QDQT + Ω

(43)

whence W∗ = QWQT + Ω,

D∗ = QDQT .

(44)

The spin W satisfies (20) whereas D is a tensor. Another operator obeying (20) arises by arguing as follows. By the polar decomposition theorem, the deformation gradient F is factorized as F = RU,

(45)

where U ∈ Sym while R is a rotation. Since F∗ = QF and U is invariant then R∗ U = F∗ = QF = (QR)U.

(46)

As a consequence R∗ = QR.

(47)

Let ˙ T ∈ Skw. Z = RR

(48)

It follows that ˙ T ˙ + QR)R ˙ T QT Z∗ = QR(QR) = (QR T T T ˙ ˙ Q = QZQT + Ω. = QQ + QRR

(49)

Accordingly Z satisfies (20). Since D is a tensor and tr D = ∇ · v is invariant then infinitely many operators Λ occur in the forms Λ = W + λD + ν(∇ · v)1

(50)

Λ = Z + λD + ν(∇ · v)1,

(51)

and for any pair of scalar invariants λ, ν ∈ R, all of them being related to the velocity field of the continuum. Moreover, since W ̸= Z then (50) and (51) represent two distinct operators. By means of the representations (50) and (51) we can now characterize some well-known (objective) time derivatives occurring in the literature of continuum physics. They are given in the following table. Derivative of vectors

Derivative of tensors ◦

Jaumann–Zaremba

u = u˙ − Wu

◦

˙ − WK − KWT K =K

Cotter–Rivlin

u = u˙ + LT u

△

˙ + LT K + KL K =K

Oldroyd

u = u˙ − Lu

▽

˙ − LK − KLT K =K

Truesdell

u = u˙ − Lu + (∇ · v)u

□

˙ − LK − KLT + (∇ · v)K K =K

⊙

˙ − ZK − KZT K =K

Green–Naghdi

u = u˙ − Zu

△ ▽ □

⊙

In the Jaumann–Zaremba derivative, also called co-rotational derivative, Λ = W. In the Cotter–Rivlin derivative Λ = W − D while in the Oldroyd derivative Λ = W + D, that is λ = −1, 1, respectively. In the Green–Naghdi derivative Λ = Z. Curiously, in the Truesdell derivative Λ = L − (∇ · v)1 for vectors and Λ = L − 21 (∇ · v)1 for tensors. Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.

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5. Objective derivatives and fluids of higher-order grade There are many substances which are capable of flowing but which exhibit flow characteristics that cannot be adequately described by the classical linearly viscous fluid model. To improve the Navier–Stokes model, a rough ˙ of the stretching D. However, D ˙ is not objective idea would be to let the stress depend also on the time derivative D ˙ in the stress constitutive equation. As a way to overcome and hence we cannot merely add a term proportional to D this difficulty material models have been developed that are commonly referred to as fluids of differential type or Rivlin–Ericksen fluids (see [5] and refs therein). The Rivlin–Ericksen kinematical tensors {An } are defined recursively through △

A1 = 2D,

˙ n + LT An + An L, An+1 = An = A

n ∈ N.

(52)

In words, An+1 is the Cotter–Rivlin derivative of An . Likewise we might define any family of kinematical tensors by replacing the Cotter–Rivlin derivative with any objective time derivative. For definiteness, by means of the Oldroyd derivative we define the tensors {An } in the form ▽

˙ n − LAn − An LT , An+1 = An = A

A1 = 2D,

n ∈ N.

(53)

By means of the co-rotational derivative we can write ◦

◦

An+1 = An − DAn − An D.

An+1 = An + DAn + An D,

(54)

This shows that while, in both families of tensors, the recursive definition involves the co-rotational derivative, a different sign occurs in front of DAn + An D and DAn + An D. Much research has been addressed to the restrictions on the constitutive equation T = − p1 + µA1 + α1 A2 + α2 A21

(55)

for the stress tensor [5], relative to incompressible fluids. For compressible fluids we simply add to (55) a term accounting for bulk viscosity so that T = − p1 + µA1 + 12 λ(tr A1 )1 + α1 A2 + α2 A21 .

(56)

It seems of interest to contrast the restrictions on (56) with those on the analogous relation, T = − p1 + µA1 + 21 λ(tr A1 )1 + β1 A2 + β2 A21 .

(57)

To this end it is convenient to express A1 , A2 and A1 , A2 in terms of the stretching tensor D. Now, A1 = A1 = 2D while ◦

A2 = 2D + 4D2 ,

◦

A2 = 2D − 4D2 .

(58)

To distinguish the two cases denote by Tα and Tβ the stress as given by (56) and (57). It follows that ◦

Tα = − p1 + 2µD + λ(tr D)1 + 2α1 D + 4(α1 + α2 )D2 ,

(59)

and ◦

Tβ = − p1 + 2µD + λ(tr D)1 + 2β1 D + 4(β2 − β1 )D2 .

(60)

In so doing we let the fluid be compressible whereas the literature restricts attention to incompressible fluids (see [5]). Let ρ be the mass density, θ the (absolute) temperature, and g the temperature gradient. Owing to the form of Tα and Tβ , a thermodynamic scheme for the corresponding viscous and heat-conducting fluid should involve ◦

Γ = (ρ, θ, D, D, g)

(61)

as the set of independent variables. Let ψ be the free energy and η the entropy, per unit mass. The second law inequality takes the standard form 1 ˙ + T · D − q · g ≥ 0. −ρ(ψ˙ + ηθ) (62) θ Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.

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We let η, T, q be continuous functions of Γ . Moreover we assume ψ is differentiable and p is independent of g. Further, we assume µ, λ, α1 , α2 , β1 , β2 be functions of ρ and θ . Let T = Tα . Upon evaluation of ψ˙ and substitution we find that ◦

˙ − ρ∂ ◦ ψ · (D)˙− ρ∂g ψ · g˙ −ρ(∂θ ψ + η)θ˙ + (ρ 2 ∂ρ ψ − p)tr D − ρ∂D ψ · D D

◦

+ 2µD · D + λ(tr D)2 + 2α1 D · D + 4(α1 + α2 )(DD) · D −

1 q · g ≥ 0. θ

(63)

◦

The linearity and the arbitrariness of (D)˙, g˙ , θ˙ imply that ∂ ◦ ψ = 0,

∂g ψ = 0,

D

η = −∂θ ψ,

(64)

ψ then being a function of ρ, θ, and D. Now, since ◦

˙ · D + (DW − WD) · D = D ˙ ·D D·D=D

(65)

then inequality (63) reduces to ˙ + 2µD · D + λ(tr D)2 + 4(α1 + α2 )(DD) · D − 1 q · g ≥ 0. (ρ 2 ∂ρ ψ − p)tr D + (2α1 D − ρ∂D ψ) · D θ 2 First let g = 0. The linear dependence on tr D implies that ρ ∂ρ ψ − p has to vanish and hence p = ρ 2 ∂ρ ψ.

(66)

(67)

˙ imply The linearity and the arbitrariness of D ρ∂D ψ = 2α1 D.

(68)

The third-order term (DD)·D is predominant as |D| → ∞ and has no definite sign. As a consequence the inequality requires that α1 + α2 = 0.

(69)

Hence we are left with the classical quadratic form of linear viscosity, 2µD · D + λ(tr D)2 which is non-negative if and only if µ ≥ 0,

2µ + 3λ ≥ 0.

(70)

The quantity −q · g/θ is the only term dependent on g. The inequality then requires the classical heat conduction inequality q · g ≤ 0.

(71)

The requirement (68) in turn implies that ψ(ρ, θ, D) = α1 D · D + Φ(ρ, θ).

(72)

No thermodynamic restriction follows on α1 unless we let α1 ≥ 0

(73)

so that ψ is minimal at D = 0 or in view of stability arguments [4]. Accordingly, we have µ ≥ 0,

α1 ≥ 0,

α2 = −α1 .

(74)

The thermodynamic restrictions on Tβ can be derived in the same way. It follows in particular that µ ≥ 0,

β1 ≥ 0,

β2 = β1 .

(75)

Some comments are in order about (74) and (75). The same result on the sign of α1 and β1 is due to the fact △

▽

◦

that both derivatives D and D embody the co-rotational derivative D. The occurrence of both α1 A21 and β1 A21 is required to avoid the trivial case α1 = 0, β1 = 0. Further, the selection of a different derivative leads to a different sign for α2 and β2 , i.e. α2 ≤ 0, β2 ≥ 0. Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.

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6. Objective rate equations in solids The selection of the objective time derivative is a problem of interest in modelling constitutive rate equations where the response function is given in fact by a differential equation. This topic is investigated in [9,10], in connection with the constitutive equation for the heat flux. It emerges that the selection of the time derivative for the heat flux affects the constitutive equation of the stress tensor. Rheological models of solids are prototypes of rate equations. In particular, the classical Maxwell model and Wiechert model are given in the forms τ T˙ + T = kE,

(76)

˙ τ T˙ + T = k1 E + k2 E.

(77)

We now look for objective rate equations. We let F, θ, D, g, T be the set of independent variables, T being an appropriate tensor to be identified and characterized by a rate equation. The invariance of ψ(F, θ, D, g, T ) under SO(3) requires that the free energy ψ depends on invariants of F, θ, D, g, T . Let η, T, q be continuous functions of F, θ, D, g, T . Upon evaluation of ψ˙ and substitution in (62) we find ˙ − ρ∂g ψ · g˙ − ρ∂T ψ · T˙ + T · D − 1 q · g ≥ 0. −ρ(∂θ ψ + η)θ˙ − ρ(∂F ψFT ) · L − ρ∂D ψ · D (78) θ ˙ g˙ , and θ˙ imply The arbitrariness and the linearity of D, ∂D ψ = 0,

∂g ψ = 0,

η = −∂θ ψ,

(79)

ψ then being a function of F, θ, T . Though L = D + W, at this stage we cannot exploit the arbitrariness of W because, as we see in a moment, T˙ too might involve a dependence on W. We assume ∂F ψFT ∈ Sym. This is so if ψ depends on F via C = FT F or B = FFT or their invariants. To show this property we observe, in suffix notation, ∂ F p Q C H K = F pK δ H Q + F p H δ K Q ,

∂ F p Q Bi j = δi p F j Q + Fi Q δ j p ,

(80)

whence (∂F ψ FT ) pr = ∂ F p Q ψ Fr Q = 2(F ∂C ψ FT ) pr ,

(81)

(∂F ψ FT ) pr = ∂ Bi j ψ∂ F p Q Bi j Fr Q = ∂ B pj ψ B jr + ∂ Bi p ψ Bir = 2[sym(∂B ψ B)] pr ,

(82)

and the symmetry is apparent. Hence the reduced inequality becomes 1 q · g ≥ 0. (83) θ Further restrictions follow by specifying the (constitutive) rate equation for T . First we identify T by letting [T − ρ∂F ψFT ] · D − ρ∂T ψ · T˙ −

T := T − ρ∂F ψFT ∈ Sym.

(84)

Accordingly T = ρ∂F ψFT + T .

(85)

We then view T as the sum of an elastic part, ρ∂F ψF , and a dissipative part, T . The simplest rate-type model for T is based on the equation T

◦

τ T + T = 2µD, ◦

τ > 0,

(86) ◦

where T is the co-rotational derivative, T = T˙ − WT + T W. Since ˙ = 2FT DF ≃ 2D C

(87)

in small deformations, Eq. (86) can be viewed as the model of a viscoelastic material in that it is a superposition of a viscous character, T = 2µD, and an elastic character, T˙ = (2µ/τ )D, T ≃ (µ/τ )(C − 1). Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.

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To check the thermodynamic consistency of (86) we replace 1 T˙ = WT − T W + (2µD − T ) (88) τ in (83) to obtain 2µρ ρ 1 (T − ∂T ψ) · D + ∂T ψ · T − 2ρ(∂T ψ T ) · W − q · g ≥ 0. (89) τ τ θ The arbitrariness and the linearity of D and W imply that 2µρ T − ∂T ψ = 0, ∂T ψ T ∈ Sym, (90) τ the second condition being a consequence of the first one. Finally, since ψ is independent of g it follows ρ ∂T ψ · T ≥ 0, q · g ≤ 0. (91) τ Now, integration of τ ∂T ψ = T (92) 2µρ gives τ ψ = Ψ (F, θ) + T ·T (93) 4µρ and hence 1 ρ T ·T (94) 0 ≤ ∂T ψ · T = τ 2µ implies µ > 0 as is the case for the classical (shear) viscosity coefficient. This completes the consistency of the whole thermodynamic scheme with the rate equation (86) for the stress tensor T . 6.1. Remarks about some rate equations Since the equation of motion involves the divergence of the stress tensor then it is of interest to examine the invariance of the time derivative of ∇·T. This is examined in [2] in connection with the Truesdell (or upper-convected Oldroyd) derivative. In suffix notation □

T i j = ∂t Ti j + vk ∂xk Ti j − ∂x p vi T pj − Ti p ∂x p v j + (∂xk vk )Ti j .

(95)

It follows that □

∂x j T i j = (∂t + vk ∂xk )∂x j Ti j − L i p ∂x j T pj + (∂xk vk )∂x j Ti j − (∂x p ∂x j vi )T pj

(96)

whence □

□

∇ · T = ∇ · T − (∇∇v)T.

(97)

The desired invariance holds if ∇∇v = 0, that is if v depends linearly on x. A further question about rate equations is pointed out in [3]. Let T be given by T = µ0 (1 + µ1 ∂t )E,

µ0 , µ1 > 0,

(98)

which is viewed as the model of a “pure-retardation fluid”. It is then asserted that the relaxational analogue has the constitutive relation (1 − µ1 ∂t )T ≃ µ0 E.

(99)

Eq. (99) would be a Maxwell-type model with a negative relaxation time, −µ1 , which is said to be unphysical. Now, τ > 0 is assumed to guarantee bounded solutions of T induced by constant strains. Setting aside the physical meaning of τ as a (relaxation) time, it is natural here to ask about the thermodynamic consistency of τ < 0. Let us go back to (86) and let merely τ ∈ R. The procedure holds unchanged and again we find µ > 0. Moreover, no restriction is found about the sign of τ . We might conclude that the condition τ > 0 is justified by the physical meaning of τ and the required boundedness of T but not by thermodynamics. Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.

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7. Conclusions The paper develops two topics related to the application of rate equations in the modelling of continuum physics. By the viewpoint underlining the procedure, any constitutive equation, and hence any constitutive rate equation, has to be objective and consistent with the second law of thermodynamics. In this sense, the structure of objective time derivatives ∂ is reviewed. It follows that if the derivative is considered in the form (16) then the tensor Λ can take the forms Λ = W + λD + ν(∇ · v)1,

Λ = Z + λD + ν(∇ · v)1,

(100)

where W is the spin tensor whereas Z is the spin associated with the rotation of the polar decomposition. The objective derivatives applied in the literature turn out to be particular cases. Second, the thermodynamic consistency of some models involving objective time derivatives is investigated. In particular, attention is addressed to two topics. One is the improvement of the constitutive equation of viscous fluids by adding the contribution of the derivative of the stretching tensor. While the literature has investigated mainly the use of the Cotter–Rivlin derivative for incompressible fluids, here the analysis is developed with both Cottler–Rivlin and Oldroyd derivatives applied to compressible fluids. Next the thermodynamic consistency is proved for a rate equation of the dissipative stress T governed by the co-rotational derivative. Acknowledgement The research leading to this paper has been developed under the auspices of INDAM, Italy. References [1] F. Bampi, A. Morro, Objectivity and objective time derivatives in continuum physics, Found. Phys. 10 (1980) 905–920. [2] C.I. Christov, Frame indifferent formulation of Maxwell’s elastic-fluid model and the rational continuum mechanics of the electromagnetic field, Mech. Res. Commun. 38 (2011) 334–339. [3] I.C. Christov, C.I. Christov, Stress retardation versus stress relaxation in linear viscoelasticity, Mech. Res. Commun. 72 (2016) 59–63. [4] J.E. Dunn, R.L. Fosdick, Thermodynamics, stability, And boundedness of fluids of complexity 2 and fluids of second grade, Arch. Ration. Mech. Anal. 56 (1974) 191–252. [5] J.E. Dunn, K.R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (1995) 689–729. [6] M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2010. [7] D. Gutierrez-Lemini, Engineering Viscoelasticity, Springer, Berlin, 2013 (Chapter 3). [8] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. [9] A. Morro, A thermodynamic approach to rate equations in continuum physics, J. Phys. Sci. Appl. 7 (2017) 15–23. [10] A. Morro, Thermodynamic consistency of objective rate equations, Mech. Res. Commun. 84 (2017) 72–76. [11] I. Müller, On the frame dependence of stress and heat flux, Arch. Ration. Mech. Anal. 45 (1972) 241–250. [12] I. Müller, T. Ruggeri, Extended Thermodynamics, Springer, Berlin, 1993. [13] W. Muschik, Objectivity and frame indifference revisited, Arch. Mech. 50 (1998) 541–547. [14] W. Muschik, Is the heat flux density really non-objective? Cont. Mech. Thermodyn. 24 (2012) 333–337. [15] W. Noll, A mathematical theory of the mechanical behavior of continuous media, Arch. Ration. Mech. Anal. 2 (1958) 197–226. [16] B. Straughan, Heat Waves, Springer, Berlin, 2011. [17] C. Truesdell, W. Noll, The non-linear field theories of mechanics, in: S. Flügge (Ed.), Encyclopedia of Physics, third ed., Springer, Berlin, 1965.

Please cite this article as: A. Morro and C. Giorgi, Objective rate equations and memory properties in continuum physics, Mathematics and Computers in Simulation (2019), https://doi.org/10.1016/j.matcom.2019.07.014.