1.13 Constitutive Equations

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1.13.1 Summary of the balance equations

In Section 1.10, the balance equations were derived in material coordinates. They amount to

ρJ = ρ0 (1 equation) (1.352)

∇0(S · FT) + ρ0f = ρ0˙v (3 equations) (1.353)

S = ST (3 equations) (1.354)

44 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

ρ0˙ε = ˙E : S −∇0 ·Q+ ρ0h (1 equation) (1.355)

ρ0

_˙η −

˙ε

θ

_ + 1

θ

˙E

: S − 1

θ2Q· (∇0θ) ≥ 0. (1.356)

In sum, there are eight equations and one inequality. The unknowns are ρ (1), J (1), S (9),

F (9), v (3), ε (1), E(6), Q (3), η (1) and θ (1) which yields 35 unknowns. The variables

J , F, v and E can be reduced to x (3 unknowns) since

J = det(xk

,K ) (1.357)

F = xk

,Kgk

⊗ GK (1.358)

v = ˙x (1.359)

E = 12

(xk

,Kxl

,Lgkl − GKL)GK ⊗ GL. (1.360)

In that way, 19 unknowns remain. Accordingly, we need another 11 equations to solve the

problem for x(X, t) and θ(X, t). This is not surprising, since the material properties were

not considered so far. All balance equations apply to steel as well as to wood, water or

air. It is well known that these materials behave quite differently and it is the task of the

constitutive equations to describe these different kinds of behavior. It looks like a huge

task to tackle but luckily there are some physical principles that may guide us. Here I wish

to adhere to a simplified form of the axiomatic formulation found in (Eringen 1980) since

it leads us in a systematic way to the constitutive equations of widely different materials.

1.13.2 Development of the constitutive theory

The constitutive equations bridge the gap between physically observable quantities (independent

variables in the constitutive equations) and the quantities arising in the balance

laws (dependent variables in the constitutive equations). For thermomechanical processes,

the observable quantities are the location x(X, t) and the temperature θ(X, t). All other

variables such as the stress S, the flux Q, the internal energy ε and the entropy η are

measured indirectly by the effect they produce on the displacements and the temperature.

For instance, the stress is usually measured through strain gauges. Accordingly, the value

of the dependent variables (S, Q, ε, and η – the density ρ is not considered as a dependent

variable but rather immediately eliminated through Equation (1.352)) at X at time t is

assumed to be a function of the value of the independent variables (x, θ) at all former times

and in the complete body. This can be written in the form of the following functionals:

S(X, t) = S[x(X

_

, t

_

), θ(X

_

, t

_

),X, t] (1.361)

Q(X, t) = Q[x(X

_

, t

_

), θ(X

_

, t

_

),X, t] (1.362)

ε(X, t) = ε[x(X

_

, t

_

), θ(X

_

, t

_

),X, t] (1.363)

η(X, t) = η[x(X

_

, t

_

), θ(X

_

, t

_

),X, t] (1.364)

t

_ ≤ t,X

_ ∈ V0.

Hence, a priori it is assumed that the deformation and temperature in the complete

body at all former times can have an impact on the value of any dependent variable at

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 45

X and t . This formulation includes memory effects (e.g. viscosity) and nonlocal effects

(atomic forces).

There are two major postulates that must be obeyed by the constitutive equations.

First, there is the principle of objectivity, which states that the constitutive equations must

not depend on the spatial motion of the observer. This principle has already been briefly

discussed in Section 1.6. There, it was emphasized that only objective tensors should be

used in constitutive equations. What does this translate to in Equations (1.361) to (1.364)?

Since the left-hand side of these equations is formulated in terms of material quantities,

objectivity is no problem. What about the right-hand side? In general, a time-dependent

rigid body motion combined with a time-shift maps x(X, t) into

x(X

_

, t _) = Q(t

_

) · x(X

_

, t

_

) + b(t

_

) (1.365)

Q·QT = QT ·Q = I , t _ = t

_ − a. (1.366)

This mapping can be split into a time-dependent translation, a time-shift and a timedependent

rotation.

A time-dependent translation must not change the constitutive equation. Taking the

translation to be x(X, t

_

) one obtains, Equation (1.361),

S(X, t) = S[x(X

_

, t

_

) − x(X, t

_

), θ(X

_

, t

_

),X, t], (1.367)

which means that only the relative position with respect to x(X, t

_

) is kept.

A time shift must not influence Equation (1.367) either. Taking the shift to be t one obtains

S(X, t) = S[x(X

_

, t

_ − t) − x(X, t

_ − t), θ(X

_

, t

_ − t),X, 0]. (1.368)

Consequently, the explicit dependence on t drops out:

S(X, t) = S[x(X

_

, t

_

) − x(X, t

_

), θ(X

_

, t

_

),X]. (1.369)

Finally, a time-dependent rotation of the spatial frame of reference should also leave the

constitutive equation unaltered. One obtains

S(X, t) = S _Q(t) · (x(X

_

, t

_

) − x(X, t

_

)), θ(X

_

, t

_

),X_ (1.370)

for an arbitrary rotation Q(t).

The second postulate states that the constitutive equations must be form-invariant with

respect to a certain class of rotations Q and translations B of the material frame, which

are the result of material symmetries and material homogeneities. A lot of materials exhibit

symmetries with respect to a specific class of rotations due to the intrinsic crystallographic

structure. For example, single crystals are frequently orthotropic, which means that mutually

orthogonal planes exist in the material frame with respect to which the material properties

are symmetric. A usual assumption in polycrystals is the form-invariance with respect to

the full group of rotations: this means that the material properties are independent of the

direction and this is called isotropy. For a homogeneous material, the properties are not

changed by an arbitrary translation. Once the classes {Q} and {B} are determined on

46 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

the basis of observations of the material behavior, the constitutive equations should be

form-invariant with respect to transformations of the type

X = Q· X + B, QT ·Q = Q·QT = I , detQ = ±1 (1.371)

mapping the material frame X into X. Notice that the axiom of objectivity involves

transformations of the spatial frame, whereas the axiom of material invariance concerns

transformations of the material frame.

In order to obtain further simplifications, the expressions for the independent quantities

are expanded in a Taylor series. Taylor expansion of x(X

_

, t

_

) − x(X, t

_

) in space yields

x(X

_

, t

_

) − x(X, t

_

) = x,K1 (X, t

_

) _X

_K1 − XK1

+ 1

2!

x,K1K2 (X, t

_

) _X

_K1 − XK1 _X

_K2 − XK2 +· · · (1.372)

Similarly,

θ(X

_

, t

_

) = θ(X, t

_

) + θ,K1 (X, t

_

) _X

_K1 − XK1

+ 1

2!

θ,K1K2 (X, t

_

) _X

_K1 − XK1 _X

_K2 − XK2 +· · · (1.373)

and Equation (1.98) can be replaced by

S(X, t) = S[Q(t) · x,K1 (X, t

_

),Q(t) · x,K1K2 (X, t

_

), . . . ,

θ(X, t

_

), θ,K1 (X, t

_

), θ,K1K2 (X, t

_

), . . . ,X]. (1.374)

Notice that the dependent variables are explicitly dependent on θ(X, t

_

) but not on x(X, t

_

).

Materials satisfying Equation (1.374) are said to be of mechanical grade N and thermal

grade M if the spatial derivatives are at most of Nth order and the thermal derivatives of

at most Mth order.

Taylor expanding the remaining independent variables in time yields

x,K1 (X, t

_

) = x,K1 (X, t)+ ˙x,K1 (X, t )(t

_ − t) + 1

2!

¨x

,K1 (X, t )(t

_ − t)2 +· · · (1.375)

and similar for the other variables. Hence, one can replace Equation (1.374) by

S(X, t) = S[Q(t) · x,K1 (X, t),Q(t) · ˙x,K1 (X, t), . . . ,

Q(t) · x,K1K2 (X, t),Q(t) · ˙x,K1K2 (X, t), . . . ,

· · ·

Q(t) · x,K1K2...KN (X, t),Q(t) · ˙x,K1K2...KN (X, t), . . . ,

θ(X, t), ˙ θ(X, t), ¨ θ(X, t), . . . ,

θ,K1 (X, t), ˙ θ,K1 (X, t), ¨ θ,K1 (X, t), . . . ,

· · ·

θ,K1K2...KM (X, t), ˙ θ,K1K2...KM (X, t), ¨ θ,K1K2...KM (X, t), . . . ,X]. (1.376)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 47

In what follows, we will concentrate on materials of mechanical grade 1 and thermal

grade 1. Hence,

S(X, t) = S[Q(t) · x,K (X, t),Q(t) · ˙x,K (X, t), . . . ,

θ(X, t), ˙ θ(X, t), ¨ θ(X, t), . . . ,

θ,K (X, t), ˙ θ,K (X, t), ¨ θ,K(X, t), . . . ,X]. (1.377)

The principle of objectivity implies that the right-hand side of Equation (1.377) must be

invariant with respect to spatial rotations. This means that the list of independent variables

can be replaced by the invariants of {x,K , ˙x ,K , ¨x,K, . . . } with respect to an arbitrary rotation.

The theory of invariants (Spencer 1971) shows that an integrity basis for the invariants

of the above set subject to proper transformations (i.e. detQ = +1) consists of the scalar

product of any two vectors in the set, for example,

x,K · x,L = xk

,Kxl

,Lgkl = CKL (1.378)

and triple products of the form

eklmxk

,Kxl

,Lxm

,M. (1.379)

For K _= L,K _= M and L _= M the expression in Equation (1.379) is the Jacobian determinant

J. For K = L, K = M or K = M Equation (1.379) is zero since this amounts to

the determinant of a matrix with two equal rows or columns. Consequently, the dependence

on {x,K , ˙x,K , ¨x ,K, . . . } in Equation (1.377) can be replaced by a dependence on

{CKL, ˙CKL, ¨CKL, . . . , J, ˙ J, ¨ J, . . . } (1.380)

or

{CKL, ˙CKL, ¨CKL, . . . , ρ

−1, ˙ ρ, ¨ρ, . . . } (1.381)

since J = ρ0ρ

−1. Equation (1.377) now reads

S(X, t) = S[CKL(X, t), ˙CKL(X, t), . . . ,

ρ

−1(X, t), ρ˙(X, t), . . . ,

θ(X, t), ˙ θ(X, t), ¨ θ(X, t), · · · ,

θ,K (X, t), ˙ θ,K (X, t), ¨ θ,K(X, t), . . . ,X]. (1.382)

This also applies to Q, ε and η yielding the missing 11 equations.

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