1.7 Balance Laws

Back

Balance laws are important statements describing the conservation of some physical quantities.

These quantities and the conservation thereof will be defined in the present section.

1.7.1 Conservation of mass

Each object in space is assigned a strictly positive scalar quantity called the mass. The

mass is assumed to be continuously distributed, which allows for the definition of density

ρ0(X, t) by letting the volume containing particle X go to zero:

ρ0(X) := lim

_V0→0

_M

_V0

, X ∈ _V0. (1.195)

_V0 is the volume the particle occupies in the reference configuration at time t = t0. The

density can change during the motion of a body. The density of a particle at time t originally

at X is

ρ(X, t) := lim

_V→0

_M

_V

, x(X, t) ∈ _V. (1.196)

_V is the volume the particle occupies at time t in the spatial configuration. The axiom

of the conservation of mass now states that “the time rate of change of the total mass of a

body is zero”. Accordingly,

D

Dt

__

V

ρ dv

_ = 0. (1.197)

1.7.2 Conservation of momentum

The momentum (also called linear momentum) of an infinitesimal mass dm moving with

a velocity v is defined as

v dm = ρv dv. (1.198)

The principle of conservation of momentum states that “the time rate of change of linear

momentum is equal to the total force F acting on a body”. Forces acting on a body are

either body forces Fb resulting from distant actions such as gravity, surface tractions Fs

resulting from immediate contact such as classical friction forces, or concentrated forces

Fc. Enough continuity is assumed such that the body force per unit volume f and the force

per unit area t (n) can be defined as follows:

dFb =: ρf dv (1.199)

dFs =: t (n) da. (1.200)

Accordingly,

D

Dt

_

V

ρv dv = _

A

t (n) da + _

V

ρf dv +_Fc (1.201)

where A denotes the surface of the body at stake. This principle is also known as Newton’s

second law.

26 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

1.7.3 Conservation of angular momentum

The angular momentum of a particle with mass dm, velocity v and location x is defined as

x × v dm (1.202)

where × symbolizes the vector product (also called the cross product) of two vectors. The

vector product of two vectors a and b is a one-form c satisfying

c · a = c · b = 0 (1.203)

and

c · c = (a · a)(b · b) − (a · b)2. (1.204)

Accordingly, gi

× gj is proportional to gk. The proportionality constant λ can be determined

from Equation (1.204):

λ2gkk = giigjj − (gij )2 = cofactor(gkk). (1.205)

Since g_ is the inverse of g_, one finds

gkk = cofactor(gkk)

det g_

(1.206)

leading to

gi

× gj

= eij kgk_det g_. (1.207)

Similarly, the moment of a force F acting at a location x is defined as x × F. The

principle of conservation of angular momentum states that “the time rate of change of

angular momentum is equal to the total moment due to forces and couples acting on the

body”. Hence,

D

Dt

_

V

ρx × v dv = _

A

x × t (n) da + _

V

ρx × f dv +_x × Fc +_Mc. (1.208)

Here Mc represents concentrated moments. It is assumed that there are no distributed

moments, which essentially means that this treatise is limited to nonpolar theories. Readers

interested in polar theories (used, for example, for the description of liquid crystals) are

referred to (Eringen 1980).

1.7.4 Conservation of energy

This principle states that “the time rate of change of the sum of the kinetic energy K and

internal energy E is equal to the sum of the work rate of all forces and couples W acting

on the body and all other energies U that enter or leave the body per unit time”. The total

kinetic energy of a body is defined by

K = 1

2

_

V

ρv · v dv (1.209)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 27

and the rate of work of all forces and couples by

W = _

A

t (n) · v da + _

V

ρf · v dv +_Fc · vc +_Mc · ωc (1.210)

where ωc is the angular velocity of the particle Mc is acting.

The internal energy is a new quantity. It is assumed that it is continuously distributed

such that the energy density or energy per unit mass ε can be defined as

E = _

V

ρε dv. (1.211)

Other energies can, for example, be of thermal, chemical or electromagnetic origin. Here

we limit the discussion to thermal energy. In that case, U amounts to

U = −_

A

q · da + _

V

ρh dv +_Hc (1.212)

where q is the heat flux through area da (the minus sign implies that the body is losing

energy if q points outwards), h is the body heat density and Hc is the heat due to

concentrated heat sources. Consequently, the principle of conservation of energy reads

D

Dt

_

V

_

ρε + 1

2

ρv · v_ dv = _

A

(t (n) · v − q · n) da

+ _

V

(ρf · v + ρh) dv +_Hc +_Fc · vc +_Mc · ωc. (1.213)

This is also called the first law of thermodynamics.

1.7.5 Entropy inequality

This principle, also called the second law of thermodynamics or Clausius–Duhem inequality,

states that “the time rate of change of the entropy H of a body is never less than the

sum of the entropy s entering the body through its surface and the entropy B generated by

body sources”. Hence,

DH

Dt

≥ B + _

A

s · da. (1.214)

Defining the entropy density η and the entropy source density b by

H = _

V

ρη dv (1.215)

and

B = _

V

ρb dv (1.216)

one finds

D

Dt

_

V

ρη dv ≥ _

V

ρb dv + _

A

s · da. (1.217)

Notice that this is an inequality. If other phenomena are considered such as electromagnetic

actions, additional laws apply. Here we concentrate on thermomechanical processes.

28 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

1.7.6 Closure

At first sight, the formulation of the balance laws does not look very promising for our

primary goal, that is, the determination of x(X, t). Indeed, a lot of extra unknowns have

been defined: ρ, t (n), ε, η, . . . On the other hand, some of the new variables are formulated

in terms of previously defined unknowns such as K(v). The relevance of the balance laws

is based on the relationship they establish with the physical world through quantities such

as f and h. They are fundamental axioms based on physical observations and as such

indispensable. The extra unknowns will be taken care of later on by the material description

(constitutive equations).

Навигация