Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

4 Balance Principles

In this ch~pter we provide the classical balance principles and -discuss some of their important consequences. The fundamentaJ balance principles, i.e. conservation of mass, the momentum balance principles and balance of energy, are valid .in all branches of continuum mechanics. They are applicable to any particular material and must be satisfied for all times.

We also discuss another fundamental set of laws that are expressed as inequalities, such as the second law of thermodynamics. The section devoted to continuum thermo- dynamics specifically addresses balance of energy and the entropy .inequality principle. Finally, the structure -of principles is summarized as the ·master balance (inequality) principle.

4.1Conservation of Mass

Every continuum body B possesses mass, denoted by tn. It .is a fundamental physical

.property com.manly de·fined to be a measure of the amount of a material contained in the body B. In order to perform a macroscopic study we assume .that mass is continuously

(or at .least piecewise continuously) distributed over an arbitrary region n (of physical space) with boundary surface an at time t. The mass .is a scalar measure (a positive number) which is invariant during a motion. We exclude concentrated masses such as those used in class"ical Newtonian mechanics.

~losed and ope.a systems. We define a system as a quantity -of mass or .a particular c~llection of matter in space. The complement of a system, i.e. the mass or region outside the system, we call the surroundings, while the surface that separates the system from its surroundings we can the boundary or wall of the system (se.e Figure 4. l ).

A closed system (or control mass) consists of a fixed amount of mass in a properly selected region n .in space with boundary surface .an which depends on ti.me t (see Figure 4.1). No mass can -cross (enter or leave) its boundary, but energy, in the form can cross the boundary. The volume of a closed system does not have

13.1

configuration it must stay the same during a motion. Considering a closed system, obviously that holds for the total mass too. We write

which holds for all times :t. Relation (4.1) is a statement of a fundamental mechanical

law known as the conservation of mass. The ·boundary surfaces in the reference and configurations, with volume ll and v, are denoted by n 0 and !1, respectively. Note that the mass -rn is independent of the mot.ion and of the region occupied by the body.

Hence, the material time derivative of the mass 1n gives

denote physic.a·1 properties of the same particle. Property p0 is called the .reference mass density .(or just density) and pis -called the spatial ·mass dens.ity during a motion j( = x(X, -t) .. The spatial muss density, a1so known as the density in the motion, depends on place x E n and ti.met throughout the body. Note that {Jo is time-independent

depends only on the position X chosen in configuration 0 0 • If the density does not depend ·on X ·E n0 , i.e. if Gradp0 === o, the configuration is said to =be homogeneous.

The mass densities at the points X and x are defined by the limit

\Vhere ~1n denotes a continuous function of incremental mass of an :incremental vol-

:·ume element in the reference and current configurations, which we have denoted by :·Liv·and ~'IJ, respectively.

Note that ~l ..~(.n0 ), il·v (n), actually must not tend :to zero since then the limit of p0 , p, would show a discrete distribution according to the atomistic stmcture of matter. Therefore, to obtain representative averages, .6. ll(n0 ) and ~v(n) must be large in terms of an atomistic scale and s.mall in lenns of a length scale of a certain physical ·problem. Usually the ratio of the length scale of a physical problem and the length sea-le of an incremental volume ele~nent -il ll(!10 ) and ilv(n) is of the order 1oa -or

more.

which means that volume increases when density decreases. By integrating the infinitesimal mass over the entire region, we .find the total mass rn of that region. He.nee, an alternative expression for (4.6) reads

n

Hence, conservation of mass requires that the material time derivative of m.. is zero for all regions n of a body which change with time (mass remains unchanged during the motion of .n).

An equation which holds at .every point of a continuum and for all times, for ex.ample, eq. {4.6), is referred to us the local (or differential) form of that equation (local means pointwise). An equation in which physical quantities over a certain region of space are integrated is referred to as the global (or integral) form of that equation; see, for example., eq .(4.. 7). Consequently we may say that (4.6) is the local form and (4.7) is the global form of conservation of mass.

.In general, local forms are .ideally suited for approximation techniques such as the finite difference method while global forms are the best to start with when the finite

·element method is employed.

Continuity mass equation. We want to find a relationsh.ip between the reference mass density Po (X) E n0 and the spatial mass density p( x, t) E n.

By recalling eqs. (2.50) .and (2.51), i.e. dv = J(X;t)d l/, J = detF(X, f) > 0, we may change the variable of integration in eq. (4.7) from x = x(X, t) to X and we obtain the identity

C')

.l .. 0

136" 4 .Balance Principles

Since J > 0 we deduce from (4.16)2 the desired result (4.12), which is the correspond-

ing local 2. With the material time de.rivative of the spatial density function (4.17) and by means of identity (I ~287) we may obtain from (4.12) the two equivalent

forms (4.13) and (4..14). ·•

lf the density of a -continuum body is constant at any particle, then from relation (4.12) we find with p = 0 a kine:matical restriction which characterizes un isochoric (volume-preserving) motion, i.e. divv = 0 (compare also with eqs. (2.177)5). A continuum body is inco:mpressible if every motion it undergoes satisfies

The rate forms of continuity mass equations (4.12)-(4. "I 4) show how the spatial mass dens.ity p changes as time changes. They represent the continuity mass equation in the spatial description, which is the appropriate description in.fluid dynamics.

App.lying the divergence theorem for a fixed amount of volume f2c we have using

(l .297)

where n denotes the outward unit vector field perpendicular to the boundary control surface anc. The tenn .l~nc pv. nds determines the flux of {JV out of nc across Drlc.

Integrating the continuity mass equation in the form of (4.1.4) over a certain region nc and using eqs. (4.18) and (4.19), we obtain the conservation of mass for a control

volume in the global form, .i.e.

Relation (4.20) .asserts that the material ti.me derivative of the .mass inside a control volume nc is equal to the flux -of pv ente.ring f1e across Df1c:. The global form (4.20) -is

widely used in fluid dynamics.