Momentum Balance Equation

@t C r .v/D PCr vD0 onC.t /†.t /;

ŒŒ.v_nc_n/D0 on†.t /: (5.22)

5.3 Momentum Balance Equation

Let S be a continuous system and let .x; t / be its mass density in the current configuration. In the following discussion any dependence on timetwill be omitted for the sake of simplicity.

A fundamental axiom of continuum mechanics is the momentum balance principle:

For any arbitrary material volumecofSit holds that

Q.c/P DF.c; c^e/; (5.23)

is the momentum of candF.c; c^e/is the resultant of all the forces acting oncfrom its exteriorc^e.

In a first simplified approach to continuum mechanics, the external actions onc are divided intomass forces, continuously distributed overc, and contact forces, acting on the boundary@cofc; therefore,

F.c; c^e/D

144 5 Balance Equations wherebis thespecific force, defined onc, andtis thetractionor thestress, defined on@c.

With such an assumption, the first integral in (5.25) represents all (gravitational, electromagnetic, etc.) forces acting on the volumecfrom theexterior of Sand the vectorb is an a priori assigned field depending onx andt. On the other hand,t is a field ofcontact forcesacting on@cwhich come from the molecular attraction between particles ofc^eandcat the boundary@c.

The contact forces, which are strictly influenced by the deformation ofS, are unknown, as are reactions in rigid body mechanics. The main difference between reactions and contact forces is the fact that for the latter, we can provideconstitutive lawsthat specify their link with the motion of the system.

At this stage, some remarks are necessary to show how restrictive the previous hypotheses are. First, the mass forces acting over c could originate from other portions of S, external to c. This is just the case for mutual gravitational or electromagnetic attractions among parts ofS. In fact, mass forces are unknown a priori, and the need arises to add to the motion equations the other equations governing the behavior of those fields which generate such forces.

As an example, ifb is the gravitational field produced by the system itself, it is necessary to introduce Poisson’s equation for this field (see Volume II for other examples).

Furthermore, the assumption made about contact forces means that molecular actions can beuniquely represented by the vectortd. According to this assumption the approximation of asimple continuumis usually introduced. But when the need arises to capture essential features linked to the microstructure of the body, contact actions are better represented by a vectortdas well as a torquemd. In this case, the approximation of apolar continuum(E. and F. Cosserat, 1907) is used, and the related model is particularly useful in describing liquid crystals, due to the fact that molecules or molecule groups behave as points of a polar continuum.

Anyway, in the following discussion and within the scope of this first volume, the classical assumption ofmD0will be retained, so thatS will be modeled as a simple continuum. Since molecular actions have a reduced interaction distance, the force acting on the surfaced depends on the particles of S adjacent to d and not on particles far away. This remark justifies theEuler–Cauchy postulatethat the vectort, acting on the unit area atx, depends on the choice of the surface only through itsorientation, i.e.,

tDt.N/; (5.26)

whereNis the outward unit vector normal tod.³

This assumption is equivalent to saying that the traction t depends on the boundary @c only to first order; i.e.,t depends on the orientation of the tangent plane atx2@cand not on the curvature of@c.

3See the footnote of Sect.5.1.

It is customary to introduce the decomposition tDtnNCt;

where the componenttnDtNis called thenormal stressand the componenttis called theshear stress.

By applying (5.23) to amaterial volume cand using definitions of momentum and external forces, we obtain

d dt

Z

c

vdcD Z

@c

t.N/ dC Z

c

bdc: (5.27)

Then, applying Theorem5.1to (5.27), we have:

Theorem 5.2. Ift;band the accelerationaare regular functions, then the action-reaction principle also holds for stresses, i.e.,

t.N/D t.N/: (5.28) There exists a second-order tensorT, calledCauchy’s stress tensor, such thatt is a linear function ofN, so that

t.N/DTN: (5.29)

The tensorTdepends on.x; t /, but is independent ofN.

In the next section, it will be proved that Tis a symmetric tensor; therefore, its eigenvalues are real and there is at least one orthonormal basis of eigenvectors.

Eigenvalues are known asprincipal stressesand eigenvectors ofTgiveprincipal directions of stresscharacterized by the following property: the tractiont, acting on the areadnormal to the principal directionu, is normal tod, or, equivalently, on this area there are no shear stress components (see Exercise 2).

When (5.29) is substituted into (5.27), the balance equation with the general structure (5.1) is recovered, where

fDv; ˆD T; rDb:

Taking into account (5.3) as well as the local form of the mass conservation (5.22), the local expression of the momentum balance and the jump conditioncan be easily derived:

PvD r TCb; (5.30)

ŒŒv.cnvn/CTnD0: (5.31)

146 5 Balance Equations In the case of amaterial surface, the jump condition (5.31) reduces toPoisson’s condition

ŒŒTnD0;

on which are based thestress boundary conditions.

We observe, in fact, that one of the most interesting material surfaces is the boundary of a body, so that the mechanical interaction of two sub-bodies is uniquely determined by tractions on this surface, of equal magnitude and opposite sign.