## Introduction

1 We have thus far studied the stress tensors (Cauchy, Piola Kirchoff), and several other tensors which describe strain at a point. In general, those tensors will vary from point to point and represent a tensor field.

2 We have also obtained only one differential equation, that was the compatibility equation.

3 In this chapter, we will derive additional differential equations governing the way stress and deformation vary at a point and with time. They will apply to any continuous medium, and yet we will not have enough equations to determine unknown tensor field. For that we need to wait for the next chapter where constitututive laws relating stress and strain will be introduced. Only with constitutive equations and boundary and initial conditions would we be able to obtain a well defined mathematical problem to solve for the stress and deformation distribution or the displacement or velocity fields.

4 In this chapter we shall derive differential equations expressing locally the conservation of mass, momentum and energy. These differential equations of balance will be derived from integral forms of the equation of balance expressing the fundamental postulates of continuum mechanics.

### 6.1.1 Conservation Laws

5 Conservation laws constitute a fundamental component of classical physics. A conservation law establishes a balance of a scalar or tensorial quantity in voulme V bounded by a surface S. In its most general form, such a law may be expressed as d J adV + J adS = I AdV (6.1)

Rate of variation Exchange by Diffusion where a is the volumetric density of the quantity of interest (mass, linear momentum, energy, ...) a, A is the rate of volumetric density of what is provided from the outside, and a is the rate of surface density of what is lost through the surface S of V and will be a function of the normal to the surface n.

6 Hence, we read the previous equation as: The input quantity (provided by the right hand side) is equal to what is lost across the boundary, and to modify a which is the quantity of interest. The dimensions of various quantities are given by dim(a) = dim(AL 3)

7 Hence this chapter will apply the previous conservation law to mass, momentum, and energy. the resulting differential equations will provide additional interesting relation with regard to the imcom-pressibiltiy of solids (important in classical hydrodynamics and plasticity theories), equilibrium and symmetry of the stress tensor, and the first law of thermodynamics.

8 The enunciation of the preceding three conservation laws plus the second law of thermodynamics, constitute what is commonly known as the fundamental laws of continuum mechanics.

9 Prior to the enunciation of the first conservation law, we need to define the concept of flux across a bounding surface.

10 The flux across a surface can be graphically defined through the consideration of an imaginary surface fixed in space with continuous "medium" flowing through it. If we assign a positive side to the surface, and take n in the positive sense, then the volume of "material" flowing through the infinitesimal surface area dS in time dt is equal to the volume of the cylinder with base dS and slant height vdt parallel to the velocity vector v, Fig. 6.1 (If v-n is negative, then the flow is in the negative direction). Hence, we Figure 6.1: Flux Through Area dS

define the volume flux as

Figure 6.1: Flux Through Area dS

where the last form is for rectangular cartesian components.

11 We can generalize this definition and define the following fluxes per unit area through dS:

6.2 Con

Mass Flux =

Momentum Flux =

Kinetic Energy Flux =

Heat flux =

Electric flux =

/— pv2(v^n)dS = i — pviViVj nj dS (6.6) s 2 J s 2

0 0