Chapter 2
Governing Equations

 

The purpose of this chapter is to summarize the equations that govern the response of solids to mechanical or thermal loading.  The following topics will be addressed in turn:

 

1. The assumptions and approximations that we make when describing the large-scale behavior of matter;

 

2. The mathematical description of shape changes in a solid;

 

3. The mathematical description of internal forces in a solid;

 

4. Equations of motion for deformable solids;

 

5. Concepts of mechanical work and power for deformable solids; and the important principle of virtual work;

 

6. The laws of thermodynamics for a continuum;

 

7. How to transform the variables describing deformation and forces from one reference frame to another.

 

 

2.1 Basic Assumptions

Continuum mechanics is a combination of mathematics and physical laws that approximate the large-scale behavior of matter that is subjected to mechanical loading.   It is a generalization of Newtonian particle dynamics, and starts with the same physical assumptions inherent to Newtonian mechanics; and adds further assumptions that describe the structure of matter.  Specifically:

 

· The Newtonian reference frame: In classical continuum mechanics, the world is idealized as a three dimensional Euclidean space (a vector space consisting of all triads of real numbers $(x_1,x_2,x_3)$ ).  A point in space is identified by a unique set of three real numbers.   A Euclidean space is endowed with a metric, which defines the distance between points: $d=\sqrt{x_1^2+x_2^2+x_3^2}$.  Vectors can be expressed as components in a basis - $\{e_1,e_2,e_3\}$ of mutually perpendicular unit vectors.  Physical quantities such as force, velocity, acceleration are expressed as vectors in this space.  A Cartesian Coordinate Frame is a fixed point O together with a basis.   A Newtonian reference frame is a particular choice of Cartesian coordinate frame in which Newton’s laws of motion hold.  

 

· The Continuum:  Matter is idealized as a continuum, which has two properties: (i) it is infinitely divisible (you can subdivide some region of the solid as many times as you wish); and (ii) it is locally homogeneous $–$ in other words if you subdivide it sufficiently many times, all sub-divisions have identical properties (eg mass density).  A continuum can be thought of as an infinite set of vanishingly small particles, connected together.

 

Both the existence of a Newtonian reference frame, and the concept of a continuum, are mathematical idealizations.   Experimental evidence suggest that the laws of motion based on these assumptions accurately approximate the behavior of most solid and fluid materials at length scales of order mm-km or so in engineering applications.  In some cases, continuum models can also approximate behavior at much shorter length scales (for volumes of material containing a few 1000 atoms), but models at these length scales often require different relations between internal forces deformation measures in the solid to those used to model larger volumes.