430 The finite element method for partial differential equations
Figure 13.3. Two types of deformation, compression (left) and shear (right).particular when the deformation is relatively small so that the total energy of the
deformed system can be well approximated by a second order Taylor expansion.
There are two types of deformation. The first is compression or expansion of
the system, and the second is shear. These effects are shown in Figure 13.3. We
restrict ourselves to homogeneous isotropic systems in two dimensions. Then the
resistance of a material to the two types of deformation is characterised in both
cases by an elastic constant – in the literature either the Lamé constantsλandμ
are used, or the Young modulusEand Poisson ratioν. They are related by
E=
μ( 3 λ+ 2 μ)
λ+μ(13.25a)ν=λ
2 (λ+μ). (13.25b)
To formulate the equations of deformation, consider the displacement fieldu(r).
This vector field is defined as the displacement of the pointras a result of external
forces acting on the system. These forces may either be acting throughout the
system (gravity is an example) or on its boundary, like pushing with a finger on
the solid object. In the equilibrium situation, the forces balance each other inside
the material. So, if we identify a small line (a planar facet in three dimensions) with
a certain orientation somewhere inside the object, the forces acting on both sides of
this line should cancel each other. These forces vary with the orientation of the line
or facet, as can be seen by realising that in an isotropic medium and in the absence
of external forces, the force is always normal to the line (it is due to the internal,
isotropic pressure). Another way to see this is by considering gravity. This acts on
a horizontal facet from above but not on a vertical facet. Therefore it is useful to
define thestress tensorσijwhich gives thejth component of the force acting on a
small facet with a normal along theith Cartesian axis.