4.4.1 Constant Strain Tetrahedron

Figure 4.10 shows the natural coordinates system for a four-noded tetrahedron element with the face 1-2-3 on the local x- y plane. ^{[11] , [12]} The relationship between element local and natural coordinate systems may be expressed as

which, referring to Fig. 4.10, reduces to

Figure 4.10: Four-noded tetrahedron and other solid elements.

In the volume coordinate system ? _i = V _i/V, in which V _i is the tetrahedron bound by the center of volume point P and surface opposite to the node i of the element. The volume V of the element is defined as

where

The inverse transformation of Eq. (4.103) is obtained as

or

A linear displacement distribution is assumed within the element. Thus typically

which with substitution of nodal boundary conditions yields

and similarly for u _y and u _z displacements, N _y = N _z = N _x = N. Using the chain rule of differentiation, e.g., in the x direction for a function f

Now for f = u _x, it follows that

because ? N/ ? ? is a unit matrix of order four. Similar expressions can be obtained for other displacement derivatives. The strain-displacement relationships for the three-dimensional case are

which can be written in matrix form:

or

Then the stiffness matrix is given by

because B is constant. The inertia...