my "stiffness" matrix comes out singular?

Hi there,

I am running into a problem where for a simple, 2D, axisymmetric, single element problem, I get a "singular" matrix for my stress.  Consequently my force equilibrium equations cannot be solved for the displacements at the nodes.

The principle stresses for the axisymmetric case are as usual, a 4x1 matrix sigma given as:

[sig1, sig2, tao, sig3]^T and obtained as:

[sigma]=[D][S][B3][u]

(my problem is that [D][S][B3] a 4x8 matrix for a four node axisymmetric element always comes out singular for the first load increment)

where [u] is the unknown nodal displacement matrix, a 8x1 column matrix [u1,v1, u2,v2, u3,v3, u4,v4]^T

D is the material properties for my constitutive model is a constant 4x2 matrix defined as:

[d11 d12
 d21 d22
 d31 d32
 d41 d42]

[S] is a 2x1 stress matrix and for the first load increment is the initial shear and hydrostatic matrices given by [qo po]^T where qo and po are known at the start of each iteration.


shear strain is given by [B3][u] where [B3] the 3rd row of the derivative matrix.  For the case of a 4 node element in axisymetry [B3] is:

[dN1/dz  dN1/dr  dN2/dz  dN2/dr  dn3/dz  dn4/dr]


Now since [B]^T[sigma]=-[F], knowing [B]^T and [sigma] in terms of knowns [D], [S], and [B3] and the unknown [u], I can solve for the unknown [u] (the displacements at the nodes).

The problem I run into is that [D][S][B3], a 4x8 matrix, comes out singular for my first increment of loading on the cylinderical element.  Consequently,  [B]^T[D][S][B3], a 8x8 matrix for the 4 node element, also turns out also to be singular .

It seems I can put any values in for [D] or for [S], (I have no control on [B] and so also [B3]) but it always turns out that the 4x8 matrix [D][S][B3] is singular.  As a result, even though I know both [B]^T and [F], I cannot solve for [u].

What is it that ensures that whatever I try with [D] or [S] (as an experiement), I keep getting a singular matrix for [D][S][B3]?

Any help much appreciated (and will be acknowledged in any paper that results).

thx.
Paul