sq
2005-09-15, 13:27
Assume the following:
I have an effectively incompressible elastomer such that if in uniaxial tension or compression, the first principal stretch lambda(1) = lambda, then lambda(2) = lambda(3) = lambda^(-1/2), obviously.
Deviatoric stress is expressed as a derivative of the energy function:
t(i) = lambda(i) * d(W)/d lambda(i) + p
Where p is indeterminate. For example, for Gaussian statistical theory, W = G/2 *(lambda(1)^2+lambda(2)^2+lambda(3)^2-3), which yields
t(i) = G*lambda(i)^2-p
Now, here's the trick: If we use p to set t(2) = t(3) = 0, which holds for uniaxial tension, then
t(1) = G*(lambda^2-1/lambda), the familiar result.
However, in the numerical implementation (such as in a VUMAT), we do not know for a given stress field how to properly set p. If p is set to a hydrostatic pressure stress as a function of J (as is often done), then the stress field for the uniaxial case illustrated above will be offset by (t2 - t1).
Does anybody have insight into how this discrepancy may be resolved?
I have an effectively incompressible elastomer such that if in uniaxial tension or compression, the first principal stretch lambda(1) = lambda, then lambda(2) = lambda(3) = lambda^(-1/2), obviously.
Deviatoric stress is expressed as a derivative of the energy function:
t(i) = lambda(i) * d(W)/d lambda(i) + p
Where p is indeterminate. For example, for Gaussian statistical theory, W = G/2 *(lambda(1)^2+lambda(2)^2+lambda(3)^2-3), which yields
t(i) = G*lambda(i)^2-p
Now, here's the trick: If we use p to set t(2) = t(3) = 0, which holds for uniaxial tension, then
t(1) = G*(lambda^2-1/lambda), the familiar result.
However, in the numerical implementation (such as in a VUMAT), we do not know for a given stress field how to properly set p. If p is set to a hydrostatic pressure stress as a function of J (as is often done), then the stress field for the uniaxial case illustrated above will be offset by (t2 - t1).
Does anybody have insight into how this discrepancy may be resolved?