The capability to do a thermoplastic analysis exists in the following elements:

These elements support the thermoplastic effect which manifests itself as an increase in temperature during plastic deformation due to the conversion of some of the plastic work into heat.

In a thermoplastic analysis, the stress equation of motion (Equation 2–51) and heat flow conservation equation (Equation 6–1) are coupled by the plastic heat density rate defined as:

(10–40)

where:

β = fraction of plastic work coefficient (input as QRATE on MP command)

= plastic work rate =

where:

= stress vector =

= plastic strain vector =

The coupled-field finite element matrix equation for the thermoplastic analysis is:

where:

[M] = element mass matrix (defined by Equation 2–58)

[C] = element structural damping matrix (discussed in Damping Matrices)

[K] = element stiffness matrix (defined by Equation 2–58)

{u} = displacement vector

{F} = sum of the element nodal force (defined by Equation 2–56) and element pressure (defined by Equation 2–58) vectors

[Ct] = element specific heat matrix (defined by Equation 6–22)

[Kt] = element diffusion conductivity matrix (defined by Equation 6–22)

{T} = temperature vector

{Q} = sum of the element heat generation rate load and element convection surface heat flow vectors (defined by Equation 6–22)

= element plastic heat generation rate load =

where:

= element plastic heat density rate at substep n (output as NMISC,5)

{N} = element shape functions