The evolution of microstructure outside of the thin layer is computed using the methodology described in chapter IV. The microstructure evolution is assumed to be in equilibrium during heating, i. e., no super heating occurs. In austenite, grain growth begins after Nb and V carbo-nitrides dissolve and ceases when either delta ferrite forms on heating or ferrite, pearlite, bainite or martensite forms on cooling. Formation of ferrite, pearlite and bainite on cooling is modeled by ODEs, (see the general form in equation 4-21 and the special form for the austenite transformation to ferrite in equation 2-19). Martensite formation on cooling is modeled by the Koisten-Marburger equation.

Thermal Stress Analysis

The thermal stress analysis computes an FEM approximate solution to the conservation of momentum and mass. The loads, material properties and geometry are time dependent. Analogy to equation (5-12), chapter V, we say:

V-(T + / = 0

—+ pV-v = 0 (6-5)

Dt

where g is the Cauchy stress,/is the body force and p is the density. Inertial forces are ignored.

For each time step, a displacement increment dU is computed using a Newton-Raphson iterative method in an updated Lagrangian formulation. The deformation gradient is in agreement with equation (5-6) for a time step F = I + VdU. The Green-Lagrange strain increment for a time step is ds = (FT F - Y) / 2 . For details describing the finite strain theory, see chapter V and also Goldak et al [16] .The deformation gradient is decomposed into elastic, initial and plastic deformation gradients, F - FElFMtFpl, where Fhlit includes deformations due to thermal expansion and phase transformations.

For stress analysis, F, nit is made piecewise constant in an element.

The increment in the 2nd Piola-Kirchoff stress is do2PK - DdeEl

where D is the elasticity tensor. At temperatures below ~1100 °K rate independent plasticity is used. At temperatures above ~1700 °K a linear viscous constitutive model is used.

The residual for the Newton-Raphson iteration is:

<R = P-^BTa1PK (6-6)

where В is the usual FEM matrix that maps nodal displacements to the strain at a point and P is the nodal vector of external loads. Each Newton-Raphson iteration computes a correction du to the current trial displacement dUby solving:

Kdu = -91 (6-7)

where К is the global stiffness matrix. For each Newton-Raphson iteration, the linear equation (6-7) is solved to a tolerance of 10”9 in the energy norm. When the 1-norm of the residual 9t is less than 1 (Г4 P, the time step is considered to have converged.