SUBROUTINE UMATHT(U,DUDT,DUDG,FLUX,DFDT,DFDG,
     1 STATEV,TEMP,DTEMP,DTEMDX,TIME,DTIME,PREDEF,DPRED,
     2 CMNAME,NTGRD,NSTATV,PROPS,NPROPS,COORDS,PNEWDT,
     3 NOEL,NPT,LAYER,KSPT,KSTEP,KINC)
C
      INCLUDE 'ABA_PARAM.INC'
C
      CHARACTER*80 CMNAME
      DIMENSION DUDG(NTGRD),FLUX(NTGRD),DFDT(NTGRD),
     1 DFDG(NTGRD,NTGRD),STATEV(NSTATV),DTEMDX(NTGRD),
     2 TIME(2),PREDEF(1),DPRED(1),PROPS(NPROPS),COORDS(3)

      user coding to define U,DUDT,DUDG,FLUX,DFDT,DFDG,
      and possibly update STATEV, PNEWDT

      RETURN
      END

To use more than one user-defined thermal material model, the variable CMNAME can be tested for different material names inside user subroutine UMATHT, as illustrated below:

IF (CMNAME(1:4) .EQ. 'MAT1') THEN
   CALL UMATHT_MAT1(argument_list)
ELSE IF(CMNAME(1:4) .EQ. 'MAT2') THEN
    CALL UMATHT_MAT2(argument_list)
END IF

UMATHT_MAT1 and UMATHT_MAT2 are the actual user material subroutines containing the constitutive material models for each material MAT1 and MAT2, respectively. Subroutine UMATHT merely acts as a directory here. The argument list can be the same as that used in subroutine UMATHT.

As a simple example of the coding of user subroutine UMATHT, consider uncoupled heat transfer analysis in a material. The equations for this case are developed here, and the corresponding UMATHT is given. This problem can also be solved by specifying thermal conductivity, specific heat, density, and internal heat generation directly.

First, the equations for an uncoupled heat transfer analysis are outlined.

The basic energy balance is

where is the volume of solid material with surface area , is the density of the material, is the material time rate of the internal thermal energy, is the heat flux per unit area of the body flowing into the body, and is the heat supplied externally into the body per unit volume.

A heat flux vector is defined such that

where is the unit outward normal to the surface . Introducing the above relation into the energy balance equation and using the divergence theorem, the following relation is obtained:

The corresponding weak form is given by

where is the temperature gradient and is an arbitrary variational field satisfying the essential boundary conditions.

Introducing the backward difference integration algorithm:

the weak form of the energy balance equation becomes This nonlinear system is solved using Newton's method.

In the above equations the thermal constitutive behavior of the material is given by

and where are state variables.

The Jacobian for Newton's method is given by (after dropping the subscripts on )

The thermal constitutive behavior for this example is now defined. We assume a constant specific heat for the material. The heat conduction in the material is assumed to be governed by Fourier's law.

The internal thermal energy per unit mass is defined as

with where is the specific heat of the material and Fourier's law for heat conduction is given as where is the thermal conductivity matrix and is position, so that and

The assumption of conductivity without any temperature dependence implies that

No state variables are needed for this material, so the allocation of space for them is not necessary.

A thermal user material definition can be used to read in the two constants for our simple case, namely the specific heat, , and the coefficient of thermal conductivity, , so that

SUBROUTINE UMATHT(U,DUDT,DUDG,FLUX,DFDT,DFDG,
     1 STATEV,TEMP,DTEMP,DTEMDX,TIME,DTIME,PREDEF,DPRED,
     2 CMNAME,NTGRD,NSTATV,PROPS,NPROPS,COORDS,PNEWDT,
     3 NOEL,NPT,LAYER,KSPT,KSTEP,KINC)
C
      INCLUDE 'ABA_PARAM.INC'
C
      CHARACTER*80 CMNAME
      DIMENSION DUDG(NTGRD),FLUX(NTGRD),DFDT(NTGRD),
     1 DFDG(NTGRD,NTGRD),STATEV(NSTATV),DTEMDX(NTGRD),
     2 TIME(2),PREDEF(1),DPRED(1),PROPS(NPROPS),COORDS(3)
C
      COND = PROPS(1)
      SPECHT = PROPS(2)
C
      DUDT = SPECHT
      DU = DUDT*DTEMP
      U = U+DU
C
      DO I=1, NTGRD
         FLUX(I) = –COND*DTEMDX(I)
         DFDG(I,I) = –COND
      END DO
C
      RETURN
      END