Obtaining stress invariants, principal stress/strain values and directions, and rotating tensors in an Abaqus/Standard analysis

Utility routines are available for calculating stress invariants, principal stress/strain values, and principal stress/strain directions from the relevant tensors, as well as for transforming tensors to a new basis.

These utility routines are available for Abaqus/Standard user subroutines that store stress and strain components according to the convention presented in Conventions. They are most commonly called from user subroutine UMAT.

The following topics are discussed:

SINV (calculate stress invariants)
SPRINC (calculate principal values)
SPRIND (calculate principal values and directions)
ROTSIG (rotate a tensor)

SINV (calculate stress invariants)

Utility routine interface

      CALL SINV(STRESS,SINV1,SINV2,NDI,NSHR)

Variables to be provided to the utility routine

STRESS: A stress tensor.
NDI: Number of direct components.
NSHR: Number of shear components.

Variables returned from the utility routine

SINV1

First invariant.

SINV1 = \frac{1}{3} trace σ,

where $σ$ is the stress tensor.

SINV2

Second invariant.

SINV2 = \sqrt{\frac{3}{2} S : S},

where $S$ is the deviatoric stress tensor, defined as

S = σ - \frac{1}{3} trace σ I .

SPRINC (calculate principal values)

Utility routine interface

      CALL SPRINC(S,PS,LSTR,NDI,NSHR)

Variables to be provided to the utility routine

S: Stress or strain tensor.
LSTR: An identifier. LSTR=1 indicates that S contains stresses; LSTR=2 indicates that S contains strains.
NDI: Number of direct components.
NSHR: Number of shear components.

Variables returned from the utility routine

PS(I), I=1,2,3: The three principal values.

SPRIND (calculate principal values and directions)

Utility routine interface

      CALL SPRIND(S,PS,AN,LSTR,NDI,NSHR)

Variables to be provided to the utility routine

S: A stress or a strain tensor.
LSTR: An identifier. LSTR=1 indicates that S contains stresses; LSTR=2 indicates that S contains strains.
NDI: Number of direct components.
NSHR: Number of shear components.

Variables returned from the utility routine

PS(I), I=1,2,3: The three principal values.
AN(K1,I), I=1,2,3: The direction cosines of the principal directions corresponding to PS(K1).

ROTSIG (rotate a tensor)

Utility routine interface

      CALL ROTSIG(S,R,SPRIME,LSTR,NDI,NSHR)

Variables to be provided to the utility routine

S: A stress or strain tensor.
NDI: Number of direct components.
NSHR: Number of shear components.
R: Rotation matrix.
LSTR: An identifier. LSTR $= 1$ indicates S contains stresses; LSTR $= 2$ indicates S contains strains.

Variables returned from the utility routine

SPRIME: The rotated stress or strain tensor.

Typical usage

In user subroutine UMAT it is often necessary to rotate tensors during a finite-strain analysis. The matrix DROT that is passed into UMAT represents the incremental rotation of the material basis system in which the stress and strain are stored. For an elastic-plastic material that hardens isotropically, the elastic and plastic strain tensors must be rotated to account for the evolution of the material directions. In this case S is the elastic or plastic strain tensor and R is the incremental rotation DROT.