Equivalent Stresses and Strains


Parameter Name	Formula
SIG_1	$σ_{1}$
SIG_2	$σ_{2}$
SIG_3	$σ_{3}$
SIG_11	$σ_{11}$
SIG_22	$σ_{22}$
SIG_33	$σ_{33}$
SIG_12	$σ_{12}$
SIG_23	$σ_{23}$
SIG_13	$σ_{13}$
SIG_ABS_123	$max (\|σ_{1}\|, \|σ_{2}\|, \|σ_{3}\|)$
SIG_ABS_3	$\|σ_{3}\|$
SIG_BELTRAMI	$\sqrt{σ_{1} σ_{1} + σ_{2} σ_{2} + σ_{3} σ_{3} - 2 ν (σ_{1} σ_{2} + σ_{2} σ_{3} + σ_{3} σ_{1})}$
SIG_DRUCKER_PRAGER	${\frac{ν - 1}{2 ν} + \frac{1 + ν}{2 ν} (σ_{1} + σ_{2} + σ_{3})} \sqrt{\frac{1}{2} {(σ_{1} - σ_{2})^{2} + (σ_{2} - σ_{3})^{2} + (σ_{3} - σ_{1})^{2}}}$
SIG_KUHN	$\frac{1 + ν}{3} {(σ_{1} - σ_{1})^{2} + (σ_{2} - σ_{3})^{2} + (σ_{3} - σ_{1})^{2}} + \frac{1 - 2 ν}{3} \frac{(σ_{1} + σ_{2} + σ_{3})^{3}}{\| σ_{1} + σ_{2} + σ_{3} \|}$
SIG_MARIOTTE	$\max {(σ_{1} - ν (σ_{2} + σ_{3})), (σ_{2} - ν (σ_{3} + σ_{1})), (σ_{3} - ν (σ_{1} + σ_{2}))}$
SIG_MISES	$\sqrt{\frac{1}{2} {{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{1} - σ_{3})}^{2}}}$
SIG_SIGNED_MISES	$sgn(1, σ_{11} + σ_{22} + σ_{33}) σ_{vM}$
SIG_SANDEL	$\sqrt{σ_{1}^{2} + σ_{2}^{2} + σ_{3}^{2} - 2 ν \frac{2 - ν}{1 + 2 ν^{2}} (σ_{1} σ_{2} + σ_{2} σ_{3} + σ_{1} σ_{3})}$
SIG_SAUTER	$\sqrt[n]{{\| σ_{1} - σ_{2} \|}^{n} + {\| σ_{2} - σ_{3} \|}^{n} + {\| σ_{3} - σ_{1} \|}^{n}}$
SIG_TRESCA	$max (\|σ_{1} - σ_{2}\|, \|σ_{2} - σ_{3}\|, \|σ_{3} - σ_{1}\|)$

Contact and Strain Measures


Parameter Name	Formula
ABQ_ND_PEEQ	PEEQ, see Abaqus documentation
SIG_CONTACT_PRESSURE	$p$
SIG_CONTACT_SHEAR	$\sqrt{τ_{1}^{2} + τ_{2}^{2}}$
SIG_CONTACT_SHEAR_X	$τ_{1}$
SIG_CONTACT_SHEAR_Y	$τ_{2}$
SIG_CONTACT_TOTAL	$\sqrt{p^{2} + τ_{1}^{2} + τ_{2}^{2}}$
STRAIN_ELASTIC	${\frac{2}{3} [{(ε_{e})}_{11}^{2} + {(ε_{e})}_{22}^{2} + {(ε_{e})}_{33}^{2} + 2 ({(ε_{e})}_{12}^{2} + {(ε_{e})}_{13}^{2} + {(ε_{e})}_{23}^{2})]}^{\frac{1}{2}}$
STRAIN_PLASTIC	${\frac{2}{3} [{(ε_{p})}_{11}^{2} + {(ε_{p})}_{22}^{2} + {(ε_{p})}_{33}^{2} + 2 ({(ε_{p})}_{12}^{2} + {(ε_{p})}_{13}^{2} + {(ε_{p})}_{23}^{2})]}^{\frac{1}{2}}$
STRAIN_TOTAL	${\frac{2}{3} [ε_{11}^{2} + ε_{22}^{2} + ε_{33}^{2} + 2 (ε_{12}^{2} + ε_{13}^{2} + ε_{23}^{2})]}^{\frac{1}{2}}$

Important:

Elastic and plastic strain values (TYPE = STRAIN_ELASTIC and TYPE = STRAIN_PLASTIC) are available only in a nonlinear analysis with plastic materials. Using these DRESPs for linear analysis results in an error.

Analysis Types: Any Analysis Type with Stress Output or Fatigue Result

For all design responses from the table above (except NEUBER, GLINKA and SIG_SIGNED_MISES ), the following table shows the allowed combinations between the strategy and the items OBJ_FUNC and CONSTRAINT with C for controller-based optimization and S for sensitivity-based optimization.


	TOPO	SHAPE	BEAD	SIZING
OBJ_FUNC		C
CONSTRAINT

The equivalent stresses are the main input for the shape optimization controller algorithm. All values, whether nodal, from gauss points or elements, are interpolated to the nodes.

In addition, SIG_1, SIG_3 , and SIG_MISES are allowed for the following optimization types. SIG_MISES in only supported in optimization types with a *.


	TOPO	SHAPE	BEAD	SIZING
OBJ_FUNC	S	S*	S*	S
CONSTRAINT	S	S*	S*	S

The NEUBER, GLINKA, and SIG_SIGNED_MISES design responses are supported with the following optimization types.


	TOPO	SHAPE	BEAD	SIZING
OBJ_FUNC	S	S	S	S
CONSTRAINT	S	S	S	S

Typical optimization tasks:

Minimize maximal von Mises stress
Minimize maximal damage (fatigue analysis)

Equivalent stresses are always read for ALL_NODES to give the user the most appropriate output. This gives some warnings for nodes that do not have any equivalent stress values. For example, nodes on elements that are not in contact when evaluating SIG_CONTACT_... .

Analysis Types: Any Analysis Type with Contact and Strain Output

For equivalent strain and contact measures, the following table shows the allowed combinations between the strategy and the items OBJ_FUNC and CONSTRAINT with C for controller-based optimization and S for sensitivity-based optimization.


	TOPO	SHAPE	BEAD	SIZING
OBJ_FUNC		C
CONSTRAINT

Typical optimization tasks:

Minimize maximal strain in a given region
Minimize contact pressure in a contact region

Important:

Plastic strain values (TYPE = STRAIN_PLASTIC) are calculated by Tosca Structure and might differ slightly from your solver results.