Equivalent stress

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	$\|σ_{2}\|$
SIG_MISES	$\sqrt{\frac{1}{2} {{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{1} - σ_{3})}^{2}}}$
SIG_TRESCA	$max (\|σ_{1} - σ_{2}\|, \|σ_{2} - σ_{3}\|, \|σ_{3} - σ_{1}\|)$
SIG_BELTRAMI
SIG_GALILEI
SIG_KUHN
SIG_MARIOTTE
SIG_SANDEL	${[σ_{1}^{2} + σ_{2}^{2} + σ_{3}^{2} - 2 ν \frac{2 - ν}{1 + 2 ν^{2}} (σ_{1} σ_{2} + σ_{2} σ_{3} + σ_{1} σ_{3})]}^{\frac{1}{2}}$
SIG_SAUTER	$\sqrt[n]{{\| σ_{1} - σ_{2} \|}^{n} + {\| σ_{2} - σ_{3} \|}^{n} + {\| σ_{3} - σ_{1} \|}^{n}}$
SIG_DRUCKER_PRAGER
STRAIN_ENERGY	$c = \frac{1}{2} u^{T} K u$
STRAIN_ENERGY_DENSITY	$c = \frac{1}{2 V} u^{T} K u$
DAMAGE_LC	user / program dependent
DAMAGE	user / program dependent

Contact and Strain Measures

Parameter Name	Formula
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	${\frac{2}{3} [ϵ_{11}^{2} + ϵ_{22}^{2} + ϵ_{33}^{2} + 2 (ϵ_{12}^{2} + ϵ_{13}^{2} + ϵ_{23}^{2})]}^{\frac{1}{2}}$
ABQ_ND_PEEQ	PEEQ, see Abaqus documentation

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

({4 π}^{2} f^{2} M - K) φ = 0

For equivalent stress the following table shows the allowed combinations between strategy and the items OBJ_FUNC and CONSTRAINT with C for controller 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. Equivalent stress is only allowed in the objective function by controller-based shape optimization.

Typical optimization tasks:

Minimize maximal von Mises stress
Minimize maximal damage (fatigue analysis)
Minimize contact pressure in a contact region

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

Important:

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