Tsai and Wu (Ref. 13, Ref. 22) proposed a stress-dependent criterion intended at modeling failure in composites. Under Tsai-Wu criterion, failure occurs when a given quadratic function of stress is grater than zero. The failure criterion is given by

here, σ is the stress tensor, F a fourth rank tensor (SI unit: 1/Pa^2) and f is a second rank tensor (SI unit: 1/Pa). For Tsai-Wu criterion, failure occurs when g(σ) ≥ 0.

Due to the symmetry of these tensors, the fourth rank tensor can be represented by a symmetric 6-by-6 matrix, and the second rank tensor by a 6-by-1 vector (see Voigt order in the section Tensor vs. Matrix Formulations).

Certain constraints ensure that the failure surface g(σ) = 0 forms a closed ellipsoid in the stress space. Also, thermodynamic considerations restrict the value of some components of the fourth rank tensor to be positive only. These restrictions are summarized as (no summation of the indexes)

The failure index is computed from the failure criterion as

The damage index is given by a boolean expression based on the failure criterion

here di = 1 means damage, and di = 0 represents a healthy material.

The safety factor, also called reserve factor or strength ratio, is computed by scaling the stress tensor such as the failure criterion is equal to zero

For a isotropic criteria, such as von Mises criterion, g(σ) = σmises/σts − 1, and the safety factor is given by sf = σts/σmises.

The margin of safety (Ref. 23) is then computed from the safety factor

Use the Safety subnode to set up variables which can be used to check the risk of failure according to various criteria. It can be used in combination with Linear Elastic Material or Nonlinear Elastic Materials.

Following Tsai-Wu formalism, different orthotropic criteria can be defined by setting appropriate values for the coefficients in F and f tensors.

For this criterion, enter twenty one coefficients to define the 6-by-6 matrix F, and six coefficients to define the vector f. The failure criterion is evaluated from the expression

here, σij are the stress tensor components given in the local coordinate system of the parent node.

For this criterion, enter nine coefficients corresponding to the tensile strengths σtsi, compressive strengths σcsi, and shear strengths σssij given in the local coordinate system of the parent node. Tsai-Wu coefficients are then computed from

all the other coefficients in F and f tensors are set to zero.

For this criterion, enter six coefficients corresponding to the tensile strengths σtsi and shear strengths σssij given in the local coordinate system of the parent node. Tsai-Wu coefficients are then computed from

all the other coefficients in F and f tensors are set to zero. See also Hill Orthotropic Plasticity.

For this criterion, enter nine coefficients corresponding to the tensile strengths σtsi, compressive strengths σcsi, and shear strengths σssij given in the local coordinate system of the parent node. Tsai-Wu coefficients are then computed from

all the other coefficients in F and f tensors are set to zero.

Enter nine coefficients corresponding to the tensile strengths σtsi, compressive strengths σcsi, and shear strengths σssij given in the local coordinate system of the parent node. The failure criterion is then computed from

here, εsi is either the tensile strength or the compressive strength depending whether the stress in the i-direction, σi, is positive or negative. The absolute value of the shear stress σij in the ij-plane is compared to the corresponding shear strength σssij.

Waddoups criterion is similar to Jenkins criterion, but the failure criterion is given in terms of strains, not strengths. For this criterion, enter nine coefficients corresponding to the ultimate tensile strains εtsi, ultimate compressive strains εcsi, and ultimate shear strains γssij given in the local coordinate system of the parent node. The failure criterion is then computed from

here, εsi is either the ultimate tensile strain or the ultimate compressive strain depending whether the strain in the i-direction, εi, is positive or negative. The absolute value of the shear strain γij in the ij-plane is compared to the corresponding ultimate shear strain γssij.

This criterion is derived from Tsai-Wu theory for two-dimensional plane stress problems (Ref. 23). It is available in 2D for the Pate interface and the Solid Mechanics interface in plane stress, and for the Shell interface in 3D. Enter the coefficients corresponding to the tensile strengths σtsi, compressive strengths σcsi, and shear strengths σssij given in the local coordinate system of the parent node. The failure criterion is then computed from the in plane stresses

all the other coefficients in F and f tensors are set to zero.

This criterion is derived from Tsai-Wu theory for two-dimensional plane stress problems. It is available in 2D for the Plate interface and the Solid Mechanics interface in plane stress, and for the Shell interface in 3D. Enter the coefficients corresponding to the tensile strengths σtsi, compressive strengths σcsi, and shear strengths σssij given in the local coordinate system of the parent node. The failure criterion is then computed from the in plane stresses

all the other coefficients in F and f tensors are set to zero.

This criterion is derived from Tsai-Wu theory for two-dimensional plane stress problems. It is available in 2D for the Plate interface and the Solid Mechanics interface in plane stress, and for the Shell interface in 3D. Enter the coefficients corresponding to the tensile strengths σtsi, compressive strengths σcsi, and shear strengths σssij given in the local coordinate system of the parent node. The failure criterion is then computed from the in plane stresses

all the other coefficients in F and f tensors are set to zero.

The effective von Mises stress σmises is defined from the deviatoric stress tensor, see the section about plasticity and von Mises Criterion. For ductile materials the tensile strength corresponds to the yield stress, while for brittle materials it corresponds to the failure strength.

here, Tresca effective stress is defined in terms of principal stresses, σtresca = σ1 − σ3, see Tresca Criterion. For ductile materials the tensile strength corresponds to the yield stress, while for brittle materials it corresponds to the failure strength.

St. Venant criterion is similar to Tresca criterion, as the failure criterion is given in terms of principal stresses. For this criterion, enter the tensile strength σts, and the compressive strength σcs. The failure criterion is then computed from

here, σs is either the tensile strength or the compressive strength depending whether the principal stress, σpi, is positive or negative. For ductile materials the tensile strength corresponds to the yield stress, while for brittle materials it corresponds to the failure strength.

The St. Venant criterion is similar to the Waddoups criterion, as the failure criterion is given in terms of strains, not strengths. For this criterion, enter the ultimate tensile strains εts, and the ultimate compressive strains εcs. The failure criterion is then computed from

here, εs is either the ultimate tensile strain or the ultimate compressive strain depending whether the principal strain, εpi, is positive or negative. For ductile materials the ultimate tensile strain corresponds to the strain at yielding, while for brittle materials it corresponds to the strain at failure.

The Mohr-Coulomb criterion is similar to the Tresca criterion, as the failure criterion is given in terms of principal stresses, see Mohr-Coulomb Criterion for soil plasticity. For this criterion, enter the cohesion c, and the angle of internal friction φ. The failure criterion is then computed from

The Drucker-Prager criterion approximates the Mohr-Coulomb criterion by a smooth function (a cone in the stress space), see Drucker-Prager Criterion for soil plasticity. The failure criterion is computed from the stress invariants I1 and J2, and two material parameters, α and k,

The material parameters α and k are related to the cohesion c and angle of internal friction φ in the Mohr-Coulomb criterion, see Drucker-Prager Criterion for details. Also, the cohesion and the angle of internal friction can be related to the tensile and compressive strengths, see Mohr-Coulomb Criterion for details. The failure index is computed from

The Bresler-Pister criterion was originally devised to predict the strength of concrete under multiaxial stresses. This failure criterion is an extension of the Drucker-Prager Criterion to brittle materials. The failure criterion is computed from the stress invariants I1 and J2, and three parameters, k1, k2, and k3,

The parameters k1, k2, and k3 are computed from the uniaxial compressive strength σc, the uniaxial tensile strength σt, and the biaxial compression strength σb, see Bresler-Pister Criterion for details. The failure index is computed from

The Willam-Warnke criterion is used to predict failure in concrete and other cohesive-frictional materials such as rock, soil, and concrete. Just as the Bresler-Pister Criterion, failure is computed from the stress invariants I1 and J2, and the Lode angle θ, and three material parameters

here, σc is the uniaxial compressive strength, σt is uniaxial tensile strength, and σb is the biaxial compressive strength. The function r(θ) describes the segment of an ellipse on the octahedral plane, see Willam-Warnke Criterion for details. The failure index is computed from

In this formulation, the parameters a and b are positive and dimensionless, and σc is the uniaxial compressive strength for concrete (also with a positive sign). The dimensionless function λ(θ) depends on the Lode angle θ and two positive parameters k1 and k2, see Ottosen Criterion for details. The failure index is computed from

You can add any number of Safety nodes to a single material model. The contents of this feature will not affect the analysis results as such, as this feature does not account for post-failure analysis. You can add Safety nodes after having performed an analysis and just do an Update Solution in order to access to the new variables for result evaluation.