Tsai and Wu (Ref. 94, Ref. 107) proposed a stress-dependent criterion intended at modeling failure in composites. Under the Tsai–Wu criterion, failure occurs when a given quadratic function of stress is greater than zero. The failure criterion is given by

where, σ is the stress tensor, F a fourth rank tensor (SI unit: 1/Pa2) and f is a second rank tensor (SI unit: 1/Pa). For the 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. These restrictions are summarized as (no index summation)

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

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

The margin of safety (Ref. 107) is then computed from the safety factor sf as

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

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, Linear Elastic Material, Layered, or Nonlinear Elastic Materials.

Add any number of Safety subnodes to a single material model; the contents of these features will not affect the analysis results because they do not account for postfailure analysis.

Add Safety subnodes after having performed an analysis and just do an Update Solution in order to access to the new variables for results evaluation.

Add a Safety subnode to calculate the safety factor for a specified criterion. When multiple Safety subnodes are added, additional built-in variables become available. These additional variables—combined safety factor, combined failure index, combined damage index, and combined margin of safety—offer a conservative approach to safety assessment.

For this anisotropic criterion, enter 21 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 orthotropic 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. The Tsai–Wu coefficients are then computed from

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

For this orthotropic criterion, enter six coefficients corresponding to the tensile strengths σtsi and shear strengths σssij given in the local coordinate system of the parent node. The equivalent coefficients for the Anisotropic Tsai–Wu Criterion are then computed from

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

and the F12 term is modified as follows

For this orthotropic 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. The equivalent coefficients for the Anisotropic Tsai–Wu Criterion are then computed from

For Jenkins orthotropic 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. 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.

The Waddoups orthotropic criterion is similar to the 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 the Tsai–Wu theory for two-dimensional plane stress problems. It is available in 2D for the Plate interface, for 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

This criterion is derived from the 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

The equivalent 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, the Tresca equivalent 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.

The Rankine criterion is similar to the Tresca criterion, as the failure criterion is given in terms of principal stresses. For this isotropic 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 isotropic 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 on 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 isotropic 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 isotropic 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

Griffith criterion (Ref. 108) is intended to study the fracture of brittle materials.

The convex failure surface is given in terms of the principal stresses, the compressive strength, σcs, and the tensile strength, σts

where the parameter m is computed from the ratio

Griffith’s criterion is a paraboloids that opens into the compressive region in the stress space. Authors in Ref. 108 combined it with a Rankine Criterion to avoid tensile stresses higher than the tensile strength σts, see Combined Failure Criterion.

The Bresler–Pister criterion was originally devised to predict the strength of concrete under multiaxial stresses. This isotropic failure criterion is an extension of 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 σcs, the uniaxial tensile strength σts, and the biaxial compressive strength σbc, see Bresler–Pister Criterion for details. The failure index is computed from

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

here, σcs is the compressive strength, σts is tensile strength, and σbc 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 σcs is the 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

To replicate the von Mises Isotropic criterion with a tensile strength of 350 MPa, define g(S) as solid.mises/350[MPa]-1 and sf(S) as 350[MPa]/(solid.mises+eps).

The Safety subnode calculates the failure index, damage index, safety factor, and margin of safety for a specified criterion. When multiple Safety subnodes are added, additional built-in variables become available. The combined safety factor, combined failure index, combined damage index, and combined margin of safety, offer a conservative approach to safety assessment for multiple failure criteria.

The combined failure criterion is estimated by selecting the most critical failure criterion for a given multiaxial stress state

The combined failure index is computed accordingly

The combined damage index is given by a Boolean expression based on the combined failure index

where dicom = 1 represents a damaged material, and dicom = 0 represents a healthy material.

The combined safety factor is obtained from the smallest safety factor among the added criteria

The combined margin of safety is then computed from the combined safety factor sf,com as

The basic types of failure modes in unidirectional composites are (Figure 3-36)

Table 3-8 shows failure criterion and considered failure modes by that criterion.

This criterion was developed from a maximum stress failure theory to a well-structured set of noninteracting criteria to identify failure modes. The theory, in general, gives reasonably good failure envelopes for unidirectional laminates and a good fit to the experimental final failure envelopes for multidirectional laminates (Ref. 111). The approach is one of the most used due to its simplicity. The failure criteria for different failure modes are defined as follows.

The Hashin failure theory is an extension of Hashin–Rotem failure theory, as it includes six failure modes for fiber failure, matrix failure and interlaminar failure (Ref. 110). Stress interactions are considered for the determination of the tensile fiber failure mode, tensile matrix failure mode and compressive matrix failure mode.

The plane stress version of the Hashin criterion is obtained by setting σ13 = σ23 = σ33 = 0, however, the interlaminar failure cannot be predicted.

The Puck criterion is mainly used for predicting strength of unidirectional laminate and for predicting the initial strength of multidirectional laminates, for which the other methods do not predict the failure correctly (Ref. 111). This criterion is also recommended by World Wide Failure Exercise. It is distinguishing and treating separately failure criteria the fiber failure (FF) and interfiber failure (IFF).

where Ef1 is the Young’s modulus of the fiber in the longitudinal direction, νf12 is the in-plane Poisson’s ratio of the fiber, and mσf is the mean stress magnification factor.

where ptl is the slope of the in-plane fracture envelope in tension, and σ1D is the linear degradation stress.

where pcl is the slope of the in-plane fracture envelope in compression, RAtt is the fracture resistance against transverse shear loading, and σcss12 is the modified shear strength.

where pct is the slope of the transverse fracture envelope in compression.

This criterion is used for accurate predicting the failure of unidirectional FRP laminates with in-plane stress state. The criterion is composed of six phenomenological failure modes describing matrix and fiber failure accurately without the use of curve-fitting parameters (Ref. 112), and it assumes a fragile fracture for the matrix failure in compression. This criterion implements the action plane concept according to the Mohr–Coulomb theory. This failure theory considers failure modes based on the fiber kinking due to misalignment and on the tensile matrix cracking associated with interlaminar crack propagation.

where τeff,t and τeff,l are effective shear stresses in transverse and longitudinal directions, respectively, and σiss12 is the longitudinal in situ shear strength. The effective shear stresses are functions of the fracture plane angle which is found out by maximizing the Mohr–Coulomb effective stresses.

where σits2 is in situ tensile strength, and r is a material constant based on fracture toughness.

where σmij are the ply stresses transformed in the misalignment coordinate frame, and ηl is a nondimensional parameter based on the failure strength and fracture plane angle under uniaxial transverse compression.

where the effective shear stresses in transverse and longitudinal directions, τmeff,t and τmeff,l, are calculated from stresses in the misalignment coordinate frame.