Safety Factor Evaluation
There are many theories available in the literature for predicting material failure, these can predict, for instance whether a ductile material will yield or not, or if a brittle material will crack under a given set of loads.
Tsai and Wu (Ref. 7, Ref. 16) 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
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 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 to be positive only. These restrictions are summarized as (no summation of the indices)
and
The failure index is computed from the failure criterion as
so failure is predicted for a failure index greater than one, fi ≥ 1.
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 quadratic failure criterion, as the Tsai–Wu criterion, this means solving a quadratic equation for the safety factor variable sf
the safety factor is then obtained from the smallest positive root.
For an isotropic criterion, such as the von Mises criterion, g(σ) = σmisests − 1, and the safety factor is given by sf = σtsmises.
The margin of safety (Ref. 16) is then computed from the safety factor
Following the Tsai–Wu formalism, different orthotropic criteria can be defined by setting appropriate values for the coefficients in F and f tensors.
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, Layered Linear Elastic Material, or Nonlinear Elastic Materials.
The Anisotropic Tsai–Wu Criterion
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.
The Orthotropic Tsai–Wu Criterion
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 tensors are set to zero.
The Tsai–Hill Criterion
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. 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.
The Hoffman Criterion
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 tensors are set to zero.
The Jenkins Criterion
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 Criterion
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.
The Modified Tsai–Hill Criterion
This criterion is derived from the Tsai–Wu theory for two-dimensional plane stress problems (Ref. 17). 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 Tsai–Wu coefficients are then computed from
, or
, or
or
all the other coefficients in F and f tensors are set to zero.
The Azzi–Tsai–Hill Criterion
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 Tsai–Wu coefficients are then computed from
, or
, or
or
all the other coefficients in F and f tensors are set to zero.
The Norris Criterion
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 Tsai–Wu coefficients are then computed from
, or
, or
all the other coefficients in F and f tensors are set to zero.
The von Mises Criterion
The von Mises criterion is one of the simplest isotropic criteria to predict yielding in metals and other ductile materials. The failure criterion is computed from the isotropic tensile strength σts
The equivalent von Mises stress σmises is defined from the deviatoric stress tensor, see the section about plasticity and The von Mises Criterion. For ductile materials the tensile strength corresponds to the yield stress, while for brittle materials it corresponds to the failure strength.
The Tresca Criterion
Tresca criterion is similar to the von Mises criterion. For this isotropic criterion, the failure criterion is computed from the isotropic tensile strength σts
Here, the Tresca equivalent stress is defined in terms of principal stresses, σtresca = σ1 − σ3; see The 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
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
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 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
The Mohr–Coulomb criterion is similar to the Tresca criterion, as the failure criterion is given in terms of principal stresses, see The 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
and the failure index from
where
, , and
The cohesion and the angle of internal friction are related to the tensile and compressive strengths by the expressions
and
The Drucker–Prager Criterion
The Drucker–-Prager criterion approximates the Mohr–Coulomb criterion by a smooth function (a cone in the stress space), see The 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 in the Mohr–Coulomb criterion, see The Drucker–Prager Criterion for details. Also, the cohesion and the angle of internal friction can be related to the tensile and compressive strengths, see The Mohr–Coulomb Criterion for details. The failure index is computed from
The Bresler–Pister 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 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 The Bresler–Pister Criterion for details. The failure index is computed from
The Willam–Warnke Criterion
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 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 The Willam–Warnke Criterion for details. The failure index is computed from
The Ottosen Criterion
The Ottosen criterion is a five-parameter failure isotropic criterion developed to model short-time loading of concrete. It corresponds to a smooth convex failure surface with curved meridians
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 The Ottosen Criterion for details. The failure index is computed from
User Defined
This option allows you to write explicitly how the failure criterion and the safety factor depend on stress and/or strain. These could be analytic functions of stress or strain tensor components, principal stresses, principal strains, stress or strain invariants, or data interpolated from tables.
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.
Fiber Composite Specific Failure Criteria
Apart from Tsai–Wu family criteria, there are certain advanced composite specific criteria. These criteria account for multi-axial state of stress, different failure modes, and different criterion for tensile and compressive loading. For example, in fiber composites, failure can happen because of fiber rupture in tension or fiber buckling in compression, matrix failure in tension or matrix failure in compression.
Figure 3-28: The fiber and matrix failure modes and respective failure planes in a unidirectional composite.
The basic types of failure modes, under various loadings, in unidirectional composites are as follows (Figure 3-29)
Figure 3-29: Different fiber failure (FF) modes and interfiber failure (IFF) modes of a unidirectional composite under different loading conditions. The plane where the brittle fracture occurs is also shown.
The list of fiber composite specific failure criteria includes:
Table 3-7 shows failure criterion and considered failure modes by that criterion. .
Zinoviev Criterion
This criterion is 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 multi-directional laminates (Ref. 19). The approach is one of the most used due to its simplicity, and it is recommended by World Wide Failure Exercise. The failure criteria for different failure modes are as follows.
The longitudinal failure criterion in tension is
for
The longitudinal failure criterion in compression is
for
The transverse failure criterion in tension is
for
The transverse failure criterion in compression is
for
The in-plane shear failure criterion is
By picking up the most critical failure mode, the failure criterion for composite is written as
The Hashin–Rotem Criterion
This criterion is based on the first ply failure theory and designed for unidirectional laminates. The Hashin–Rotem failure theory considers effects of tensile and compressive stresses while predicting the failure of fiber or matrix, however it does not distinguish whether the failure is exactly at the interface or inside the matrix. This criterion involves four failure modes associated with fiber failure and the matrix failure, distinguishing between tension and compression.
The fiber failure criterion in tension is
for
The fiber failure criterion in compression is
for
The matrix failure criterion in tension is
for
The matrix failure criterion in compression is
for
By picking up the most critical failure mode, the failure criterion for composite is written as
The Hashin Criterion
The Hashin failure theory is extension of Hashin–Rotem failure theory with six failure modes including fiber failure, matrix failure and interlaminar failure, distinguishing between tension and compression (Ref. 18). Stress interactions are considered for the determination of the tensile fiber failure mode, tensile matrix failure mode and compressive matrix failure mode.
The fiber failure criterion in tension is
for
The fiber failure criterion in compression is
for
The matrix failure criterion in tension is
for
The matrix failure criterion in compression is
for
The interlaminar failure criterion in tension is
for
The interlaminar failure criterion in compression is
for
By picking up the most critical failure mode, the failure criterion for composite is written as
For plane stress conditions, the Hashin criterion is modified by setting σ13 σ23 =σ33  = 0. With 2D plane stress version of Hashin criterion, the interlaminar failure cannot be predicted.
Puck Criterion
The Puck criterion is based on 3-D phenomenological models, where the experimental results are matched with theoretical formulation. Based on fracture mechanics and experimental observation, three different failure criteria called as modes A, B and C for matrix failure are considered. Mode A corresponds to tensile loading, while mode B and C correspond to compressive and shear loading.
Figure 3-30: Interfiber failure (IFF) mode using Puck criterion.
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. 19). 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).
The fiber failure criterion in tension is
for
where Ef1 and νf12 are Young’s modulus of fiber in longitudinal direction and in-plane Poisson’s ratio of fiber, respectively, and mσf is the mean stress magnification factor.
The fiber failure criterion in compression is
for
The failure criterion for interfiber failure mode A is
for
where ptl is slope of the in-plane fracture envelope in tension and σ1D is linear degradation stress.
The failure criterion for interfiber failure mode B is
for and
where pcl is slope of the in-plane fracture envelope in compression, RAtt is fracture resistance against transverse shear loading and σcss12 is modified shear strength.
The failure criterion for interfiber failure mode C is
for and
where pct is slope of the transverse fracture envelope in compression.
By picking up, the most critical failure mode, the failure criterion for composite is written as
LARC-03 Criterion
This criterion is used for accurate predicting the failure of unidirectional FRP laminates with in-plane stress state. This is one of the most advanced failure criteria composed of six phenomenological failure modes describing matrix and fiber failure accurately without the use of curve-fitting parameters (Ref. 20). This criterion assumes a fragile fracture for the matrix failure in compression and 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.
The matrix failure criterion under transverse compression (LaRC03-1) is
for
where τeff,t and τeff,l are effective shear stresses in transverse and longitudinal directions, respectively, and σiss12 is 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.
The matrix failure criterion under transverse tension (LaRC03-2) is
for
where σits2 is in situ tensile strength, and r is a material constant based on fracture toughness.
The fiber failure criterion under longitudinal compression (LaRC03-3) is
for
The fiber failure criteria with matrix tension (LaRC03-4) is
for
The fiber failure criteria with matrix compression (LaRC03-5) is
for
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.
The matrix failure criterion under biaxial compression (LaRC03-6) is
for and  
where the effective shear stresses in transverse and longitudinal directions, τmeff,t and τmeff,l, are calculated from stresses in the misalignment coordinate frame.
By picking up the most critical failure mode, the failure criterion for composite is written as