Governing differential equation for a general laminate:
Cross Ply Laminate:
A laminate in which the ply orientations are oriented at right angles to each other, with ply orientations limited to 0° and 90° only. It is usually best to arrange stacking sequences with fibers oriented in different directions.
Angle Ply Laminate:
One possessing equal plies with positive and negative angles. This is a bidirectional orthotropic laminate, such as a [±45°].
Composite Failure Criteria:
Failure criteria for composite materials are significantly more complex than yield criteria for metals because composite materials can be strongly anisotropic and tend to fail in a number of different modes depending on their loading state and the mechanical properties of the material. While theories which reflect detailed mechanisms of failure are currently being developed, empirical criteria based on test data have been used for decades. These criteria have been incorporated in the MSC.Laminate Modeler to allow rapid evaluation of the strength of a structure according to the current industry standards. The user can also define custom criteria using PCL functions for use in specialized applications.
Nomenclature:
Failure criteria compare the loading state at a point (stress or strain) with a set of values reflecting the strength of the material at that point (often referred to as the material allowable). Both loading and strength values should be reflected in the same material coordinate system. For unidirectional materials, this is typically in the direction of the fibres. However, for woven and knitted fabrics, this direction is not obvious, and might change as the material is formed to shape.
In general, the load is represented by a full stress or strain tensor having six independent components. By convention, for lamina materials the material X axis lies in the direction of the warp fibres while the Z axis lies in the throughthickness direction of the sheet. Note than in Patran, shear strains are stored in tensor rather than engineering notation, and any experimental failure strengths should reflect this.
STRESS 
σ_{x},σ_{y},σ_{z},τ_{xy},τ_{yz},τ_{xz} 
STRAIN 
ε_{x},ε_{y},ε_{z},γ_{xy},γ_{yz},γ_{xz} 
The strength of a composite can be expressed by an arbitrarily large number of values, depending on the complexity of the failure criterion. However, lamina materials, used in composites, are often assumed to be orthotropic; the through‑thickness stresses or strains are ignored and it is assumed that there is negligible interaction between the different failure modes. The strength of the material can therefore be represented by seven independent variables:
TX 
tensile strength along the X axis 
0 < TX 
CX 
compressive strength along the X axis 
0 < CX 
TY 
tensile strength along the Y axis 
0 < TY 
CY 
compressive strength along the Y axis 
0 < CY 
SXY 
shear strength in the XY plane 
0 < SXY 
SYZ 
shear strength in the YZ plane 
0 < SYZ 
SXZ 
shear strength in the XZ plane 
0 < SXZ 
In the TsaiWu criterion, these values have been supplemented by an interaction term which reflects the interdependence of failure modes due to loading along both the X and Y material directions.
IXY 
interaction between X and Y directions 
1< IXY <1 
Note that the above values can be applied to either stress or strain.
The form of the failure criterion is typically described as a mathematical function of the above variables which reaches the value of unity at failure as follows.
Failure Index = FI (load, strength) = 1
The strength of a structure can be given as a Strength Ratio (SR), which is the ratio by which the load must be factored to just fail. (Note that the Strength Ratio is not necessarily the reciprocal of the Failure Index.) Alternatively, the Margin of Safety (MoS), where MoS = SR – 1, is used.
Maximum Criterion
This criterion is calculated by comparing the allowable load with the actual strength for each component. Mathematically, it is defined by:
FI = max (σ_{x}/TX, σ_{x}/CX, σ_{y}/TY, σ_{y}/CY,
abs(τ_{xy})/SXY, abs(γ_{yz})/SYZ, abs(γ_{xz})/SXZ)
In this case,
SR = 1/FI
Hill Criterion
The Hill criterion was one of the first attempts to develop a single formula to account for the widely different strengths in the various principal directions:
FI = FXX σ_{x}^{2} + FYY σ_{y}^{2} + 2 FXY σ_{x} σ_{y} + FSS τ_{xy}^{2}
where
FXX = 
1/(TX TX) 
if σ_{x} >= 0 
1/(CX CX) 
if σ_{x} <0 

FYY = 
1/(TY TY) 
if σ_{y} >= 0 
1/(CY CY) 
if σ_{y} <0 

FXY = 
1/(2 TX TX) 
if σ_{x}σ_{y} >= 0 
1/(2 CX CX) 
if σ_{x}σ_{y} <0 

FSS = 
1 / (SXY SXY) 
Because this failure theory is quadratic:
SR = 1 / sqrt (FI)
In the Laminate Modeler, the TsaiWu criterion for inplane loads (representing fiber failure) has been supplemented by a maximum load theory for outofplane shear loads (representing matrix failure):
FI = max( abs(γ_{yz})/SYZ, abs(γ_{xz})/SXZ )
In this case,
SR = 1/FI
For every ply, the lower of the Margins of Safety for fibre and matrix failure is calculated and displayed.
TsaiWu Criterion
The TsaiWu failure criterion is an unashamed, empirical criterion based on the sum of the linear and quadratic invariants as follows:
F_{i} σ_{i} + F_{ij} σ_{i} σ_{j} = 1 
i,j = 1…6 
where F_{i} and F_{ij} are dependent on the material strengths. For the restrictions of lamina materials, this equation reduces to:
FI = FX σ_{x} + FY σ_{y} + FXX σ_{x}^{2} + FYY σ_{y}^{2} + 2 FXY σ_{x} σ_{y} + FSS τ_{xy}^{2}
where:
FX = 1/TX – 1/CX
FY = 1/TY – 1/CY
FXX = 1/(TX CX)
FYY = 1/(TY CY)
FXY = IXY sqrt(FXX FYY) = IXY / sqrt(TX CX TY CY)
FSS = 1 / (SXY SXY)
Because this failure theory is quadratic, the Strength Ratio (SR) = 1/FI. However, multiplying the failure criterion by SR and rearranging gives
a SR^{2 }+ b SR – 1 = 0
where
a = FXX σ_{x}^{2} + FYY σ_{y}^{2} + 2 FXY σ_{x} σ_{y} + FSS τ_{xy}^{2}
b = FX σ_{x} + FY σ_{y}
Therefore
SR = [b + sqrt (b^{2} + 4a)] / 2a
In the Laminate Modeler, the TsaiWu criterion for inplane loads (representing fiber failure) has been supplemented by a maximum load theory for outofplane shear loads (representing matrix failure):
FI = max( abs(γ_{yz})/SYZ, abs(γ_{xz})/SXZ )
In this case,
SR = 1/FI
For every ply, the lower of the Margins of Safety for fibre and matrix failure is calculated and displayed.