COMSOL 6.4 - Parametric Study on Cross-Ply Laminate Failure

Parametric Study on Cross-Ply Laminate Failure

Laminated composite shells made from graphite–epoxy fiber-reinforced plies are widely used in many applications due to their high strength-to-weight ratio. Evaluating their structural integrity under applied loads is essential for ensuring reliable designs.

This example demonstrates how to perform failure analysis of a laminated composite shells using the interface. The laminate is analyzed under multiple load cases and loaded to failure, considering in-plane stress states. Both classical and advanced failure criteria are described and compared.

This example analyzes a cross-ply laminate subjected to monotonic tension, monotonic compression, and biaxial tension to determine the failure loads as a function of the fiber angle. This work is inspired by a similar study presented in Ref. 1.


	Read more about the Composite Materials Module in COMSOL blog, Introduction to the Composite Materials Module.

Model Definition

The geometry is a rectangular plate measuring 450 mm in length and 400 mm in width. The laminate features a symmetric stacking sequence of four layers, each with a thickness of 0.10 mm. The fiber orientation of the cross-ply laminate is parameterized as illustrated in Figure 1.

Figure 1: Stacking sequence from bottom to top, showing fiber orientation in each ply.

Material Properties

All material properties and strengths are given in the layer coordinate system, where the first axis is aligned with the fiber orientation. The material parameters and failure coefficients are summarized in Table 1 and Table 2.

Table 1: Elastic material parameters.
Elastic parameters	Value
{E1, E2}	{130.0, 11.0} GPa
G12	5.0 GPa
{υ12, υ23}	{0.280, 0.350}

Table 2: Failure coefficients.
Failure coefficients	Value
{σts1, σts2, σts3}	{1700, 70.5, 70.5} MPa
{σcs1, σcs2, σcs3}	{1050, 240, 240} MPa
{σss23, σss13, σss12}	{96,90,90} MPa

Boundary Conditions

The laminate is subjected to three different load cases: monoaxial compression, monoaxial tension, and biaxial tension. For monoaxial compression and tension, Edge 1 is fixed in the X and Z directions, and a boundary load is applied to Edge 3. For the biaxial tension case, Edge 1 is fixed in the X and Z directions, Edge 4 is fixed in the Y and Z directions, and a boundary load is applied to Edge 2 and Edge 3.

Failure Criteria

Three failure criteria are employed to determine the failure loads: LaRC03, Tsai–Hill, and Hashin. The latter two are applied in their plane-stress formulations

LaRC03

This criterion is used to accurately predict the failure of unidirectional FRP laminates under in-plane stress. It is one of the most advanced failure models, comprising six phenomenological failure modes that describe matrix and fiber failure without relying on curve-fitting parameters. The model assumes brittle fracture for matrix compression failure and applies the action-plane concept from Mohr–Coulomb theory. It accounts for fiber kinking due to misalignment and tensile matrix cracking associated with interlaminar crack propagation.

The matrix failure criterion under transverse compression (LaRC03-1) is

for

and

where τeff,t and τeff,l are the effective shear stresses in the transverse and longitudinal directions, respectively, σiss12 is the in situ in-plane shear strength, σss23 is the shear strength in the transverse direction, and σcs2 is the compressive strength in the transverse direction. The effective shear stresses are functions of the fracture plane angle which is found by maximizing the Mohr–Coulomb effective stresses.

The matrix failure criterion under transverse tension (LaRC03-2) is

for

where σits2 is the in situ transverse tensile strength, and r is a dimensionless parameter based on fracture toughness.

The fiber failure criterion under longitudinal tension (LaRC03-3) is

for

The fiber failure criterion under matrix tension (LaRC03-4) is

for

The fiber failure criterion under matrix compression (LaRC03-5) is

for

where σmij are the ply stresses transformed to 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 the transverse and longitudinal directions, τmeff,t and τmeff,l, are calculated from the stresses in the misaligned coordinates.

The failure criterion for the composite is then computed by selecting the most critical failure mode

Tsai–Hill

The Tsai-Hill criterion follows a polynomial description

where σi is the 6-by-1 stress vector (sorted using Voigt notation), Fij is a 6-by-6 symmetric matrix that contains the coefficients for the quadratic terms. For the plane stress case, many of the coefficients are zero except for the following:

, or

Hashin

The Hashin failure theory is an extension of the Hashin–Rotem model that incorporates six failure modes, including fiber, matrix, and interlaminar failure, with distinctions between tension and compression. Stress interactions are considered in determining the tensile fiber failure mode, tensile matrix failure mode, and compressive matrix failure mode.

The fiber failure criterion in tension is defined by

for

The fiber failure criterion in compression is defined by

for

The matrix failure criterion in tension is defined by

for

The matrix failure criterion in compression is defined by

for

The interlaminar failure criterion in tension is defined by

for

The interlaminar failure criterion in compression is defined by

for

The failure criterion for the composite is computed by selecting the most critical failure mode

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

Results and Discussion

The mechanical loading of the laminate is systematically increased until damage is detected, at which point the load is defined as the failure load. For each load case in this study, the plots in Figure 2, Figure 3, and Figure 4 show the failure loads for the three failure criteria as functions of the fiber angle.

Figure 2: Monoaxial compression test.

Figure 3: Monoaxial tension test.

Figure 4: Biaxial tension test.

Notes About the COMSOL Implementation

•

Modeling a composite laminate as a layered shell requires a surface geometry, in general referred to as a base surface, and a node which adds an extra dimension (1D) to the base surface geometry in the surface normal direction. You can use the functionality to model several layers stacked on top of each other having different thicknesses, material properties, and fiber orientations. You can optionally specify the interface materials between the layers, and control the number of through-thickness mesh elements for each layer.

•

The and have an option to transform given into a symmetric or antisymmetric laminate. A repeated laminate can also be constructed using a transform option.

•

The third direction for the selected coordinate system in the , , or represents the normal direction of the or physics. This is also the direction in which the layer stacking is interpreted from bottom to top, and therefore, it is crucial to know it during modeling. There are two ways to achieve this:

Using physics symbols: Go to the physics settings, find the section, and select the checkbox. Then go to the material feature, for instance, , to see the normal direction represented by green arrows in the geometry.

Using result templates: When a solution dataset is available, use the result template to plot the normal direction.

•

From a constitutive model point of view, you can either use the Layerwise (LW) theory based interface, or the Equivalent Single Layer (ESL) theory based node in the interface. The laminated composite presented in the current model is modeled using the interface.

•

Since we are only interested in the failure loads, the node is included in our solver configurations to prevent the load from increasing further once damage is detected at the current fiber angle.

Reference

1. P. Nali and E. Carrera, “A numerical assessment on two-dimensional failure criteria for composite layered structures,” Composites Part B: Engineering, vol. 43, no. 2, pp. 280–289, 2012; doi.org/10.1016/j.compositesb.2011.06.018.

Composite_Materials_Module/Verification_Examples/parametric_study_on_crossply_laminate_failure

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file parametric_study_on_crossply_laminate_failure_parameters.txt.

Define the load case parameters.

Load Case Parameters

In the toolbar, click

and choose > .

In the window for , type Load Case Parameters in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
xLoad	1	1	Load magnitude in x-direction
yLoad	0	0	Load magnitude in y-direction

In any node in the Model Builder, you can add comments to explain the settings. Right click on the node to select the option to add the comment.

In the toolbar, click

In the window for , type Monoaxial Compression in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Description
xLoad	-1	Load magnitude in x-direction

In the toolbar, click

In the window for , type Monoaxial Tension in the text field.

In the toolbar, click

In the window for , type Biaxial Tension in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Description
yLoad	1	Load magnitude in y-direction

In this model, three different failure criteria will be applied to determine failure. Define the relevant parameters to create safety cases for each criterion and to establish the stop condition in the study node.

Safety Case

In the toolbar, click

and choose > .

In the window for , type Safety Case in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
safety1	1	1	Used to turn on the LaRC03 criterion in the stop condition
safety2	0	0	Used to turn on the Tsai-Hill criterion in the stop condition
safety3	0	0	Used to turn on the Hashin criterion in the stop condition

In the toolbar, click

In the window for , type LaRC03 in the text field.

In the toolbar, click

In the window for , type Tsai-Hill in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Description
safety1	0	Used to turn on the LaRC03 criterion in the stop condition
safety2	1	Used to turn on the Tsai-Hill criterion in the stop condition

In the toolbar, click

In the window for , type Hashin in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Description
safety1	0	Used to turn on the LaRC03 criterion in the stop condition
safety3	1	Used to turn on the Hashin criterion in the stop condition

Unidirectional Fiber Composite

In the window, under right-click and choose .

In the window, click .

In the window for , click to expand the section.

In the tree, select > .

Click

In the text field, type Unidirectional Fiber Composite.

Locate the section. In the tree, select > > .

Click

In the tree, select > > .

Click

In the window, under > > click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Expression	Unit	Size
Density	rho	Rho	kg/m³	1x1

In the window, under > > click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Expression	Unit	Size
Young’s modulus	{Evect1, Evect2}	{E1, E2}	Pa	2x1
Poisson’s ratio	{nuvect1, nuvect2}	{nu12, nu23}	1	2x1
Shear modulus	Gvect1	G	N/m²	1x1

In the window, under > > click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Expression	Unit	Size
Tensile strengths	{sigmats1, sigmats2, sigmats3}	{XT, YT, YT}	Pa	3x1
Compressive strengths	{sigmacs1, sigmacs2, sigmacs3}	{XC, YC, YC}	Pa	3x1
Shear strengths	{sigmass1, sigmass2, sigmass3}	{ST, SL, SL}	Pa	3x1
Ultimate tensile strain in longitudinal direction	epsilont1	epsT11	1	1x1

Now add a node and assign appropriate thickness and rotation angles to each layer. The laminate is symmetric. It is sufficient to define only half the laminate in the node. The transformation into a full laminate is performed through the layered material settings in the node.

Laminate

In the window, right-click and choose .

In the window for , type Laminate in the text field.

Locate the section. In the table, enter the following settings:


Layer	Material	Rotation	Value	Thickness	Mesh elements
Layer 1	Unidirectional Fiber Composite (mat1)	theta	0.7854 rad	th	1

Click

In the table, enter the following settings:


Layer	Material	Rotation	Value	Thickness	Mesh elements
Layer 2	Unidirectional Fiber Composite (mat1)	-theta	-0.7854 rad	th	1

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click

Work Plane 1 (wp1) > Plane Geometry

In the window, click .

Work Plane 1 (wp1) > Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type H.

Click

Definitions

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Define the stop conditions in the node.

Variables 1

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
stopCondition1	safety1*aveop1(lshell.xdintopall(lshell.lemm1.sf1.d_i))>eps		Stop condition for LaRC03 criterion
stopCondition2	safety2*aveop1(lshell.xdintopall(lshell.lemm1.sf2.d_i))>eps		Stop condition for Tsai-Hill criterion
stopCondition3	safety3*aveop1(lshell.xdintopall(lshell.lemm1.sf3.d_i))>eps		Stop condition for Hashin criterion