Finite Element Analysis of Composite Layered Structures

Finite Element Analysis of Composite Layered Structures

___________________________________

A B S T R A C T

A complete formulation of three-dimensional finite element analysis for multi-layered structures was examined using MATLAB. Throughout the investigation, standard weak form expressions, model discretization, local node-numbering, global matrix partitioning, and solution procedures were followed in modeling a uniaxially-loaded, symmetric laminate. Modeling each discretized element as a tri-linear element, the effect of mesh refinement was observed to be significant, and agreed well with the previously hypothesized possibility of a stress singularity. Furthermore, the results obtained from the three-dimensional analysis were compared with that of Classical Laminated Plate Theory, as well as ANSYS. For code validation purposes, a force-reaction balance was performed a posteriori to verify that it was implemented correctly.

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A U T H O R S

Connor Kaufmann

Neola Putnam

Ethan Seo

Ju Hwan (Jay) Shin

___________________________________

Sibley School of Mechanical & Aerospace Engineering

Cornell University, Ithaca, NY, 14853

Submitted on May 1, 2014 to Prof. N. Zabaras

Shin et al.

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1. Introduction

Composite materials are known to be more favorable for optimizing weight of the structure than monolithic materials. This is due to the anisotropic nature of the fiber-reinforced laminate, which allows engineers to utilize the composite materials without overdesigning in certain orientations. While these multidirectional laminates were shown to be effective at reducing the overall payload of the structure, they introduced a new level of complexity in the proper understanding of their mechanics and led to investigative research in this field.

One particular area of such research is devoted to the study of failure by delamination (Fig. 1). Of the many mechanisms by which a layered structure fails due to delamination, one notable cause of this is the development of interlaminar stresses. Interlaminar stresses are out-of-plane stresses, such as 𝜎𝑧𝑧 , 𝜏𝑦𝑧 , and 𝜏𝑥𝑧 , that are induced at the ply

interface near the free-edges of a laminated structure. Though completely neglected in the framework of 2-D CLPT, or Classical Laminated Plate Theory, these through-thickness stresses can be quite severe in some cases, as there can exist a state of stress singularity from the abrupt mismatch of the Poisson’s effects along the layers. The present study sought to confirm this behavior numerically by performing a complete three-dimensional finite element analysis, as the Kirchhoff-Love’s assumptions used in CLPT was limited to estimating the membrane stresses only, or in-plane stresses. It is noted, however, that to model the physical mechanics with even greater accuracy, one should take into account the material nonlinearity in the analysis. Material nonlinearity plays an important role in failure by delamination because failure is often not governed only by the magnitude of the interlaminar stress (which is infinite in linear elastic theory). Nonlinearity of material properties causes the crack tip, formed at the onset of progressive fracture to experience a blunting effect at the material’s yield point. Nevertheless, it is worth acknowledging that this singular stress is important for understanding the failure mechanism of a laminated structure.

Fig. 1: Failure due to delamination

2. FEM Formulation

In developing the FEM code, the analysis was divided into three main stages: pre-processing, processing, and post-processing. Using MATLAB, any necessary inputs, including the anisotropic material properties, the applied load, the laminate configuration, and the laminate dimensions, were specified. Following the standard solution procedures to be discussed in

more detail in the following paragraphs, the nodal displacements, strain field, and the stress field were outputted.

In the pre-processing stage, the model was discretized into finite elements. In doing so, bias-factor was considered to allow for a greater mesh density (Fig. 2) near the free-edge. This optional step is helpful for obtaining more accurate results without having to increase the number of elements.

A tri-linear element was employed throughout the project. A first-order element was considered the easiest to implement among the general p-order Lagrangian elements. The element used was an eight-node hexahedron element (Fig. 3) with the local node-numbering scheme as given, and each node contained three translational degrees-of-freedom: 𝑢, 𝑣, and 𝑤. The appropriateness of the node-numbering convention was confirmed by checking that the determinants of the elemental Jacobian matrices were positive. This ensured that the mapping between physical and natural coordinates was correct.

Moreover, the elasticity, or the stiffness matrix, which relates the strain to stress, was also defined for each layer of the laminate. This constitutive relationship, {𝜎} = [�̅�]{𝜀} , is also referred to as the generalized Hooke’s Law (App. A-1).

The rest of the components required to formulate the

elemental stiffness equation were calculated in the pre-processing stage. These included the [𝑵𝑒], [𝑩𝑒], [𝑱𝑒], and [𝑳𝑒] matrices evaluated at the Gauss integration points. Here it is noted that [𝑵𝑒] is the shape function matrix, [𝑩𝑒] = [𝛁s𝑵

𝑒] is the symmetric gradient of [𝑵𝑒], and [𝑱𝑒] and [𝑳𝑒] are referred to as the Jacobian matrix and the connectivity matrix, respectively (App. A-2).

The last step of the pre-processing stage involved specifying the boundary conditions. In the case of uniaxially-loaded laminate, three symmetry planes within the model were observed, such that only one octant of the model was analyzed. The essential boundary conditions were that the appropriate degrees-of-freedom on each symmetry plane were set to zero directional displacements. In addition, the applied pressure load boundary condition was specified as follows.

𝑢(𝑥 = 0, 𝑦, 𝑧) = 0

𝑣(𝑥, 𝑦 = 0, 𝑧) = 0

𝑤(𝑥, 𝑦, 𝑧 = 0) = 0

𝑡̅(𝑥 = 𝐿, 𝑦, 𝑧) = 𝑃0

Fig. 2: Mesh profile of a laminated structure

Ply interface

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Fig. 3: 8-node hexahedron element with 3 degrees-of-freedom per node

Once the pre-processing stage was completed, the

elemental stiffness equations were assembled to construct the global stiffness equation. Recalling that the weak form of the governing equation is as shown below, the stiffness matrix, [𝑲𝑒], and the load vector, {𝑓𝑒}, were simply summed for all elements.

To solve for the unknown components of the global degrees-of-freedom, {𝑑𝐹} , the global stiffness equation was rearranged and partitioned by applying the appropriate transformation, as shown below.

𝑲𝑑 = 𝑓

[𝑲𝐸 𝑲𝐸𝐹𝑲𝐸𝐹⊤ 𝑲𝐹

] [𝑑𝐸𝑑𝐹] = [

𝑓𝐸𝑓𝐹]

𝑑𝐹 = 𝑲𝐹−1𝑓𝐹 , ∀ 𝑑𝐸 = 0⃑

It is noted that the method of Gaussian elimination

implemented in MATLAB was used to solve the above set of equations; this is done by setting d_F = K_F \ f_F for greater efficiency than direct matrix inversion. The known components of the degrees-of-freedom, {𝑑𝐸} were appended to the solved solutions, or {𝑑𝐹}, in order to construct the full set of solutions.

Finally, the strain field and the stress field were calculated in the post-processing stage. Using the kinematic relationship, the element-averaged strain was calculated, as follows.

{𝜀} = [𝑩𝑒]{𝑑}, where [𝑩𝑒] was computed as the weighted average of the values obtained at each Gauss integration point.

Similarly, the stress field was calculated by applying Hooke’s Law using the [𝑫𝑒] matrices that were previously computed.

{𝜎} = [𝑫𝑒]{𝜀}

While there are other ways of displaying the strain/stress fields, such as an interpolated field that would result in a smooth variation over the model, as opposed to a discontinuous set of elemental results, the latter was chosen for its slightly greater efficiency. Furthermore, a number of useful features were built into the code, including the options to show the deformed model, display the elemental boundaries, and generate a comprehensive solution file. 3. Numerical Results

Several salient path-wise and contour plots (App. B) are illustrated in this section. A number of observations with emphasis on the free-edge effects are also briefly mentioned.

Fig. 4: Path-wise distribution of 𝜏𝑥𝑧 along y

Fig. 5: Path-wise distribution of 𝜏𝑥𝑧 along z

∑(𝑳𝑒⊤ ∫ ∫ ∫ 𝑩𝑒⊤𝑫𝑒𝑩𝑒|𝑱𝑒| d𝜉

+1

−1

d𝜂

+1

−1

d𝜁

+1

−1

𝑳𝑒)

nel

𝑒=1⏟ 𝑲

𝑑

=∑(𝑳𝑒⊤ ∫𝑵𝑒⊤𝑡̅𝑒 dΓ

Γt𝑒

)

nel

𝑒=1⏟ 𝑓

𝜏𝑥𝑧

Possibility

of stress

singularity

(Ply interface)

𝜏𝑥𝑧

(Free-edge)

𝜉

𝜂

𝜁

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Fig. 6: 𝜎𝑥𝑥 distribution along 𝑦 for various mesh profiles

Fig. 7: 𝜎𝑦𝑦 distribution along 𝑦 for various mesh profiles

Fig. 8: 𝜎𝑧𝑧 distribution along 𝑦 for various mesh profiles

Fig. 9: 𝜏𝑦𝑧 distribution along 𝑦 for various mesh profiles

Fig. 10: 𝜏𝑥𝑧 distribution along 𝑦 for various mesh profiles

Fig. 11: 𝜏𝑥𝑦 distribution along 𝑦 for various mesh profiles

𝜎𝑧𝑧

(Free-ed

ge)

𝜏𝑥𝑦

(Free-ed

ge)

𝜏𝑥𝑧

(Free-ed

ge)

𝜎𝑦𝑦

(Free-ed

ge)

𝜎𝑥𝑥

(Free-ed

ge)

𝜏𝑦𝑧

(Free-ed

ge)

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It is observed (Fig. 4-11) that the magnitude of 𝜎𝑦𝑦

begins to decay rapidly near the free-edge, which is consistent with the behavior for a traction-free surface. Similarly, 𝜏𝑦𝑧 ,

which is assumed to be zero for a cross-ply laminate under the framework of CLPT, displayed a gradual decay toward the free-edge. On the other hand, 𝜎𝑧𝑧 exhibited a possible state of stress singularity. It is noted that this observation was consistent with findings from prior research.

Furthermore, the validity of CLPT for various laminates was investigated. The table below indicated that CLPT was an accurate assumption only for a laminate with small mismatch in Poisson effect between the neighboring plies. Moreover, it was noted that the mesh refinement had a significant effect on the axial stress results, especially for a laminate with a large mismatch in fiber orientation.

4. Further Verification

After the model was implemented in MATLAB, a force- reaction balance test was conducted as a sanity check to confirm that the code had solved the mathematical model correctly. The configuration being modeled was in static equilibrium, and as such, summation of forces in each coordinate direction should be equal to zero. The following figure (Fig. 12) shows how the reaction forces at nodes with essential boundary conditions specified should equal the externally applied traction forces.

Fig. 12: Free-body-diagram of the force-reaction balance

Laminate width, 𝐵 10 m

Laminate thickness, 𝐻 3.6 m

Applied load, 𝑃0 5 × 107 Pa

Total reaction force -1.8 × 109 N

Normalized traction -5 × 107 Pa

∑𝐹𝑥 = 𝑓ext + 𝑓𝑟 0

The reaction traction was computed by dividing the

external force by the cross-sectional area of the plane

orthogonal to the direction of the applied load. The table above summarizes a sample computation. Since the applied load is equal and opposite to the corresponding reaction traction, their sum is zero and the force balance is satisfied.

A simple error analysis is shown in the following paragraph. Due to the difficulty in solving for the exact solution for a layered structure, the theoretical L2 and energy norm errors were computed to demonstrate the error convergence rate with mesh refinement.

||𝑒||L2= (∑∫ (𝑢(𝑥 ) − 𝑢(𝑥 ))

2 dΩ

Ω𝑒nel

)

12⁄

≤ 𝐶ℎ𝑝+1

||𝑒||en= (∑

1

2∫ 𝐸𝑒 (𝜀(𝑥 ) − 𝜀ℎ(𝑥 )) dΩΩ𝑒nel

)

12⁄

≤ 𝐶ℎ𝑝

Here, it is noted that 𝐶 is a constant that is dependent on

the interpolation function used for the FE solution, 𝑝 refers to the order of the element (𝑝 = 1 for a linear element), and ℎ ≡

√ℎ𝑥2 + ℎ𝑦

2 + ℎ𝑧2 is the largest diameter (Fig. 13) that can

inscribed in each element.

Fig. 13: Schematic illustrating element size, ℎ

Thus by reducing the element parameter, ℎ, by a factor of

2, the L2 and energy errors are expected to be reduced to ¼ and ½ the original values, respectively. As successively refined meshes were tested with the developed code, mesh convergence was observed, and it could be reasonably be assumed that the FE solution approached the exact solution.

Lastly, the effect of varying the aspect ratio of an element was also investigated in the present study. Here, the aspect ratio, 𝛼 ≡ 𝑟max/𝑟min , is defined as the ratio of the largest dimension that can be inscribed in the element to the smallest dimension. While the reasoning behind how the aspect ratio has an effect on the solution will not be explained in detail, it is noted that this is due to the shear locking phenomena. The first-order element’s inability to capture the effect of large shear deformation leads to an inaccurate result as the stiffness will be too high. Thus it is recommended that the element be discretized appropriately so as to attain a moderate aspect ratio. 5. Concluding Remarks

This paper discusses the formulation methodology for implementing a 3-D finite element analysis of a layered structure using MATLAB. Following the standard procedures of FEM for an eight-node tri-linear element, several path-wise and contour plots were examined. The results exhibited our expected behavior for a uniaxially-loaded, symmetric laminate.

fext fr

(%)

Shin et al.

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This includes the state of stress singularity at the free-edges near the ply interface, as observed from successive mesh refinements. As an a posteriori verification of our code, a force-reaction balance was performed to confirm that our model satisfied static equilibrium. Finally, the results obtained from the 3-D FE analysis using ANSYS showed strong correlation with our results. In addition, all simulations were run with an Intel Xeon dual-processor @ 2.27 GHz and 30.0 Gb of RAM. For the results reported, a model with approximately 9K degrees-of-freedom was solved and took roughly 15 minutes to process. The finest mesh had ≈33K degrees-of-freedom and the computation time was 13 hours. 6. References

[1] O. O. Ochoa and J. N. Reddy, ‘Finite element analysis of composite laminates’, Kluwer Academic Publishers (1992). [2] N. J. Pagano, ‘Exact solutions for rectangular bidirectional composites and sandwich plates’, J. Compos. Mater., 4, 20- 34 (1970). [3] B. Peseux and S. Dubigeon, ‘Equivalent homogenous finite element for composite materials via Reissner principle. Part I: finite element for plates’, Int. j. numer. methods eng., 31, 1477-1495 (1991). [4] J. R. Rice, ‘The Aspect Ratio Significant for Finite Element Problems’ Computer Science Technical Reports, Paper 454, (1985). 7. Appendices

Appendix A

Constitutive equation:

{𝜎} = [𝑻1(−𝜃)][𝑪][𝑻2(𝜃)]{𝜀}

{𝜎} = [𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 𝜏𝑦𝑧 𝜏𝑥𝑧 𝜏𝑥𝑦]⊤

{𝜀} = [𝜀𝑥𝑥 𝜀𝑦𝑦 𝜀𝑧𝑧 𝛾𝑦𝑧 𝛾𝑥𝑧 𝛾𝑥𝑦]⊤ [𝑪]

=

[ 1 − 𝜈23𝜈32𝐸2𝐸3Δ

𝜈21 + 𝜈23𝜈31𝐸2𝐸3Δ

𝜈31 + 𝜈21𝜈32𝐸2𝐸3Δ

0 0 0

𝜈21 + 𝜈23𝜈31𝐸2𝐸3Δ

1 − 𝜈13𝜈31𝐸1𝐸3Δ

𝜈32 + 𝜈12𝜈31𝐸1𝐸3Δ

0 0 0

𝜈31 + 𝜈21𝜈32𝐸2𝐸3Δ

𝜈32 + 𝜈12𝜈31𝐸1𝐸3Δ

1 − 𝜈12𝜈21𝐸1𝐸2Δ

0 0 0

0 0 0 𝐺23 0 00 0 0 0 𝐺13 00 0 0 0 0 𝐺12]

Δ ≡1 − 𝜈12𝜈21 − 𝜈23𝜈32 − 𝜈13𝜈31 − 2𝜈21𝜈32𝜈13

𝐸1𝐸2𝐸3

Calculation of [𝑵𝑒] and [𝑩𝑒] matrices:

[𝑵𝑒] = [

𝑁1𝑒 0 0 𝑁2

𝑒 0 0 ⋯ 𝑁nen𝑒 0 0

0 𝑁1𝑒 0 0 𝑁2

𝑒 0 ⋯ 0 𝑁nen𝑒 0

0 0 𝑁1𝑒 0 0 𝑁2

𝑒 ⋯ 0 0 𝑁nen𝑒

]

𝑁𝑖𝑒 = 𝐿𝐼

𝑒(𝜉)𝐿𝐽𝑒(𝜂)𝐿𝐾

𝑒 (𝜁)

𝐿𝑚𝑒 (𝜉) =∏

𝜉 − 𝜉𝑗𝑒

𝜉𝑚𝑒 − 𝜉𝑗

𝑒

𝑝+1

𝑗≠𝑚

𝐿𝑚𝑒 (𝜂) =∏

𝜂 − 𝜂𝑗𝑒

𝜂𝑚𝑒 − 𝜂𝑗

𝑒

𝑝+1

𝑗≠𝑚

𝐿𝑚𝑒 (𝜁) =∏

𝜁 − 𝜁𝑗𝑒

𝜁𝑚𝑒 − 𝜁𝑗

𝑒

𝑝+1

𝑗≠𝑚

[𝑩𝑒] ≡ [𝜵𝑠𝑵

𝑒]

=

[ 𝜕𝑁1

𝑒

𝜕𝑥0 0

𝜕𝑁2𝑒

𝜕𝑥0 0 ⋯

𝜕𝑁nen𝑒

𝜕𝑥0 0

0𝜕𝑁1

𝑒

𝜕𝑦0 0

𝜕𝑁2𝑒

𝜕𝑦0 ⋯ 0

𝜕𝑁nen𝑒

𝜕𝑦0

0 0𝜕𝑁1

𝑒

𝜕𝑧0 0

𝜕𝑁2𝑒

𝜕𝑧⋯ 0 0

𝜕𝑁nen𝑒

𝜕𝑧

0𝜕𝑁1

𝑒

𝜕𝑧

𝜕𝑁1𝑒

𝜕𝑦0

𝜕𝑁2𝑒

𝜕𝑧

𝜕𝑁2𝑒

𝜕𝑦⋯ 0

𝜕𝑁nen𝑒

𝜕𝑧

𝜕𝑁nen𝑒

𝜕𝑦𝜕𝑁1

𝑒

𝜕𝑧0

𝜕𝑁1𝑒

𝜕𝑥

𝜕𝑁2𝑒

𝜕𝑧0

𝜕𝑁2𝑒

𝜕𝑥⋯

𝜕𝑁nen𝑒

𝜕𝑧0

𝜕𝑁nen𝑒

𝜕𝑥𝜕𝑁1

𝑒

𝜕𝑦

𝜕𝑁1𝑒

𝜕𝑥0

𝜕𝑁2𝑒

𝜕𝑦

𝜕𝑁2𝑒

𝜕𝑥0 ⋯

𝜕𝑁nen𝑒

𝜕𝑦

𝜕𝑁nen𝑒

𝜕𝑥0

]

𝑑𝐿𝑚𝑒 (𝜉)

𝑑𝜉=∑

1

𝜉𝑚𝑒 − 𝜉ℎ

𝑒 ( ∏𝜉 − 𝜉𝑗

𝑒

𝜉𝑚𝑒 − 𝜉𝑗

𝑒

𝑝+1

𝑗≠ℎ ∧ 𝑗≠𝑚

)

𝑝+1

ℎ≠𝑚

𝑑𝐿𝑚𝑒 (𝜂)

𝑑𝜂=∑

1

𝜂𝑚𝑒 − 𝜂ℎ

𝑒 ( ∏𝜂 − 𝜂𝑗

𝑒

𝜂𝑚𝑒 − 𝜂𝑗

𝑒

𝑝+1

𝑗≠ℎ ∧ 𝑗≠𝑚

)

𝑝+1

ℎ≠𝑚

𝑑𝐿𝑚𝑒 (𝜁)

𝑑𝜁=∑

1

𝜁𝑚𝑒 − 𝜁ℎ

𝑒 ( ∏𝜁 − 𝜁𝑗

𝑒

𝜁𝑚𝑒 − 𝜁𝑗

𝑒

𝑝+1

𝑗≠ℎ ∧ 𝑗≠𝑚

)

𝑝+1

ℎ≠𝑚

Appendix B

Parameters Values

𝐸1 132 × 109 Pa 𝐸1 10.8 × 109 Pa 𝐸1 10.8 × 109 Pa 𝜈12 0.24 𝜈23 0.24 𝜈13 0.24 𝐺12 5.65 × 109 Pa 𝐺23 3.38 × 109 Pa 𝐺13 5.65 × 109 Pa

Load (pressure) 2 × 109 Pa 𝐿 (longitudinal length) 500 m 𝐵 (transverse length) 100 m 𝐻 (normal length) 36 m

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Fig. 14: 𝜎𝑥𝑥 distribution along 𝑧 for various mesh profiles

Fig. 15: 𝜎𝑦𝑦 distribution along 𝑧 for various mesh profiles

Fig. 16: 𝜎𝑧𝑧 distribution along 𝑧 for various mesh profiles

Fig. 17: 𝜏𝑦𝑧 distribution along 𝑧 for various mesh profiles

Fig. 18: 𝜏𝑥𝑧 distribution along 𝑧 for various mesh profiles

Fig. 19: 𝜏𝑥𝑦 distribution along 𝑧 for various mesh profiles

𝜏𝑥𝑦

𝜎𝑥𝑥 (Ply interface)

𝜎𝑦𝑦 (Ply interface)

𝜏𝑦𝑧 (Ply interface)

𝜏𝑥𝑧 (Ply interface)

(Ply interface)

𝜎𝑧𝑧 (Ply interface)

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