USING FINITE ELEMENT ANALYSIS TO DETERMINE COMPOSITE LAMINATE DESIGN ALLOWABLES Dan Milligan Firehole Composites 203 South 2nd Street, Suite A Laramie WY 82070

ABSTRACT This paper presents how to use advanced finite element analysis (FEA) techniques to determine an open hole compression (OHC) design allowable. The advanced FEA techniques include using progressive failure and multicontinuum theory in a non-linear analysis to simulate ultimate failure of an open hole compression test coupon. Progressive failure is an analysis technique that identifies elements in a FEA model that have sustained damage according to a predefined composite ply failure criterion and then degrades the material stiffness of the composite plies within the element in order to simulate the progression of damage in a laminate due to material softening. The failure criterion used in this study, MCT, is provided through the Helius:MCT commercial FEA add-on. MCT uses multicontinuum theory, which is a form of multi-scale analysis that separates composite ply averaged stresses into averaged stresses in the fiber and matrix constituents. Failure can then be evaluated using constituent stresses, and appropriate stiffness degradation can be applied at the constituent level. An experimental correlation will compare simulation results against experimental test data. Conclusions will be proposed for how this approach can be used to create carpet plots for determining OHC design allowables across a wide range of lamination schemes.

1. INTRODUCTION It has always been a goal of the finite element analysis (FEA), industry to supplement and perhaps one day replace expensive and time consuming testing that composite materials undergo in order to gain an understanding of their mechanical behavior. One of the building blocks in the design process for composite parts is to understand the mechanical behavior of the material that is intended to be used in a design. This is conventionally achieved though coupon testing to determine the modulus and strength of the composite material. There are typically two levels in which a composite material is tested: at the lamina (ply) level and at the laminate level. It is reasonable to assume that lamina level testing will always be necessary to account for manufacturing processes, but what this paper will focus on is how designers and engineers can use lamina material properties determined from testing to simulate laminate level coupon tests and in particular an open hole compression (OHC) laminate coupon test. There are two advanced FEA techniques that this paper will describe how to use in a FEA simulation of an OHC coupon: progressive failure and multicontinuum theory. Simulation results will be compared against experimental test results of an OHC coupon. Conclusions will be presented on the accuracy of the simulation and how this process can be extended to a wide range of lamination schemes to develop carpet plots for OHC design allowables. SAMPE 2013 Proceedings: Education & Green Sky – Materials Technology for a Better World. Long Beach, CA, May 69, 2013. Society for the Advancement of Material and Process Engineering.

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1.1 Finite Element Analysis Techniques The first advanced FEA technique used in this study, progressive failure, is an analysis technique that identifies elements in a FEA model that have sustained damage according to a predefined composite ply failure criterion and then degrades the material stiffness of the composite ply (or plies) within the element. Progressive failure allows the analyst to simulate the progression of damage in a laminate by reducing the stiffness of the element when the failure criterion is satisfied. When the stiffness of an element is degraded, the stress level in that element drops, and the load is routed around the damaged element(s) into the surrounding, pristine elements. This increases the stress level in the adjacent elements, causing those elements to reach a stress state that causes material failure at a lower global load than if the material softening was not included in the failed element. There is a progression of material failure from one element to the next until there is significant material softening to cause the laminate to be unable to support an increase in loading, the point of ultimate failure. The failure criterion used in this study, MCT, is provided through the Helius:MCT commercial FEA add-on. MCT uses multicontinuum theory, which is a form of multi-scale analysis. Multiscale analysis is a method of analyzing materials by looking at the material behavior at multiple scales. Multicontinnum theory uses two different scales to predict the mechanical response of a composite material: the lamina (ply) scale and the constituent (fiber and matrix) scale. Multicontinuum theory has been discussed in detail through the works of Hill [1], Garnich and Hansen [2], Mayes and Hansen [3], [4], and Nelson, Hansen and Mayes [5]. To summarize, MCT evaluates failure at the constituent (fiber and matrix) level. To do this, the lamina stresses calculated by a FEA solver are separated into fiber stresses and matrix stresses and a constituent specific failure criterion is evaluated using the constituent stresses (matrix failure criterion is evaluated using matrix stresses and fiber failure criterion is evaluated using fiber stresses). The advantage in using matrix and fiber stresses to determine failure is the added accuracy resulting from capturing the mechanical interactions between the fiber and matrix. Fibers and the surrounding matrix have very different material properties, and therefore the differences in stiffness and Poisson ratios cause internal stresses to develop that are not captured when looking at the overall ply stress state. Using MCT in a progressive failure analysis, appropriate stiffness degradation can be applied at the constituent level. Once the matrix failure criterion has been satisfied, the matrix stiffness values are reduced and once the fiber failure criterion has been satisfied, the fiber stiffness values are reduced. These reduced constituent values are re-combined though micromechanics into reduced stiffness lamina properties which are then used in the FEA solver. This efficiently allows the FEA solver to model material softening resulting from matrix and fiber failure as loading increases on a composite lamina. 1.2 Experimental Benchmark To determine the accuracy of the proposed FEA techniques an experimental open-hole compression (OHC) benchmark was used. For this study, OHC test data was provided by the Advanced General Aviation Transport Experiments (AGATE) material database [6]. As shown in Figure 1, OHC Strength values are provided for three layups: 25/50/25, 10/80/10, and 50/40/10 (described in detail in section 2.2). For this study, the Normalized, room temperature dry (RTD) test conditions were simulated.

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Figure 1. Experimental benchmark data for OHC simulation.

2. FINITE ELEMENT ANALYSIS 2.1 Composite Material The composite material used in this study was IM7/8552, a carbon/epoxy lamina. Lamina material properties were provided for IM7/8852 through a material qualification report [6] provided by the Advanced General Aviation Transport Experiments (AGATE) material database. The lamina material properties assigned to the composite material are listed in Table 1. Table 1. Lamina material properties for IM7/8552 (RTD Mean, Normalized). Material Property

Value

Units

Fiber Volume Fraction Longitudinal Modulus (E11) Transverse Modulus (E22) Longitudinal Poisson Ratio (ν12) Transverse Poisson Ratio (ν23) In-Plane Shear Modulus (G12) Longitudinal Tensile Strength (+S11) Longitudinal Compressive Strength (-S11) Transverse Tensile Strength (+S22) Transverse Compressive Strength (-S22) In-Plane Shear Strength (S12) Transverse Shear Strength (S23)

0.57 138 9.81 0.355 0.496 4.61 2.50 -1.72 64.1 -286 89.9 95.2

~ GPa GPa ~ ~ GPa GPa GPa MP a MP a MP a MP a

Helius:MCT calculates stiffness properties for the fiber and matrix materials based on the lamina properties provide in Table 1. The stiffness properties for the IM7 carbon fibers and 8552 epoxy matrix are provided in Tables 2 and 3 respectively.

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Table 2. Fiber material properties for IM7 carbon fibers. Material Property

Value

Units

Longitudinal Modulus (E11) Transverse Modulus (E22) Longitudinal Poisson Ratio (ν12) Transverse Poisson Ratio (ν23) In-Plane Shear Modulus (G12)

240 15.7 0.305 0.218 20.2

GPa GPa ~ ~ GPa

Table 3. Matrix material properties for 8552 epoxy. Material Property

Value

Units

Longitudinal Modulus (E11) Transverse Modulus (E22) Longitudinal Poisson Ratio (ν12) Transverse Poisson Ratio (ν23) In-Plane Shear Modulus (G12)

4.53 4.53 0.420 0.420 1.59

GPa GPa ~ ~ GPa

A progressive failure analysis using the MCT failure criterion requires two additional input values to determine the degree of softening a composite material undergoes once failure occurs. When using the MCT criterion, there are two failure modes, matrix failure (determined from the matrix stresses and a matrix specific failure criterion) and fiber failure (determined from the fiber stresses and a fiber specific failure criterion). When using the MCT multi-scale failure criterion, material softening is applied to the matrix and fiber stiffness values separately and these reduced constituent stiffnesses are combined into a reduced lamina stiffness value to use in the FEA model. For this analysis, values of 0.01 and 1E-06 were used for the post failure matrix and fiber stiffness factors respectively. This means that once matrix failure occurred, the matrix stiffness values were reduced to 1% of the pristine stiffness values in Table 3 and once fiber failure occurred, the fiber stiffness values were reduced to 0.0001% of the pristine stiffness values in Table 2. These values are the default values for using Helius:MCT. 2.2 Composite Lamination Schemes For this study, three different composite laminates were examined. The laminates are defined by the percentage of 0° plies, ±45° plies, and 90° plies through-the-thickness of the laminate. For example, a 25/50/25 laminate is a quasi-isotropic laminate with 25% 0° plies, 50% ±45° plies, and 25% 90° plies. The three laminates studied were 50/40/10, 25/50/25, and 10/80/10 laminates. Table 4 details the lamination scheme of these three laminates.

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Table 4: Lamination scheme of laminates used in this study. Laminate

Ply Schedule

50/ 40/ 1 0 25/ 50/ 25 10/ 80/ 10

[0/45/0/90/0/-45/0/45/0/-45]s [(45/0/-45/90)3]s [±45/0/±45/90/(±45)2]s

2.3 Finite Element Analysis Model Geometry and Mesh The Open Hole Compression (OHC) model geometry was created using the dimensions provided in Figures 2 and 3.

Figure 2. Topological dimensions for the OHC model.

Figure 3. Thickness dimensions for the OHC model.

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The OHC model uses what is commonly referred to as a discrete layer (DL) mesh. A DL mesh assigns a single element through-the-thickness to each composite ply. Therefore, there were 20 and 24 elements through-the-thickness for the 20 and 24 ply coupons respectively. The element type used to model the composite plies was a hexagonal 3D linear solid element with a reduced integration formulation. This type of element has eight nodes and one integration point located at the centroid of the element. A representative mesh used for this analysis is shown in Figure 4.

Figure 4. Representative mesh used for the OHC model. A DL mesh scheme allows for a more realistic progressive failure analysis than meshing the coupon using layered elements (commonly known as equivalent single layer, or ESL, modeling). The reason for this is that when a ply fails (either matrix or fiber failure), the stiffness of the element that contains that ply is reduced, accurately representing the overall stiffness reduction of the coupon. When using a layered element that contains multiple plies, when a ply fails, the stiffness of the element is only slightly reduced and does not capture the overall stiffness reduction of the coupon as accurately. 2.4 Finite Element Analysis Model Boundary Conditions To apply a uniform load to one end of the coupon, an equation constraint was used to couple the axial (X-direction) displacements of one node to the axial displacements of every other node on one end of the coupon (Figure 5). Using this type of constraint allows the analyst to apply a displacement to a single node and the displacement for all the constrained nodes is set to the

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same value. This is useful because the reaction force of a single node can be queried and this force represents the load that is applied to the entire end of the coupon.

Figure 5. Equation constraint applied to one end of the OHC coupon. As shown by the highlighted region in Figure 6, on the opposite end of the coupon, the axial displacements of each node were fixed (set to a displacement of 0.0 in the X-direction).

Figure 6. Axial displacements were fixed on the coupon edge opposite of the constraint equation.

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As shown by the highlighted region in Figure 7, the edges of the though-thickness mid-plane were fixed in the normal direction (set to a displacement of 0.0 in the Z-direction) to prevent buckling of the coupon in compression.

Figure 7. Normal displacements were fixed on the though-thickness mid-plane edges. As shown by the highlighted point in Figure 8, a single node on the axially constrained coupon end was fixed in the transverse direction (set to a displacement of 0.0 in the Y-direction) to prevent a rigid body translation in the Y-direction.

Figure 8. The transverse (Y-direction) displacement was fixed on a single, centrally located point on the axially constrained end of the coupon.

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A negative displacement in the axial direction (X-direction) was applied to a single node. Through the constraint equation described earlier, this negative displacement was applied to all nodes on the positive X end of the coupon. The displacement was applied incrementally to allow for the progressive failure analysis to gradually reduce the overall coupon stiffness due to material failure in the elements.

3. RESULTS Using a progressive failure analysis, the ultimate compressive strength of an OHC coupon can be determined when there is a large load drop from one increment to the next in a non-linear FEA simulation. Recall from section 2.4 that the displacement controlled loading was applied incrementally to allow for the gradual stiffness reduction in the composite coupon as damage occurs in the composite plies. Eventually, enough damage accumulates to cause a large failure cascade in the coupon, and this results in a significant load drop in the reaction force measured at the node in which the axial displacement was applied. This can be viewed using a loaddisplacement curve (Figure 9) that plots the reaction force as a function of time (normalized displacement, total applied displacement = 1.0)

Figure 9. Load-displacement curve for 25/50/25 laminate. By tracking the point at which a large load drop occurs and determining the compressive force just prior to this load drop, the ultimate strength of the coupon can be calculated as, 𝐹𝑥𝑜ℎ𝑐𝑢 �

𝑃𝑚𝑎𝑥 𝐴

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[1]

where, 𝐹𝑥𝑜ℎ𝑐𝑢 = ultimate open-hole compressive strength 𝑃𝑚𝑎𝑥 𝐴

= maximum force prior to failure

= gross cross-sectional area (disregarding hole)

Using this approach, Table 5 summarizes the simulation ultimate compressive strength calculated for each of the three composite laminates. Also included is the experimental ultimate compressive strength given by the AGATE material report [6] and the percent error of the simulation result. Table 5. Simulation ultimate compressive strength results compared with experimental test results. Laminate 50/40/10 25/50/25 10/80/10

Simulation Ultimate Compressive Strength [MPa] 493 366 264

Test Ultimate Compressive Strength [MPa] 436 339 267

Percent Error 13.1 8.2 -1.3

The results of the progressive failure simulation are encouraging, as good correlation is seen with test results over a wide range of lamination schemes. The largest percent error was 13.1% and typically a percent error under 15% is considered exceptional for simulation results. The results indicated that the simulations involving a lower percentage of 0° plies were more accurate. This is expected for compression testing where delamination failure modes are more typical in laminates higher concentrations of 0° plies. The simulation in this study did not account for delamination failure modes and better correlation could be achieved at higher 0° ply concentrations if delamination was included in the progressive failure simulation.

4. CONCLUSIONS The modeling techniques provided in this paper provides a method for using FEA to provide accurate OHC ultimate strength values. This is valuable because a large number of laminates can be simulated quickly and inexpensively to generate a set of design values that would not be achievable in a similar time frame or cost through testing. An example of this is the carpet plot shown in Figure 10, where the same FEA techniques used to model the three laminates in this paper are used to simulate a wide range of laminates.

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Figure 10. Carpet plot that displays the simulation results for a wide range of composite laminates. A progressive failure analysis is sensitive to many input parameters including element type, mesh density, and post-failure stiffness factors. The simulation results will vary depending on the exact modeling procedure. A study of the sensitivity of the simulation ultimate compressive strength when varying these input parameters is a good topic for future research. The progressive failure and multicontinuum analysis methods were the keys to simulating the ultimate compressive strength of the OHC coupon. Using these methods to extend the simulation beyond OHC to other design allowables such as open hole tension, filled hole tension/compression, and bolted joint analyses, and to different environments such as elevated temperature wet and cold temperature dry provide a realistic method to supplement expensive and time consuming testing with simulation to provide composite design allowables.

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5. REFERENCES 1. Hill R. “Elastic Properties of Reinforced Solids: Some Theoretical Principles.” Journal of the Mechanics and Physics of Solids 11 (1963): 357-372. 2. Garnich M. and Hansen A. “A Multicontinuum Approach to Structural Analysis of Linear Viscoelastic Composite Materials.” Journal of Applied Mechanics 64 (1997): 795-803. 3. Mayes J. and Hansen A. “Composite Laminate Failure Analysis using Multicontinuum Theory.” Composites Science and Technology 64 (2004): 379-394. 4. Mayes J. and Hansen A. “A Comparison of Multicontinuum Theory based Failure Simulation with Experimental Results.” Composites Science and Technology 64 (2004): 517-527. 5. Nelson E., Hansen A., and Mayes J. “Failure Analysis of Composite Laminate Subjected to Hydrostatic Stresses: A Multicontinuum Approach.” Accepted as part of the second world wide failure exercise. 6. National Institute for Aviation Research. “Hexcel 8552 IM7 Unidirectional Prepreg 190 gsm & 35%RC Qualification Material Property Data Report.” (2011)

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