Autodesk Helius PFA Guidelines for Determining Finite Element Cohesive Material Parameters
Contents Introduction...1 Determining Cohesive Parameters for Finite Element Analysis...2 What Test Specimens Are Best?...2 What Properties Are Needed?...2 How Should the Models be Created?...3 Determining Cohesive Properties...3 References...5
Introduction Determining the input properties for cohesive elements used to model delamination can be one of the most challenging and confusing tasks that a structural analyst faces. This document describes an easy process that can be used for determining cohesive input parameters from experimental double cantilever beam (DCB) and end notched flexure (ENF) experimental data. The process involves 3 easy steps: 1. Determine Mesh Size. 2. Calculate Cohesive Stiffness. 3. Calibrate Initiation Strength. Following this description are some specific steps and guidelines for using cohesive elements with Helius PFA.
Determining Cohesive Parameters for Finite Element Analysis The following outlines a process for using finite element models of delamination test specimens to arrive at cohesive stiffness, strength and energy properties. Properties will be calibrated such that simulated tests match the measured response of the specimens. What Test Specimens Are Best? In most composite structural components, damage evolution consists of combined delamination and intra-laminar ply failure. The evolution of these two physical damage forms is strongly coupled. For the purpose of trying to use experimental data to characterize cohesive material properties, it is necessary to identify experiments that isolate delamination behavior (i.e., tests that result in minimal or no intra-laminar material failure). Moreover, it is desirable to identify delamination tests that separate the normal and shear modes of delamination. For example, the double cantilever beam (DCB) specimen is designed to produce pure normal mode delamination without any intra-laminar material damage. Similarly, the end-notched flexural (ENF) specimen is designed to produce pure shear mode delamination without any intra-laminar material damage. This approach of utilizing experimental data to characterize cohesive behavior minimizes the number of cohesive properties that must be simultaneously determined (optimized). DCB and ENF specimen tests will isolate delamination behavior and separate normal and shear modes of delamination. Figure 1. Typical DCB (left) and ENF (right) configurations. What Properties Are Needed? Cohesive input properties include parameters that define the stiffness, strength and fracture energy of the cohesive material layer in each of its three deformation modes (e.g., a normal mode denoted by a subscript n), and two shear modes (denoted by subscript s and t respectively). Normal mode: K nn = Stiffness (F/L 2 /L), S n = Strength (F/L 2 ), G n = Fracture Energy (FL/L 2 ) Shear mode: K ss = Stiffness (F/L 2 /L), S s = Strength (F/L 2 ), G s = Fracture Energy (FL/L 2 ) Shear mode: K tt = Stiffness (F/L 2 /L), S t = Strength (F/L 2 ), G t = Fracture Energy (FL/L 2 )
How should the models be created? FE Models Should Match DCB & ENF Test Conditions The first step in this process is to create DCB and ENF finite element models with loading and dimensions that match the test conditions. The DCB model is used to characterize the normal cohesive properties (Knn, Sn, and Gn), and the ENF model is used to characterize the shear cohesive properties (Kss, Ss, and Gs). Be Aware of Mesh Dependency When determining cohesive material properties, one must be cognizant of the fact that predicted cohesive behavior is mesh-dependent, i.e., using a consistent set of cohesive properties across a wide range of cohesive mesh densities will result in a considerable range of predicted delamination responses. In other words, in order for three different cohesive mesh densities to predict the same delamination response, we must use three different sets of cohesive properties. Since cohesive solutions are mesh dependent, it is important that the meshes for the DCB and ENF specimens use cohesive elements that are approximately the same size as the cohesive elements that are anticipated to be used in subsequent progressive failure analyses of composite structural components. Select Appropriate Damage Criteria and Damage Evolution Model For the DCB and ENF models, the cohesive material definition should use the Maximum Stress damage initiation criterion and the Energy damage evolution law. For all delamination models implemented in Helius PFA, the prediction of delamination initiation is based on the tractions (t n,t s,t t ) that occur at the integration points of the cohesive elements. There are currently two different traction-based delamination initiation criteria implemented in Helius PFA Max Stress and Quadratic Stress. However, since DCB and ENF tests are designed to produce single-mode tractions, it is unnecessary to use a Quad stress criterion and Max Stress is sufficient. Also, Helius PFA allows for both a displacement-based and energy-based degradation laws. Autodesk recommends using the energy-based model. Determining Cohesive Properties Let us now consider the process of determining the cohesive properties based on matching measured test results. We will first determine the cohesive stiffness in relation to the stiffness of the surrounding composite plies. We will then use our finite element models to iteratively determine strengths. It is recommended that the experimentally determined fracture energies are used in the simulations, regardless of mesh size, cohesive stiffness, and cohesive strength. The suggestions given in this document have been adapted from material 1, 2 published by Turon, Dávila, Camanho, and Costa. Cohesive Stiffness The cohesive stiffness should be determined before the cohesive strength is determined. It is important to realize that one cannot determine a definitive value of stiffness for cohesive layers when used to simulated delamination between plies. The stiffness of the cohesive layer needs to be stiff enough so that it provides adequate load transfer between the bonded layers, but if it is too stiff, then spurious stress oscillations can occur as shown in Figure 2. As such, the following equation should be used to estimate the stiffness of the cohesive layer: K nn = K ss = K tt = αe 33 PLY /t PLY
where α is a parameter with a suggested value of 50, E 33 PLY is the normal modulus of the composite material, and t PLY is the thickness of the bonded plies. The stress distribution in the cohesive layer for the ENF model should be checked for the presence of stress oscillations. If detected, decrease the stiffness of the cohesive layer by 5% until the stress oscillation vanishes. Figure 2. Example of stress oscillation at the crack tip in the cohesive layer of an ENF model. Cohesive Strengths After setting the stiffness of the cohesive material, we are now ready to use the finite element models of the DCB and ENF specimens to iteratively determine the strengths of the cohesive material (S n, S s, S t). The DCB finite element model is used to calibrate S n, and the ENF finite element model is used to calibrate S s=s t. Initial Strength Estimate Similar to the estimation of the cohesive stiffnesses, the transverse strengths of the composite material plies can be used as starting points in the iterative determination of the cohesive strengths. In particular, the initial estimate of cohesive strength can be computed as: ppp S n = S +33 S s = S t = S ppp ppp 12 +S23 2 Note: The stress values predicted in the DCB and ENF finite element models are dependent on both the cohesive mesh density and the stiffness chosen for the cohesive material (previous step), thus it is likely that these initial strength estimates will need to be adjusted in order for the CDB and ENF models to match the measured DCB and ENF test results. Simulate Results Run the models using the above strength estimates as starting values and plot the simulated load-displacement results against the experimental measured results. Revise and Repeat If the maximum simulated loads are lower than the maximum experimental loads, increase the strengths and run again. If the maximum simulated loads are greater than the maximum experimental loads, decrease the strengths and run again. Repeat this iterative process until good agreement is obtained for the maximum measured loads in the DCB and ENF specimens.
Example As a specific example, consider the results plotted in Figure 3, which show simulated DCB results for multiple values of S n. When S n = 15 N/mm 2, the simulated maximum load was low compared to the experimental maximum load. Increasing S n to 25 N/mm 2 resulted in an over-prediction of the maximum load. Good agreement with the experimental curve was finally achieved when S n = 21 N/mm 2. The same iterative process can be used in conjunction with the ENF model to establish the cohesive shear strengths (S s= S t). Cohesive Energies Figure 3. Load-displacement results for a DCB model with variable S n values. The values for the cohesive energies (G n, G s, G t) do not need to be calibrated and should be set to the experimentally determined fracture energies (G IC, G IIC). Specifically: G n = G IC G s = G t = G IIC References 1. A Turon A., Dávila C., Camanho P., and Costa J., An Engineering Solution for using Coarse Meshes in the Simulation of Delamination With Cohesive Zone Models, NASA/TM-2005-213547, March 2005. 2. Turon A., Dávila C., Camanho P., and Costa J., An Engineering Solution for Mesh Size Effects in the Simulation of Delamination using Cohesive Zone Models, Engineering Fracture Mechanics, Vol. 74, Pg. 1665-1682, 2007.