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## Composite Structures

Volume 176,

15 September 2017

, Pages 768-779

## Abstract

A method is presented to calculate the material parameters required by the progressive damage analysis (PDA) material-model in ANSYS®. The proposed method is based on fitting results calculated with PDA to experimental data by using Design of Experiments and Direct Optimization. The method uses experimental modulus-reduction vs. strain data for only two laminates to fit all the parameters required by PDA transverse/shear damage mode. Fitted parameters are then used to predict and compare with the experimental response of other laminates. Mesh sensitivity of PDA is studied by performing p- and h-mesh refinement. PDA can then be used to predict the response with damage evolution of any layup.

## Introduction

Damage initiation and propagation in composite materials are of particular importance for the design, production, certification, and monitoring of an increasingly large variety of structures. While most models are implemented on shell-type formulations, some are implemented on solid-type finite element formulations [1].

One approach is to predict the response of each lamina using a meso-model that couples the behavior of each lamina with adjacent interlaminar layers [2]. An intralaminar damage model takes into account the effects of transverse matrix crack and the interlaminar layer takes into account delaminations. Parameter identification and validation with open–hole tensile tests on quasi-isotropic laminates is described in [3].

Another approach is to use serial-parallel mixing theory [4], [5], [6], or multi-continuum theory (MCT) [7], to simulate the onset and propagation of transverse matrix cracking and other failure mechanisms. The composite response is obtained from the constitutive behavior of its constituents, each one of them simulated with its own constitutive law. In this way, it is possible to use any given non-linear material model, such as damage or plasticity, to characterize the constituents. The accuracy of damage onset and evolution predictions depends on the accuracy of the constituent models used.

Most of the models developed specifically to address the phenomenon of transverse matrix cracking are meso-models, meaning that each lamina is considered to be a homogeneous, orthotropic material [8]. Some models, such as Progressive Damage Analysis (PDA) use strength properties to predict damage onset, and fracture mechanics properties to predict damage evolution [9], [10]. Others, such as Discrete Damage Mechanics (DDM) [11] use only fracture mechanics properties to predict both damage initiation and damage evolution.

Regardless of how initiation is predicted, evolution is commonly predicted by establishing a relation between the available strain energy in the material and the density of matrix cracks [12], [13], [14], [15], [16]. For example, [17] uses Finite Fracture Mechanics (FFM) to obtain the energy release rate required to double the crack density, by propagation of a new crack between two existing cracks, or by continuous variation of crack density [18]. Similarly, the equivalent constrained model (ECM) [19], [20], defines a law that provides the evolution of stiffness as a function of matrix crack density.

Regarding the solution of the stress and strain field in the vicinity of a crack, which is necessary to calculate the strain energy release rate (ERR), most models approximate the problem as being periodic. However, numerical models such as PDA are not constrained by this assumption. While most models calculate the ERR using the equations of elasticity, with various degrees of approximation, others use Crack Opening Displacement approximations [14], [16], [21].

Some of the above mentioned formulations provide analytical expressions that can be used to obtain the mechanical response for simple geometry and load configurations. However, it is often necessary to include the constitutive model into Finite Element Analysis (FEA) software. Some models have been included in commercial FEA software [9], [10], [22]. Others are available as plugins for existing FEA software [7], [23], or as user programmable features, including UMAT, UGENS, UserMAT [24], and VUMAT [1].

ANSYS Mechanical provides progressive damage analysis (PDA) starting with release 15. Furthermore, ANSYS Workbench allows optimization of any set of variables to any user defined objective defined in a Mechanical APDL (MAPDL) model by importing the APDL script into Workbench and using Design of Experiments (DoE) and Direct Optimization (DO). Since PDA is not implemented in the graphical user interface (GUI) of ANSYS Workbench, the user must use APDL commands to define the damage initiation criterion, damage evolution law, and material parameters, as described in this manuscript.

Although elastic moduli are available for many composite material systems, the same is not true for the additional material properties required by PDA. However, laminate modulus and Poisson’s ratio degradation of laminated composites as a function of applied strain are available for several material systems, for example [21], [25], [26]. This study shows how to use available data to infer the material parameters required by PDA. Specifically, the main purpose of this study is to find in situ values [27] of *transverse tensile strength* ${F}_{2t}$, *in-plane shear strength* ${F}_{6}$, and *energy dissipation per unit area* ${G}_{c}$ (transverse tensile/shear mode) for the material system (composite lamina) which can be used in PDA to predict damage initiation and evolution of laminated composite structures built with the same material system but different laminate stacking sequence (LSS), geometry, loads, and boundary conditions. Experimental data from [21], [25] is used for illustration in this study.

The proposed method requires experimental data in the form of laminate extensional modulus ${E}_{x}$ reduction (or laminate shear modulus ${G}_{\mathit{xy}}$ reduction, or laminate Poisson’s ratio ${\nu}_{\mathit{xy}}$ reduction) vs. damage for two laminates. One laminate should have a significant component of mode I (opening) crack propagation and the other a significant component of mode II (shear) crack propagation, such as for example ${[0/90]}_{S}$ and ${[0/\pm \theta ]}_{S}$, respectively. Extensional modulus reduction tests are the easiest to perform but the user could use laminate shear or Poisson’s ratio reduction as well.

The stated objective is achieved by minimizing the error between PDA predictions and available experimental data. Once the *input* parameters ${F}_{2t}\text{,}{F}_{6}\text{,}{G}_{c}$ are found, the accuracy of PDA predictions is checked by comparing those predictions with experimental data for other laminates that has not been used to fit the input parameters. In fact, experimental data for only two laminates are required to fit the parameters. The input parameters are fitted using an specific mesh (one element) and type of element (SHELL 181). However, several types of elements and variable mesh refinement are customarily used for the analysis of a complex structure. Therefore, sensitivity of the PDA predictions with respect to mesh refinement and element type (quadratic vs. linear) is assessed in this work by performing both p- and h-refinement.

## Section snippets

## Progressive damage analysis

To perform progressive damage analysis of composite materials, the user needs to provide linear elastic orthotropic material properties and two material models: damage initiation and damage evolution law.

## Methodology

In this section we propose a methodology using Design of Experiments (DoE) and Design Optimization (DO) to adjust the values of ${F}_{2t}\text{,}{F}_{6}$, and ${G}_{c}$, so that the PDA prediction closely approximates the experimental data.

First we use DoE to identify the laminates that are most sensitive to each parameter. The focus at this point is to identify the minimum number of experiments that are needed to adjust the parameters. In this way, additional experiments conducted with different laminate stacking

## Comparison with experiments

In this section, predicted laminate modulus ${E}_{x}({\u220a}_{x})$ with parameters listed in Table 7 are compared with experimental data for all the laminates. The error (8) for each laminate is reported in Table 6.

Laminates #1 through 5 have a cluster of 8 plies in the layer that is subject to transverse matrix cracking (Table 2). Since the in situ transverse tensile strength was adjusted with laminate #1, damage onset can be predicted well with PDA, as shown in Fig. 5, Fig. 12, Fig. 13, Fig. 14, but not on

## Mesh sensitivity

Mesh sensitivity refers to how much the solution changes with mesh density, number of elements, number of nodes, and element type used to discretize the problem under study. There are two sources of mesh sensitivity. The most obvious is type I sensitivity, where the quality of the solution, particularly stress and strain gradients, depends on mesh density; the finer the mesh, the better the accuracy of the solution. Assuming that the mesh is refined enough to capture stress/strain gradients

## Conclusions

A novel methodology is proposed to determine the material parameters for Progressive Damage Analysis (PDA) in ANSYS and the procedure is explained in detail.

It is observed that adjusted material parameters ${F}_{2t}\text{,}{F}_{6}\text{,}{G}_{c}$, can be used to predict damage initiation and evolution in laminated composites using ANSYS, and that good comparison with available experimental data can be achieved with certain restrictions.

The in situ strength values ${F}_{2t}$ and ${F}_{6}$ should be adjusted with data from laminates with

## Acknowledgments

The authors acknowledge the infrastructure support from the Energy Materials Science and Engineering (EMSE) program at West Virginia University.

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