An inverse analysis of cohesive zone model parameter values for ductile crack growth simulations

https://doi.org/10.1016/j.ijmecsci.2013.12.006Get rights and content

Highlights

  • An inverse analysis procedure is developed to identify cohesive zone parameter values for ductile crack growth simulations.

  • Key points on the Mode I load–crack extension curve are used to create the error function in inverse analysis.

  • The inverse analysis procedure is robust and converges fast.

  • Mixed-mode simulation results using identified CZM parameter values are validated by experimental measurements.

Abstract

An inverse analysis using a modified Levenberg–Marquardt method is carried out to identify cohesive zone model parameter values for use in 3D finite element simulations of stable tearing crack growth events in Arcan specimens made of 2024-T3 aluminum alloy. The triangular cohesive law is employed in the simulations. The set of cohesive parameter values is determined in the inverse analysis by minimizing the difference between simulation predictions of key points on the load–crack extension curve with experimental measurements. From three different initial values, similar cohesive parameter value sets are reached. Using these sets of values, the predicted load–crack extension curves and the variation of a generalized crack tip opening displacement (CTOD) with crack extension for mixed-mode loading cases are compared with experimental measurements, which provide a validation of the cohesive parameter values and of the finite element simulation predictions.

Introduction

Numerical simulations of stable tearing crack growth events play an important role in assessing the structural integrity and residual strength of critical engineering structures such as aircraft. For many fracture events, the cohesive zone model (CZM) concept [1], [2], [3] has found wide applications in numerical simulations (e.g. [4], [5]). CZM represents the behavior of the fracture process zone and describes the relationship between cohesive tractions and separations across the cohesive crack surfaces. Due to its strong physics basis and ease in numerical implementation, the CZM approach has been utilized for a wide range of material systems (e.g. [6], [7], [8], [9], [10], [11], [12]).

In order to apply the CZM approach in numerical simulations, the values of CZM parameters must be properly specified to define the cohesive traction and separation relationship. However, these parameters are often not readily measurable experimentally. There are yet no well-established universal rules for determining CZM parameter values. In practice, the cohesive parameter values are usually assumed or found by trial and error through matching simulation predictions with certain experimental measurements. To make this matching procedure automatic, the use of a numerical inverse analysis method is preferred.

Inverse analysis methods arise from the need to estimate unknown parameters or conditions in a physical system by matching certain system responses with measured or specified conditions. These methods are now widely employed in many fields of engineering and sciences, such as for heat conduction problems [13], medical and biological problems [14], acoustic problems [15], and in modeling explosive events [16], just to name a few. In general, an inverse analysis method is composed of two parts: (a) a forward analysis and (b) an optimization procedure. In the forward analysis, the unknown parameters are assigned initial values or given updated values from the optimization procedure and then the system response is predicted. In the optimization procedure, the predicted system response is compared with the measured response to produce updated parameter values for the forward analysis in order to minimize the difference between the predicted and measured system responses. The forward analysis and optimization procedure are iterated until the difference between the predicted and measured system responses is below a certain error tolerance.

In the optimization procedure, various methods (e.g. [17]) have been proposed for different kinds of problems, including, for example, Golden Section methods, Gauss–Newton methods, extended Kalman filters, genetic algorithms, neural networks. Among these techniques, the Levenberg–Marquardt (LM) method is one of the efficient inverse operators for solving engineering problems. This method was first proposed by Levenberg [18] and Marquardt [19] for least-squares estimation of nonlinear parameters. It is effective for dealing with ill-posed problems and small residual problems. In engineering practice, the unknown parameters or conditions should vary within reasonable physical boundaries, instead of a full range of mathematically possible values as in pure mathematical models, otherwise the inverse procedure will not be practical. To this end, the Levenberg–Marquardt method has been modified using weighted penalty functions, so that the parameters will be optimized under specified constraints (e.g. [20]). With this modification, the LM method is called the modified LM method.

There have been few studies in the literature that focus on the identification of cohesive zone model parameter values using inverse analysis. The available studies have been limited to the modeling of fracture in brittle or quasi-brittle materials. Bolzon et al. [21] used the Kalman filter method to solve parameter identification problems in a Mode I cohesive crack model, on the basis of experimental data generated by wedge-splitting tests on concrete specimens. Gain et al. [22] proposed a hybrid technique to extract cohesive fracture properties of a quasi-brittle material (PMMA) using inverse analysis and surface deformation measurement data.

The current investigation will study the viability of inverse analysis technique in estimating cohesive zone model parameter values for ductile materials in stable tearing crack growth tests by using Mode I crack growth measurement data, and will apply the estimated CZM parameter values to simulate both Mode I and mixed-mode I/II stable tearing crack growth tests. After exploring the applicability of a number of inverse analysis methods, the current study finds that the modified LM method, along with experimental measurements of the load vs. crack extension curve of test specimens, can be used to determine CZM parameter values for stable tearing crack growth tests [23] in Arcan specimens made of 2024-T3 aluminum alloy. The details of the current study are presented in subsequent sections.

Section snippets

Arcan test

To study the viability of inverse analysis in estimating cohesive parameter values for simulating stable tearing crack growth events in ductile materials, the current study utilizes the Arcan test data [23]. The Arcan fixture and specimen are designed to facilitate stable tearing crack growth tests under mixed-mode I/II loading conditions ranging from pure mode I to pure mode II [23]. The Arcan fixture, shown in Fig. 1a, is made of a 15-5PH stainless steel and has a thickness of 19.05 mm. The

The modified Levenberg–Marquardt method

The inverse analysis in this study is based on the modified LM method. With respect to the input parameter vector p, the objective of the LM method is to minimize the error function Ψ below in the least squares form [20].Ψ(p)=12i=1m[ri(p)]2=12rTrr=ffwhere r is the residual vector (with m components), f is the vector of system responses calculated from the FE model, and f is the vector of corresponding experimental measurements.

To make the LM method practical, it is often necessary to

The inverse analysis for the current problem

In the simulation of stable tearing crack growth events using the CZM approach, the input parameter vector p is, in transposed form, pT=[Tmax, δsep]. As discussed in Section 2.2, the other CZM parameter, δ0 or K, has a small effect on simulation predictions within a reasonable range. Based on this observation and to reduce computation time, the value of K is set to be 105 MPa/mm, which is large enough to avoid influencing the overall structural stiffness of the specimen but is not too large to

Load–crack extension curve and peak load

At the start of the inverse analysis, the input parameter vector is assigned an initial value of pT(0)=[600 MPa, 0.04 mm], which is selected without any preference in the feasible range. Following the algorithm shown in the flow chart (Fig. 6), iterations are carried out automatically between the inverse analysis code and the forward analysis code ABAQUS. At the end of each iteration the load–crack extension curve is predicted and compared with experimental measurements, and the values of the

Conclusions

The current study has investigated the applicability of inverse analysis to the identification of cohesive parameter values in cohesive zone model based simulations of stable tearing crack growth in ductile materials. It is proposed that key points on the load–crack extension curve can be used to create the error function in the inverse analysis. It is found that a modified Levenberg–Marquardt method works well in identifying cohesive parameter values with the feasible constraints. A Mode I

Acknowledgments

The authors gratefully acknowledge the financial support provided by AFOSR (Grant No. FA9550-09-0543; program manager Dr. David S. Stargel) and by the GM R&D Center.

References (36)

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