Finite Element Analysis of Progressive Degradation versus Failure Stress Criteria on Composite Damage Mechanics

J. L. Curiel Sosa

Materials and Engineering Research Institute, Sheffield Hallam University

United Kingdom

I. Introduction

It is well known that the engineering applications using composite materials is in constant growth, mainly because of the large strength/weight ratio that they provide. The modelling of these materials has been of interest for a long time, due to the experimental costs that can be saved by means of computer simulations. However, the mixed mode of failure in composite materials makes it a complicated task to deal with, resulting often in sophisticated damage models.

There have been numerous techniques proposed for the simulation or prediction of the failure of composites. Many of these techniques were integrated on analytical methods that were subsequently implemented on major simulation software packages or in-house finite element method programs. This is the case in failure models based on stress quadratic functionals, such as those by Tsai & Wu (1971), and implemented within ANSYS (Swanson, 2007) or by Hoffman (1967) and included within ABAQUS (Hibbit et al., 2007). Such functionals imply the disappearing of bearing capability to outstanding loads once the stress criteria are satisfied. From a strict numerical point of view, a finite element satisfying the criteria may potentially be removed from the mesh as it does not experience further loading. This possibility is available in major software packages such as LS-DYNA. The removal of a finite element frequently causes certain numerical oscillations when using explicit solvers. This may degenerate into instabilities and, hence, in divergence of the numerical procedure. A significant number of these criteria have been proposed in the last decades. For instance, the models by Tsai & Wu (1971), Hoffman (1967), Yamada and Sun (1978) or Puck and Schurmann (1998) amongst many others have been very popular. A worldwide assessment failure exercise (WWFE) of a number of these criteria is described in references (Hinton and Soden, 1998; Hinton et al., 2004). Also, Soden et al. (1998a) presented the result for fibre-reinforced composite laminates and their correlation to a set of shared-by-participants experimental data (Soden et al., 1998b). It is clear that considerable efforts have been done in the searching of a general criteria that may be applied in a wide range of problems. However, Daniel (2007) reveals discrepancies of up to 200-300% in the WWFE results shown by Soden et al. (1998a). Unawareness of the numerical consequences that carry the use of these criteria within a finite element method, such as instability and, finally, divergence of the numerical procedure, result in unrealistic solutions. On the other hand, different

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computational techniques for modelling damage progressively were developed to adapt to the finite element methodology. The progressive damage causes the degradation of the stiffness in the damaged zone. Thus, a damaged finite element does not lose completely its loading bearing capacity but the latter is decreased inversely proportional to the degree of damage. The progressive damage models derive from the thermodynamical approaches proposed by Kachanov (1958) initially and, most famously acknowledged by Lemaitre (1992); Lemaitre and Chaboche (1990) and Chaboche (1981). Proposals in this field by Matzenmiller et al. (1995), Maimi et al. (2007a;b), Barbero & De Vivo (2001) or Schipperen (2001) have contributed to extend the number of techniques available for damage evolution on composite materials.

The progressive damage models are attractive as they are readily implemented either in major codes or in-house finite element programs. Nevertheless, these models have the uncertainty on when the onset of damage is reproduced. Some authors coupled it to stress criteria as initiation criteria for developing damage to solve this drawback. For instance, Lapczyk & Hurtado (2007) combined a progressive damage model with the stress criteria proposed by Hashin (1980) as a damage initiation criteria. The formulation is based on the fracture energy for representation of fibre failure and matrix failure. Hufenbach et al. (2004) have successfully shown how interactive criteria – combining progression and failure criteria – may be applied for the prediction of failure in textile reinforced composites assuming that they are formed by unidirectional layers. However, Cuntze & Freund (2004) state that the conditions of initiation of failure are not as relevant as the evolution of the stiffness degradation, due to the fact that its influence is decreasing with the damage progression. This is in agreement with other theories that defend the inelastic behaviour of a range of composite materials, (Barbero & Lonetti, 2002). Chow and Yang (1998) developed an inelastic model for the description of damage in composite laminates and its implementation into an incremental displacement-based Finite Element Method(FEM). The stress strain relationship is incorporated into a modified Newton-Raphson iterative method. More recently, Zobeiry et al. (Camanho et al., 2008) presented a progressive damage model with special attention to the nonlocal regularisation of the damage computations. A significant number of these last approaches are limited to plane stress models, such as those by, for example, Allen et al. (1987); Edlun and Volgers (2004); Harris et al. (1995); Hochard et al. (2001); Talreja (1987); Tan (1991) and McCartney (2003). In the best of knowledge, pioneering works on nonlinear behaviour of composites were developed by Chang and Chang (1987) and by Shahid and Chang (1995). Both works are dedicated to the analysis of composite plates. Lessard and Shokrieh (1995) state that two-dimensional analysis may produce sensibly different results as a consequence of the anisotropy induced by distinct modes of damage in the originally orthotropic composite. Nowadays, three dimensional models for laminates are readily implemented in computational techniques due to advances in computer power and programming facilities.

New techniques have been explored for assessing damage and, in some cases, healing on composite laminates. Such are the cases of the Virtual Crack Closure Technique (VCCT) or the use of cohesive elements -(interface elements)- on finite element procedures. Both of these techniques have links to the Fracture Mechanics field. VCCT was proposed by Rybicki and Kanninen (1977) and Rybicki et al. (1977) derived from the Irwint’s theory (Irwin, 1948) for crack analysis. Xie & Biggers (2006) and Leski (2007) coded successfully VCCT within a finite element program. VCCT relies on the calculation of the J-integral without the restriction

Finite Element Analysis of Progressive

Degradation versus Failure Stress Criteria on Composite Damage Mechanics

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of having excessive refinement of the mesh in the proximities of the crack tip which is also an advantage for computational saving. VCCT has the problem, like progressive damage models based in thermodynamical theory, of not having an initiation criteria for propagation of the fracture. The use of cohesive elements has been recently boosted. An excellent works by Camanho & Mathews (1999); Camanho et al. (2003) or Iannucci and Willows (2006) show a damage progression scheme combined with interface elements to couple the damage evolution with the mechanics of the fracture. An excellent review of these technique is provided by Wisnom (2010). Cohesive models are preferred to VCCT technique for a growing number of authors, such as Dugdale (1960), Xie & Waas (2006), Turon et al. (2007)), Tvergaard & Hutchinson (1996), Allen & Searcy (2000), or Cox & Yang (2006). Camanho et al. (2008) and also Hallet (1997) have used interface elements for the prediction of delamination on laminates.

In this chapter, finite element analysis is applied to laminates, and the formulation of the model is developed at lamina scale. The laminate is a stack of laminae of different, in general, fibre orientations. An explicit integration strategy for the finite element analysis is used due to the simplicity and robust convergence that provide[1]. The model is adapted in order to be included into an explicit FEM (see explicit formulation in Curiel Sosa et al. (2006) for implementation details) whereby the transient response may be conveniently simulated. This chapter is outlined as follows: firstly, a general discussion over damage modes is performed; secondly, the main theoretical aspects of the model are shown; thirdly, the computational algorithm, for implementation of the damage model as an individual module into an in-house FEM program as well as in major commercial software packages such as Abaqus or Ansys, is provided; and finally, a set of numerical examples including the low velocity impact on a composite laminate [0,90] a.