A coupled viscoplastic rate-dependent damage model for the simulation of fatigue failure of cement–bone interfaces
Introduction
Aseptic loosening of cemented hip prostheses is the most frequent cause of revision arthroplasty. It seems clear that both biological and mechanical effects contribute to implant loosening and cement–bone interface and cement–stem interface mechanical properties are among the most important mechanical factors involved in the process of failure (Jasty et al., 1991).
Few works have aimed at the mechanical role of the cement–bone interface. However, radiographs and retrieved specimens from loose cemented hip components have shown radiolucencies on the cement–bone interface, indicating that this is a main site of debonding initiation (Gruen et al., 1979). The mechanical behavior of this interface under static and cyclic loading has been experimentally well characterized in the works of Mann et al., 1997a, Mann et al., 1998, Mann et al., 2001, Kim et al., 2004a, Kim et al., 2004b among others. Moreover, correlations were found between the mechanical properties of the cement–bone interface and the amount of cement interdigitated within bone. The influence of the surgical preparation and corresponding bone surface topography on the fatigue response of cement–bone interfaces (Arola et al., 2006) and on the apparent volume available for interdigitation (Arola et al., 2001) has also been studied.
Several authors have simulated the bone–cement interface as totally bonded (Stolk et al., 2002, Lennon and Prendergast, 2001, Lennon et al., 2003). Others incorporated the non-linear behavior only along the normal direction by means of cohesive zone models (Clech et al., 1985) or nonlinear fracture mechanics (Mann et al., 1997b). In particular, Mann and Damron (2002) developed a model with independent damage in each direction using a piece-wise linear constitutive law. However, Moreo et al. (2006) showed that the use of a cohesive zone model with a combined damage law and exponential softening can improve the accuracy of the results. Nevertheless, none of the aforementioned models takes into account plastic deformations, which have demonstrated to play an important role in the process of failure of the interface, even higher than the loss of stiffness (Kim et al., 2004b). Therefore, plasticity or viscoplasticity (if rate effects need to be considered) should be incorporated in a cement–bone interface model.
The theories of damage mechanics and plasticity are profusely used for the modeling of nonlinear material behavior.
With regard to damage, continuum damage mechanics (CDM) is one of the most common approaches. Since the initial work of Kachanov (1958), the CDM framework was further developed by Lemaitre, 1985, Lemaitre, 1992, Chaboche, 1984. Scalar damage variables have been proposed by Kachanov, 1958, Lemaitre, 1985, among many others. When it is necessary to differentiate between two damage isotropic mechanisms, models with two scalar damage variables have been proposed (Comi and Perego, 2001, Marfia et al., 2004, Contrafatto and Cuomo, 2006). Moreover, in order to take into account damage induced material anisotropy, fourth or, more frequently, second order damage tensors have been introduced (Kachanov, 1980, Ju, 1990, Al-Rub and Voyiadjis, 1988, Menzel et al., 2005, Brünig and Ricci, 2005).
As for as plasticity, the reader is referred to Hill, 1950, Lubliner, 1990, Khan and Huang, 1995 for a detailed description of various plasticity theories and to Simo and Hughes, 1998, Simo, 1998 for a detailed summary for integration algorithms for classical plasticity. Moreover, viscoplastic models that take into account rate effects have also been developed for different types of materials (Lion, 2000, Khan et al., 2000, Khan et al., 2006, Tashman et al., 2005, Ganapathysubramanian and Zabaras, 2005, Kontou and Spathis, 2006). Perhaps the most known viscoplastic model is that of Perzyna (1966). Perzyna-type models are still frequently used (Mähler et al., 2001, Tashman et al., 2005, Helm, 2006, Pierard and Doghri, 2006, Voyiadjis and Abed, 2006) and models based on a generalization of the “overstress” concept have also been proposed (Krepml and Khan, 2003, Colak and Krempl, 2005, Colak, 2005, Menzel et al., 2005, Yaguchi et al., 2002, Yaguchi and Takahashi, 2005, Shenoy et al., 2006). Another concept of viscoplastic model is the so-called Duvaut–Lions format (Duvaut and Lions, 1972), in which the direction of viscoplastic flow is determined by the gradient of the yield surface evaluated at the state defined by the closest-point projection of the current stress onto the yield surface. The use of this type of model was brought into focus by Simo et al. (1988) and later in Simo and Hughes, 1998, Simo, 1998, where the model was generalized and several numerical aspects were discussed. Usually the evolution laws for the Duvaut–Lions model are postulated and not derived from a potential function. However, Ristinmaa and Ottosen (1998) showed how this model can also be derived from potential functions such that the dissipation inequality is fulfilled.
Moreover, for the continuum modeling of failure, models that couple damage and plasticity have been commonly used: Bonora, 1997, Park and Voyiadjis, 1998, Contrafatto and Cuomo, 2002, Pirondi and Bonora, 2003, Menzel et al., 2005, among many others. Discussions about how to couple damage and elastoplasticity can be found in Brünig, 2003, Menzel et al., 2001, Menzel and Steinmann, 2003.
However, classical continuum formulations of damage/plasticity when applied to strain softening type problems become ill-posed. From the point of view of finite element analysis, this results in a pathological dependence of the solution upon the spatial discretization. Enhanced models that include higher order gradients (de Borst and Pamin, 1996, Brünig, 2001, Dorgan and Voyiadjis, 2003, Voyiadjis et al., 2004) or non-local terms (Bažant and Lin, 1988, Strömberg and Ristinmaa, 1996, Brünig and Ricci, 2005) have successfully overcome these limitations. However, this kind of models suffer from some shortcomings such as difficulties in the treatment of boundary conditions or non-trivial computational implementations.
An alternative possibility is to directly incorporate the discontinuities that arise as a result of the strain localization into the displacement field (strong discontinuity). Within this approach, one possibility is the use of the so-called cohesive zone models, where a traction/jump of displacement constitutive law is introduced for the discontinuity. This approach, pioneered by Dugdale, 1960, Barenblatt, 1962, has been profusely used during the last two decades (see de Borst et al. (2004) for a review). When the crack path is known in advance from experimental evidence or when the crack is highly localized because of the structure of the material (such as in laminated composites or adhesive joints), interface elements with a cohesive zone model have proved to be efficient (Allix and Corigliano, 1999, Wang and Siegmund, 2006, Moreo et al., 2006). A comprehensive review of interface element-based models for delamination can be found in Alfano and Crisfield (2001). Moreover, there exist different approaches to incorporate inelastic deformations into interface models: plastic interface models (Caron et al., 2006, Carpinteri et al., 2006, Dmitriev et al., 2006, Liu et al., 2006), mixed cohesive zone/friction interface models (Tsai et al., 2005) or molecular dynamics-based models (Yamakov et al., 2006, Spearot et al., 2007).
The goal of this study is to extend the previous cohesive zone model developed by the authors (Moreo et al., 2006) in order to take into account plastic elongations at cement–bone interfaces. We show that the model is able to predict the evolution of total and plastic deformations, mechanical properties degradation and fatigue life under static and cyclic mixed-mode loading with a sufficiently good agreement with experimental data.
Section snippets
Interface coupled viscoplastic model
In this section an outline of the interface model is presented. Initially, a brief overview of the mechanical behavior of the interface is presented. A thorough description of the mechanical behavior of the interface can be found in the previously cited works of Mann and coworkers. Next, viscoplasticity and damage are characterized. Finally, the numerical implementation of the model by means of the finite element method is discussed.
Finite element model
In this section the experimental tests performed by Kim et al., 2004a, Kim et al., 2004b have been reproduced. In those experiments, several cement–bone specimens were fixed in test jigs and tested under pure normal or shear cyclic loading with a periodic pattern at a frequency of 5 Hz with a stress ratio of 0.1 and a maximum apparent stress level between 50% and 80% of the maximum strength. Apparent stress was computed as the applied load divided by the cross-sectional area of the specimen. In
Discussion
The main goal of this work was the development of a bone–cement interface constitutive model that accounts for static and cyclic damage and rate-dependent plastic deformations. The motivation of this paper was the experimental works of Mann et al., 1997a, Mann et al., 1998, Mann et al., 2001, Kim et al., 2004a, Kim et al., 2004b, where the mechanical properties and the static and cyclic behavior of bone–cement interfaces were well characterized. It is specially important to highlight that
Acknowledgements
The authors thank Professor K. A. Mann for helpful advice and useful suggestions during the execution of this work. Research support of the Spanish Ministry of Science and Technology through the Research Project DPI2006-14669 and the FPU graduate research fellowship program are gratefully acknowledged.
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