A nonlocal coupled damage-plasticity model for the analysis of ductile failure
Introduction
The approaches to the issue of structural analysis in the aeronautical industry have evolved over the decades. Original approaches relied on the principle of avoiding crack initiation in order to ensure fatigue resistance and durability. Later, damage tolerant design principles were introduced, in which the presence of small defects (cracks) was accepted as unavoidable, and emphasis was placed on ensuring that they do not grow to critical lengths within exploitation intervals between inspections. In the current view, not only crack initiation, but also crack propagation and trajectory analysis are very important for ensuring safe design and operation. Aeroengine components are subjected to complex loading induced by the combination of mechanical loading, changing temperatures and thermal gradients, inducing plastic deformation and creep that ultimately may lead to crack initiation and propagation. High temperature components are mostly made from ductile materials such as nickel-base superalloys. In these materials rupture (material separation) is preceded by damage that finds its physical representation in void nucleation, growth and coalescence. Development of populations of these defects in ductile materials is responsible for material softening that is played out in competition with the common strain hardening behaviour observed when ductile metal alloys are subjected to (tensile) loading. Modelling the behaviour of these materials in the post-peak softening regime and the correct and reliable prediction of crack propagation even today represent a serious challenge.
In the early 1960s, the development of the finite element method (FEM) brought about a major improvement in the tools available for the study and understanding of deformation in structural components subjected to arbitrary thermo-mechanical loading. However, despite the methods of crack propagation modelling being intensively debated in the scientific community ever since, performing it efficiently and reliably remains nowadays challenging even for simple loading cases, such as pure tensile loading. Considerable effort has been expended by the scientific community in order to solve crack propagation problems in a manner that is computationally as light as possible and that gives the same results for the crack path and propagation rate whichever (suitably refined) mesh is used. For that purpose, two radically different approaches have been thought and developed throughout the years.
The discrete crack approach is performed by introducing the separation of surfaces within the original structural body. Cracks within the original structures can be modelled either through the insertion of special types elements whose boundaries lie along the faces of the advancing crack (e.g. cohesive zone element method (Barenblatt, 1962) or by the introduction of elements with enriched nodal degree of freedom. This later method, known as extended finite element method (X-FEM) (e.g. Moes et al., 1999, Sukumar et al., 2000), allows crack propagation without remeshing. The newly introduced degree of freedom accounts for the displacement jump along the crack and the singular stress field at the tip of the propagating crack.
The second approach is derived from continuum mechanics principles. It considers fracture as the natural ultimate consequence of material degradation (Lemaitre, 1984). It is more realistic than the discrete crack approach in the sense that it accounts for the degradation of the material in the area surrounding the crack and can capture phenomena such as strain localisation. The load-bearing properties of the material (stiffness and strength) are modified through a special state variable referred to as damage. Damage is typically represented by a scalar or a higher dimension object (such as vector or tensor) with values between zero for virgin material and unity for the material that lost all its bearing capacity. Considered in this way, damage becomes an additional field quantity that needs to be considered along with strain and stress, and can be computed either incrementally, or as a certain function of a suitable physical parameter such as inelastic strain. The advantage of enriching the formulation of a continuum deformation problem with the damage parameter is that it allows considering the material post-critical behaviour, i.e. its response under deformations exceeding those when the maximum load-bearing capacity has been reached. Typically, this is associated with strain localisation, formation of discontinuities and fracture. Within a Continuum Damage Mechanics (CDM) framework (Kachanov, 1958, Lemaitre, 1992), cracks are represented by diffuse regions of material damaged so that it lost all its strength in at least one direction. Complete or partial loss of stress-carrying capacity of the material belonging to the “cracked” region provokes stresses redistribution and results in deformation and crack growth concentration in a relatively small region ahead of the crack tip. The rate and direction of the crack propagation are thus determined directly by the damage growth in the process zone. As opposed to the discrete crack approach, no separate fracture criteria are needed. Crack initiation and propagation follow directly from continuum mechanics theory. CDM approach allows treating cracks problems in any nonlinear FE code by simply implementing a routine for a new material constitutive behaviour. Therefore, the method is particularly attractive. In contrast to the discrete crack approach, it allows to take into consideration the material degradation in the vicinity of the crack and strain localisation phenomena. Moreover, in theory, no re-meshing techniques are needed.
However, soon after the introduction of CDM models based on the local formulation (i.e. such that the damage state at a point within the continuum only depends on the stress–strain state and history of that point), it was shown (Bazant et al., 1984) that these finite elements solutions do not converge upon mesh refinement. Due to the stress singularity at the crack tip, the damage field obtained from a local CDM model tends to localise on a plane, even though CDM assumes an homogeneous (or, at least smooth) damage distribution. The total energy dissipated during the cracking process is found to be proportional to the element size, and when the mesh size becomes infinitesimally small the dissipated energy vanishes. This physically unrealistic phenomenon is at the origin of strong mesh-dependency (as for the discrete crack approach). Moreover, strong damage localisation also leads, in some cases, to numerical instabilities (Benallal et al., 1989). Strain softening associated with the material degradation results in the loss of positive definiteness of the tangent material stiffness and ill-posedness of the boundary value problems. This phenomenon is more commonly described as the loss of ellipticity of the governing equation.
Over the years it has become clear that extensions of CDM models incorporating spatial interaction terms (whether they are gradient-based models (Peerlings et al., 1996) or models based on the spatial integral of one of the internal variables (Pijaudier-Cabot and Bažant, 1987) are the best way to overcome the above-listed difficulties. Nonlocal models make use of a material characteristic length to smooth the deformation and/or damage fields and thus prevent localisation of strain and damage within a plane. The newly introduced material characteristic length is particularly interesting because it also turns out to be related to a particular manifestation of the strain localisation phenomenon that accompanies fracture in materials: the so-called size effects, i.e. dependence of material mechanical properties (notably, its strength) on the structure size.
The majority of nonlocal models proposed up to now were intended to describe the behaviour of brittle and quasi-brittle materials (e.g. Pijaudier-Cabot and Bažant, 1987, Jirásek and Rolshoven, 2003, Grassl and Jirásek, 2006, Nguyen and Korsunsky, 2008, Peerlings et al., 1996) whereas ductile rupture was primarily addressed using micro-mechanically based models formulated using the considerations of void growth within a perfectly plastic matrix. A very detailed review of the subject can be found in Besson (2009) and recent contributions to the field include Dunand and Mohr, 2011, Lecarme et al., 2011, Malcher and Reis, 2013, Scheyvaerts et al., 2011, Tekoglu et al., 2012. Like all local or discrete descriptions of fracture, micro-mechanically-based models exhibit mesh-dependency and numerical instabilities when material softening occurs. At the microscopic scale, quasi-brittle rupture is mostly caused by micro-cracking of the matrix, while damage in ductile materials originates in void nucleation, growth and coalescence (Hosokawa et al., 2013a, Hosokawa et al., 2013b, Kadkhodapour et al., 2011). At the macroscopic level and as opposed to quasi-brittle materials, ductile materials exhibit very significant yield behaviour. Their tensile stress–strain diagram thus shows first a linear behaviour; after the yield point has been reached, a nonlinear ascending curve characteristic of material hardening is observed; the curve finally starts to descend after the peak nominal stress is attained. The decreasing curve at the end of the deformation process is a manifestation of necking (reduction of the effective cross-sectional area) and of strain softening. As for the damage in quasi-brittle materials, strain softening is accompanied by the localisation of deformation and damage into a band of a non-zero width. Size effects that follow from strain localisation have been observed in ductile materials for various loading modes and specimen geometries: in bending of thin films (Stolken and Evans, 1998), torsion of thin wires (Fleck et al., 1994), micro-indentation (Ma and Clarke, 1995, Poole et al., 1996), tensile tests on dog-bone specimens (Korsunsky and Kim, 2005, Korsunsky et al., 2005), micro scale flanged upsetting experiments (Ran et al., 2013). Strain gradient plasticity modelling (Fleck and Hutchison, 2001, Fleck et al., 1994) has been introduced in order to capture the nonlocality of physical response and the attendant size-effects. Therefore, it appears necessary to introduce this aspect into numerical modelling as well (Aifantis, 1987). As explained above, nonlocal models are able to treat fracture problems in a way that avoids mesh-dependency, and overcomes the numerical instabilities arising from the strong strain softening, and are also able to capture size effects. They thus appear to be a perfect candidate to describe ductile rupture.
The interest generated by nonlocal description of crack propagation in application involving ductile materials is only relatively recent (Engelen et al., 2003, Lorentz and Andrieux, 1999, Nilsson, 1998, Saczuk et al., 2003) and attention so far has been mainly devoted to provide physically valid (Abu Al-Rub and Voyiadjis, 2004, Abu Al-Rub and Voyiadjis, 2006, Abu Al-Rub et al., 2007, Andrade et al., 2011, Belnoue et al., 2007, Marotti de Sciarra, 2009, Marotti de Sciarra, 2012, Mediavilla et al., 2006a, Mediavilla et al., 2006b, Peerlings et al., 2012, Saanouni and Hamed, 2013) and numerically efficient models (Belnoue et al., 2010, Mediavilla et al., 2006a, Mediavilla et al., 2006b, Poh and Swaddiwudhipong, 2009, Seabra et al., 2013). In addition, the effect of stress triaxiality on ductile damage has been the subject of a number of experiments (Driemeier et al., 2010, Farbaniec et al., 2013, Hosokawa et al., 2012, Luo et al., 2012) and theoretical works (Bai and Wierzbicki, 2008, Brünig et al., 2008, Brünig et al., 2013, Ebnoether and Mohr, 2013, Khan and Liu, 2012, Luo et al., 2012, Rousselier and Luo, 2014, Malcher et al., 2012, Abu Al-Rub and Voyiadjis, 2004, Andrade et al., 2011). Despite a large volume of contributions, the correlation between experimental results and theoretical modelling together with the calibration of model parameters are not always addressed at length. This ends up at having some “empirical” parameters which either have no physical meaning or are difficult to determine from experimental tests (Abu Al-Rub and Voyiadjis, 2004, Andrade et al., 2011, Abu Al-Rub and Voyiadjis, 2004, Andrade et al., 2011). In our opinion, parameter calibration is the critical issue that determines model applicability and its predictive ability. This is a motivation for our current approach, which places much focus on an overall framework within which the constitutive model is developed, enhanced with nonlocal features, followed by a detailed calibration procedure that serves both to identify the parameters and to demonstrate the model features. The strategy is first to build a consistent approach for nonlocal constitutive modelling of ductile failure, while at the same time addressing outstanding issues, and then to propose the improvements for ingredients of this approach in the next steps.
The paper is organised as follows. The thermodynamic formulation of the proposed model is presented in the next section. Damage mechanics and plasticity theory aspects are taken into account, insofar they are both present in the coupled model. The approach shares some similarity with Lemaitre’s approach, in terms of using an explicit evolution law for the damage variable. However, the thermodynamic framework (Houlsby and Puzrin, 2000) used for the model formulation makes it rigorous in the sense that both the damage function and yield function are explicitly linked to a dissipation potential, while most existing models are based on implicitly defined dissipation potential. In the proposed approach, the model definition needs only two potentials: energy potential and explicitly defined dissipation potential. In Section 3, the identification and calibration of model parameters controlling the constitutive responses are addressed. The nonlocal enhancement of the model, together with implementation aspects and a localisation analysis to prove analytically the regularisation effects, are presented in Section 3. Since nonlocal model involves a length scale, a calibration procedure is developed following closely the experimental counterparts (Korsunsky and Kim, 2005) for a robust and consistent calibration of the internal length of the proposed model. Numerical examples are presented to demonstrate features of the proposed approach.
Section snippets
A thermodynamics-based formulation
The thermodynamic framework by Houlsby and Puzrin (2000) is used as the basis for the model formulation. This thermodynamic framework provides a consistent and rigorous basis for the development of rate-independent models, while the generalisation of the formulation to include rate-dependent effects is also straightforward (Houlsby and Puzrin, 2002). For the sake of simplicity, isotropic damage is assumed and coupled with a Von Mises type yield function. Isotropic hardening rules with
Nonlocal regularisation and implementation
It has been well known that softening is the cause of material instability in the analysis of boundary value problems (Jirásek and Bazant, 2002). A constitutive model devoted to modelling the behaviour of a softening material must be properly “regularised” to deal with such an instability issue. In this paper, the nonlocal regularisation technique will be used for the enhancement of the proposed model. With damage being the only cause of softening, the regularisation is applied directly to the
Calibration of model parameters
The determination of the spatial parameter of the proposed nonlocal model is briefly presented here. This is based on the calibration procedure for local parameters described in Section 2.2 and a procedure to numerically establish the relationship between the effective width h of the fracture process zone, and the internal length of nonlocal continua. The latter procedure has been developed and used for nonlocal models of concrete in Nguyen and Houlsby (2007), and Nguyen and Korsunsky (2008).
Conclusions
We proposed a consistent approach to the modelling of ductile failure. A thermodynamics-based model incorporating plastic deformation for the ductile behaviour and damage effects for post-peak softening and separation was developed and enhanced with nonlocal regularisation. In conjunction with this theoretical development, we pursued the objective of developing a calibration procedure that follows closely the experimental setup, but possesses special features for nonlocal analysis. Our focus is
Acknowledgements
Giang Nguyen gratefully acknowledges support from the Australian Research Council via projects DP1093485 and FT140100408. Alexander M. Korsunsky acknowledges the support through EU FP7 project 604646 iSTRESS.
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