A generalized anisotropic damage interface model for finite strains
Introduction
Interfaces play an important role in numerous engineering applications, such as adhesive bonding layers and laminate composite structures. Interfaces are of zero thickness and of lower dimension than the surrounding bulk. They can be considered as two-sided surfaces which represent a thin layer of material. The interface’s properties differ from the surrounding bulk. Generally, they are weaker which causes the deformation to localize in the interface layer. Accordingly, it is crucial to accurately model the interface’s response, in particular during failure, in order to simulate the behavior of the overall continuum.
In general, mechanical imperfect interfaces, e.g. isotropic and anisotropic cohesive zone models (CZM), interface (in)elasticity and generalized interface models, allow for displacement jumps and/or traction jumps across the interface with the jump in the bulk Piola stress , and the interface normal vector , see Table 1. CZM were introduced by Barenblatt, 1959, Barenblatt, 1962, Dugdale, 1960 and widely adopted in literature, e.g. Needleman, 2014, Rice, 1968, Needleman, 1990a, Needleman, 1990b, Nguyen et al., 2001. Ahead of the crack tip, a nonlinear process zone is considered. It is assumed that previously identical material points can be separated by a displacement jump . Within the process zone, cohesive tractions , which depend via traction–separation laws on the displacement jump, arise at the interface surfaces and . For isotropic CZM, the direction of the cohesive tractions coincides with that of the displacement jump. These models show no response to in-plane stretch, since the material between the two surfaces of the interface is neglected. The decomposition of the cohesive tractions in normal and shear contributions (with different stiffnesses) results in anisotropic cohesive laws. For the geometrically nonlinear case considered here, these violate thermodynamic consistency and the balance of angular momentum if no further stress contributions at the interface than the cohesive tractions are taken into account, see e.g. Mergheim and Steinmann, 2006, Mosler et al., 2011, Vossen et al., 2013, Ottosen et al., 2016. When expanding the CZM to anisotropy, a dependence on structural tensors is introduced to comply with physical requirements, cf. Mosler and Scheider (2011). As a consequence, an additional shear-like stress is introduced within the interface, which we denote as the cohesive Piola stress , and which results in a jump in the tractions across the interface . However, in-plane stretches do not evoke any stresses within the interface if only the anisotropic CZM is used.
In contrast, interface (in)elasticity characterizes interfaces with a continuous displacement across the interface, but discontinuous tractions. The concept of interface elasticity within solids was developed by Murdoch, 1976, Gurtin and Struthers, 1990, Angenent and Gurtin, 1989, Gurtin et al., 1998 based on the surface elasticity theory of Gurtin and Murdoch, 1975b, Gurtin and Murdoch, 1975a, Gurtin and Murdoch, 1978. Its finite element implementation was presented in Javili and Steinmann, 2009, Javili and Steinmann, 2010, Javili et al., 2014, Esmaeili et al., 2016. Numerous applications, especially at nanometer scale, exist in literature, e.g. Benveniste and Miloh, 2001, Sharma et al., 2003, Duan et al., 2005a, Duan et al., 2005b, Duan et al., 2009, Fischer and Svoboda, 2010, Wang et al., 2005, Sharma and Ganti, 2004, Sharma and Wheeler, 2007, Cammarata, 1994, Dingreville et al., 2005, He and Lilley, 2008, Miller and Shenoy, 2000, Lu et al., 2021. Pursuant to the interface (in)elasticity theory, a coherent interface obtains its own resistance by means of energetic and dissipative structures on the interface, i.e. membrane Piola stresses occur. This model captures the (in)elastic material behavior in the interface’s tangential plane when loading parallel to the interface.
A mechanical interface can be considered as generalized if it allows for a jump in both, the displacement and the traction . The decohesion may be anisotropic and the interface can account for in-plane resistance, i.e. the interface Piola stress consists of a cohesive and a membrane part . A generalized interface model combining anisotropic decohesion and interface (in)elasticity is suitable for applications subjected to combined shear, tensile and stretch loading, e.g. for delamination processes of adhesive bonds or composite materials. Modeling interfaces in metals, i.e. grain boundaries, requires an additional microtractions model, compare Gurtin, 2008, Spannraft et al., 2020.
Although there are many publications on CZM and interface (in)elasticity, few works exist on anisotropic CZM or generalized interface models for finite strains. The requirement to include structural tensors for describing material anisotropy in CZM at large deformations was pointed out by Mergheim and Steinmann (2006). Mosler and Scheider (2011) introduced a variationally consistent anisotropic CZM depending not only on the displacement jump, but also on the ‘average deformation gradient’. As a consequence, out-of-plane shear forces act in the interface. Esmaeili et al. (2017) implemented a generalized mechanical interface model which couples a hyperelastic interface law accounting for (non-local) in-plane degradation with an isotropic CZM. The placement of an interface layer within its associated interphase via weighted averages was discussed by Saeb et al. (2020). Javili et al. (2016) and Saeb et al. (2019) emphasized the potential of capturing size effects in computational homogenization by applying generalized interfaces on the microscopic scale. The four latter contributions account for isotropic decohesion and in-plane stretch by providing the interface with energetic quantities in tangential direction. Ottosen et al. (2016) presented a framework for geometrically nonlinear anisotropic CZM, noting the possibility of an additive decomposition of the interface Piolas stress tensor for a generalized interface accounting for normal, shear and in-plane loading. The derived thermodynamically consistent framework satisfies the dissipation inequality, the balance equations, objectivity, and spatial covariance. The constitutive modeling was exclusively discussed for anisotropic cohesive behavior providing an expansion of hyperelasticity to quasi-brittle damage and elasto-plasticity. Heitbreder et al. (2018) and Heitbreder (2019) presented a finite element implementation of this framework. Thereby, computational homogenization is included to analyze size effects. The presented interface model accounts for exponential normal and shear damage which is coupled, and in addition, for interface hyperelasticity. Heitbreder et al. (2021) enhanced the generalized interface law with gradients of the displacement jumps and conducted experiments on a 3d-printed specimen with a soft interface.
To the authors’ knowledge, no literature on generalized interface models combining an anisotropic cohesive law and interface inelasticity at finite strains exists. As discussed, only a few models were developed based on anisotropic cohesive laws and interface elasticity not accounting for in-plane damage, e.g. Heitbreder et al., 2018, Heitbreder et al., 2021 and Heitbreder (2019). The mentioned publications provided some numerical examples, but no extensive numerical analysis is given to demonstrate and analyze the effects of anisotropy. Furthermore, in none of the presented models, a coupling between the damage variables of normal, shear and in-plane degradation is taken into account.
This contribution presents a generalized damage interface law derived from the interfacial strain energy density which is composed of a membrane part and a cohesive part . The cohesive strain energy density’s dependence on the normal vector via the interface deformation gradient allows for anisotropic decohesion. The cohesive and membrane degradations are coupled via damage variables to analyze their combined effect on the overall degradation. The presented interface law is thermodynamically consistent, material frame indifferent, and fulfills the balance of angular momentum. As consequence of a variationally consistent derivation, the dissipation is physical, the stiffness matrix is symmetric, and the same model describes loading, as well as unloading. The model is implemented as an Abaqususer element (UEL) to have a flexible, numerical implementation. The influence of the additional shear stress in the interface which results from the anisotropy of the cohesive law is numerically analyzed. Furthermore, the coupling conditions of the damage law are studied by means of a peel test. Finally, a more complex example, comparing uncoupled and coupled cohesive and membrane damage in a buckling simulation of pre-stretched substrates, is provided.
Subsequently, the following notations are used. Scalars are denoted by standard letters, e.g. , first-order and second-order tensors by lower or upper-case bold letters, e.g. and , whereas fourth-order tensors are denoted by upper-case sans-serif letters, e.g. . In terms of their Cartesian components, the non-standard contraction and outer product are introduced as The jump and the average value of a quantity are specified, whereby the following relation holds .
Section snippets
Kinematics
A continuum body in the Euclidean space is partitioned into two subdomains, and by an interface , i.e. , see Fig. 1. The material and spatial placements of bulk particles are labeled and . The outward material and spatial unit normal vectors to the bulk boundary are specified as and . The bulk deformation map and deformation gradient are and . All geometrical operations of the interface are performed on fictitious reference points and ,
Generalized interface law for nonlinear kinematics
In this section, the relevant balance equations for continua with generalized interfaces are recapitulated and a generalized interface law is developed. The balance equations are formulated in the material configuration and mainly limited to the interface part of the equations. Further discussions of the balance equations with generalized interfaces can be found in Ottosen et al., 2016, Javili et al., 2016, Esmaeili et al., 2017, Javili, 2012.
Computational framework
This section describes the implementation of the previously defined generalized mechanical interface model in a three-dimensional, geometrically nonlinear Finite Element Method framework.
Numerical examples
The following numerical examples illustrate and analyze the material behavior which can be modeled by means of the introduced generalized interface model. Thereby, the effects of various parameters on resulting reaction forces, bulk and interface stresses, cohesive tractions and on the fracture length are analyzed and depicted. In the first example, the impact of the cohesive Piola stress for different degrees of anisotropy, damage and membrane stretches is discussed for a simple academic
Summary and conclusion
This contribution presents a generalized mechanical interface model accounting for irreversible anisotropic decohesion and membrane degradation, whereby the normal, shear and membrane damage variables can be fully coupled. The model is thermodynamically consistent, material frame indifferent and fulfills the balance of angular momentum because of its variationally consistent derivation. The model is embedded in a geometrically nonlinear finite element framework and implemented for non-zero
CRediT authorship contribution statement
Lucie Spannraft: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Writing – review & editing, Visualization. Paul Steinmann: Resources, Writing – review & editing, Supervision, Funding acquisition. Julia Mergheim: Conceptualization, Methodology, Formal analysis, Writing – original draft, Writing – review & editing, Supervision, Project administration, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We sincerely thank Gunnar Possart from the Institute of Applied Mechanics, Universität Erlangen-Nürnberg, Germany, for the in-depth discussions on applying the presented model to realistic applications, including suggestions for set-ups and discussions of results.
Funding
Support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 377472739/GRK 2423/1-2019.
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