A generalized anisotropic damage interface model for finite strains

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Highlights

  • Generalized interface (cohesive zone and interface elasticity) for finite strains.

  • Anisotropic decohesion induces shear-like stresses for thermodynamical consistency.

  • Cohesive and membrane damage variables are coupled for an interaction.

  • Presented model is implemented in a user subroutine in Abaqus.

  • Numerical examples illustrate the influence of anisotropy and damage coupling.

Abstract

This contribution presents a generalized mechanical interface model for nonlinear kinematics. The interface’s response is non-coherent, i.e. allows for a jump in the deformations and for cohesive failure, and also includes interfacial (in)elasticity, which means that an additional membrane stiffness is introduced in the interface. This can result in a jump in the tractions across the interface and induces membrane stresses for in-plane stretches. An anisotropic cohesive law, i.e. dependent on the spatial interface normal, is formulated which induces for nonlinear kinematics additional shear-like stresses within the interface to satisfy the balance of angular momentum. The cohesive and membrane degradations are coupled via damage variables of the different deformation modes to account for an interaction. It is shown that the model is thermodynamically consistent, fulfills the balance equations and material frame indifference. An interface element with the generalized interface model is implemented as a user element in Abaqus. Numerical examples illustrate the influence of damage coupling on the mechanical response of adhesive layers.

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 φ and/or traction jumps T̄=PM̄ across the interface with the jump in the bulk Piola stress P, and the interface normal vector M̄, 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 φ. Within the process zone, cohesive tractions Tc, which depend via traction–separation laws on the displacement jump, arise at the interface surfaces I+ and I. 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 Tcm and shear Tct 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 P̄c, and which results in a jump in the tractions across the interface T̄0. 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 P̄=P̄m 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 φ0 and the traction T̄=PM̄0. 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 P̄=P̄c+P̄m. 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 P̄=P̄m+P̄c 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 ψ̄ which is composed of a membrane part ψ̄m and a cohesive part ψ̄c. 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. M, first-order and second-order tensors by lower or upper-case bold letters, e.g. a and A, whereas fourth-order tensors are denoted by upper-case sans-serif letters, e.g. C. In terms of their Cartesian components, the non-standard contraction and outer product are introduced as (a¯C)ikl=(a)j(C)ijkl,(B¯a)ijk=(a)j(B)ik=a(i¯B).The jump =+ and the average value {}=12++ of a quantity are specified, whereby the following relation holds ={}+{}.

Section snippets

Kinematics

A continuum body in the Euclidean space BE3 is partitioned into two subdomains, B+ and B by an interface I, i.e. BB+B, see Fig. 1. The material and spatial placements of bulk particles are labeled X and x. The outward material and spatial unit normal vectors to the bulk boundary B are specified as N and n. The bulk deformation map and deformation gradient are φ(X,t) and F(X,t)φ(X,t). All geometrical operations of the interface are performed on fictitious reference points X̄ and x̄,

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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