A self-adaptive finite element approach for simulation of mixed-mode delamination using cohesive zone models

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Abstract

Oscillations observed in the load–displacement response of brittle interfaces modeled by cohesive zone elements in a quasi-static finite element framework are artifacts of the discretization. The typical limit points in this oscillatory path can be traced by application of path-following techniques, or avoided altogether by adequately refining the mesh until the standard iterative Newton–Raphson method becomes applicable. Both strategies however lead to an unacceptably high computational cost and a low efficiency, justifying the development of a process driven hierarchical extension of the discretization used in the process zone of a cohesive crack. A self-adaptive enrichment scheme within individual cohesive zone elements driven by the physics governing the problem, is an efficient solution that does not require further mesh refinements. A two-dimensional mixed-mode example in a general framework with an irreversible cohesive zone law shows that an enriched formulation restores the smoothness of the solution in structures that are discretized in a relatively coarse manner.

Introduction

Accumulation of damage at interfaces in laminated structures results in the formation and growth of interlaminar cracks through a non-linear and irreversible process which is known as delamination and constitutes one of the most common failure modes in such structures. Interfacial stresses orthogonal or parallel to laminae cause delamination in mode I (normal opening) or modes II (sliding) and III (tearing), respectively. Continuum models with strong discontinuities seem to yield the best description for deformation of a structure undergoing this failure mechanism.

Linear elastic fracture mechanics (LEFM)-based techniques have often been adopted for the modeling of discrete crack propagation problems when the size of the failure process zone is small compared to structural dimensions which is the case in large structures or brittle interfaces [1], [2], [3], [4], [5], [6]. In this class of methods, the mesh has to be sufficiently fine near the crack tip to capture the stress singularity in that region [7], [8]. Using the partition of unity property of finite elements [9], the extended finite element method (X-FEM) has been introduced to eliminate the need for a mesh refinement by enriching the elements adjacent to the crack with a discontinuous function or near-tip asymptotic functions [10], [11]. The concept has also been extended to cohesive crack propagation analyses [12], [13]. However, additional criteria are still required for crack initiation and propagation [14], [15].

Eliminating the stress singularity at the crack tip, cohesive zone models (CZMs) were introduced by Barenblatt [16] for perfectly brittle materials and by Dugdale [17] for a perfectly plastic material. These models were extensively used after being applied in a finite element framework to predict crack initiation and growth [18]. Later, an interfacial potential was defined to apply the methodology to interface problems [19]. Thereafter, these models have been improved and used in a wide variety of applications [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] while their different numerical aspects have also been investigated [22], [31], [32], [33], [45].

Simulation of interfacial delamination in a finite element framework is carried out by placing interface (decohesion) elements along interfaces between laminae where the interfacial damage in the structure is reflected. The damage free continuum can then be described with an arbitrary material law. Being characterized by a traction–separation law (TSL) which describes the variation of traction with respect to separation along the cohesive zone, interface models have the appealing feature of combining stress-based and fracture mechanics-based approaches to predict delamination onset and propagation [34].

Using CZMs for the modeling of delamination in brittle interfaces in a quasi-static finite element framework suffers from an intrinsic discretization sensitivity. A large number of interface elements are needed for the discretization of the process zone, i.e. the region entering the softening regime [35], [36]. Without this very fine discretization, sudden release of energy in large CZ elements results in a sequence of snap-through or snap-back points to appear in the global load–displacement response of the system [37]). Further mesh refinement may still be necessary to ensure a stable crack growth in case of physically stable crack growth problems [38].

Application of the standard iterative Newton–Raphson method under load or displacement control fails to converge in the case of snap-through or snap-back. Path-following techniques have been proposed to follow the discretization-induced oscillations, requiring expensive computations accompanying the small increments needed to trace this unphysical path [39], [40], [41], [42], [43], [44], [45], [46]. Since these oscillations are artifacts of the discretization, a persistent mesh refinement finally results in a smooth path which can be solved by the standard Newton–Raphson scheme [47]. However, for realistic interface parameters with a small process zone size, the element size has to be extremely small, which results in unacceptably high computational costs. In some cases, a coarse mesh is used by artificially reducing the interface strength without changing the fracture toughness [48], which can underestimate the loading capacity of the structure [49].

The efficiency and robustness of brittle CZMs can be significantly increased by reducing the oscillations observed in the global load–displacement behavior without further mesh refinement. In line with this purpose, a local enrichment of the elements in the vicinity of the softening process zone with hierarchical polynomial shape functions has been proposed in [50]. The partition of unity property of finite element interpolation functions has been employed to enrich the basis functions of CZ and bulk elements with the analytical solutions of a beam bending problem [51].

Recently, the authors proposed an approach to enrich the separation approximation in the process zone of a cohesive crack by an adaptive hierarchical extension [52]. The linear separation approximation throughout the CZ element was enriched with a bi-linear function, where the enrichment peak position and the magnitude of the enrichment were regarded as additional degrees of freedom. The effect of the enrichment on the global load–displacement response of a simple mode I delamination problem discretized by a relatively coarse mesh was found to be a reduction of the discretization-induced oscillations. However, the adaptivity of the enrichment peak was not optimal; moreover, defining the enrichment peak position in terms of an auxiliary parameter lowered the convergence rate due to introduction of an extra non-linearity.

In this paper, the formulation of the process driven enrichment scheme proposed in [52] is extended to account for mixed-mode delamination while taking the irreversibility of the delamination process into account. The formulation of enriched CZ elements and their adjacent bulk elements – that are also enriched for continuity of the displacement field – are derived in a full two-dimensional finite element framework. All numerical aspects of the enrichment are discussed first on the basis of a simple mode I delamination problem which results in a considerable improvement of the enrichment adaptivity. Benefiting from the new developments, the enrichment function acts in a self-adaptive manner within the interface; that is, its peak position is controlled by the moving process zone. To achieve the latter, the penalty formulation used in [52] has been modified to improve the adaptivity of the enrichment in the elements within the process zone. This ensures a maximum benefit to be gained from the enrichment adaptivity regarding the elimination of discretization-induced oscillations. An additional penalty condition near the boundaries of enriched CZ elements is introduced to keep the bi-linear enrichment function confined within element boundaries. This eliminates the need for a transformation to an auxiliary parameter as done in [52] and prevents convergence problems due to the non-linearity added to the system. The new set of developments significantly improves the efficiency and robustness of an enriched CZM while allowing it to be used in general applications. The advantage of the self-adaptive enrichment is illustrated through a full two-dimensional peel-off example using a mixed-mode irreversible bi-linear traction–separation law.

Section snippets

Interface model

The most important parameters of an interface constitutive law are the area under the traction–separation curve (known as the fracture toughness) and the maximum strength. However, different shapes have been adopted to describe the variation of traction with respect to separation (or TSL) for which an overview can be found in [32], [53]. In this work, a bi-linear TSL is used.

At an interfacial point between two materials in a two-dimensional continuum, separation and traction vectors are defined

Cohesive zone enrichment

As delamination grows in brittle interfaces discretized with a coarse mesh, a characteristic non-smooth behavior is observed in the global load–displacement response of the system. To remedy this problem, the finite element formulation of the interface is enriched. The finite element formulation of the neighboring bulk is also enriched to ensure compatibility of the displacement field in the process zone of a cohesive crack which propagates in mixed-mode loading conditions.

Numerical aspects of the enrichment

A simplified peel-off test, for which an analytical solution exists [52], is revisited to illustrate important numerical aspects of the enrichment which leads to a deeper insight in its efficiency. The adaptivity of the proposed enrichment and the penalty formulation used to ensure a unique solution are discussed in detail.

A linear elastic bulk material is pulled from a rigid substrate as shown in Fig. 5. A bi-linear TSL as introduced in Section 2 and characterized by Gc = 0.25 N/mm, t0 = 100 MPa, λ0 =

Mixed-mode application

In order to evaluate the performance of the improved enriched CZM in a mixed-mode delamination problem, a full 2-D peel-off test under plane strain conditions is considered. In this example, a linear elastic bulk strip with length L, height H, and width W is pulled from a rigid substrate, see Fig. 9a. In the interface between the strip and the substrate, an initial crack with length Lc is inserted. A normal displacement up to u = 3 mm is prescribed at the pre-cracked end of the strip.

The bulk

Discussion and conclusion

Cohesive zone models are commonly used to simulate initiation and propagation of delamination, whereby enough cohesive zone (CZ) elements should be used in the process zone to ensure a correct description of the opening profile which results in a more stable solution. Hence, a fine mesh is necessary for brittle interfaces, where the process zone size is small compared to other structural dimensions. A fine mesh typically increases the computational cost whereas a coarse discretization results

Acknowledgement

This research was carried out within the project WP IV-C-3 of the MicroNed program.

References (53)

  • Y.J. Wei et al.

    Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

    J Mech Phys. Solids

    (2004)
  • S. Li et al.

    Mixed-mode cohesive-zone models for fracture of an adhesively bonded polymer-matrix composite

    Engng Fract Mech

    (2006)
  • B.A.E. van Hal et al.

    Cohesive zone modeling for structural integrity analysis of IC interconnects

    Microelectron Reliab

    (2007)
  • M.R. Wisnom

    Modelling discrete failures in composites with interface elements

    Composites: Part A

    (2010)
  • M. Elices et al.

    The cohesive zone model: advantages, limitations and challenges

    Engng Fract Mech

    (2002)
  • N. Chandra et al.

    Some issues in the application of cohesive zone models for metal–ceramic interfaces

    Int J Solids Struct

    (2002)
  • R. de Borst

    Numerical aspects of cohesive-zone models

    Engng Fract Mech

    (2003)
  • W. Cui et al.

    A combined stress-based and fracture-mechanics-based model for predicting delamination in composites

    Composites

    (1993)
  • P.W. Harper et al.

    Cohesive zone length in numerical simulations of composite delamination

    Engng Fract Mech

    (2008)
  • J.L. Chaboche et al.

    Interface debonding models: a viscous regularization with a limited rate dependency

    Int J Solids Struct

    (2001)
  • J.W. Foulk

    An examination of stability in cohesive zone modeling

    Comput Methods Appl Mech Engrg

    (2010)
  • M.A. Crisfield

    A fast incremental/iterative solution procedure that handles snap-through

    Comput Struct

    (1981)
  • R. de Borst

    Computation of post-bifurcation and post-failure behavior of strain-softening solids

    Comput Struct

    (1987)
  • A. Turon et al.

    An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models

    Engng Fract Mech

    (2007)
  • I. Guiamatsia et al.

    Decohesion finite element with enriched basis functions for delamination

    Compos Sci Technol

    (2009)
  • M.J. van den Bosch et al.

    An improved description of the exponential Xu and Needleman cohesive zone law for mixed-mode decohesion

    Engng Fract Mech

    (2006)
  • Cited by (0)

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