Thin film delamination: A discrete dislocation analysis

https://doi.org/10.1016/j.jmps.2006.02.008Get rights and content

Abstract

Interface delamination during indentation of micron-scale ceramic coatings on metal substrates is modeled using discrete dislocation (DD) plasticity to elucidate the relationships between delamination, substrate plasticity, interface adhesion, elastic mismatch, and film thickness. In the DD method, plasticity in the metal substrate occurs directly via the motion of dislocations, which are governed by a set of physically based constitutive rules for nucleation, motion and annihilation. A cohesive law with peak stress σ^ characterizes the traction–separation response of the metal/ceramic interface. The indenter is a rigid flat punch and plane strain deformation is assumed. A continuum plasticity model of the same problem is studied for comparison. For low interface strengths (e.g. σ^<σy), DD and continuum plasticity results are quantitatively similar, with delamination being nearly independent of interface strength, and easier for thinner, lower-modulus films. For higher interface strengths (σ^/σy>2), continuum plasticity predicts no delamination up to very high loads while the DD model shows a smooth increase in the critical indentation force for delamination with increasing interface strength. Tensile delamination in the DD model is driven by the accumulation of dislocations, and their associated high stresses, at the interface upon unloading. The DD model is thus capable of predicting the nucleation of cracks, and its dependence on material parameters, in realms of realistic constitutive behavior and/or small length scales where conventional continuum plasticity fails.

Introduction

The failure, by cracking or delamination, of thin protective films on ductile substrates is of great technological interest. The delamination of such films is often an initial failure mode that eventually leads to failure of an entire component. Thus, the design of coatings—their modulus, hardness, thickness, and adhesion with the substrate—represents an important materials engineering problem. Coating/substrate systems are often evaluated under contact loading conditions of indentation, sliding, or scratch testing. The effective interfacial toughness or adhesive energy has been measured by evaluating the total delaminated area upon complete unloading (e.g. Kriese et al., 1998), and is based on overall energy considerations. However, the conditions necessary for the nucleation of damage, e.g. the critical indentation depth at which a delamination crack will first appear, are far less well understood. For films on the order of microns in thickness, the influence of size-dependent plasticity on interfacial failure is also unclear.

The most comprehensive continuum-level work on thin film failure modes is that of Abdul-Baqi and Van der Giessen, which covers three important aspects of thin film mechanics during indentation: shear crack nucleation during indentation loading (Abdul-Baqi and Van der Giessen, 2001b); nucleation of normal cracks during indentation unloading (Abdul-Baqi and Van der Giessen, 2001a); and the possibility of cracking in the film, perpendicular to the film–substrate interface (Abdul-Baqi and Van der Giessen, 2002). In each of these studies the substrate is modeled with standard continuum plasticity, the film is completely elastic, a rigid spherical indenter is modeled, finite-strain axisymmetric deformation is considered, and a cohesive zone model is employed to permit nucleation and propagation of interfacial cracks. Of interest here is the nucleation of normal cracks during unloading. Standard continuum models (e.g. Abdul-Baqi and Van der Giessen, 2001a, Gao and Bower, 2004b) predict, however, that tensile delamination does not occur if the strength of the coating/substrate interface, i.e. the maximum cohesive stress σ^, is larger than about twice the yield stress of the substrate, for materials with minimal hardening. Essentially, upon unloading, reverse plasticity occurs in the substrate, preventing the development of the high stresses required to overcome the interface strength. From a system design perspective, the standard continuum models imply a fail-safe design: if the interface adhesion is above a critical value then failure is absolutely precluded. A similar situation arises in fracture problems, for which cracks cannot propagate if the interfacial adhesion is more than four to five times the yield stress (Tvergaard and Hutchinson, 1992). Such a non-conservative design strategy is dangerous and, for micron-scale films, relies on the applicability of standard continuum plasticity at scales where many studies now indicate that it is inadequate.

Size-dependent plastic flow, with smaller being stronger, arises due the underlying dislocation structure. Ashby (1970) classified dislocations as either “geometrically necessary (GND)”, those that must exist in the presence of strain gradients to maintain material compatibility, or “statistically stored”, that contribute to the flow. Size effects in plasticity arise directly from the increasing density of GNDs as strain gradients become steeper. Classical continuum plasticity theories have no material parameter with dimensions of length, and hence are unable to reproduce such size-dependent behavior. This has led to the development of strain-gradient and “non-local” theories (e.g. Fleck et al., 1994, Aifantis, 1999, Acharya and Bassani, 2000, Gurtin, 2002), and other methods (Gao and Huang, 2001, Guo et al., 2001) for modeling size-dependent plasticity (SDP) at the micron scale. The theories introduce at least one length parameter that is determined by fitting to experimental results, and can then reproduce size-dependent phenomena in problems such as bending (Guo et al., 2001) and torsion (Aifantis, 1999, Guo et al., 2001). Modeling of indentation has largely focussed on homogeneous materials (Begley and Hutchinson, 1998, Nix and Gao, 1998, Guo et al., 2001, Wei and Hutchinson, 2003). Indentation of a metal film on a glass substrate was analyzed using strain-gradient plasticity by Saha et al. (2001), but no mechanism to allow delamination was included. Wei and Hutchinson (1997a) used standard continuum plasticity to model residual stress-driven delamination in thin films, even though the film thickness is on the order of a micrometer. While Wei and Hutchinson (1997b) have studied crack propagation in a homogeneous strain-gradient solid, no studies to date have examined crack nucleation within a SDP framework.

To directly tackle plasticity at small scales via dislocation mechanics, the discrete dislocation (DD) method of Van der Giessen and Needleman (1995) has been developed. This technique solves boundary value problems where the plasticity arises from the nucleation and motion of edge dislocations embedded in an isotropic, linearly elastic continuum and so, in contrast to the SDP and non-local gradient theories discussed above, can be considered a more mechanism-based theory. No material constitutive law is required; rather, a set of rules governing short-range dislocation interactions are used to approximate atomic-scale effects while long-range elastic dislocation interactions occur through the known elastic stress fields. There is no explicit separation of GND and statistically stored dislocations, but GNDs form and organize as necessary for any particular loading geometry. The DD method also includes a length parameter (the dislocation Burgers vector), and thus naturally exhibits size-dependent plastic flow. Applications of the DD method now include: characterization of plastic flow in composites (Cleveringa et al., 1997); bending of beams (Cleveringa et al., 1999); crack growth (Cleveringa et al., 2000); stationary crack tip fields in single crystals (Van der Giessen et al., 2001) and at single crystal–rigid material interfaces (Nakatani et al., 2003); fatigue crack growth (Deshpande et al., 2002, Deshpande et al., 2003c); size effects in model Al/Si alloys (Benzerga et al., 2001), thin films (Nicola et al., 2003) and sandwich structures (Chng et al., 2005); and crack growth along bimaterial interfaces (O’Day and Curtin, 2004b). It is also worth noting the recent encouraging comparisons of DD plasticity predictions to experiments (Chng et al., 2005, Nicola et al., 2006). The DD method has also proven useful in acting as a numerical experiment for validating various SDP theories (e.g. Bassani et al., 2001, Shu et al., 2001, Bittencourt et al., 2003). While both three-dimensional (Weygand et al., 2002) and finite strain (Deshpande et al., 2003b) extensions of DD plasticity have recently been developed, the present work follows the majority of existing literature and uses the two-dimensional plane strain, small-strain idealization. An important feature of the DD method is that, since the dislocation stress fields are explicitly included, the development of high stresses in small local regions can drive crack growth under monotonic or cyclic loading (Cleveringa et al., 2000, Deshpande et al., 2002, O’Day and Curtin, 2004b, Chng et al., 2005), situations where standard continuum plasticity models do not predict cracking.

In light of the above background, the present work investigates the nucleation of tensile delamination cracks under indentation of a hard micron-scale ceramic coating on a metallic substrate. The system is idealized as an elastic film bonded to a substrate characterized by DD plasticity, and the film is indented with a rigid flat punch modeled with simple displacement boundary conditions; the two-dimensional framework is used for computational convenience. A cohesive zone law describes the traction–separation relationship along the interface, so that crack nucleation and growth are outcomes of the boundary value problem solution. Crack nucleation has yet to be investigated within a SDP or DD framework. Since elastic mismatch between the film and substrate is typical of real thin film coatings, we use the DD superposition framework of O’Day and Curtin (2004a) for the efficient solution of elastically inhomogeneous problems. Because of the scarcity of literature results, and the variety of possible material constitutive laws and parameters, we also perform standard continuum plasticity analyses of the same coating indentation problem, with identical material properties; this permits us to make quantitative comparisons between DD and conventional continuum models. In qualitative agreement with classical continuum plasticity (Abdul-Baqi and Van der Giessen, 2001a, Abdul-Baqi and Van der Giessen, 2001b; Gao and Bower, 2004b), the DD model predicts that weak interfaces can delaminate in shear during indentation loading while stronger interfaces exhibit tensile delamination during unloading. However, while standard continuum plasticity predicts no tensile delamination for strong interfaces, as indicated above, the DD model shows delamination for interface strengths well above the continuum threshold value and with no indication of an impending threshold. Wei and Hutchinson (1997b) strain-gradient simulations of mode I crack growth can predict delamination of strong interfaces, depending on the specific choice of the length scale parameter. While SDP approaches have well-known computational benefits over the DD method, it is worth noting that DD plasticity requires no phenomenological length parameters and any plasticity size effects emerge naturally. Finally, within the DD framework we quantify the effect of various material and interfacial parameters on crack nucleation to head toward the ultimate goal of a more-fundamental, material-properties-based design framework for engineered coatings.

The remainder of this work is organized as follows. The thin film geometry, material properties, cohesive zone law and a brief overview of DD plasticity are presented in Section 2. A detailed quantitative description of crack nucleation, including the effects of film thickness and interface strength is presented in Section 3; comparisons to classical continuum plasticity simulations are also made. Section 4 summarizes the main conclusions of this work.

Section snippets

Geometry

The two-dimensional idealization of the thin film indentation problem studied here is shown in Fig. 1. An elastic film on an elastic–plastic substrate is intended to represent a typical hard ceramic coating on a ductile metal substrate. The origin of an x1-x2 coordinate frame is located on the film/substrate interface, directly below the centerline of the rigid punch. The left and right edges of the specimen are located at x1=-75 and 75μm, respectively. The substrate thickness is 100μm and the

Results and discussion

This section presents the main results of our thin film delamination study. The general mechanics of crack nucleation, which are similar to the classical continuum plasticity simulations, are presented first. Then, direct comparisons are made with continuum plasticity, showing the inadequacy of the classical continuum model when the interface strength becomes large relative to the substrate yield stress. Finally, we present the dependence of crack nucleation on various coating and interface

Summary

The mechanics of crack nucleation at a coating/substrate interface during the indentation of a coated material has been studied within the DD plasticity framework. In qualitative agreement with classical continuum plasticity, tensile cracking is possible during indentation unloading. For stronger interfaces, tensile delamination occurs by spontaneous and unstable crack nucleation. Although qualitatively similar to continuum plasticity results for weak interfaces, the DD results depart

Note added in proof

After submitting our corrected draft the following reference was brought to the author's attention. Tang et al. (2004) studied cleavage of single crystals in the presence of considerable plastic deformation and found crack growth for interface strengths up to forty times greater than the initial yield stress. The development of the large tractions required for crack propagation is directly related to the particular gradient plasticity constitutive description, which incorporates the hardening

Acknowledgments

The authors gratefully acknowledge the support of this work by US Air Force Office of Scientific Research through the MURI program “Virtual design and testing of materials: a multiscale approach”, Grant #F49620-99-1-0272, and by General Motors through the GM/Brown Collaborative Research Laboratory on Computational Materials Research. The authors thank Professor Alan Needleman and Professor Vikram Deshpande for many useful discussions pertaining to this work, and for providing the DD code used

References (46)

  • V.S. Deshpande et al.

    Discrete dislocation modeling of fatigue crack propagation

    Acta Mater.

    (2002)
  • V.S. Deshpande et al.

    Scaling of discrete dislocation predictions for near-threshold fatigue crack growth

    Acta Mater.

    (2003)
  • V.S. Deshpande et al.

    Finite strain discrete dislocation plasticity

    J. Mech. Phys. Solids

    (2003)
  • V.S. Deshpande et al.

    Discrete dislocation plasticity modeling of short cracks in single crystals

    Acta Mater.

    (2003)
  • N.A. Fleck et al.

    Strain gradient plasticity: theory and experiment

    Acta Metall. Mater.

    (1994)
  • H. Gao et al.

    Taylor-based nonlocal theory of plasticity

    Int. J. Solids Struct.

    (2001)
  • Y. Guo et al.

    Taylor-based nonlocal theory of plasticity: numerical studies of the micro-indentation experiments and crack tip fields

    Int. J. Solids Struct.

    (2001)
  • M.E. Gurtin

    A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations

    J. Mech. Phys. Solids

    (2002)
  • M.D. Kriese et al.

    Nanomechanical fracture-testing of thin films

    Eng. Fract. Mech.

    (1998)
  • W.D. Nix et al.

    Indentation size effects in crystalline materials: a law for strain gradient plasticity

    J. Mech. Phys. Solids

    (1998)
  • R. Saha et al.

    Indentation of a soft metal film on a hard substrate: strain gradient hardening effects

    J. Mech. Phys. Solids

    (2001)
  • J.Y. Shu et al.

    Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity

    J. Mech. Phys. Solids

    (2001)
  • V. Tvergaard

    Cohesive zone representations of failure between elastic or rigid solids and ductile solids

    Eng. Fract. Mech.

    (2003)
  • Cited by (15)

    • The role of atomistic simulations in probing hydrogen effects on plasticity and embrittlement in metals

      2019, Engineering Fracture Mechanics
      Citation Excerpt :

      A crack can grow along the cohesive surface when the stresses exceed the cohesive strength and the energy input at the crack tip exceeds the cohesive energy. The surrounding material is deforming plastically, as modeled by continuum J2 plasticity theory, strain gradient plasticity theory [104], or 2d discrete dislocation plasticity [105–107]. In small-deformation continuum formulations, the crack tip remains sharp.

    • Material size effects on crack growth along patterned wafer-level Cu-Cu bonds

      2013, International Journal of Mechanical Sciences
      Citation Excerpt :

      One reason for the lack of attention is the challenge of simulating a process that extends from the atomic scale at the interface, through a metal layer of thickness in the micron to sub-micron range, to an overall structure at an even larger scale that imposes the load or displacement. Even if the atomic separation process is modeled by a continuum cohesive law, as will be the case here, the scale at which the cohesive law operates is the atomic scale and this scale must be resolved by the numerical method of Wei and Hutchinson [8], and O'Day et al. [9]. The present paper has been heavily motivated by the experimental studies of interface fracture of Tadepalli and Turner [13] and Tadepalli et al. [14–15] who created fracture specimens comprising thin Cu films (≈400 nm) sandwiched between “thick” Si substrates.

    • A two-dimensional dislocation dynamics model of the plastic deformation of polycrystalline metals

      2010, Journal of the Mechanics and Physics of Solids
      Citation Excerpt :

      Two-dimensional dislocation dynamics (2D-DD) models have been applied to study plasticity at small scales (Buehler et al., 2004; Deshpande et al., 2005; Han et al., 2006; Espinosa et al., 2006), at crack-tips (Deshpande et al., 2003; Hartmaier and Gumbsch, 2005; Broedling et al., 2006; Zeng and Hartmaier, 2010; Bhandakkar et al., 2010), delamination of thin films (O’Day et al., 2006), size effects in single crystals (Guruprasad and Benzerga, 2008), on void growth (Hussein et al., 2008; Segurado and Llorca, 2009) or in other situations where understanding the collective behavior of a rather small number of dislocations is essential to describe deformation correctly (Zbib and Diaz de la Rubia, 2002).

    • Effect of source and obstacle strengths on yield stress: A discrete dislocation study

      2010, Journal of the Mechanics and Physics of Solids
      Citation Excerpt :

      In many high-strength alloys where nanoscale precipitation strengthening is operative, capturing both local dislocation/precipitate and larger-scale dislocation/dislocation interactions is also a challenge in 3d. The 2d plane-strain DD method has been used to provide fundamental insight into a variety of problems including fatigue crack growth in single crystals (Deshpande et al., 2002), delamination of thin films (O’Day et al., 2006), fracture of bimaterial interfaces (O’Day and Curtin, 2005), size effects in single crystals (Guruprasad and Benzerga, 2008), on void growth (Segurado and Llorca, 2009) in thin film interconnects (Nicola et al., 2002) and polycrystalline thin films (Nicola et al., 2005). However, all these studies have been limited to cases where the yield stress of the material is relatively low, and is partially controlled by the strength of dislocation sources.

    • Analyses of crack growth along interface of patterned wafer-level Cu-Cu bonds

      2009, International Journal of Solids and Structures
    View all citing articles on Scopus
    View full text