Vapor pressure and residual stress effects on failure of an adhesive film

Surface-mount plastic encapsulated microcircuits (PEM) are susceptible to temperature- and moisture-induced failures during reflow soldering. Adhesive failures in PEMs are studied using a model problem of a ductile adhesive joining two elastic substrates. The polymeric adhesive contains a centerline crack. The adhesive film is stressed by remote loading and residual stresses. Voids in the adhesive are pressurized by rapidly expanding water vapor. The computational study addresses three competing failure mechanisms: (i) extended contiguous damage zone emanating from the crack; (ii) multiple damage zones forming at distances of several film thicknesses ahead of the crack; and (iii) extensive damage developing along film–substrate interfaces. The second failure mechanism is found in low porosity adhesives, while the first is dominant in high porosity adhesives. The first is also the likely failure mode when voids in the adhesive are subjected to high vapor pressure. The third damage mechanism is operative in low porosity adhesives subjected to high residual stress. In general, both residual stress and vapor pressure exert pronounced effects on failure modes. Vapor pressure, in particular, accelerates voiding activity and growth of the damage zone offering insights into the catastrophic nature of popcorn cracking.     

Two mechanisms of ductile fracture: void by void growth versus multiple void interaction

Two distinct mechanisms of crack initiation and advance by void growth have been identified in the literature on the mechanics of ductile fracture. One is the interaction a single void with the crack tip characterizing initiation and the subsequent void by void advance of the tip. This mechanism is represented by the early model of Rice and Johnson and the subsequent more detailed numerical computations of McMeeking and coworkers on a single void interacting with a crack tip. The second mechanism involves the simultaneous interaction of multiple voids on the plane ahead of the crack tip both during initiation and in subsequent crack growth. This mechanism is revealed by models with an embedded fracture process zone, such as those developed by Tvergaard and Hutchinson. While both mechanisms are based on void nucleation, growth and coalescence, the inferences from them with regard to crack growth initiation and growth are quantitatively different. The present paper provides a formulation and numerical analysis of a two-dimensional plane strain model with multiple discrete voids located ahead of a pre-existing crack tip. At initial void volume fractions that are sufficiently low, initiation and growth is approximately represented by the void by void mechanism. At somewhat higher initial void volume fractions, a transition in behavior occurs whereby many voids ahead of the tip grow at comparable rates and their interaction determines initiation toughness and crack growth resistance. The study demonstrates that improvements to be expected in fracture toughness by reducing the population of second phase particles responsible for nucleating voids cannot be understood in terms of trends of one mechanism alone. The transition from one mechanism to the other must be taken into account.     

Pressure-sensitive ductile layers – I. Modeling the growth of extensive damage

Polymeric adhesives sandwiched between two elastic substrates are commonly found in multi-layers and IC packages. The non-elastic deformation and flow stress of such adhesive joints are highly pressure-sensitive. In this work, we study the effects of pressure-sensitivity, α, and plastic dilatancy, β, on void growth and coalescence ahead of a crack in ductile adhesive joints. To this end, a single layer of discrete voids is placed ahead of the crack in a pressure-sensitive dilatant adhesive sandwiched between two elastic substrates. The adhesive joint is subjected to small-scale yielding conditions. Using an associated flow rule (α = β), we show that pressure-sensitivity not only intensifies damage levels but also increases its spatial extent several fold. The damage level as well as its spatial extent is found to be even greater when a non-associated flow rule (β < α) is deployed. A reduction in the damage process zone’s thickness further increases the voiding activity in the adhesive, thereby resulting in brittle-like failure. This work also examines the fracture toughness trends using a material failure criterion for crack growth.     

Cohesive zone laws for fatigue crack growth: Numerical field projection of the micromechanical damage process in an elasto-plastic medium

Cohesive zone failure models are widely used to simulate fatigue crack propagation under cyclic loading, but the model parameters are phenomenological and are not closely tied to the underlying micromechanics of the problem. In this paper, we will inversely extract the cohesive zone laws for fatigue crack growth in an elasto-plastic ductile solid using a field projection method (FPM), which projects the equivalent tractions and separations at the cohesive crack-tip from field information outside the process zone. In our small-scale yielding model, a single row of discrete voids is deployed directly ahead of a crack in an elasto-plastic medium subjected to cyclic mode I K-field loading. Damage accumulation under cyclic loading is captured by the growth of voids within the micro-voiding zone ahead of the crack, while the evolution of the cohesive zone law representing the micro-voiding zone is inversely extracted via the FPM. We show that the field-projected cohesive zone law captures the essential micromechanisms of fatigue crack growth in the ductile medium: from loading and unloading hysteresis caused by void growth and plastic hardening, to the softening damage locus associated with crack propagation via a void by void growth mechanism. The results demonstrate the effectiveness of the FPM in obtaining a micromechanics-based cohesive zone law in-place of phenomenological models, which opens the way for a unified treatment of fatigue crack problems.     

Mode mixity and nonlinear viscous effects on toughness of interfaces

This paper examines steady-state crack growth at interfaces between polymeric materials and hard substrates under quasi-static conditions. The polymeric material is taken to be an elastic nonlinear viscous solid while the substrate is treated as a rigid material. Void growth and coalescence in the rate-dependent fracture process zone is modeled by a nonlinear viscous porous strip of cell elements. In the first part of this paper, the polymeric background material surrounding the process zone is assumed to be purely elastic. Under fixed mode mixity, the computed interface toughness is found to be a monotonically increasing function of crack velocity; toughness also increases rapidly with higher rate sensitivity. This behavior can be explained in terms of voids growing in a strain-rate strengthened process zone. In the second part of the paper, the background material is also treated as an elastic nonlinear viscous solid. The competition between work of separation in the process zone and energy dissipation in the background material leads to a U-shaped toughness–crack velocity curve. Effects of mode mixity, initial porosity, rate sensitivity, as well as the initial yield strain on toughness are studied. The simulations produce trends that agree with interface toughness vs. crack velocity data reported in experimental studies for rubber toughened epoxy-paste adhesive and urethane acrylate adhesive.     

Pressure-sensitive ductile layers – II. 3D models of extensive damage

The mechanisms of void growth and coalescence in ductile polymeric layers, taking into account the effects of pressure-sensitivity, α, and plastic dilatancy, β, are explored in this two-part paper. In Part I, a two-dimensional model containing discrete cylindrical voids was used to simulate void growth and coalescence ahead of a crack. This paper extends the previous work by explicitly modeling initially spherical voids in a three-dimensional configuration. Damage predictions from the present 3D model for low yield strain adhesives are found to be in good agreement with both the 2D model in Part I and the computational cell element model. Significant discrepancies in the damage predictions, however, exist among all three models for high yield strain adhesives (e.g. polymers). The present 3D study also discusses the increasing damage level and its spatial extent with pressure-sensitivity, as well as the exacerbation of these effects arising from the deviation from an associated flow rule. In fact, both high porosity and high pressure-sensitivity promote void interaction. In addition, pressure-sensitivity increases the oblacity of the voids and reduces the intervoid ligament spacing over a wide range of load levels. These effects are compounded as the fracture process zone thickness decreases relative to the adhesive thickness. Results further show that both the adhesive toughness levels and the critical porosity governing the onset of void coalescence are significantly lowered with increasing pressure-sensitivity.     

Rock burst of deep circular tunnels surrounded by weakened rock mass with cracks

Rock masses containing pre-existing cracks are considered as non-homogeneous geomaterials. During excavation of tunnels, pre-existing cracks may nucleate, grow and propagate through rock matrix, then secondary cracks may appear. The stress concentration at the tips of secondary cracks is comparatively large, which may lead to the unstable growth and coalescence of secondary cracks, and consequently the occurrence of fractured zones. For brittle rocks, the dissipative energy of slip and growth of pre-existing cracks and secondary crack growth is small, but the elastic strain energy storing in rock masses may be larger than the dissipative energy of slip and growth of pre-existing macrocracks and secondary crack growth. The sudden release of the residual elastic strain energy may lead to rock burst in crack-weakened rock masses. Based on this understanding, the criteria of rock burst in crack-weakened rock masses are established. The influences of the in situ stresses, micromechanical parameters and physico-mechanical parameters on the distribution of rock burst zones and area of rock burst zones are investigated in detail.     

Cohesive zone laws for void growth — II. Numerical field projection of elasto-plastic fracture processes with vapor pressure

Modeling ductile fracture processes using Gurson-type cell elements has achieved considerable success in recent years. However, incorporating the full mechanisms of void growth and coalescence in cohesive zone laws for ductile fracture still remains an open challenge. In this work, a planar field projection method, combined with equilibrium field regularization, is used to extract crack-tip cohesive zone laws of void growth in an elastic-plastic solid. To this end, a single row of void-containing cell elements is deployed directly ahead of a crack in an elastic-plastic medium subjected to a remote K-field loading; the macroscopic behavior of each cell element is governed by the Gurson porous material relation, extended to incorporate vapor pressure effects. A thin elastic strip surrounding this fracture process zone is introduced, from which the cohesive zone variables can be extracted via the planar field projection method. We show that the material's initial porosity induces a highly convex traction-separation relationship — the cohesive traction reaches the peak almost instantaneously and decreases gradually with void growth, before succumbing to rapid softening during coalescence. The profile of this numerically extracted cohesive zone law is consistent with experimentally determined cohesive zone law in Part I for multiple micro-crazing in HIPS. In the presence of vapor pressure, both the cohesive traction and energy are dramatically lowered; the shape of the cohesive zone law, however, remains highly convex, which suggests that diffusive damage is still the governing failure mechanism.     

THE EFFECTIVE ELECTROMECHANICAL PROPERTIES OF CELLULAR PIEZOELECTRET FILM:FINITE ELEMENT MODELING

Cellular space-charge polymer film,also called cellular piezoelectret,has very large piezoelectric effect due to their unique microvoid structure.In this article,the cellular piezoelectret film is considered to be a periodic composite material with closed-cell microvoids aligned periodically.Three dimensional finite element modeling is carried out to obtain the effective elastic modulus and piezoelectric coefficients.Sensitivity analysis was presented by modeling the effective electromechanical properties with different individual variable,including material constants and void shape parameters.By assuming a relation between void shape and void volume fraction,the finite element model can simulate quite well the inflation experiments of the voided charged Polypropylene film published in literature.Finally,the finite element model is used to explore the voided charge polymer film with non-uniform distribution of the trapped charges on the internal surface of voids.It was found that the resultant overall piezoelectric coefficients will be more significant if charges are closely gathered in the central area,and sparse in the around area of the internal surface of voids.     

A discrete-fibers model for bridged cracks and reinforced elliptical voids

A model of crack bridging and reinforced elliptical voids is proposed, in which the fibers joining the surfaces of the void or crack are modelled as discrete, linear elastic bars. We show that a theory recently developed by us to analyze structural interfaces permits analytical attack and solution of multiple important previously unsolved problems of stress concentration and fracture. In particular, an analytical solution is provided for a reinforced elliptical void, which, by superposition, allows treatment of arbitrary fiber distributions, which can be even randomly distributed and oriented. In the special case of small or null ratio between a void's axes, new stress intensity factor expressions are obtained, which account for fibers’ inclination and geometry.     

伴随微孔洞生长的裂纹弹塑性定常扩展

裂纹在韧性材料中扩展时,将们随着微孔洞的萌生和生长,孔洞的萌生和深化将直接影响着材料的总体断裂韧性和强度,以往的研究主要集中在将裂纹的扩展刻划为微孔洞的萌生、生长和汇合这样一个过程。从传统的断裂过程区模型出发研究微孔洞的萌生和生长对材料总体断裂韧性的影响,通过采用Ｇｕｒｓｏｎ模型,建立塑性增量本构关系,然后针对定常扩展情况直接进行分析,孔洞对材料断裂韧性的影响由本构关系刻划,而在孔洞汇合模型中,上     

Experimental and theoretical study on time dependence of the quasi-piezoelectric d33 coefficients of cellular piezoelectret film

The voided charged polymer film, also called piezoelectret, has a very large quasi-piezoelectric coefficient in the thickness direction, and has emerged as a new kind of electromechanical transducer materials. Piezoelectret film is usually prepared from polymer by means of biaxial stretching and electric charging process. Due to the inherent viscosity of polymer, the quasi-piezoelectric d₃₃ coefficient of cellular piezoelectret film usually depends on the pressure and time of the measurement. In this article, experiments were carried out on the time spectra of quasi-piezoelectric d₃₃ coefficient in the thickness direction for cellular linear Polypropylene piezoelectret film. To study the effect of void microstructures on the time-dependence of quasi-piezoelectric d₃₃ coefficient, samples of three different thicknesses were tested under two different pressures. The micromechanical theory for viscoelastic composite was extended to predict the electromechanical properties of voided charged polymer film. The voids with surplus charges, which can be piezoelectriclike under deformation, are considered as ellipsoidal heterogeneous piezoelectric inclusions, while the viscous polymer is taken as the matrix. In Laplace transformed space, the generalized Eshelby tensor is formulated for the isotropic nonpolar matrix as well as for the anisotropic matrix. The Mori–Tanaka average scheme is used to find the overall electromechanical properties. Time dependence of the effective properties in real space can be studied by Laplacian inversion. Sensitivity analysis to various parameters is investigated for time dependence of the effective properties, including effective elastic moduli and the quasi-piezoelectric coefficients. Theoretical simulation was presented and comparison with experimental results was conducted. Both qualitative analysis and quantitative comparison with experiments show that this theoretical formulation can predict the time dependence of the effective properties of voided charged piezoelectret film.     

Modeling hydrogen attack effect on creep fracture toughness

The effect of high temperature hydrogen attack on creep crack growth rates in steels is studied by modeling the interaction between creep deformation and gaseous pressures generated by hydrogen and methane. The equilibrium methane pressure as a function of hydrogen pressure, temperature and carbide types for carbon steels and Cr–Mo steels is calculated. This gaseous driving force is incorporated into a micromechanics model for void growth along grain boundaries of a creeping solid. Growth and coalescence of voids along grain boundaries is modeled by a microporous strip of cell elements, referred to as the fracture process zone. The cell elements are governed by a nonlinear viscous constitutive relation for a voided material. Two rate sensitivities as well as two types of grain boundaries are considered in this computational study. Simulations of creep crack growth accelerated by gaseous pressures are performed under conditions of small-scale and extensive creep. The computed crack growth rates at elevated temperatures are able to reproduce the trends of experimental results.     

受有两级空洞损伤时韧性材料的力学行为

本文利用大应变有限元方法研究了两级空洞对韧性材料的损伤作用.模型是以轴对称圆柱基体作为胞元,内含一初始的球型空洞.基体内的应力/应变随胞元外载的增大而达到临界状态,从而在围绕初级空洞的基体内将萌生次级空洞.后者是由空单元实现的.两级空洞的交互作用被证明将促进材料中的空洞化现象从而加速损伤并导至材料的总体弹性模量值在临近破断时急剧下降.     

非均匀损伤介质中波传播的数值解

对弹性波在非均匀损伤介质中的传播理论进行了研究。通过将非均匀损伤区域离散成分层均匀的区域,结合相邻区域交界面处的连续条件,推导出了以右行波、左行波为状态向量的波动方程和传递矩阵。对几种非均匀损伤介质中波的传播进行了实例数值计算,并和其解析解的结果进行了比较,讨论了弹性波在非均匀损伤介质中传播的一般性质。     

Modeling of crack growth in ductile solids: a three-dimensional analysis

A population of several spherical voids is included in a three-dimensional, small scale yielding model. Two distinct void growth mechanisms, put forth by [Int. J. Solids Struct. 39 (2002) 3581] for the case of a two-dimensional model containing cylindrical voids, are well contained in the model developed in this study for spherical voids. A material failure criterion, based on the occurrence of void coalescence in the unit cell model, is established. The critical ligament reduction ratio, which varies with stress triaxiality and initial porosity, is used to determine ligament failure between the crack tip and the nearest void. A comparison of crack initiation toughness of the model containing cylindrical voids with the model containing spherical voids reveals that the material having a sizeable fraction of spherical voids is tougher than the material having cylindrical voids. The proposed material failure determination method is then used to establish the fracture resistance curve (J–R curve) of the material. For a ductile material containing a small volume fraction of microscopic voids initially, the void by void growth mechanism prevails, which results in a J–R curve having steep slope. On the other hand, for a ductile material containing a large volume fraction of initial voids, the multiple voids interaction mechanism prevails, which results in a flat J–R curve. Next, the effect of T-stress on fracture resistance is examined. Finally, nucleation and growth of secondary microvoids and their effects on void coalescence are briefly discussed.     

Formulation of a damage internal state variable model for amorphous glassy polymers

The following article proposes a damage model that is implemented into a glassy, amorphous thermoplastic thermomechanical inelastic internal state variable framework. Internal state variable evolution equations are defined through thermodynamics, kinematics, and kinetics for isotropic damage arising from two different inclusion types: pores and particles. The damage arising from the particles and crazing is accounted for by three processes of damage: nucleation, growth, and coalescence. Nucleation is defined as the number density of voids/crazes with an associated internal state variable rate equation and is a function of stress state, molecular weight, fracture toughness, particle size, particle volume fraction, temperature, and strain rate. The damage growth is based upon a single void growing as an internal state variable rate equation that is a function of stress state, rate sensitivity, and strain rate. The coalescence internal state variable rate equation is an interactive term between voids and crazes and is a function of the nearest neighbor distance of voids/crazes and size of voids/crazes, temperature, and strain rate. The damage arising from the pre-existing voids employs the Cocks–Ashby void growth rule. The total damage progression is a summation of the damage volume fraction arising from particles and pores and subsequent crazing. The modeling results compare well to experimental findings garnered from the literature. Finally, this formulation can be readily implemented into a finite element analysis.     

Ultrasonic-image evaluation of microstructural damage accumulation in materials

Ultrasonic imaging techniques for portraying and evaluating cumulative internal microstructural damage in engineering materials are described. A quantitative delineation of the damage is made in terms of acoustic attenuation obtained from computer analyses of digitized ultrasonic images. Acoustic attenuation data are a basic ingredient in previously developed models of damage processes in materials. The ultrasonic imaging methodology has been developed using filled polymer (inert solid rocket-propellant) samples subjected to progressive uniaxial tensile strain. Successive ultrasonic images taken at various levels of applied strain display dewetting and the evolving microvoid formation/growth which occurs. Both initially intact material and that with pre-existing cracks are of interest. Changes in acoustic attenuation with strain, derived from the processing of digital images, have provided results as to the degree of preferential damage accumulation at sites of filler particle agglomerations appearing on the ultrasonic images. Also, the quantitative extent of an asymmetry in the damage-field distribution near the tips of an extending crack was determined in precracked material. Iso-attenuation type contours generated by computer reveal that kidney-shaped damage zones occur in the neighborhood of the propagating crack tips, reminiscent of the plastic-zone shapes near crack tips in ductile metals under strain. Ultrasonic images of precracked samples show that before crack extension begins, the material damage in the neighborhood of the crack already extends over a relatively large volume of the specimen. G.C. Knollman is Senior Staff Scientist and Senior Member, Mechanics and Maternals Engineering Laboratory     

A finite element study of the near crack tip deformation of a ductile material under mixed mode loading

In ductile fracture, voids near a crack tip play an important role. From this point of view, a large deformation finite element analysis has been made to study the deformation, stress and strain, and void ratio near the crack tip under mixed mode plane strain loading conditions, employing Gurson's constitutive equation which has taken into account the effects of void nucleation and growth. The results show that: (i) one corner of the crack tip sharpens while the other corner blunts, (ii) the stress and strain distributions except for the near crack tip region, can be superimposed by normalizing distance from the crack tip by a crack tip deformation length, i.e., a steady-state solution under a mixed mode condition has been obtained, (iii) the field near a crack tip can be divided into four characteristic fields (K field, HRR field, blunted crack tip field, and damaged region), and (iv) the strain and void volume fraction become concentrated in the sharpened part of a crack tip with increasing Mode II component.     

Cylindrical void in a rigid-ideally plastic single crystal. Part I: Anisotropic slip line theory solution for face-centered cubic crystals

The fracture toughness of ductile materials depends upon the ability of the material to resist the growth of microscale voids near a crack tip. Mechanics analyses of the elastic–plastic deformation state around such voids typically assume the surrounding material to be isotropic. However, the voids exist predominantly within a single grain of a polycrystalline material, so it is necessary to account for the anisotropic nature of the surrounding material. In the present work, anisotropic slip line theory is employed to derive the stress and deformation state around a cylindrical void in a single crystal oriented so that plane strain conditions are admitted from three effective in-plane slip systems. The deformation state takes the form of angular sectors around the circumference of the void. Only one of the three effective slip systems is active within each sector. Each slip sector is further subdivided into smaller sectors inside of which it is possible to derive the stress state. Thus the theory predicts a highly heterogeneous stress and deformation state. In addition, it is shown that the in-plane pressure necessary to activate plastic deformation around a cylindrical void in an anisotropic material is significantly higher than that necessary for an isotropic material. Experiments and single crystal plasticity finite element simulations of cylindrical voids in single crystals, both of which exhibit a close correspondence to the analytical theory, are discussed in a companion paper.