Calculation Of Energy Release Rate Research Articles

Initiation and arrest of cracks from corners in multi-chip semiconductor devices

A contemporary semiconductor device often contains multiple chips. Corners of the chips concentrate stress, and are principal sites to initiate failure. Here we propose to characterize the corners using a double cantilever beam, in which two silicon beams sandwich a row of chips. As the two beams are pulled open, a crack initiates at the corner of a chip, and runs unstably on the interface between the chip and a beam. The crack may or may not arrest, depending on various experimental conditions. We calculate energy release rate as a function of crack length by using a combination of finite element method and an analytical solution of the singular field around a corner. At a fixed applied displacement, the energy release rate is low for a short crack, peaks for a crack of intermediate length, and drops for a long crack. This non-monotonic behavior explains how a crack initiates, grows unstably, and possibly arrests. If the crack does arrest, as the two beams open further, the crack grows stably. We relate the initiation and arrest of the crack to machine compliance, specimen geometry, and flaw size. The force at which the crack initiates can be used to characterize the manufacturing process, whereas the stable growth of the crack can be used to measure interfacial toughness. It is hoped that this work will aid the development of multi-chip semiconductor devices.

Damage history in composite laminates: Matrix cracks leading to delaminations

Strain energy release rate calculations for various cases of delaminations emanating from matrix cracks are developed and used to predict the onset of delaminations and their growth size as a function of applied tension and shear loads in composite laminates. The method determines the matrix crack spacing, the delamination onset load, the delamination size at onset and, through the use of a newly proposed delamination resistance curve, the size of delaminations as they grow under load. The method can be applied to any symmetric laminate. Comparisons to test results in the literature for a variety of layups and materials shows very good agreement with the exception of cases where significant edge delaminations appear before delaminations caused by matrix cracks.

A node-splitting lattice spring model coupled with a J-integral formulation as a fracture criterion

A global energy minimization criterion based on Griffith’s theory is introduced for the node-splitting lattice spring model. The fracture criterion is computed by both direct numerical simulations of energy release rate G and through a J-integral formulation for comparison and validation. For mode I fractures, the standard implementation of J-integral formulation yields very good estimations of the energy release rate, but for mixed mode fracture the estimations deviates from the direct calculated energy release rate. The reasons for this discrepancy are elucidated and an approach to best approximate the J value is given. This method is compared with the more standard maximum tip stress threshold crack criterion, and shows a much better prediction of the energy release rate and is more robust under grid refinement.

An Analytical Solution of the Total Strain Energy Release Rate and Mode Partitioning of a Beam Type Delamination Specimen under High Speed Mixed-Mode Loading

In this work an analytical solution for the calculation of the total Stain Energy Release Rate and mode partitioning of beam type delamination specimen, at crack initiation, under high speed, mixed-mode loading is obtained for the first time. The analytical model is based on the dynamic response of a Mixed-Mode End Load Split type specimen subjected to loading with a constant velocity. The Timoshenko beam theory is used to model the specimen's kinematics and the mode partitioning method used is based on the Virtual Crack Closure Technique. To model the bonded region a method based on Lagrangian multipliers is utilized. The specimen is considered as a waveguide and the waves that are allowed to propagate are analytically obtained. Due to the complexity of the analytical form of the wavenumbers for the bonded region of the specimen, which makes the calculation of eigenfrequencies intractable, a series approximation method is used. The results of the proposed analytical solution are verified using a 2D Finite Element Model. The results show the dependency of the Strain Energy Release Rate and the mode mixity on the eigenfrequencies and the eigenmodes of the specimen rendering the value of the mode mixity highly inconsistent through time.

A hierarchical quadrature element method for energy release rate calculation in combination with the virtual crack closure technique

In this paper, the virtual crack closure technique (VCCT) is combined with a hierarchical quadrature element method (HQEM) with p-convergence for the evaluation of strain energy release rate (SERR) for the first time. A universal SERR formula suitable for HQEM is proposed, regardless of the node arrangements and the number of nodes per element boundary. The validity of the present formula is evaluated by the typical cases of mode I and II fracture of homogeneous materials, as well as the bimaterial interface crack problem. A comprehensive comparison with several widely-acknowledged h-version elements shows that the present approach can significantly improve the accuracy and efficiency of SERR calculation even with a relatively coarse mesh and thus be readily used as a viable and effective tool for fracture analysis.

Calculating energy release rates of MD non-symmetric laminate MMELS and C-ELS specimens

Calculating energy release rates of MD non-symmetric laminate MMELS and C-ELS specimens

Design of thermal and environmental barrier coatings for Nb-based alloys for high-temperature operation

The design of thermal and environmental barrier coatings (T/EBCs) applied to Nb-based alloys for high-temperature operation (>1200 °C) was studied. Different multilayer T/EBCs consisting of rare earth aluminates, silicates, and zirconates (with Y 3 Al 5 O 12 (YAG), Y 2 SiO 5 (YMS), and Gd 2 Zr 2 O 7 (GZO) as exemplars) are proposed. Mechanics models were used to determine the magnitude of thermal stresses developed and the tendency for delamination or penetration cracking. The calculated energy release rates (ERRs) for crack formation show that the T/EBC composed of a porous YAG layer (porosity > 0.15) and a dense YAG + YMS layer is effective in preventing the delamination and can potentially arrest cracks penetrating from the surface before reaching the substrate. Phase equilibria calculations indicate that all combinations are stable to above their expected use temperatures. The greatest likelihood for melting is the scenario where a YAG + YMS solid mixture contains excess SiO 2 due to process variability, but even in this case the melting temperature, 1430 °C, exceeds the corresponding temperature in the prescribed temperature profile. • Multilayer thermal/environmental barrier coatings inhibit crack growth. • Rare earth aluminate and silicate coatings are suitable for Nb-based alloy. • Phase mixtures have better properties but lower melting temperatures. • Columnar porosity in coating reduces energy release rate for crack formation.

Strain energy release rate under dynamic loading considering shear and crack tip root rotation effects

A closed form solution for the calculation of the strain energy release rate (SERR), at the initiation of a crack, for a double cantilever beam (DCB) test, under dynamic loading conditions, is presented. The dynamics of a Timoshenko beam are used in the analytical formulation. The equations of motion are obtained using a variational approach and the adhesively bonded region is modelled via a Lagrangian multiplier method. The SERR is calculated based on the contour integral method and is a function of the vibrational mode and the beam's eigenfrequencies. The analytical formulation is applied on two representative joint cases. One consists of isotropic and the other of composite sub-laminates. Finite element analysis was used to validate the analytical formulation. The results demonstrate the advantages of the use of the Timoshenko beam theory and the consideration of crack tip root rotations in dynamic fracture mechanics and emphasize that the localized vibration cannot be neglected in the SERR calculation.

Comparison of calculations of energy release rates for DCB multi-directional laminate specimens

Comparison of calculations of energy release rates for DCB multi-directional laminate specimens

A Force-Based Rectangular Cracked Element

A rectangular plane element containing an internal open stable crack is formulated. Using the well-known local flexibility approach, the element flexibility matrix is derived by the addition of the supplementary flexibility caused by cracking to the intact element flexibility. For utilization of the force formulation method, the calculation of the strain energy release rate is a necessary parameter for computing the additional flexibility matrix due to crack. The developed cracked element can be used for static and vibration analysis of structures and also vibration-based crack detection. Some numerical examples evaluate the accuracy of the suggested formulation.

Homogenized model of environmental barrier coatings for evaluation of energy release rate

AbstractFor efficient calculation of energy release rate (ERR) for interface cracks in environmental barrier coatings (EBCs) with a columnar layer, we propose a simple and versatile homogenization method of the columnar layer. The columnar layer consisting of a number of fine columns is modeled as an anisotropic homogenized layer whose elastic properties are isotropic in the plane perpendicular to the layer thickness direction. The homogenization was achieved by a finite element method (FEM) analysis for a single columnar model to obtain elastic parameters that characterize the homogenized layer. For validation of the homogenization method, we conduct an FEM analysis for a typical EBC structure and obtain the ERRs of an interface crack with various crack length and columnar size. We demonstrate that the homogenization method provides estimation of ERR with a good accuracy on the safer side, which is advantageous for EBC structure design. In addition, the method can reduce the calculation time by more than 50%. From these results, we conclude that the proposed homogenization method can be applied generally for estimation of ERR in EBCs with a columnar layer.

Computation of mode I strain energy release rate of symmetrical and asymmetrical sandwich structures using mixed finite element

The use of composite materials is on the rise in different engineering fields, the main advantage of these materials for the aerospace industry is their low weight for excellent mechanical qualities. The analysis of failure modes, such as delamination, of these materials has received great attention from researchers. This paper proposes a method to evaluate the mode I Strain Energy Release Rate (SERR) of sandwich structures. This method associated a two-dimensional mixed finite element with virtual crack extension technique for the analysis of interfacial delamination of sandwich beams. A symmetrical Double Cantilever Beam (DCB) and asymmetrical Double Cantilever Beam (UDCB) have been analyzed in this study. The comparison of the results obtained by this method and those found in the literature shows efficiency and good precision for the calculation of Strain Energy Release Rate (SERR).

An Enhanced Virtual Crack Closure Technique for Stress Intensity Factor Calculation along Arbitrary Crack Fronts and the Application in Hydraulic Fracturing Simulation

The virtual crack closure technique (VCCT) is widely used for calculating energy release rates along crack fronts and modeling the propagation of cracks in solid materials. Although the VCCT formulation for smooth crack fronts has been sufficiently addressed in the literature, the application of VCCT to a nonsmoothed crack front with sharp corners warrants further investigation. The present study describes an enhanced VCCT to calculate energy release rates and stress intensity factors for cracks with arbitrary shapes in 3D domains discretized on structured grids. The formulations of the enhanced VCCT were developed and implemented into a multiphysics simulation environment capable of simulating crack propagation in the framework of linear elastic fracture mechanics. Comparisons with existing analytical/numerical solutions and other VCCT approaches were performed to verify the enhanced VCCT in terms of SIF calculation along nonsmoothed crack fronts. We then applied the enhanced VCCT to a hydraulically driven penny-shaped fracture problem to further demonstrate its capability to simulate nonsmoothed fracture propagation.

Computation of energy release rate for interfacial crack in orthotropic bimaterials

The interfacial crack in bimaterials is a very interesting problem for composite materials and which has received particular attention from several researchers. In this study, we will propose a numerical modeling of the interfacial crack between two orthotropic materials using a special mixed finite element. For the calculation of the energy release rate, a technique, based on the association of the present mixed finite element with the virtual crack extension method, was used. The numerical model proposed, in this work, was used to study a problem of interfacial crack in bimaterials. Two cases were treated: isotropic and orthotropic bimaterials. The results obtained, using the present element, were compared with the values of the analytical solution and other numerical models found in the literature.

Virtual crack closure technique in peridynamic theory

In this paper, the virtual crack closure technique (VCCT), commonly used in the numerical finite element method (FEM), is for the first time reformulated in nonlocal peridynamic theory, and a new peridynamics-based virtual crack closure technique (PD_VCCT) is proposed for the energy release rate calculation in the framework of peridynamics. The mode I, mode II, and their mixed mode fracture cases are considered. Two typical tests using the single edge-notched tension (SENT) and asymmetric double cantilever beams (ADCB) specimens are considered for implementation and verification, and the numerical energy release rates calculated by the present PD_VCCT are compared with those by the FEM-based VCCT. The close comparisons demonstrate that the proposed PD_VCCT can successfully predict the energy release rates for the cases of mode I, mode II and their mixed mode fracture and it can be used as a viable and effective tool for fracture analysis.

On energy release rates in Peridynamics

This work is concerned with the calculation of fracture energy release rates in Peridynamics. The existing crack growth criterion and the related damage model in Peridynamics, e.g. the critical bond stretch in the prototype micro-elastic brittle (PMB) material model, are based on the energy release rate of an idealized infinite crack model. Therefore, applications of such damage model in Peridynamics to arbitrary fracture patterns, e.g. cracks with finite increment, kinked cracks, and penny-shaped cracks etc., are problematic, because that the idealized crack model assumes an infinitely large crack surface, and the crack growth direction is always assumed being along the direction of the old crack surface.In all engineering applications, we only encounter cracks of finite size. In this work, based on the celebrated Griffith theory that interprets the energy release rate as the surface energy density, we have systematically investigated the fracture energy release rate in Peridynamics for a number of finite crack models: two-dimensional (2D) plane strain finite size horizontal crack, 2D edge crack in a plate, 2D kinked crack, and three-dimensional (3D) penny-shaped crack.The highlights of this work are a set of elegant mathematical solutions: (1) We have proved that for 2D horizontal finite cracks, the energy release rate is exactly the same as that of 2D infinite large crack, i.e. Silling’s original result; (2) We have derived the exact expressions of the energy release rates as well as the related critical bond stretch formulas for 2D finite edge cracks and kinked cracks, and (3) We have found an asymptotic solution of the energy release rate for 3D penny-shaped cracks of finite size in Peridynamics, showing that Silling’s energy release formula for 3D crack model is only the zero-th order approximation of the exact solution.In addition, we have presented some numerical solutions of finite increment crack growths in Peridynamics, and discussed their implications, ramifications, and applications in nonlocal continuum fracture mechanics.

On the computation of energy release rates by a peridynamic virtual crack extension method

A new peridynamic virtual crack extension (PVCE) method is developed to predict the energy release rate G in fracture mechanics, and the directionally and virtually broken bonds are uniquely applied to constitute the virtual crack for crack extension in the numerical implementation. An algorithm of the PVCE method considering serial crack extensions and quasi-static analysis is proposed, and the modified PVCE method is further introduced to reduce the computational error. As validation and application examples, the developed PVCE method is applied to compute the energy release rates of single edge-notched tension (SENT), double cantilever beam (DCB), and end loaded split (ELS) tests, and the results are compared with the existing analytical solutions and numerical finite element analysis. The new PVCE method builds a natural “bridge” for peridynamic fracture analysis between the peridynamics and classical fracture mechanics, and it is capable of accurately and effectively computing the energy release rates for both mode I and mode II fracture cases as well as predicting the critical load and displacement in fracture analysis.

Influence of the viscoelastic material behaviour on the delamination in multilayered beam

An analysis of the influence of the viscoelastic behaviour of the material on the delamination fracture in multilayered beam structures subjected to creep is carried-out. The fracture is studied in terms of the strain energy release rate. For this purpose, an approach for calculation of the strain energy release rate by analyzing the energy balance in the beams is developed. Four viscoelastic models consisting of combinations of linear dashpots and springs for a constant applied stress are used for treating the creep behaviour. The approach is applied to analyze the strain energy release rate in a multilayered cantilever beam with a delamination crack subjected to creep. The analysis indicates that the creep leads to increase of the strain energy release rate. The effects of the ratios of coefficients of viscosity and the moduli of elasticity in the beam layers on the delamination fracture behaviour are evaluated by using the four viscoelastic models.

Effects of the beam cross-section geometry on the lengthwise fracture of inhomogeneous beams

The present paper analyzes the effects of the beam cross-section on the lengthwise fracture behaviour of inhomogeneous beam configurations in terms of the strain energy release rate by applying linear-elastic fracture mechanics. The material is continuously inhomogeneous along height and length of the beams. A methodology for calculation of the strain energy release rate is developed and used for analyzing lengthwise fracture of a simply supported inhomogeneous beam loaded by one vertical force applied at the lower crack arm. The solutions to the strain energy release rate are verified by using the compliance method. Four beam cross-section geometries (isosceles triangle, antiparallelogram, T-shape and circle) and their effects on the lengthwise fracture behaviour are considered in detail. The lengthwise fracture behaviour of the inhomogeneous beam is analyzed also for the case when material has different moduli of elasticity in tension and compression. The results reported in the present paper can be applied in fracture mechanics based design of inhomogeneous structural members and components.

A new approach to model delamination growth in fatigue using the Virtual Crack Closure Technique without re-meshing

A modeling approach is proposed to simulate delamination propagation in fatigue that combines the Virtual Crack Closure Technique (VCCT) with a progressive nodal release strategy. The progressive nodal release alleviates the artificial stress concentrations found when using the VCCT with instantaneous release to model 3D delamination without re-meshing. This enables crack shapes that do not conform to the underlying mesh to be readily simulated.The progressive release is formulated such that no artificial growth or healing occurs as a result of a change in loading conditions. Furthermore, expressions for calculating energy release rates at partially released nodes are proposed and applied to arbitrarily shaped delamination fronts. The shape of the delamination front is considered explicitly in the energy release rate calculations, and is determined by considering the normalized crack positions associated with the partial nodal release. Contrary to previous implementations, the local delamination propagation direction is not assumed to follow mesh lines, but instead is computed as part of the iterative procedure proposed to determine the maximum energy release along the crack front. The results obtained suggest the approach can accurately simulate Mode I, Mixed Mode I/II and Mode II fatigue delamination growth and be used to simulate growth of arbitrarily shaped cracks that do not conform to the underlying mesh.