Energy Release Rate Of Crack Research Articles

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.

Lap shear of a soft and elastic adhesive

This paper studies the lap shear, in which both the adhesive and adherends are elastic, but the adhesive is much softer than the adherends. The shear lag model identifies a length, called the shear lag length Ls. The energy release rate of a debond crack is affected by the elasticity of both the adhesive and adherends. Their relative importance is characterized by the ratio of the length of the remaining joint, L, to the shear lag length, Ls. In the short-joint limit, L/Ls→0, the adherends do not deform, and the elasticity of the adhesive gives the energy release rate. In the long-joint limit, L/Ls→∞, the interior of the adhesive does not deform, and the elasticity of the adherends gives the energy release rate. The shear lag model gives an approximate expression of the energy release rate for all values of L/Ls. This expression is in excellent agreement with the results obtained by finite element calculations, so long as the crack is long compared to the thickness of the adhesive.

Longitudinal vertical crack analysis in beam with relaxation stresses

PurposeThis paper is concerned with analysis of the time-dependent strain energy release rate for a longitudinal crack in a beam subjected to linear relaxation. A viscoelastic model with an arbitrary number of parallel units is used for treating the relaxation. Each unit has one dashpot and two springs. A stress-strain-time relationship is derived for the case when the coefficient of viscosity in each unit of the viscoelastic model changes continuously with time. The beam exhibits continuous material inhomogeneity along the thickness. Thus, the moduli of elasticity and the coefficients of viscosity vary continuously in the thickness direction. The aim of the present paper is to obtain time-dependent solutions to the strain energy release rate that take into account the relaxation when the coefficient of viscosity changes with time.Design/methodology/approachTime-dependent solutions to the strain energy release rate are derived by considering the time-dependent strain energy and also by using the compliance method. The two solutions produce identical results.FindingsThe variation of the strain energy release rate with time due to the relaxation is analysed. The influence of material inhomogeneity and the crack location along the beam width on the strain energy release rate are evaluated. The effects of change of the coefficients of viscosity with time and the number of units in the viscoelastic model on the strain energy release rate are assessed by applying the solutions derived.Originality/valueThe time-dependent strain energy release rate for a longitudinal vertical crack in a beam under relaxation is analysed for the case when the coefficients of viscosity change with time.

Lengthwise fracture of functionally graded beams: a study of stress relaxation effects

The present paper is focussed on analyzing the strain energy release rate for a lengthwise crack in a cantilever beam that exhibits stress relaxation. The beam is continuously inhomogeneous in the thickness direction. A linear viscoelastic model is used for treating the stress relaxation. The balance of the energy is analyzed in order to derive a time-dependent solution to the strain energy release rate. The equilibrium of the lower crack arm and the un-cracked beam portion is considered to determine the curvatures and the coordinates of the neutral axes which are involved in the solution. The compliance method is applied in order to verify the solution to the strain energy release rate. A time-dependent solution to the strain energy release rate is derived also for the case of material exhibiting different viscoelastic behaviour in tension and compression. The solutions derived are used to investigate the evolution of the strain energy release rate with the time. The influence of the crack location along the thickness of the beam and the material inhomogeneity on the time-dependent strain energy release rate under stress relaxation is also investigated.

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.

Fractal damage and crack propagation in decoupled charge blasting

Decoupling coefficient and filling medium are the two key influencing factors that affect the effect of decoupled charge blasting. Using the model experiment method, combined with the fractal damage theory and the energy release rate theory, the fractal damage of the breakage zone, the fracture zone and the crush zone in coupled charge blasting and decoupled charge blasting (respectively using air and clay as the filling medium) is studied, and the energy release rate of the blast-induced crack is analyzed. In decoupled charge blasting, the filling medium plays a role in buffering and weakening the transfer of blasting energy, which inhibits the formation and development of the crush zone and reduces the fractal damage in the breakage zone. However, different filling media (air and clay) have significant differences in the formation of the fracture zone. Reasonable decoupling coefficient can make the energy carrying of the blast-induced crack more concentrated and increase the propagation length, thereby increasing the range of the fracture zone and improving the fragmentation effect.

Performance degradation due to anodic failure mechanisms in lithium-ion batteries

We report a mechano-chemical model for anodic degradation during fast-charging of nickel-manganese-cobalt (NMC)/graphite (C) cell due to SEI growth, lithium plating/stripping, dead lithium storage, and film fracture of composite SEI and plated lithium film. Degradation of the battery is analyzed for a range of charging rates from 1 to 6 C-rates, and the influence of plating mechanisms – lithium plating and dead lithium deposition and recovery during stripping – on the film resistance of the anode are accounted for in the model. Dynamic evolution of the interfacial properties is modeled using rule-of-mixture approach. Model predictions of plating associated stress fields are used to compute critical energy release rate for film cracking. The results indicate an increased tendency of fracture for thinner SEI film with lithium plating at higher charging rates. The process of reforming the cracked film absorbs a significant portion of the electrode current thereby reducing the cell capacity and plating efficiency. The mechano-chemical model provides an extensive analytical framework for understanding the synergistic coupling of anodic degradation mechanisms, prognosticating conditions of SEI failure, and evaluating the capacity fade and efficiency of lithium-ion battery.

Investigating inhomogeneity effects on the crack driving force in two-phase electrode particles using a planar composite core-shell model

It is well known that phase separation occurs in many intercalation materials as a result of solute intercalation. In lithium storage materials, phase separation induced by electrochemical cycling is known to intensify mechanical stresses in the host, and results in the emergence of a variety of defects causing mechanical degradation of the electrode particles. Treating two-phase electrode particles as a homogeneous material, fracture mechanics models have been previously developed for studying fracture of two-phase electrode particles. Recent theoretical and experimental studies, however, reveal that the elastic properties of the coexisting phases in the material could significantly differ due to their markedly different solute contents. This work aims to present a fracture mechanics study of the two-phase electrode particles accounting for the different elastic moduli of the phases. To this end, a planar composite core-shell model is utilized to examine how different elastic moduli of the phases could impact the stress intensity factors and the energy release rates associated with the pre-existing cracks in the particle. Our results reveal that although softening of the material as a result of phase change could result in stress relief in the electrode particle, however, it could still raise the energy release rate for pre-existing cracks, since a softened material behaves like a weakened deformational constraint ahead of a pre-existing crack. The model presented is also applied to the material systems exhibiting modulus change as a result of phase separation, and implications of the modulus change on the energy release rate for crack growth are also discussed. It is shown that although significant changes in the toughness of the material could arise due to the phase transformation, however, changes in elastic moduli could still raise the energy release rate for a crack whose tip is embedded in the phase with the lower toughness.

The effect of materials and design on the reliability of through-glass vias for 2.5 D integrated circuits: a numerical study

PurposeThis study aims to investigate the factors responsible for substrate cracking reliability problem in through-glass vias (TGVs), which are critical components for glass-based 2.5 D integration.Design/methodology/approachNumerical models were used to examine the driving force for substrate cracking in glass interposers due to stress coupling during heating. An analytical solution was used to demonstrate how the energy release rate (ERR) for the glass substrate cracking is affected by the via design and the mismatch in thermal strain. Then, the numerical models were implemented to investigate the design factors effects, such as the pitch distance, via diameter, via pattern, via design, effect from a stress buffer layer and the interposer materials selection on the susceptibility to substrate cracking.FindingsERR for substrate cracking was found to be directly proportional to the via diameter and the thermal mismatch strain. When a via pattern is implemented for high-density integration, a coupling in the stress fields was identified. This coupling effect was found to depend on the pitch distance, the position of the vias, and the via arrangement, suggesting a via pattern-dependent reliability behavior for glass interposers. Changing the design of the via to an annular shape or a substrate-cored via was found to be a promising approach to reduce the susceptibility to substrate cracking compared to a fully filled solid via. Also, the use of a stress buffer layer, an encouraging design prospect presented for the first time for TGVs in this study, was found to significantly reduce cracking. Finally, alternative via and substrate materials showed lower tendency for substrate cracking, indicating that the reliability of glass interposers can be further enhanced with the implementation of such new materials.Originality/valueThis study signifies the first attempt to comprehensively evaluate the susceptibility to crack formation in glass interposers during heating. Therefore, this study provides new perspectives on how to achieve a significant potential reliability improvement for TGVs.

Magnetic-yielding zone model for assessment of two mode-III semi-permeable collinear cracks in piezo-electro-magnetic strip

A magnetic-yielding zone model for an infinitely long piezo-electro-magnetic strip weakened by two mode-III semi-permeable collinear cracks is proposed. Fourier series method and integral equation technique are adopted to solve the problem. The closed-form analytical expressions are obtained for several fracture parameters, namely crack opening potential drop, crack-sliding displacement, crack opening induction drop, intensity factors, and energy release rates. Also, the explicit analytical expressions are derived for developed magnetic-yielding zones. An illustrative numerical study is presented for poled BaTiO3-CoFe2O4 material to observe the impact of inter-crack distance, volume fraction, prescribed loadings, electro-magnetic crack-face boundary conditions on fracture parameters.

Closed-form analysis of interlaminar crack initiation in angle-ply laminates

The layerwise discontinuous stiffness properties in composite laminates may cause highly localized, singular concentrations of interlaminar stresses. These stress singularities occur in the vicinity of free edges and may induce interfacial cracks, often referred to as delaminations. Since conventional design processes, based on classical laminate plate theory, only consider inplane stress components, they do not cover the so-called free-edge effect and the associated delamination onset and, therefore, disregard premature laminate failure. In the present study, interlaminar crack initiation in symmetric angle-ply laminates under tensile loading is investigated assuming the spontaneous formation of delaminations of finite length. A coupled stress and energy criterion is used requiring only two fundamental material parameters, the strength and the fracture toughness. The mechanical behaviour of the laminate is modeled by a highly efficient analytical closed-form approach providing results for the interlaminar shear stress in the interface and the energy release rates of potential cracks. The challenge of unknown interface fracture toughness is addressed by a parameter fit within physically reasonable limits. The resulting effective failure loads are compared to experimental data and show the high potential of the coupled criterion on the one hand and of the proposed analytical model on the other hand.

Correlation of nanoscale interface debonding and multimode fracture in polymer carbon composites with long-term hygrothermal effects

The correlation between nanoscale interface debonding and crack energy release rate for modes I, II, and I/II for extended exposure to hygrothermal conditions in polymer matrix composites (PMCs) needs to be understood. In this study, the effects on PMCs for two years of exposure, to a heat of 60 °C and relative humidity of 90%, were investigated. The interphase properties were studied using PF QNM. During hygrothermal treatment, the interphase region's thickness around the crack tip increases, and the morphology of the interphase changes. Nanoscale debonding of the fiber and matrix in artificially aged samples is a mix of cohesive and adhesive failure. Degradation at the nanoscale interphase and the macroscale crack initiation toughness slows after one year of aging. Interface debonding and nanoscale cracks cause intralaminar cracks, fiber bridging, and an increase in crack energy. Toughness in mode I fracture was increased in the artificially aged samples.

A novel method for measuring AFO stiffness using motion capture

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

Poling angle effect on two mode-III semi-permeable collinear cracks in a piezoelectric strip: Strip-saturation model

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

Crack Growth and Energy Release Rate for an Angled Crack under Mixed Mode Loading

The evaluation of energy release rate with angle is still a challenging task in metal crack propagation analysis, especially for the mixed Mode I-II-III loading situation. In this paper, the energy release rate associated with stress intensity factors at an arbitrary angle under mixed mode loadings has been investigated using both a numerical method and theoretical derivation. A relatively simple and precise numerical method was established through a series of spatial-inclined ellipses in Mode I-II and ellipsoids in Mode I-II-III, with different propagation angles computed from simulation. Meanwhile, a theoretical expression of the energy release rate with angle for a crack tip under a I-II-III mixed mode crack was deduced based on the propagation mechanism of the crack tip under the influence of a stress field. It is confirmed that the theoretical expression deduced could provide results as accurately as the present numerical method. The present results were confirmed to be effective and accurate by comparison with experimental data and other literature.

Longitudinal fracture analysis of inhomogeneous beams with continuously changing radius of cross-section along the beam length

This paper focusses on deriving a general solution to the strain energy release rate for a longitudinal circular cylindrical crack in non-linear elastic inhomogeneous beams with circular cross-section. The radius of the cross-section changes continuo

Thermal fracture characteristics of an interface crack subjected to temperature variations

Abstract Thermal fracture characteristics – the thermal energy release rate and thermal stress intensity factor of a semi-infinite crack at an interface between the two elastic isotropic materials, subjected to the temperature variations, are considered in this paper. Those characteristics are determined based on application of the linear elastic fracture mechanics (LEFM) concept. Expressions for obtained theoretical solutions are compared to solutions from literature and they are found to be more concise. Influence of the materials change on these two thermal fracture properties were observed, as well as the influence of the thickness ratio of the two layers constituting the interface.

Fracture Behavior of Cracked Ring Specimen at Different Crack Positions

Abstract The previous research review of piping systems revealed that the plastic pipe companies suffered from many problems in natural gas pipeline systems. One of the most significant problems that appeared in the piping systems are external cracks due to manufacturing processes, welding technique, and installation processes. The principal goal of the present experimental study is to predict the crack growth behavior and energy release rate of cracked ring specimens made from high-density polyethylene (HDPE) under different crack position angles and various crosshead speeds. The effect of loading rate on the external radial crack at different crack position angles plays an important role in the prediction of fracture behavior of plastic pipe materials. For this reason, it is necessary to conduct a study for the fracture analysis of pipe ring specimens under tension loading with double external cracks at constant radial crack length to width ratio equal a/W = 0.5. A precracking machine is designed especially in the present experimental study to simulate the actual radial cracks at outer surface of pipe ring specimens. The effects of crosshead speed and crack position angle revealed a significant effect on the energy release rate and maximum applied load under tensile load.

Thermal effect on delamination and buckling of superconducting film-substrate structures

High-temperature superconducting (HTS) film-substrate structures have great potential applications in magnets and superconducting cables that have great potential for high temperature superconducting wires in space solar power stations. The most important issue in the successful application of those structures is the delamination of interfaces. Therefore, in this paper, the delamination induced by electromagnetic force and temperature change in the superconducting film-substrate structure is studied. Energy release rates of delamination are given. It is found that thermal effect on the energy release rate is more significant when the electromagnetic force increases. If without electromagnetic force, as the crack length reaches at its critical value, the film will be buckling and the crack becomes a mixed crack. The closed-form solution of the energy release rate of the mixed crack is provided. On the other hand, when the crack length is smaller than its critical buckling length and the crack is the mode II crack. The influence of the stress intensity factor of the mode II crack is more significant for a thicker film. This suggests that the film should be designed to be thinner. This study is useful for engineers to design HTS film-substrate structures.

Modeling snow slab avalanches caused by weak-layer failure – Part 1: Slabs on compliant and collapsible weak layers

Abstract. Dry-snow slab avalanche release is preceded by a fracture process within the snowpack. Recognizing weak-layer collapse as an integral part of the fracture process is crucial and explains phenomena such as whumpf sounds and remote triggering of avalanches from low-angle terrain. In this two-part work we propose a novel closed-form analytical model for a snowpack under skier loading and a mixed-mode failure criterion for the nucleation of weak-layer failure. In the first part of this two-part series we introduce a closed-form analytical model of a snowpack accounting for the deformable layer. Despite the importance of persistent weak layers for slab avalanche release, no simple analytical model considering weak-layer deformations is available. The proposed model provides deformations of the snow slab, weak-layer stresses and energy release rates of cracks within the weak layer. It generally applies to skier-loaded slopes as well as stability tests such as the propagation saw test. A validation with a numerical reference model shows very good agreement of the stress and energy release rate results in several parametric studies including analyses of the bridging effect and slope angle dependence. The proposed model is used to analyze 93 propagation saw tests. Computed weak-layer fracture toughness values are physically meaningful and in excellent agreement with finite element analyses. In the second part of the series (Rosendahl and Weißgraeber, 2020) we make use of the present mechanical model to establish a novel failure criterion crack nucleation in weak layers. The code used for the analyses in both parts is publicly available under https://github.com/2phi/weac (last access: 6 January 2020) (2phi, 2020).