Mechanical Energy Release Rate Research Articles

Numerical solution for the interior electric displacement of the moving DBY‐PS model for semi‐permeable cracked piezoelectric material

AbstractIn this study, we analysed a moving crack at the interface of an infinitely long piezoelectric bilayer using the Dugdale–Barenblatt yield (DBY) model and the polarisation saturation (PS) model. To model the moving crack problem, a Yoffe‐type crack moves at a constant subsonic speed on the interface of an infinitely long piezoelectric bilayer. The crack faces are assumed to be semi‐permeable, and at the boundary of the bilayer, in‐plane electrical and out‐of‐plane mechanical stresses are applied. Due to the application of electro‐mechanical loads, cracks propagate, mechanical yielding zones and electric saturation zones are developed. To arrest the crack from further propagation, mechanical yield stress and saturation electric displacement are applied at the developed zones. To address this problem analytically and numerically, the mixed boundary value problem is transformed into a set of coupled Fredholm integral equations (FIEs) of the second kind using the Fourier transform and the Copson method. The closed‐form analytical expressions for the length of the electrical saturation zone (ESZ), whether longer, shorter or equal to the mechanical yielding zone (MYZ), show dependence on external electro‐mechanical loads under semi‐permeable crack conditions. The algorithm to solve the electric crack condition parameter (ECCP) has been defined using numerical discretization and the bisection method. Illustrative examples demonstrate the proposed technique's effectiveness and suitability for Yoffe‐type moving cracks. The numerical results show the convergence of the ECCP. Furthermore, the numerical results show how mechanical and electrical zone lengths and energy release rate (ERR) are affected by electrical and mechanical loads, strip thickness and crack velocity. In addition, the size of the mechanical yielding zone is consistently promoted by electrical load, while the promotion or prevention of the electrical saturation zone by mechanical load depends on the relative sizes of the nonlinear zones.

Effects of fluid composition and salinity on subcritical crack growth in granite

Using the double-torsion technique, the fracture mechanical response of granite specimens was investigated under the influence of fluid with different electrolyte types and salinities. The fracture toughness (KIC) and critical mechanical energy release rate (G0) of the granite in electrolyte solution are weakened compared to those in distilled water. The KIC in all electrolyte solutions is 2.08–2.28 MPa·m1/2, which is 8.4 % − 16.5 % lower than that in distilled water, and it is insensitive to ion type and salinity. In contrast, the subcritical crack growth index (n) shows different salinity dependence in three salt (AlCl3, CaCl2, and NaCl) solutions. The slight increase in n with rising salinity of the AlCl3 solution is attributed to crack tip blunting caused by over-dissolution of minerals. The n and G0 decrease and then increase with rising salinity of CaCl2 solution, which may be related to the salinity-dependent dissolution rate of quartz and the weakening of repulsive force between crack surfaces. The n and G0 shows a negative correlation with salinity and decrease by 26.4 % and 15.4 %, respectively as the NaCl salinity increases from 0.1 M to 1.0 M. Limitations in the dissolution reactions of the main minerals (albite, biotite and diopside) is accountable for enhancement of n in silica solutions. Experimental results emphasize the critical and subcritical fracture behavior of the granite in response to chemically reactive fluids with implications for both hydraulic fracture stimulations in geothermal fields and caprock integrity for CO2 sequestration. In these applications, the fracture properties of the reservoir rock can be enhanced or weakened by changing the electrolyte type and salinity of the fluid, depending on the actual requirements.

Tracing the energetic outflows from galactic nuclei: observational evidence for a large-scale bipolar radio and X-ray-emitting bubble-like structure in M106

ABSTRACT The role of energetic outflows from galactic nuclei in shaping galaxy formation and evolution is still shrouded in uncertainty. In this study, we shed light on this complex phenomenon by presenting evidence for a large-scale bipolar radio/X-ray-emitting bubble-like structure emanating from the central region of the nearby disc galaxy M106 (NGC 4258). Our findings, based on Low-Frequency Array survey data and Chandra observations, provide a glimpse into the underlying physical processes driving this enigmatic structure. Similar to the eROSITA/Fermi bubbles in our own Galaxy, the M106 bubbles enclose diffuse hot plasma and are partially bounded by prominent radio/X-ray-emitting edges. We constrain the magnetic field and cosmic-ray properties of the structure. The analysis of the X-ray data gives an estimate of the thermal energy of the bubbles as ∼8 × 1056 erg. This energy can be supplied by the jets and perhaps by the wind from the accretion flow of the galaxy’s low-luminosity AGN, which most likely has been much more powerful in the recent past, with an average mechanical energy release rate of ∼4 × 1042 erg s−1 over the last ∼8 × 106 yr – the estimated age of the structure. We also show evidence for diffuse X-ray emission on larger scales, indicating the presence of a hot galactic corona. Our results provide a clear manifestation of galactic nuclear feedback regulating the gas content and energetics of the circumgalactic medium of disc galaxies similar to our own.

Phase-field modeling of electromechanical fracture in piezoelectric solids: Analytical results and numerical simulations

The optimal design, reliable manufacture and durable utilization of electromechanical systems demand objective modeling of failure under coupled electric field and mechanical actions. This is still a challenging and largely an open issue despite the large volumes of contributions from the discontinuous approach and the phase-field community for brittle fracture. In this work we propose a length scale insensitive phase-field cohesive zone model (PF-CZM) for electromechanical fracture in linear piezoelectric solids, overcoming all the issues of mesh dependence, crack tracking complexity, length scale sensitivity and crack nucleation inconsistency. More specifically, in the electromechanically coupled phase-field constitutive relations, the evolution law for the crack phase-field is established from the geometric regularization of sharp cracks and the energetic criterion in global form. In order to capture the unsymmetric mechanical and electrical fracture behavior, the mechanical energy release rate contributed from the positive cone of the effective stress in energy norm is employed as the crack driving force, resulting in a hybrid formulation. With a set of phase-field characteristic functions optimal for cohesive fracture, the critical stress upon crack nucleation and the post-peak softening regime are derived in close-form for the first time, and the experimentally observed effects of the electric field on the fracture load are predicted qualitatively. The proposed electromechanical PF-CZM is numerically implemented in the multi-field finite element method and applied to several representative examples. In all cases, as those for purely mechanical problems, the predicted crack patterns and fracture loads are insensitive to the phase-field length scale and the experimental results are well reproduced. In particular, the effect of electric field on the deflection of crack paths can be captured. These merits make the proposed PF-CZM promising in the modeling of electromechanical fracture in piezoelectric solids.

Interfacial crack growth in piezoelectric-piezomagnetic bi-layered structures with a modified mechanical energy release rate criterion

This paper investigates the interfacial crack growth in the piezoelectric-piezomagnetic (PE-PM) bi-layered composite structures under quasi-static assumption. The extended finite element method (X-FEM) is utilized to solve the fracture parameters and simulate the growth paths, in which the suitable enrichment functions are chosen according to different singular properties of the crack tip. Based on the crack-tip fields of interfacial crack in PE-PM bimaterials, the mechanical energy release rate (MERR) criterion is modified and extended to the PE-PM composites where the relation of fracture toughness and poling direction should be considered. In the numerical examples, the validity of the present scheme is firstly verified by compared with the three point bending test for PE material. And then, with the modified MERR criterion, the effects of the geometries of cracked model, electro-magneto-mechanical coupling loadings and fracture toughness on interfacial crack growth are emphatically studied.

The Proposal of The Method to Predict Delamination Growth Rate between Resin and Metal

The method to predict delamination growth rate between resin and metal was proposed. This method was used static mechanical properties and energy release rate of the interface crack between Resin and metal.

A simplified evaluation of the mechanical energy release rate of kinked cracks in piezoelectric materials using the boundary element method

Based on the concept of the hoop field intensity factors of an initial crack prior to any kink, an apparent hoop mechanical (strain) energy release rate (MERR) is defined to approximate the MERR of a piezoelectric crack with an infinitesimal kink at any arbitrary angle. The validity and the efficiency of the simplified approximation are examined by numerical examples using the boundary element method (BEM). The generalized crack-opening-displacements or displacement jumps are computed by the traction boundary integral equations (BIEs). By using the displacement extrapolation method, the crack-tip field intensity factors of any arbitrarily kinked crack in linear piezoelectric materials are obtained and the BEM results are validated by comparing them with the available reference analytical results. Then, the differences between the conventional field intensity factors and MERR of an infinitesimally kinked crack and the hoop field intensity factors and hoop MERR of the main crack prior to any kink are numerically analyzed. Finally, the crack propagation in an infinite linear piezoelectric material is numerically simulated. The paths of the crack growth are predicted by adopting four different fracture criteria, namely, the maximum hoop stress intensity factor (SIF) and MERR fracture criteria for the main crack-tip before the next propagation, and the maximum KI and MERR fracture criteria for the kinked tip of the main crack with an infinitesimal branch at an arbitrary kinking angle evaluated by using a trial crack extension technique. The comparisons among these results show that the present simplified approximation can efficiently provide a sufficient accuracy for numerical simulation of crack growth in linear piezoelectric materials.

An energy-consistent fracture model for ferroelectrics

The fracture behavior of ferroelectrics has been intensively studied in recent decades, though currently a widely accepted fracture mechanism is still lacking. In this work, enlightened by previous experimental observations that crack propagation in ferroelectrics is always accompanied by domain switching, we propose a micromechanical model in which both crack propagation and domain switching are controlled by energy-based criteria. Both electric energy and mechanical energy can induce domain switching, while only mechanical energy can drive crack propagation. Furthermore, constrained domain switching is considered in this model, leading to the gradient domain switching zone near the crack tip. Analysis results show that stress-induced ferroelastic switching always has a toughening effect as the mechanical energy release rate serves as the driving force for both fracture and domain switching. In comparison, the electric-field-induced switching may have either a toughening or detoughening effect. The proposed model can qualitatively agree with the existing experimental results.

Toughening effect of ferroelectric ceramics induced by domain switching and dislocations

The multi-scale analysis of fracture toughness of ferroelectric ceramics under complicate mechanical–electrical coupling effect is carried out in this paper. The generalized stress intensity factor (SIF) arising from spontaneous strains and polarization transformation in switching domain zones is accurately obtained by using an extended Eshelby theory. Taking BaTiO3 ferroelectric ceramic for example, it is discovered that the crack propagation can be induced by domain switching arising from negative electrical field when the crack surface is parallel to the isotropic plane, and the obtained critical electric displacement intensity factor (EDIF) approximates closely to that obtained by the Green’s function method. Additionally, as pinning dislocations and slip dislocations can strongly influence properties of ferroelectric devices and induce the property degradation, it is necessary to investigate the dislocation toughening effects on fatigue and fracture mechanisms. The results show that the dislocation shielding and anti-shielding effects on mode II SIF, mode I SIF and EDIF are obviously different when a dislocation locates at a position near the crack tip. Through the calculation of the critical applied EDIF for crack propagation by using mechanical energy release rate (MERR) theory, it is discovered that the slip angles obviously influence fracture toughness, and the mode II SIF arising from dislocation has little influence on fracture toughness, however, the mode I SIF and EDIF arising from dislocation have great influences on fracture toughness.

Modeling and Reliability Analysis of TSVs for High Frequency Applications

This paper presents electrical modeling of thermo-mechanical delamination in TSV. Previously, several papers in electrical modeling of TSV have been published in open literature; however, these do not consider the thermal-mechanical delamination in TSV. Stress-induced thermo-mechanical delamination in TSV has been analyzed by considering the thermo-mechanical stress loading effects of high aspect ratio copper electroplated TSVs. Two-dimensional mechanical finite element models have been built to analyze the stress distribution in TSV structures. The mechanical models and energy release rate at the crack are then translated to electrical defect models to evaluate the functional electrical performance of TSVs with delaminations. Simulation was performed on 7 μm by 60 μm TSVs using an RF signal with −10 dBm power on a crack length of 0.2 um and width of 0.15 um. We observed output power of −11.8 dB at a frequency of 77 GHz. When the crack length was increased, the output power decreased to −11.2 dBm due to increased parasitics resulting in more attenuation of RF output. This reliability analysis enables us to evaluate the electrical performance of TSVs for high frequency applications.

Application of X-FEM to study two-unequal-collinear cracks in 2-D finite magnetoelectoelastic specimen

A quasi-static and propagating analysis is simulated for two-unequal-collinear cracks in a 2-D finite magnetoelectroelastic specimen using X-FEM. The intensity factors (IFs) are calculated using interaction integral in conjugation with the near tip behavior given by the Stroh formalism. In quasi-static case energy release rate (ERR) variations are investigated with respect to inter-crack space, crack lengths and mechanical/electrical/magnetic loads. Two-collinear-unequal cracks in an infinite domain problem are simulated, analyzed and validated. Further, effects of asymmetric orientation and symmetric orientation of quasi-static two-unequal-collinear cracks vis-à-vis specimen boundaries on total energy release rate (TERR) and mechanical energy release rate (MERR) are investigated. The case of one edge and one internal quasi-static crack is obtained as a corollary and the case study is presented. Next, the crack growth study for two-unequal-collinear edge cracks is simulated using maximum modified hoop stress intensity factor criterion and considering anisotropic fracture toughness behavior of polarized ceramics. The effect of volume fraction, electrical and magnetic loadings are similar to the strain energy density function criterion of crack propagation for magnetoelectroelastic ceramics. Lastly dependence of volume fraction and material constants of magnetoelectroelastic ceramics are observed on the set of standard eight basis functions for these ceramics. Hence, a more generalized set of basis functions is also defined here for these ceramics.

Numerical Solution of an Edge Cracked 2-D Piezoelectric Media Using Extended Finite Element Method

The numerical solution for an edge crack problem in a two-dimensional (2-D) finite piezoelectric media has been discussed using extended finite element method. The four-fold standard enrichment functions are taken in conjugation with the interaction integral to evaluate the intensity factors (IFs). The intensity factors as well as the mechanical energy release rate and the total energy release rate has been analyzed for different electro-mechanical boundary conditions. It is observed that the IFs results are coupled and contrary to analytic solution which shows uncoupled behaviour.

“Driving forces” and radiated fields for expanding/shrinking half-space and strip inclusions with general eigenstrain

A half-space constrained Eshelby inclusion (in an infinite elastic matrix) with general uniform eigenstrain (or transformation strain) is analyzed when the plane boundary is moving in general subsonic motion starting from rest. The radiated fields are calculated based on the Willis expression for constrained time-dependent inclusions, which involves the three-dimensional dynamic Green’s function in an infinite traction-free body, and they constitute the unique elastodynamic solution, with initial condition the Eshelby static fields obtained as the unique minimum energy solutions by a limiting process from the spherical inclusion. The mechanical energy-release rate and associated “driving force” to create dynamically an incremental region of eigenstrain (due to any physical process) is calculated for general uniform eigenstrain. For dilatational eigenstrain the solution coincides with the one obtained by a limiting process from a spherically expanding inclusion, while for shear eigenstrain the fields are due to the propagation of the rotation. The “driving force” has the same expression both for expanding and shrinking motions, resulting in expenditure of the energy rate for motion of the boundary in both cases. By superposition from the half-space inclusions, the fields and “driving force” for a strip inclusion with both boundaries moving are obtained. The “driving force” consists also of a contribution from the other boundary when it has time to arrive. The presence of applied loading contributes the counterpart of the Peach-Koehler force of dislocations, in addition to the self-force.

Acoustic Emission Characteristics of Mode I Delamination in Glass/Polyester Composites

This article proposes an experimental study on the mode I interlaminar fracture of glass/polyester composites by using acoustic emission (AE) to analyze the damage evolution and evaluate the interlaminar performance of polymeric composites. A delamination process simulated with a double cantilever beam in opening mode (Mode I) coupled with an AE technique has been employed. The microscopic observation (scanning electron microscopy) is used to determine the correlation between different fracture mechanisms and their corresponding AE signal frequency content. Selected emissions are classified as matrix cracking, fiber breakage, or interface processes (fiber—matrix debonding) based on their total power in defined frequency intervals of the spectral power density. A correlation was established between the mechanical energy release rate and the AE energy rate. Analysis in the frequency domain shows AE parameters are powerful indicators of the intensity of the damage.

Anomalies associated with energy release parameters for cracks in piezoelectric materials

For a central crack in a piezoelectric plate, the mode-I stress intensity factor ( K I ), electric displacement intensity factor ( K D ), energy release rates ( G, G M ) and energy density factor ( S) are obtained from the finite element results. For the impermeable crack, the numerical results of K I and K D are coupled; this error is contrary to the uncoupled analytical solutions. The error has little effect on the total energy release rate G and energy density factor S, but in some cases, large errors in the mechanical energy release rate G M are observed. G is global while SED is local. Also G is negative which defies physics where energy cannot be created while crack attempts to extend as implied by G. Computations should be made for the J-integral and also show that J becomes negative. What this shows is that the global fracture energy criterion is not suitable to address the local release of energy because it includes the overall energy which are irrelevant to fracture initiation being a local behavior. In addition, the case study shows that the energy density theory is the better fracture criterion for the piezoelectric material. According to the results of S, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees with analytical and experimental evidence in the past references.

횡등방 압전재료의 면외하중 모서리 균열에 대한 그린함수

Surface edge crack in transversely isotropic piezoelectric material is analyzed. The concentrated antiplane mechanical and inplane electrical loadings are applied to an arbitrary point of the surface, where the impermeable crack boundary condition is imposed. Using Mellin transform the problem is formulated, from which Wiener-Hopf equations are derived. By solving the equations the solution is obtained in a closed form. Mechanical and electric intensity factors and energy release rate are obtained and discussed. This problem could be used as a Green's function to generate the solutions of other problems with the same geometry but of different loading conditions.

Approximation of curved cracks under mixed-mode loading

The calculation of stress intensity factors or mechanical energy release rate for non-straight cracks can be complicated. Approximation to equivalent crack shapes can simplify calculations considerably, but this requires an understanding of the influence of key shape parameters on crack-tip stresses. A simple analytical model has been developed, based on the concept of a relaxed volume, to predict mechanical energy release rate and deflection angle for a range of crack shapes under mixed-mode loading. Results from this model compared well with those obtained from finite element (FE) simulations, and with predictions from previous analytical models. It was found that the crack length and orientation of the crack-tip with respect to loading direction are the key influences on fracture parameters, whilst curvature near the crack-tip can also affect results.

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.

On energy changes due to the formation of a circular hole in an elastic plate

The appearance of holes in crystals, developed in stressed biological structures, has motivated us to study the energy release of an initially undisturbed plate subjected to a biaxial stress state which is disturbed by a growing hole. An energy balance allows for a kinetic equation of the hole radius a. The main emphasis is placed on the calculation of the total mechanical energy-release rate by different independent concepts. Specifically, path-independent integrals represent the most convenient approach.

Fracture analysis of magnetoelectroelastic solids by using path independent integrals

A solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The Ji and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.