Fourth-order phase-field modeling for brittle fracture in piezoelectric materials

Failure analyses of piezoelectric structures and devices are of engineering and scientific significance.In this paper,a fourth-order phase-field fracture model for piezoelectric solids is developed based on the Hamilton principle.Three typical electric boundary conditions are involved in the present model to characterize the fracture behaviors in various physical situations.A staggered algorithm is used to simulate the crack propagation.The polynomial splines over hierarchical T-meshes(PHT-splines)are adopted as the basis function,which owns the C1continuity.Systematic numerical simulations are performed to study the influence of the electric boundary conditions and the applied electric field on the fracture behaviors of piezoelectric materials.The electric boundary conditions may influence crack paths and fracture loads significantly.The present research may be helpful for the reliability evaluation of the piezoelectric structure in the future applications.

Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method

The local arc-length method is employed to control the incremental loading procedure for phase-field brittle fracture modeling.An improved staggered algorithm with energy and damage iterative tolerance convergence criteria is developed based on the residuals of displacement and phase-field.The improved staggered solution scheme is implemented in the commercial software ABAQUS with user-defined element subroutines.The layered system of finite elements is utilized to solve the coupled elastic displacement and phase-field fracture problem.A one-element benchmark test compared with the analytical solution was conducted to validate the feasibility and accuracy of the developed method.Our study shows that the result calculated with the developed method does not depend on the selected size of loading increments.The results of several numerical experiments show that the improved staggered algorithm is efficient for solving the more complex brittle fracture problems.

Characterizing the influence of stress-induced microcracks on the laboratory strength and fracture development in brittle rocks using a finite-discrete element method-micro discrete fracture network FDEM-μDFN approach

Heterogeneity is an inherent component of rock and may be present in different forms including mineralheterogeneity, geometrical heterogeneity, weak grain boundaries and micro-defects. Microcracks areusually observed in crystalline rocks in two forms： natural and stress-induced; the amount of stressinducedmicrocracking increases with depth and in-situ stress. Laboratory results indicate that thephysical properties of rocks such as strength, deformability, P-wave velocity and permeability areinfluenced by increase in microcrack intensity. In this study, the finite-discrete element method （FDEM）is used to model microcrack heterogeneity by introducing into a model sample sets of microcracks usingthe proposed micro discrete fracture network （mDFN） approach. The characteristics of the microcracksrequired to create mDFN models are obtained through image analyses of thin sections of Lac du Bonnetgranite adopted from published literature. A suite of two-dimensional laboratory tests including uniaxial,triaxial compression and Brazilian tests is simulated and the results are compared with laboratory data.The FDEM-mDFN models indicate that micro-heterogeneity has a profound influence on both the mechanicalbehavior and resultant fracture pattern. An increase in the microcrack intensity leads to areduction in the strength of the sample and changes the character of the rock strength envelope. Spallingand axial splitting dominate the failure mode at low confinement while shear failure is the dominantfailure mode at high confinement. Numerical results from simulated compression tests show thatmicrocracking reduces the cohesive component of strength alone, and the frictional strength componentremains unaffected. Results from simulated Brazilian tests show that the tensile strength is influenced bythe presence of microcracks, with a reduction in tensile strength as microcrack intensity increases. Theimportance of microcrack heterogeneity in reproducing a bi-linear or S-shape failure envelope and itseffects on the mechanisms leading to spalling damage near an underground opening are also discussed.

PROBABILISTIC MODELING OF BRITTLE FRACTURE WITH PRIOR DUCTILE CRACK EXTENSION

A computation framework for brittle fracture which incorporates weakest link statistics and a microme- chanics model reflecting reflecting local damage of the material is described.The Weibull stress W emerges as a probabilistic fracture parameter to define the condition leading material failure. Unstable crack propa- gation occurs at a critical value of W which may be attained paior to or following some amount of duc- tile crack extension. A realistic model of ductile crack growth using the computation cell methodology is used to define the evolution of near tip stress fields during crack extension. An application of proposed framework to predict the measured geometry and ductile tearing effects on the statistical distributio of fracture toughness for the pipe line steel welded joint is described.

ANALYSIS OF BRITTLE FRACTURE DUE TO COMPRESSIVE LOADING USING CENTER-CRACKED BRAZILIAN DISK

The center cracked Brazilian disk subjected to diametral compressive stress uniformly distributed along parts of its cylindrical surface is used to investigate combined mode fracture of brittle material. A fracture analysis is made of this specimen configuration. Explicit formulae for mode Ⅰ and mode Ⅱ stress intensity factor calculation are derived based on boundary integral equation method and related numerical solution given by Atkinson. The proposed formulae are valid in wide range of crack length a/R . This configuration can avoid splitting along load line usually occuring in Brazilian test and permit one to achieve easily pure mode Ⅱ crack growth (crack coplanar extension) and any combination of K Ⅰ and K Ⅱ by a simple alignment of crack orientation with respect to load line. SIF values from the present calculation and finite element solution are also given for comparison.

Effect of weld microstructure on brittle fracture initiation in the thermallyaged boiling water reactor pressure vessel head weld metal

Effects of the weld microstructure and inclusions on brittle fracture initiation are investigated in a thermally aged ferritic high-nickel weld of a reactor pressure vessel head from a decommissioned nuclear power plant.As-welded and reheated regions mainly consist of acicular and polygonal ferrite,respectively.Fractographic examination of Charpy V-notch impact toughness specimens reveals large inclusions(0.5-2.5μm)at the brittle fracture primary initiation sites.High impact energies were measured for the specimens in which brittle fracture was initiated from a small inclusion or an inclusion away from the V-notch.The density,geometry,and chemical composition of the primary initiation inclusions were investigated.A brittle fracture crack initiates as a microcrack either within the multiphase oxide inclusions or from the debonded interfaces between the uncracked inclusions and weld metal matrix.Primary fracture sites can be determined in all the specimens tested in the lower part of the transition curve at and below the 41-J reference impact toughness energy but not above the mentioned value because of the changes in the fracture mechanism and resulting changes in the fracture appearance.

INFLUENCE OF LOCAL BRITTLE ZONE ON THE FRACTURETOUGHNESS OF HIGH-STRENGTH LOW-ALLOYEDMULTIPASS WELD METALS

In this paper, toughness properties and microstructurc of low-alloyed multipass welds with yield strength above 700MPa have 6een studied using the weld thermal simulation and throughout thickness CTOD fracture mechanics tests. Impact testing of thermal simulated specimens showed that the primary weld metal and the fine gmmed weld metal had good toughness, while the coarse grained weld metal had the lowest toughness value as the local brittle zone (LBZ) in multipass weld metals. Cleavage fracture in CTOD testing of thick multipass weld metals was initiated from martensite-austenite (MA) phases in the LBZ. MA phases were distributed at the prior austenite grain boundaries and around ferrite grains. As the size of the local brittle zone along the fatigue crack front increases, CTOD frncture toughness of multipass weld metals decreases. The weakest link theory was used to evaluate effect of the local brittle zone on fracture toughness of thick multipass weld metals. The estimated curves agree well with the eaperimental data.

Thermo-hydro-mechanical-chemical(THMC)coupling fracture criterion of brittle rock

Based on analysis of thermo-hydro-mechanical-chemical(THMC)coupling mechanism for brittle rock,THMC coupling indicator in terms of rock porosity was introduced to represent the influencing degree of THMC coupling field on stress field in order to establish THMC coupling fracture criterion.A novel real-time measurement method of permeability(related to porosity)was proposed to determine the THMC coupling indicator,and self-designed THMC coupling tests and scanning electron microscope tests were conducted on pre-cracked red sandstone specimens to study the macroscopic and microscopic fracture mechanism.Research results show that the higher the hydraulic pressure is,the smaller the crack initiation load is and the easier the Mode I fracture occurs.Test results are in good agreement with prediction results(crack initiation load and angle,and fracture mode),which can verify the effectiveness of the newly established THMC coupling fracture criterion.This new fracture criterion can be also further extended to predict THMC coupling fracture of multi-crack problem.

The Influence of Welding Parameters on Brittle Fracture of Liquefied Natural Gas Storage Tank Welded Joint

Many applications operate at sufficiently low temperature conditions where most structural steels become very brittle and, therefore, unsuitable for use in safety-critical structures. So the materials used in the vessels or storage tanks which keep the natural gas at liquefaction temperatures need to remain ductile and crack resistant with a high level of safety. The material also needs to have high strength in order to reduce the wall thickness of the container and it must permit welding without any risk of brittle fracture. 9% Ni steel plates are one of most common used materials in the LNG storage tank application. However, the welding procedure for 9% Ni steel plates requires high level of skills of welding that is strictly controlled welding parameter for balancing avoidance of cold and hot cracking and maintenance of high strength. Mechanical properties are important characteristics of the weldment that must confirm to the application feasibility as well as functional requirements of the welded joint. The only way to enhanced the mechanical properties of welded joint by controlling the parameters of using welding process. From the main variables of the arc welding process are the heat input and interpass temperature where the two variables control the thermal cycle of welding process. The experiment show that for thin test specimen with thickness ≤ 14 mm, the heat input range from 1.4 to 2 KJ/mm and controlling interpass temperature within 80°C give high tensile strength with improving the toughness properties of welded joint and reduce the probability of brittle fracture happened by increase the ductility and reduce the yield strength and increased the transition temperature.

Analysis of fracture process zone in brittle rock subjected to shear-compressive loading

An analytical expression for the prediction of shear-compressive fracture process zone(SCFPZ) is derived by using a proposed local strain energy density criterion, in which the strain energy density is separated into the dilatational and distortional strain energy density, only the former is considered to contribute to the brittle fracture of rock in different loading cases. The theoretical prediction by this criterion shows that the SCFPZ is of asymmetric mulberry leaf in shape, which forms a shear-compression fracture kern. Dilatational strain energy density along the boundary of SCFPZ reaches its maximum value. The dimension of SCFPZ is governed by the ratio of K_Ⅱ to (K_Ⅰ.) The analytical results are then compared with those from literatures and the tests conducted on double edge cracked Brazilian disk subjected to diametrical compression. The obtained results are useful to the prediction of crack extension and to nonlinear analysis of shear-compressive fracture of brittle rock.

Nanoscale Morphology in Tensile Fracture of a Brittle Amorphous Ribbon

The paper reports on the observation of nanoscale morphology on the tensile fracture surface of a brittle amorphous Fe-based ribbon. The formation of nanoscale damage cavity structure is a main characteristic morphology on the fracture surfaces. Approaching the ribbon boundary, these damage cavities assemble and form the nanoscale periodic corrugations, which are neither Wallner lines nor crack front waves. The periodic corrugations result from the interactions between the reflected elastic waves by the boundaries of amorphous ribbon and the stress fields of the crack tip.

Two Modes and Criterion of Brittle Fracture

The plastic zone at the tip of the flaw (including acute cark and common notch) was investigated. Forthe notch specimen, a formula of toughness K was proposed, and its physical meaning was emphasized.Twomodes of brittle fracture was identified and the evaluating criterion was established.

MAGNETIC STRIPS TO SIMULATE LAYERED BRITTLE SOLIDS IN CLEAVAGE AND FRACTURE EXPERIMENTS

A characteristic of the fracture and cleavage experiments is that they are usually intrinsically destructive. Cracks do not completely heal in an unstressed system, even in crystals such as mica. Here, we used magnetic solids composed of magnetic strips for the non-destructive cleavage and brittle fracture experiments. Between the magnetic strips materials with different mechanical characteristics can be inserted, such as Teflon or foam strips, to change the mechanical properties of the solid. For the cleavage experiments, we developed an apparatus where parameters such as the main involved force can be measured easily. By inserting flaws, the magnetic solid can be used in dynamic fracture experiments, with the advantages of simulating macroscopically a non-destructive experiment in an easier way, that happen in real materials with much higher velocities. The apparatus and the used magnetic solid may be useful for demonstrations of fractures in classes.

A NONLOCAL THEORY FOR BRITTLE FRACTURE

In this paper,a nonlocal theory of fracture for brittle materials has been systematically devel- oped,which is composed of the nonlocal elastic stress fields of Griffith cracks of mode-Ⅰ,Ⅱ and Ⅲ,the asymptotic forms of the stress fields at the neighborhood of the crack tips,and the maximum tensile stress criterion for brittle fracture.As an application of the theory,the fracture criteria of cracks of mode-Ⅰ,Ⅱ, Ⅲ and mixed mode Ⅰ-Ⅱ,Ⅰ-Ⅲ are given in detail and compared with some experimental data and the theoretical results of minimum strain energy density factor.

Peridynamic Modeling of Brittle Fracture in Mindlin-Reissner Shell Theory

In this work,wemodeled the brittle fracture of shell structure in the framework of Peridynamics Mindlin-Reissener shell theory,in which the shell is described by material points in themean-plane with its drilling rotation neglected in kinematic assumption.To improve the numerical accuracy,the stress-point method is utilized to eliminate the numerical instability induced by the zero-energy mode and rank-deficiency.The crack surface is represented explicitly by stress points,and a novel general crack criterion is proposed based on that.Instead of the critical stretch used in common peridynamic solid,it is convenient to describe thematerial failure by using the classic constitutive model in continuum mechanics.In this work,a concise crack simulation algorithm is also provided to describe the crack path and its development,in order to simulate the brittle fracture of the shell structure.Numerical examples are presented to validate and demonstrate our proposed model.Results reveal that our model has good accuracy and capability to represent crack propagation and branch spontaneously.

Prediction of Quasi-Brittle Fracture of Concrete and Rock Structures Using Laboratory-Sized Specimens

A simple method is developed for predicting the fracture behaviour of structures made of quasi-brittle materials such as concrete and rock using the data from laboratory-sized specimens. The method is based on the recently-developed boundary effect concept and associated asymptotic model. It is demonstrated that the “apparent” size dependence of fracture behaviour of concrete and rock is in fact due to the influence of specimen boundaries. Various size effect phenomena that are often observed in fracture mechanics tests of concrete and rock are related to each other, and the asymptotic boundary effect model can explain all the observed “size” effect phenomena. Four types of experimental results available in the literature (including the data measured on (1) the specimens of identical size with different crack-to-size (%α%) ratios, (2) specimens of different sizes with different %α%-ratios, (3) different types of specimens and (4) geometrically similar specimens) are used to verify the asymptotic boundary effect model, and it is found that the predictions of the model agree very well with the experimental results. Furthermore, the important fracture properties, fracture toughness %K_{IC}% and strength %f_t% of quasi-brittle materials such as concrete and rock can also be calculated using the formulae provided in the model.

A fracture model for assessing tensile mode crack growth resistance of rocks

Evaluating the fracture resistance of rocks is essential for predicting and preventing catastrophic failure of cracked structures in rock engineering.This investigation developed a brittle fracture model to predict tensile mode(mode I)failure loads of cracked rocks.The basic principle of the model is to estimate the reference crack corresponding to the fracture process zone(FPZ)based on the maximum normal strain(MNSN)ahead of the crack tip,and then use the effective crack to calculate the fracture toughness.We emphasize that the non-singular stress/strain terms should be considered in the description of the MNSN.In this way,the FPZ,non-singular terms and the biaxial stress state at the crack tip are simul-taneously considered.The principle of the model is explicit and easy to apply.To verify the proposed model,laboratory experiments were performed on a rock material using six groups of specimens.The model predicted the specimen geometry dependence of the measured fracture toughness well.More-over,the potential of the model in analyzing the size effect of apparent fracture toughness was discussed and validated through experimental data reported in the literature.The model was demonstrated su-perior to some commonly used fracture models and is an excellent tool for the safety assessment of cracked rock structures.

Peridynamic Study on Fracture Mode and Crack Propagation Path of a Plate with Multiple Cracks Subjected to Uniaxial Tension

How to simulate fracture mode and crack propagation path in a plate with multiple cracks is an attractive but difficult issue in fracture mechanics.Peridynamics is a recently developed nonlocal continuum formulation that can spontaneously predict the crack nucleation,branch and propagation in materials and structures through a meshfree discrete technique.In this paper,the peridynamic motion equation with boundary traction is improved by simplifying the boundary transfer functions.We calculate the critical cracking load and the fracture angles of the plate with multiple cracks under uniaxial tension.The results are consistent with those predicted by classical fracture mechanics.The fracture mode and crack propagation path are also determined.The calculation shows that the brittle fracture process of the plate with multiple cracks can be conveniently and correctly simulated by the peridynamic motion equation with boundary conditions.

Experimental study on 3D internal penny-shaped crack propagation in brittle materials under uniaxial compression

Fractures are widely present in geomaterials of civil engineering and deep underground engineering.Given that geomaterials are usually brittle,the fractures can significantly affect the evaluation of underground engineering construction safety and the early warning of rock failure.However,the crack initiation and propagation in brittle materials under composite loading remain unknown so far.In this study,a three-dimensional internal laser-engraved cracking technique was applied to produce internal cracks without causing damage to the surfaces.The uniaxial compression tests were performed on a brittle material with internal cracks to investigate the propagation of these internal cracks at different dip angles under compression and shear.The test results show that the wing crack propagation mainly occurs in the specimen with an inclined internal crack,which is a mixed-ModeⅠ–Ⅱ–Ⅲfracture;in contrast,ModeⅠfracture is present in the specimen with a vertical internal crack.The fractography characteristics of ModeⅢfracture display a lance-like pattern.The fracture mechanism in the brittle material under compression is that the internal wing cracks propagate to the ends of the whole sample and cause the final failure.The initial deflection angle of the wing crack is determined by the participation ratio of stress intensity factors KII to KI at the tip of the internal crack.

Fatigue fracture mechanism of AZ31B magnesium alloy and its welded joint

Fatigue test was carried out on AZ31B magnesium alloy. Under 2×10 6 cycle times, the fatigue strengths of base metal （BM）, butt joint （BJ）, transverse cross joint （TJ）, lateral connection joint （LJ） are 66.72, 39.00, 24.38 and 24.40 MPa, respectively. The crack propagation behavior of the alloy was analyzed by optical microscopy. The AZ31B magnesium alloy base metal has a smooth crack propagation macroscopic path. However, the microscopic path is twisted and some cracks have two forks, and the crack propagation is transgranular. The crack initiates in the weld toe and the crack propagates along the HAZ for the BJ and TJ; for the LJ crack initiates in the fillet weld leg. The fatigue fracture mechanisms were analyzed by SEM. The fatigue fracture surface consists of quasi-cleavage patterns or cleavage step and a brittle fracture occurs. Numerous secondary cracks are observed; some fatigue striations exist in butt joint and its size is about 5 μm.