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.

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.

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.

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.

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.

Value of Charpy impact test results with sub-and half-size specimens extracted from overage steel bridge members

For the maintenance of steel bridges,the mechanical properties of steel used in the bridges must be elucidated.When enough dimensions of specimens cannot be extracted from the actual members,miniaturized specimens are used for evaluation.In the case of the Charpy impact test,sub-and half-size specimens are specified instead of full-size specimens of which the thickness is 10 mm.The value of absorbed energy and energy transient temperature obtained by Charpy impact tests with sub-size and half-size specimens were investigated from the viewpoint of maintenance of bridges in this study.The absorbed energy was not in proportion to the thickness of specimens of steel used in the actual overage bridges.The tendency of energy transient temperature obtained by thin specimens of the overage steel differed from that of the present steel.A method for evaluating the performance against brittle fracture occurrence based on the WES3003 criterion was examined.The results show the significance of evaluation based on the energy transient temperature for reasonable maintenance of bridges.

Weldability of steel used in an aged bridge

A series of experiments was performed to investigate the weldability of steel used in an aged bridge.A steel material used in an aged railway bridge constructed in 1912 was extracted for this investigation.The chemical compositions of this steel were suitable for welding.However,the aged steel contained much sulfur.Cruciform welded joints were fabricated with this aged steel.Welding defects or cracks were not observed in the joints.The Vickers hardness test on the welded part did not confirm extreme hardening or softening.After yielding by the static tensile test,the cruciform joints were fractured at the welded parts.One of the specimens was fractured in the middle of the thickness of the aged steel.The Sulfur contained in the aged steel might cause this type of fracture.The results show that there may be a risk of brittle fracture not only from the welded part but also from the base metal.The chemical compositions of aged steel must be analyzed when repair welding is applied to the steel.

Mechanical properties and fracture toughness of rail steels and thermite welds at low temperature

Brittle fracture occurs frequently in rails and thermite welded joints, which intimidates the security and reliability of railway ser- vice. Railways in cold regions, such as Qinghai-Tibet Railway, make the problem of brittle fi＇acture in rails even worse. A series of tests such as uniaxial tensile tests, Charpy impact tests, and three-point bending tests were carried out at low temperature to investigate the mechanical properties and fracture toughness of U71Mn and U75V rail steels and their thermite welds. Fracture micromechanisms were analyzed by scanning electron microscopy （SEM） on the fracture surfaces of the tested specimens. The ductility indices （percentage elongation aider frac- ture and percentage reduction of area） and the toughness indices （Charpy impact energy Ak and plane-strain fracture toughness Kic） of the two kinds of rail steels and the corresponding thermite welds all decrease as the temperature decreases. The thermite welds are more critical to fracture than the rail steel base metals, as indicated by a higher yield-to-ultimate ratio and a much lower Charpy impact energy. U71Mn rail steel is relatively higher in toughness than U75V, as demonstrated by larger Ak and Klc values. Therefore, U71Mn rail steel and the corresponding thermite weld are recommended in railway construction and maintenance in cold regions.

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.

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.

Study on fracture behavior of SiCp/A356 composites

The fracture behavior of SiCp/A356 composite at room and high temperatures was studied. Under tensile stress condition at room temperature, the fracture is mostly a combination of the brittle fracture of SiC particles and ductile fracture of A356 matrix. As the tensile temperature increases, the composite changes the main fracture behavior to the separation fracture of the bonding surface between SiC particles and A356 matrix. When the tensile temperature reaches 573 K, the fracture behavior of the composites is almost the whole separation fracture of the bonding surface, which is the main strengthening mechanism at high temperature. Under the cycle stress condition at room and high temperatures, the main fracture behavior of the composites is always a combination of the brittle fracture of SiC particles and ductile fracture of A356 matrix. However, under the cycle stress at high temperature, cycle behavior of the composites changes from cycle hardening at room temperature to the cycle softening at high temperature.

Fracture Characterization of High-Density Polyethylene Materials Using the Energetic Criterias

Impact behavior of polymers has received considerable attention in recent years,and much work based on fracture mechanic approaches has been carried out.In this paper,fracture behavior in large deformation of a high density polyethylene(HDPE)materials was investigated through experimental impact testing on single edge notched specimen(SENB)and by using theoretical and analytical fracture criteria concepts.Moreover,a review of the main fracture criteria is given in order to characterize the toughness of this polymer in the both cases(static and dynamic).The fractured specimens obtained from the Charpy impact test were characterized with respect to their fracture surfaces.Characteristic zones of the fracture surface can be assigned to different stages and mechanisms of the fracture process.Finally,for a better understanding of fracture and damage mechanisms and to provide the best estimation of fracture toughness in impact,an experimental approach based on microscopic observations(SEM)was used.

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.

CATASTROPHE FRACTURE OF THIN-WALL PRESSURE TUBES

Catastrophe theory was used to investigate the fracture behavior of thin-wall cylindrical tubes subjected to internal explosive pressure. Based on the energy theory and catastrophe theory, a cusp catastrophe model for the fracture vas established, and a critical condition associated with the model is given.

Energy consumption for creep fracture of metallic materials

The energy conservation law is applied to formulate the ductile and brittle creep fracture criterion for metallic materials. The criterion contains a summary of heat and latent energies. Assuming that the heat energy is given out so it has no effect on the fracture process, the ductile creep fracture criterion is simplified. To take into account the evaluation of the damage state of materials the compressibility condition is introduced and the brittle creep fracture law is formulated.

Investigation on the Tensile Properties and Fracture Characteristics of Amorphous Ribbons Fe81.50B1.40Si7.95Nb7.37Cu1.73P0.05

A number of Fe-Si-B amorphous ribbons are made by using melt spinning method. The microstructure, mechanical and other relevant properties of thin amorphous ribbons of Fe81.50B1.40Si7.95Nb7.37Cu1.73P0.05 alloy at room temperature were studied with several equipment including Differential scanning calorimetry (DSC), X-ray diffraction (XRD),Scanning electron microscope (SEM), and tensile machine. Significantly different microstructures exist between the free and wheel face of the thin amorphous ribbons. The free face is smooth while the wheel face is coarse with a great number of micro voids on the surface. Experimental results show that the tensile strength and elastic modulus of thethin amorphous ribbons at room temperature are 1951 MPa and 70 GPa. In addition, the amorphous ribbons possess reasonable tensile elongation (2.46%). The fracture appearance of amorphous ribbons of Fe81.50B1.40Si7.95Nb7.37Cu1.73P0.05 alloyis a mixed mode of ductile and brittle fracture which includes dimples and partial cleavage fracture similar to the crystalline materials. The dimple feature proves that it still has plastic characteristics on the micro scale.

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.

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.