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A. Cemal Eringen 《Engineering Fracture Mechanics》1979,12(2):211-219
Field equations of nonlocal elasticity are solved to determine the state of stress in a plate with a line crack subject to a constant anti-plane shear. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. By equating the maximum shear stress that occurs at the crack tip to the shear stress that is necessary to break the atomic bonds, the critical value of the applied shear is obtained for the initiation of fracture. If the concept of the surface tension is used, one is able to calculate the cohesive stress for brittle materials. 相似文献
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G. Block M. B. Rubin J. Morris J. G. Berryman 《International Journal of Fracture》2007,144(3):131-147
Experimental data indicates that the limiting crack speed in brittle materials is less than the Rayleigh wave speed. One reason
for this is that dynamic instabilities produce surface roughness and microcracks that branch from the main crack. These processes
increase dissipation near the crack tip over a range of crack speeds. When the scale of observation (or mesh resolution) becomes
much larger than the typical sizes of these features, effective-medium theories are required to predict the coarse-grained
fracture dynamics. Two approaches to modeling these phenomena are described and used in numerical simulations. The first approach
is based on cohesive elements that utilize a rate-dependent weakening law for the nodal cohesive forces. The second approach
uses a continuum damage model which has a weakening effect that lowers the effective Rayleigh wave speed in the material surrounding
the crack tip. Simulations in this paper show that while both models are capable of increasing the energy dissipated during
fracture when the mesh size is larger than the process zone size, only the continuum damage model is able to limit the crack
speed over a range of applied loads. Numerical simulations of straight-running cracks demonstrate good agreement between the
theoretical predictions of the combined models and experimental data on dynamic crack propagation in brittle materials. Simulations
that model crack branching are also presented. 相似文献
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In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory. 相似文献
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The paper presents a computational approach and numerical data which facilitate the use of the smeared-tip method for cohesive fracture in large enough structures. In the recently developed K-version of the smeared tip method, the large-size asymptotic profile of the stress intensity factor density along a cohesive crack is considered as a material characteristic, which is uniquely related to the softening stress-displacement law of the cohesive crack. After reviewing the K-version, an accurate and efficient numerical algorithm for the computation of this asymptotic profile is presented. The algorithm is based on solving a singular Abel's integral equation. The profiles corresponding to various typical softening stress-displacement laws of the cohesive crack model are computed, tabulated and plotted. The profiles for a certain range of other typical softening laws can be approximately obtained by interpolation from the tables. Knowing the profile, one can obtain with the smeared-tip method an analytical expression for the large-size solution to fracture problems, including the first two asymptotic terms of the size effect law. Consequently, numerical solutions of the integral equations of the cohesive crack model as well as finite element simulations of the cohesive crack are made superfluous. However, when the fracture process zone is attached to a notch or to the body surface and the cohesive zone ends with a stress jump, the solution is expected to be accurate only for large-enough structures. 相似文献
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A mesoscopic fracture model of equilibrium slit cracks in brittle solids, including inter-surface cohesive tractions acting near the crack tip, is analyzed and the effects of the cohesive tractions on the in-plane stress fields, crack-opening displacement profiles, and crack driving forces examined quantitatively for linear and penny-shaped cracks. The (numerical) analysis method is described in detail, along with results for four different cohesive forces. The assumed distribution of cohesive tractions were found to suppress the in-plane stress field adjacent to cracks in a homogeneous, isotropic medium when uniformly loaded in mode-I, and the suppression was a function of crack length. The crack-opening displacement profile was also perturbed and a new regime identified between the near-field Barenblatt zone and the far-field continuum zone. The extent of this `cohesive zone' was quantified by use of an interpolating function fit to the calculated profiles and found to be independent of crack size for a given cohesive tractions distribution. Furthermore, the crack-opening displacement at the edge of the cohesive zone was found to be independent of crack size, implying that despite significant perturbations to the stress field, the crack driving force at unstable equilibrium remains unchanged with crack size. 相似文献
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Jun Liang Zhen Gong Zhou 《International Journal of Mechanics and Materials in Design》2005,2(3-4):53-60
In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. 相似文献
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A new approach to simulating fracture, in which toughness is partitioned between the crack tip and, optionally, a process
zone, is applied to dynamic fracture processes. In this approach, classical fracture mechanics determines crack tip propagation,
and cohesive laws characterize process zone response and determine crack root and process zone propagation. The approach is
implemented in the Material Point Method, a particle method in which the fracture path is unconstrained by a body-fitted mesh.
The approach is found suitable for modeling a range of dynamic fracture processes, from brittle fracture to fracture with
crack bridging. A variety of ways of partitioning toughness are explored with the aim of distinguishing model parameters via
experimental measurements, particularly R curves. While no unique relationship exists, R curves, or effective R curves, on
a suite of materials would provide substantial insight into model parameters. Advantages to the approach are identified, both
in versatility and in regards to practical matters associated with implementing numerical fracture algorithms. It is found
to perform well in dynamic fracture scenarios. 相似文献
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基于各向异性双材料界面断裂力学理论,再根据D-B模型假设的有限裂纹尖端奇异性将消失,推导出复合材料分层裂纹尖端粘聚区长度的计算模型。结果显示复合材料分层裂纹尖端粘聚区具有振荡性(当振荡因子0时),并且粘聚区长度与裂纹长度、应力值及振荡因子有关。将新模型应用于界面单元法中,模拟了双悬臂梁(DCB)和混合型弯曲梁(MBB)分层扩展过程中的载荷-位移关系,并比较了不同的粘聚区长度对收敛性和计算精度的影响,结果表明该模型可较精确地计算复合材料的粘聚区长度,以此为基础划分网格能同时保证收敛性和计算精度要求,并可有效地节省运算时间。 相似文献
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S. Marfia E. Sacco 《International journal for numerical methods in engineering》2003,57(11):1577-1602
The aim of the paper is the development, assessment and use of suitable numerical procedures for the analysis of the crack evolution in cohesive materials. In particular, homogeneous as well as heterogeneous materials, obtained by embedding short stiff fibres in a cohesive matrix, are considered. Two‐dimensional Mode I fracture problems are investigated. The cohesive constitutive law is adopted to model the process zone occurring at the crack tip. An elasto‐plastic constitutive relationship, able to take into account the processes of fibre debonding and pull‐out, is introduced to model the mechanical response of the short fibres. Two numerical procedures, based on the stress and on the energy approach, are developed to investigate the crack propagation in cohesive as well as fibre‐reinforced materials, characterized by a periodic crack distribution. The results obtained using the stress and energy approaches are compared in order to evaluate the effectiveness of the procedures. Investigations on the size effect for microcracked periodic cohesive materials, and on the beneficial effects of the fibres in improving the composite material response, are developed. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
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The classical linear elastic fracture mechanics is not valid near the crack tip because of the unrealistic singular stress at the tip. The study of the physical nature of the deformation around the crack tip reveals the dominance of long-range atomic interactive forces. Unlike the classical theory which incorporates only short range forces, a higher-order continuum theory which could predict the effect of long range interactions at a macro scale would be appropriate to understand the deformation around the crack tip. A simplified theory of gradient elasticity proposed by Aifantis is one such grade-2 theory. This theory is used in the present work to numerically analyze plane cracks in strain-gradient elastic materials. Towards this end, a 36 DOF C1 finite element is used to discretize the displacement field. The results show that the crack tip singularity still persists but with a different nature which is physically more reasonable. A smooth closure of the structure of the crack tip is also achieved. 相似文献
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The feature size of micro-electronic, optoelectronic and biomedical devices is in the sub-micron scale and is pushing toward the nanometer scale. Defects in these small structures are correspondingly smaller such that small crack behavior is becoming an important design consideration for reliability and performance of such devices. Analyses of small cracks are complicated by the rapid variation in the deformation in the crack tip zone. Strain gradients in the near tip zone, which can be ignored in large cracks, will influence the small crack behavior when the crack tip zone is in the order of the crack size. In this paper, we consider two-dimensional crack deformations with strain gradient effect and establish the dual formulations in terms of potentials. The formulations reduce to the conventional linear elastic fracture theory, when the material length scale parameters for the higher order deformation measures are zero. One of the formulations is in terms of two complex stress functions and two pseudo potentials. The complex stress functions are harmonic and the governing equations for the pseudo-potentials are two uncoupled second order partial differential equations. The solutions for these equations are coupled through the boundary conditions.A perturbation method is used to construct the solution for mode I cracks under a K-field when only the effect of the rotational gradient is included. The perturbation solution has induced singularity for the stresses. The induced deformation decays exponentially away from the crack tip. The induced stresses become insignificant beyond 3ε, the typical characteristics of a boundary layer type. To a first order approximation, the induced deformation energy normalized by that of the classical solution under constant applied crack opening load is linearly proportional to ε. This implies that the induced energy release rate is linearly proportional to the length scale parameter. The induced energy release rate under a fixed crack opening load is negative indicating that the rotational gradient shields the crack and lowers the total deformation energy release rate for small crack. 相似文献
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Shanhu Li Somnath Ghosh 《International journal for numerical methods in engineering》2006,65(7):1028-1067
This paper introduces an extended Voronoi cell finite‐element model (X‐VCFEM) for modelling cohesive crack propagation in brittle materials with multiple cracks. The cracks are modelled by a cohesive zone model and their incremental directions and growth lengths are determined in terms of the cohesive energy near the crack tip. Extension to VCFEM is achieved through enhancements in stress functions in the assumed stress hybrid formulation. In addition to polynomial terms, the stress functions include branch functions in conjunction with level set methods, and multi‐resolution wavelet functions in the vicinity of crack tips. The wavelet basis functions are adaptively enriched to accurately capture crack‐tip stress concentrations. Conditions and methods of stability are enforced in X‐VCFEM for improved convergence with propagating cracks. Two classes of problems are solved and compared with existing solutions in the literature for validation of the X‐VCFEM algorithms. The first set corresponds to results for static cracks, while in the latter set, the propagation of cohesive cracks are considered. Comparison of X‐VCFEM simulation results with results in literature for several fracture mechanics problems validates the effectiveness of X‐VCFEM. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
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By using a nonlocal field theory, Eringer et al. [6] obtained a finite solution for the stress at the tip of a sharp crack. This solution permitted the development of a nonlocal fracture criterion for crystalline materials that is given in terms of atomic distance and theoretical cohesive strength.
The nonlocal fracture criterion is generalized for application to real materials by the introduction of a characteristic dimension (a measure of the size of the internal structures). Particleboard, a wood-based composite with controllable internal characteristics (particle dimensions and amount of resin), is used to substantiate the nonlocal fracture criterion. 相似文献
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Crack Growth Across a Strength Mismatched Bimaterial Interface 总被引:3,自引:0,他引:3
Crack growth across an interface between materials with different strength is examined by a cohesive zone model. The two materials
have identical elastic properties but different fracture process properties, or different yield stresses, which is modeled
by different cohesive stresses. The fracture criteria is a critical crack opening displacement. Load is represented by a stress
intensity factor defining a remote square root singular stress field. The results show that the ratio between the cohesive
stresses of the two materials primarily determines the behavior of the critical stress intensity factor. When the crack approaches
a material with a higher cohesive stress the crack tip is shielded, but if the crack approaches a material with smaller critical
crack opening displacement the maximum level of shielding is determined by the ratio between the critical crack opening displacements.
When a crack approaches a material with a lower cohesive stress it is exposed to an amplified load.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
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A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone. 相似文献
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In this paper, we attempt to provide a new analytical method to determine the cohesive law in the framework of nonlocal continuum mechanics. Firstly, the equivalence between the cohesive stress and the surface-induced traction (nonlocal surface residual) is established on the basis of the nonlocal stress boundary condition. Then a new cohesive stress law is derived logically from the perspective of rational mechanics, which characterizes the dependence of the cohesive stress on the crack opening displacement (COD) within the cohesive zone. Finally, we apply this new cohesive crack model to two fracture examples with different cohesive zone sizes, and investigate the stress field ahead of the crack tip in detail. The results show that the stress singularity at the crack tip is removed, and the maximum stress occurs within the cohesive zone away from the crack tip. Moreover, the stress in the large-scale cohesive zone drops rapidly to a constant approaching zero, exhibiting a stronger softening behavior. 相似文献
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A nonlinear cohesive stress distribution function is employed by relating the cohesive stress to the cohesive zone size (CZS) and the distance from the crack tip to investigate the elastic-plastic fracture behaviors. A crack-inclusion interaction problem is taken as an example to explore the fracture process in the cohesive zone area. The CZS and crack surface opening displacement are evaluated numerically. It is found that for different cohesive parameter combinations, the normalized CZS and crack surface opening displacements change drastically. By reducing the current model to the famous Dugdale model, the results obtained match well with the existing ones. 相似文献
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A treatment of interfacial cracks in the presence of friction 总被引:2,自引:0,他引:2
Frictional sliding on interface crack surfaces results in weak crack tip stress singularity and zero strain energy release rate. A fracture criterion based on finite extension strain energy release rate, is proposed to capture the intrinsic fracture toughness. The finite extension strain energy release rate is shown to represent the magnitude of the singular stress field. Numerical simulations of a center crack in a bimaterial infinite medium under remote shear as well as fiber pull-out and push-out in composite materials are presented to illustrate the frictional effect in both small and large scale contacts near the crack tip. 相似文献