The dependence of crack velocity on the critical stress in fracture

The constant velocity of crack propagation in PMMA is investigated in terms of the fracture stress for both continuously increasing loading (strain rate ?=0.59×10^?4 s^?1) and dynamic loading (strain rate ?=0.35 s^?1). It was found that the constant crack velocity increases with increasing fracture stress and that it depends on the loading conditions (continuously increasing or dynamic loading). In particular, it was found that the increase of the constant velocity for the static loading case is higher than for the dynamic one. However, in both cases, the constant velocity reaches a limiting value for stresses higher than a certain level.     

On the intensity of linear elastic high order singularities ahead of cracks and re-entrant corners

The paper deals with high order elastic singular terms at cracks and re-entrant corners (sharp V-notches), which are commonly omitted in linear elastic analyses by the argument that the strain energy and displacements in the near-tip region should be bounded. The present analysis proves that these terms are fully included in the elastic part of complete elastic–plastic stress and strain solutions.The intensities of high order singular terms are found to be linked to the linear elastic stress intensity factor and the extension of the plastic zone along the crack bisector line. The smaller the plastic radius, the smaller the intensities of high order singular terms are.A physical justification of the existence of high order singular terms is provided on the basis of the strain energy density distribution detected along the crack bisector line. Finally, the influence of the V-notch opening angle is made explicit, discussing also the relationship between the singularity orders and the solution of a Williams’ type sinusoidal eigen-equation.     

Dynamic steady crack growth in elastic-plastic solids—propagation of strong discontinuities

The near-tip field of a mode I crack growing steadily under plane strain conditions is studied. A key issue is whether strong discontinuities can propagate under dynamic conditions. Theories which impose rather restrictive assumptions on the structure of an admissible deformation path through a dynamically propagating discontinuity have been proposed recently. Asymptotic solutions for dynamic crack growth, based on such theories, do not contain any discontinuities. In the present work a broader family of deformation paths is considered and we show that a discontinuity can propagate dynamically without violating any of the mechanical constitutive relations of the material. The proposed theory for the propagation of strong discontinuities is corroborated by very detailed finite element calculations. The latter shows a plane of strong discontinuity emanating from the crack tip (with its normal pointing in the direction of crack advance) and moving with the tip. Elastic unloading ahead of and/or behind the plane of discontinuity and behind the crack tip have also been observed.The numerical investigation is performed within the framework of a boundary layer formulation whereby the remote loading is fully specified by the first two terms in the asymptotic solution of the elasto-dynamic crack tip field, characterized by K₁, and T. It is shown that the family of near-tip fields, associated with a given crack speed, can be arranged into a one-parameter field based on a characteristic length, L_g, which scales with the smallest dimension of the plastic zone. This extends a previous result for quasi-static crack growth.     

Brittle fracture dynamics with arbitrary paths. II. Dynamic crack branching under general antiplane loading

The dynamic propagation of a bifurcated crack under antiplane loading is considered. The dependence of the stress intensity factor just after branching is given as a function of the stress intensity factor just before branching, the branching angle and the instantaneous velocity of the crack tip. The jump in the dynamic energy release rate due to the branching process is also computed. Similar to the single crack case, a growth criterion for a branched crack is applied. It is based on the equality between the energy flux into each propagating tip and the surface energy which is added as a result of this propagation. It is shown that the minimum speed of the initial single crack which allows branching is equal to 0.39c, where c is the shear wave speed. At the branching threshold, the corresponding bifurcated cracks start their propagation at a vanishing speed with a branching angle of approximately 40°.     

Stable crack growth along a brittle\ductile interface—II. Small scale yielding solutions and interfacial toughness predictions

The problem of a crack growing steadily and quasi-statically along a brittle\ductile interface under plane strain, mixed mode, and small scale yielding conditions is considered. The ductile material is assumed to be characterized by the J₂-flow theory of plasticity with linear strain hardening, while the brittle material is assumed to be linear elastic. A displacement-based finite element method, exploiting the convective nature of the problem, is utilized to solve the relevant boundary value problem. In Part I of this work, the corresponding asymptotic problem was solved. This paper addresses the full-field problem in order to validate the asymptotic solutions, and to explore the physical implications of the results. The numerical full-field results are found to be in good agreement with the analytical asymptotic solutions. In particular, the full-field results strongly suggest that the stress fields in the vicinity of the crack tip are variable-separable of the power singular type; and also that the mode mix of the near-tip stress fields is, to a large extent, independent of the applied elastic mode mix. The amplitude (the plastic stress intensity factor) and the regions of validity of the asymptotic fields are estimated from the full-field results, and are observed to be strongly dependent on the applied mode mix. The remote elastic loading fields appear to influence the near-tip fields, primarily, through the plastic stress intensity factor. The present work also explores the suggestion made by Bose and Ponte Castaneda, 1992 that the solutions to the small scale yielding problem may be used in the context of a standard crack growth criterion, requiring that continued growth take place with a fixed near-tip crack opening profile, to obtain theoretical predictions for the dependence of interfacial toughness on the applied mode mix. Based on the numerical results, predictions for mixed mode toughness of the brittle\ductile interface are reported. The results, which are in qualitative agreement with available experimental data and also with some recent theoretical results, predict a strong dependence of interfacial toughness on mode mix. This suggests that ductility provides the main operating mechanism for explaining the dependence of interfacial toughness on the mode mix of the applied loading fields, during steady crack growth.     

Crack-tip fields for fast fracture of an elastic-plastic material

An asymptotic analysis of the near-tip fields is given for transient crack propagation in an elastic-plastic material. The material is characterized by J₂ flow theory together with a bilinear effective stress-strain curve. Both plane stress and plane strain conditions have been considered. Explicit results are given for the order of the crack-tip singularity, the angular position at which unloading occurs, and the angular variation of the near-tip stresses, all as functions of the crack-tip speed and the ratio of the slopes of the two portions of the bilinear stress-strain relation. It was found that the results are much more sensitive to the elastic-plastic constitutive relation than to the crack speed. This result is important for numerical analyses of dynamic crack propagation problems.     

Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material

Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.     

A numerical study of crack tip constraint in ductile single crystals

In this work, the effect of crack tip constraint on near-tip stress and deformation fields in a ductile FCC single crystal is studied under mode I, plane strain conditions. To this end, modified boundary layer simulations within crystal plasticity framework are performed, neglecting elastic anisotropy. The first and second terms of the isotropic elastic crack tip field, which are governed by the stress intensity factor K and T-stress, are prescribed as remote boundary conditions and solutions pertaining to different levels of T-stress are generated. It is found that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the crack tip are also strongly influenced by the level of T-stress, with progressive loss of crack tip constraint occurring as T-stress becomes more negative. A family of near-tip fields is obtained which are characterized by two terms (such as K and T or J and a constraint parameter Q) as in isotropic plastic solids.     

Finite-deformation analysis of the crack-tip fields under cyclic loading

The plane-strain crack subjected to mode I cyclic loading under small scale yielding was analysed. The influence of the load range, load ratio and overload on the near-tip deformation-, stress- and strain-fields was studied. Although the near-tip zones of appreciable cyclic plastic flow for all loading regimes matched closely one another, when scaled with (ΔK/σ_Y)², the activities of plastic flow within them manifested dependence on K_max and K_min, as well as on overload. Cyclic trajectories of the crack-tip opening displacement (CTOD) converged to stable self-similar loops of the sizes proportional to ΔK² and positions in CTOD-K plane dependent on the maximum K along the whole loading route, including an overload. Computed near-tip deformation evidenced plastic crack advance, this way visualising of the Laird–Smith concept of fatigue cracking. This crack growth by blunting-resharpening accelerated with rising ΔK and was halted by an overload. Crack closure upon unloading had no place. The affinities were revealed between computed near-tip stress–strain variables and the experimental trends of the fatigue crack growth rate, such as its dependence on K_max and K_min (or ΔK and K_max), and retardation by overload. Thus, the effects of loading parameters on fatigue cracking, hitherto associated with crack closure, are attributable to the stress–strain fields in front of it as the direct drives of the key fatigue constituents – damage accumulation and bond breaking.     

The surface-forming energy release rate based fracture criterion for elastic–plastic crack propagation

The J-integral based criterion is widely used in elastic–plastic fracture mechanics. However, it is not rigorously applicable when plastic unloading appears during crack propagation. One difficulty is that the energy density with plastic unloading in the J-integral cannot be defined unambiguously. In this paper, we alternatively start from the analysis on the power balance, and propose a surface-forming energy release rate (ERR), which represents the energy available for separating the crack surfaces during the crack propagation and excludes the loading-mode-dependent plastic dissipation. Therefore the surface-forming ERR based fracture criterion has wider applicability, including elastic–plastic crack propagation problems. Several formulae are derived for calculating the surface-forming ERR. From the most concise formula, it is interesting to note that the surface-forming ERR can be computed using only the stress and deformation of the current moment, and the definition of the energy density or work density is avoided. When an infinitesimal contour is chosen, the expression can be further simplified. For any fracture behaviors, the surface-forming ERR is proven to be path-independent, and the path-independence of its constituent term, so-called J_s-integral, is also investigated. The physical meanings and applicability of the proposed surface-forming ERR, traditional ERR, J_s-integral and J-integral are compared and discussed. Besides, we give an interpretation of Rice paradox by comparing the cohesive fracture model and the surface-forming ERR based fracture criterion.     

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

The determination of dynamic fracture toughness of AISI 4340 steel by the shadow spot method

Dynamic crack propagation experiments have been performed using wedge loaded double cantilever beam specimens of an austenitized, quenched and tempered 4340 steel. Measurements of the dynamic stress intensity factor have been made by means of the optical method of caustics. The interpretation of experimental data, obtained from the shadow spot patterns photographed with a Cranz-Schardin high speed camera, is based on an elastodynamic analysis. The instantaneous value of the dynamic stress intensity factor K^d_I is obtained as a function of crack tip velocity. Finally, the interaction of reflected shear and Rayleigh waves with the moving crack tip stress field is considered.     

Crack-tip fields in steady crack-growth with linear strain-hardening

Singular stress and strain fields are found at the tip of a crack growing steadily and quasi-statically into an elastic-plastic strain-hardening material. The material is characterized byJ₂ flow theory together with a bilinear effective stress-strain curve. The cases of anti-plane shear, plane stress and plane strain are each considered. Numerical results are given for the order of the singularity, details of the stress and strain-rate fields, and the near-tip regions of plastic loading and elastic unloading.     

J-integral and crack driving force in elastic–plastic materials

This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.     

Influence of Mass Ratio on Forward and Reverse Ballistic Impact Equivalence: Experiments,Simulations, and Mechanism Analysis

Reverse ballistic impact tests are widely used for studying dynamic responses because they provide more comprehensive and quantitative projectile/rod response results than forward impact tests. To examine equivalent forward and reverse conditions, a series of 8-cm length oxygen-free copper rods with varying length–diameter ratios was used in forward and reverse ballistic Taylor impact experiments with velocities and strain ratios of 104–215 m/s and 1.25?×?10³–2.5?×?10³ s^-1, respectively. Digital image correlation (DIC) and traditional optical measurements were used to determine instantaneous responses at the μs level. Based on DIC, transient structural deformation, and plastic wave propagation, the forward and reverse length difference at similar velocities ranges from 2 to 6.95 %. Rules governing deformation from the perspective of energy, along with rules for changes in energy and plastic wave propagation were determined. The relative deformation energy error was below 5 % for target projectile mass ratios above 20.     

Creep deformation at crack tips in elastic-viscoplastic solids

The evaluation of crack growth tests under creep conditions must be based on the stress analysis of a cracked body taking into account elastic, plastic and creep deformation. In addition to the well-known analysis of a cracked body creeping in secondary (steady-state) creep, the stress field at the tip of a stationary crack is calculated for primary (strain-hardening) or tertiary (strain-softening) creep of the whole specimen. For the special hardening creep-law considered, a path-independent integral C¹_h, can be defined which correlates the near-tip field to the applied load.It is also shown how, after sudden load application, creep strains develop in the initially elastic or, for a higher load level, plastic body. Characteristic times are derived to distinguish between short times when the creep-zones, in which creep strains are concentrated, are still small, and long times when the whole specimen creeps extensively in primary and finally in secondary and tertiary creep. Comparing the creep-zone sizes with the specimen dimensions or comparing the characteristic times with the test duration, one can decide which deformation mechanism prevails in the bulk of the specimen and which load parameter enters into the near-tip stress field and determines crack growth behavior. The governing load parameter is the stress intensity factor K₁ if the bulk of the specimen is predominantly elastic and it is the J-integral in a fully-plastic situation when large creep strains are still confined to a small zone. The C¹_h-integral applies if the bulk of the specimen deforms in primary or tertiary creep, and C¹ is the relevant load parameter for predominantly secondary creep of the whole specimen.     

PLASTIC BEHAVIOR AND CONSTITUTIVE MODELING OF ARMOR STEEL OVER WIDE TEMPERATURE AND STRAIN RATE RANGES

Plastic behavior of 603 armor steel is studied at strain rates ranging from 0.001 s-1 to 4500 s-1 , and temperature from 288 K to 873 K. Emphasis is placed on the effects of temperature, strain rate, and plastic strain on flow stress. Based on experimental results, the JC and the KHL models are used to simulate flow stress of this material. By comparing the model prediction and the experimental results of strain rate jump tests, the KHL model is shown to have a better prediction of plastic behavior under complex loading conditions for this material, especially in the dynamic region.     

Energy release rate and path-independent integral in dynamic fracture of magneto-electro-thermo-elastic solids

The energy release rate and associated energy flux integral in dynamic fracture of magneto-electro-thermo-elastic solids are formulated with the inclusion of multi-field fully coupled effects based on fundamental principles of thermodynamics. The difference between the global and local dynamic contour integrals is caused by unsteady state, mechanical body force, electricity conduction and thermal effect as the closed contour including crack faces is chosen. This formulation successfully captures the crack-tip singularity of coupled fields, offers the right expression for the crack driving force, and resolves the controversial issue on magneto-electro-thermo-elastic fracture criterion. Especially, for steady-state crack propagation in a magneto-electro-elastic solid, the path-independent dynamic contour integral is determined from the asymptotic near-tip field solution based on the Stroh-type formalism and the resulting dynamic energy release rate has an odd dependence on the dynamic magnetic induction intensity factor and the dynamic electric displacement intensity factor.     

冲击载荷下扩展裂纹尖端动态能量释放率分布的焦散线分析

借助高速摄影捕捉裂纹瞬态扩展过程，利用动态焦散线研究了含有裂纹的三点弯曲梁在冲击载荷作用下扩展裂纹尖端的动态能量释放率分布规律；综合分析了裂纹扩展时间、长度、速度，以及扩展裂纹尖端动态应力强度因子与它的变化关系，表明了动态能量释放率在裂纹扩展过程中的驱动作用。     

Dynamic self-similar debonding of interface at very high velocity: Anti-plane case

This paper analyzes the anti-plane problem of dynamic self-similar debonding of interface at very high velocity. The debonding is modeled as an interface crack propagating self-similarly from zero-length. The extending speed is assumed to be transonic or supersonic. We first consider the dynamic debonding under moving concentrated loads. The moving dislocation model of self-similar propagation of an interface crack is used to formulate the problem to a singular integral equation which is solved analytically. The singularity of stresses near the crack tip is discussed and the dynamic stress intensity factors are presented. Finally the solution of dynamic debonding underx ²-type loads is obtained by using the superposition method.