Deformation and the strip necking zone in a cracked steel sheet

When a tensile stress is applied to a thin cracked plate, a strip necking region results ahead of a crack tip. The relative opening displacement between the crack surfaces and between the upper and lower boundaries of the strip necking region were measured by the moiré method. The strains ahead of the strip necking region and the thickness reduction (therein) were also measured. The measured relative opening displacements were compared with the calculated values using the Dugdale strip necking model. The thickness reduction in the strip necking region is equal to the relative opening displacement.     

Limit angular velocities of variable thickness rotating disks

Elastic and plastic limit angular velocities are calculated for rotating disks of variable thickness in power function form. Analytical solution is obtained and used to calculate elastic limit angular velocities of variable thickness rotating annular disks and annular disks with rigid inclusion. An efficient numerical solution procedure is designed and used to obtain the elastic limit angular velocities of variable thickness rotating solid disks. Von Mises yield criterion and its flow rule is used to estimate plastic limit angular velocities. Both linear and nonlinear hardening material behaviors are treated numerically. The results are verified by comparing with those of uniform thickness rotating solid disks available in the literature. Elastic and plastic limit angular velocities are found to increase with the reduction of the disk thickness at the edge as well as the reduction in the disk mass due to the shape of the profile.     

The scattering of harmonic elastic anti-plane shear waves by a finite crack in infinitely long strip using the non-local theory

In this paper, the scattering of harmonic anti-plane shear waves by a finite crack in infinitely long strip is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress occurring at the crack tips. Contraty to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at 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 width of the strip and the lattice parameter. Supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province and the National Foundation for Excellent Young Investigators.     

Development of instability in a rotating elastoplastic annular disk

The paper proposes a method, based on perfect-plasticity and perturbation theories, for instability analysis of an annular flat disk tightly set on a shaft with no interference fit. The perturbed elastoplastic state of the rotating disk is analyzed by determining the stress–strain state of a fixed elastic annular plate under in-plane loading. A characteristic equation of the first order for the critical radius of the plastic zone in the disk subject to internal pressure is derived. The critical rotation rate is calculated for different parameters of the disk     

Crack solutions and weight functions for plane problems in three-dimensional quasicrystals

The plane problems of an elliptic hole and a crack in three-dimensional quasicrystals subject to far-field loadings are studied. The generalized Stroh formalism is adopted here, and the explicit solutions for the coupled fields are obtained in the closed form. When the elliptic hole reduces to a crack, the analytical expressions for both the entire fields and the asymptotic fields near the crack tip are determined. The crack theory of quasicrystals, including the determination of the field intensity factors, crack opening displacements, crack tip energy release rates and so on, is a prerequisite. Applying Betti’s theorem of reciprocity, the weight functions for a quasicrystal body with a crack are derived. The weight functions provide a means of calculating the intensity factors for the crack when both phonon and phason point forces are imposed at arbitrary locations.     

Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip

The anti-plane fracture problem for a finite crack in a one-dimensional hexagonal quasicrystal strip is analyzed. By using Fourier transforms, the mixed boundary value problems are reduced to the dual integral equations. The solution of the dual integral equations is then expressed by the complete elliptic integrals of the first and the third kinds. The expressions for stress, strains, displacements and field intensity factors of the phonon and phason fields near the crack tip are obtained exactly. The path-independent integral derived by a conservation law equals the energy release rate, which can be used as a fracture criterion for a mode III fracture problem.     

Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome 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 at the crack tip. By means of 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 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 thickness of the strip, the circular frequency of incident wave and the lattice parameter.     

Numerical analysis of fatigue striations

A computational model was developed to numerically analyse fatigue striations. The inclined strip yield model with continuous distributions of infinitesimal dislocations was utilized to express the crack tip plasticity in this model. The fatigue crack tip blunting process was approximated by sequential activation of two slip lines under loading, and crack closure during unloading was taken into account. The plastic zone at a growing fatigue crack tip at the maximum load was independent of the crack growth up to ten cycles while the reversed plastic zone decreased in a size to one twentieth of that at the maximum load as the crack grew. The ratio of these plastic zone sizes and also the crack tip opening displacement were quite different from the simple prediction by J.R. Rice for a stationary crack. The computed striation spacings were compared with the observed ones and moderate agreement between them obtained.     

Fracture of piezoelectromagnetic materials

The crack problem in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects is considered. A conservative integral is derived based on the governing equations for magnetoelectroelastic media. Closed-form solution is obtained for an anti-plane crack in an infinite medium. The conservative integral is used to obtain the path-independent integral near the crack tip. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of a crack tip are derived. It is found that the path-independent integral around the crack tip equals the energy release rate. In the absence of applied mechanical loads, the energy release rate is always negative.     

Fracture trajectories in a biaxially stressed plate

A specially designed rig has been used to examine the phenomena of crack growth in centrallynotched plates of polymethyl methacrylate under biaxial stress. Attention has been directed particularly towards investigating the phenomenon of path instability, viz. the deviation of a crack from its expected path with increasing transverse stress.Cracks grown under a biaxial stress system, whose transverse component, acting parallel to the crack, exceeds the normal component, do so in an S-shaped curve centred on the original, straight notch. Stress intensity factors for such a crack are estimated by superposition of related known solutions. On the assumption that crack extension takes place with opening displacements only at the tip, that is, the crack tip stress field remains symmetrical, the observed dependence of path geometry on the degree of stress biaxiality is explained.     

含力磁屈服区的压磁材料硬币型裂纹问题研究

本文利用Hankel 变换及Copson 求解方法,得到了在无穷远处磁荷载以及对称力荷载作用下的无限大压磁材料中,硬币型裂纹在裂纹尖端含有环状力磁双屈服区情况下的相关解析解．本文考虑了磁屈服区尺寸小于、大于、等于力屈服区尺寸的三种情况．结果表明：屈服区尺寸不仅与外荷载及材料常数相关,更与力磁屈服区尺寸的相对大小有关;其中,较大尺寸的屈服区仅受同性质单一荷载影响,与材料常数无关;而较小尺寸屈服区受力磁两荷载与材料常数共同影响;裂纹张开位移、磁势跳变均沿裂纹面径向增大而减小;裂尖张开位移、裂尖磁势跳变与材料常数相关,且随外荷载增大而增大．     

On the interaction between a dislocation and a crack in a material with a nonlinear stress-strain law

The problem of the interaction of a crack and a dislocation in a medium with a nonlinear stress-strain law is considered for the case of a semi-infinite crack in a displacement loaded strip under longitudinal shear deformation. A power law stress-strain relation is considered and the dislocation is assumed positioned so that the net effect of its interaction with the crack is to produce a zero stress intensity factor when combined with the effect of the applied displacements. Thus the Atkinson-Kay superdislocation model of a relaxed crack tip is extended to the situation where the material satisfies a power-law stress-strain relationship.     

Determining Prefracture Zones at a Crack Tip Between Two Elastic Orthotropic Bodies

The model of prefracture zones is generalized to a crack between two anisotropic materials. An exact analytical solution is found on the assumptions that the prefracture zone is localized on the crack continuation and that only the normal displacements undergo discontinuity in the zone. From this solution, equations for determining the prefracture length and crack tip opening are derived     

Von mises yield criterion and nonlinearly hardening variable thickness rotating annular disks with rigid inclusion

A computational model is developed to investigate inelastic deformations of variable thickness rotating annular disks mounted on rigid shafts. The von Mises yield condition and its flow rule are combined with Swift’s hardening law to simulate nonlinear hardening material behavior. An efficient numerical solution procedure is designed and used throughout to handle the nonlinearities associated with the von Mises yield condition and the boundary condition at the shaft–annular disk interface. The results of the computations are verified by comparison with an analytical solution employing Tresca’s criterion available in the literature. Inelastic stresses and deformations are calculated for rotating variable thickness disks described by two different commonly used disk profile functions i.e. power and exponential forms. Plastic limit angular velocities for these disks are calculated for different values of the geometric and hardening parameters. These critical angular velocities are found to increase as the edge thickness of the disk reduces. Lower plastic limit angular velocities are obtained for disks made of nonlinearly hardening materials.     

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 weight function for cracks in piezoelectrics

The weight function in fracture mechanics is the stress intensity factor at the tip of a crack in an elastic material due to a point load at an arbitrary location in the body containing the crack. For a piezoelectric material, this definition is extended to include the effect of point charges and the presence of an electric displacement intensity factor at the tip of the crack. Thus, the weight function permits the calculation of the crack tip intensity factors for an arbitrary distribution of applied loads and imposed electric charges. In this paper, the weight function for calculating the stress and electric displacement intensity factors for cracks in piezoelectric materials is formulated from Maxwell relationships among the energy release rate, the physical displacements and the electric potential as dependent variables and the applied loads and electric charges as independent variables. These Maxwell relationships arise as a result of an electric enthalpy for the body that can be formulated in terms of the applied loads and imposed electric charges. An electric enthalpy for a body containing an electrically impermeable crack can then be stated that accounts for the presence of loads and charges for a problem that has been solved previously plus the loads and charges associated with an unsolved problem for which the stress and electric displacement intensity factors are to be found. Differentiation of the electric enthalpy twice with respect to the applied loads (or imposed charges) and with respect to the crack length gives rise to Maxwell relationships for the derivative of the crack tip energy release rate with respect to the applied loads (or imposed charges) of the unsolved problem equal to the derivative of the physical displacements (or the electric potential) of the solved problem with respect to the crack length. The Irwin relationship for the crack tip energy release rate in terms of the crack tip intensity factors then allows the intensity factors for the unsolved problem to be formulated, thereby giving the desired weight function. The results are used to derive the weight function for an electrically impermeable Griffith crack in an infinite piezoelectric body, thereby giving the stress intensity factors and the electric displacement intensity factor due to a point load and a point charge anywhere in an infinite piezoelectric body. The use of the weight function to compute the electric displacement factor for an electrically permeable crack is then presented. Explicit results based on a previous analysis are given for a Griffith crack in an infinite body of PZT-5H poled orthogonally to the crack surfaces.     

Crack tip field in a linear elastic–plastic strain-hardening material

The strip necking model for strain-hardening materials is studied in this paper, in which the stress distributed over the strip necking zone is assumed to be ultimate stress. The bi-linear stress–strain relation which can model certain features of plastic flow is adopted in this model. The stress and strain fields are calculated based on this model in this paper. The size of the strip necking region is determined by balancing the stress intensity factor due to remote loading with that due to assumed closing forces equal to the ultimate tensile strength of the material distributed over the strip necking zone. It is interesting that the strip necking region size and the crack tip opening displacement depend not only on the remote load, but also the material hardening parameters, which is different from the results of strip yield model. The results agree with experiments well, and the model has wider application.     

On the elastic-plastic deformation of a sheet containing an edge crack

The solution of the planar tension and bending of an edge-cracked sheet of elastic-plastic material is given when the plastic deformation is represented by a Dugdale model. The analysis assumes conditions of generalized plane stress (for which the model of plastic relaxation is often a suitable one), but the usual transformation of elastic constants may be used to obtain the results also for plane-strain conditions. The method of solution involves the use of a Mellin transform and a Weiner-Hopf technique. Computed results for the size of the plastic zone and the opening at the crack tip are presented, and asymptotic results are obtained for small-scale and large-scale yielding. The results suggest that, when the material is constrained to fracture close to its ultimate tensile stress, the extra severity of a surface flaw compared with a corresponding internal crack is significantly greater than that predicted by linear elastic fracture mechanics.     

The computation of stress intensity factors in dissimilar materials

A reciprocal work contour integral method for calculating stress intensity factors is extended to treat the problem of two bonded dissimilar materials containing a crack along the bond. The method is based on Betti's Reciprocal work theorem from which the singular stress intensities at the crack tip may be evaluated in terms of an integral involving tractions and displacements on a contour remote from the crack tip.     

Energy release rate and contact zone in a cohesive and an interface crack by hypersingular integral equations

Solution of Cauchy-type singular integral equations permits the evaluation of the fracture parameters at the crack tips very accurately. However, it does not permit the determination of the crack opening and sliding displacements while ensuring no crack surface interpenetration unless the location of the contact zone is known a priori. In order to circumvent this shortcoming, this study presents a solution method based on the Hadamard-type singular integral equations to obtain the crack opening and sliding displacements directly while enforcing the appropriate conditions to prevent interpenetration. Furthermore, the crack opening displacements are physically more meaningful and readily validated against the finite element analysis predictions. The numerical solutions of the hypersingular integral equations provide not only crack opening and sliding displacements directly but also the stress intensity factors and energy release rates. Also, the behavior of the energy release rate is examined as the cohesive crack located parallel to the interface approaches the interface from either the soft or the stiff side of the interface. The limiting value of the energy release rate is established by considering an interface crack. As the cohesive crack approaches the interface from either side of the interface, the energy release rate approaches to that of the interface crack. However, the length of contact zone between the cohesive crack surfaces under uniform shear loading does not approach to that of the interface crack.