A model for the through-thickness elastic–plastic behaviour of paper

An elastic–plastic material model for the out-of-plane mechanical behaviour of paper is presented. This model enables simulation the elastic–plastic behaviour under high compressive loads in the through-thickness direction (ZD). Paper does not exhibit a sharp transition from elastic to elastic–plastic behaviour. This makes it advantageous to define critical stress states based on failure stresses rather than yield stresses. Moreover, the failure stress in out-of-plane shear is strongly affected by previous plastic through-thickness compression. To cover these two features, a model based on the idea of a bounding surface that grows in size with plastic compression is proposed. Here, both the bounding and the yield surfaces are suggested as parabolas in stress space. While the bounding surface is open for compressive loads, the yield surface is bordered by the maximum applied through-thickness compression.     

Elastic–plastic FE analysis of a notched shaft under multiaxial nonproportional synchronous cyclic loading

An elastic–plastic finite element analysis is presented for a notched shaft subjected to multiaxial nonproportional synchronous cyclic tension/torsion loading. The elastic–plastic material property is described by the von Mises yield criterion and the kinematic hardening rule of Prager/Ziegler. The finite element program system ABAQUS is used to solve the boundary value problem. Special emphasis is given to explore the effects of the stress amplitude, the mean-stress, and the mutual interactions on the local stress–strain responses at the notch root.     

Near crack line elastic–plastic analysis for a infinite plate loaded by two pairs of point tensile forces

The near crack line analysis method is used to investigate a center crack loaded by two pairs of point tensile forces in an infinite plate in an elastic–perfectly plastic solid, and the analytical solutions are obtained in this paper. These solutions include: the elastic–plastic stress field near the crack line, the law that the length of the plastic zone along the crack line is varied with an external loads and the bearing capacity of an infinite plate with a center crack. The results of this paper are sufficiently precise near the crack line because the assumptions of the small scale yielding theory have not been used and no other assumptions have been taken.     

Deformation, stress state and thermodynamic force for a growing void in an elastic–plastic material

The growth of a spherical void in an elastic–plastic body, subjected to external pressure or tension and a gas pressure as well as a surface stress at the void surface, is investigated. The deformation, strain and stress state in the full body is presented. In addition, the local and global energy terms are calculated. Finally the total thermodynamic force on the void surface as well as the total dissipation are evaluated and compared allowing the calculation of the mechanical contribution to void growth due to diffusion of vacancies generated by plastification or irradiation.     

Plasticity aspects of interfacial fracture for polymer/glass specimens

Experimental results suggest that the interfacial fracture resistance is minimal for approximate near tip Mode I accompanied by positive and negative near tip Mode II. Finite-strain FE analysis is made for an elastic–plastic medium bonded to an ideally elastic medium with an interface crack. Small-scale plasticity conditions are invoked and examined in relation to the elastic–plastic stress distribution along the bond line. Plasticity engenders a tendency to turn near tip biaxiality towards pure Mode I regardless of the mixed-mode loading. High levels of hydrostatic stress are attained. For different mode mixities of the applied load, the dependence of the elastic–plastic normal bond stress on load level is examined. It is found that under positive Mode II loading, the normal bond stress σ_yy tends to saturate as the load level rises. This does not occur for Mode I and negative Mode II loading. In addition, deformation patterns inside the plastic zone are examined for mixed-mode situations. A displacement criterion based on the normal bond crack opening suggests a dependence of the critical load level on the extent of mixed mode. Under positive mode II fracture, traces of the ductile material are found at the top of the elastic substrate. Some of these conclusions appear to be consistent with the fracture patterns observed for LD-polyethylene/glass interfacial mixed-mode fracture.     

Strip yield zones around ring-shaped crack in transversely isotropic thick layer

A ring-shaped crack under uniform load in an infinitely long elastic–perfectly plastic thick layer is considered. The problem is formulated for a transversely isotropic material by using integral transform technique. Due to the geometry of the configuration, Hankel integral transform technique was chosen and the problem was reduced to a singular integral equation which is solved numerically by using Gaussian Quadrature Formulae and the values were evaluated at discrete points. The plastic zone widths were obtained by using the plastic strip model after stress intensity factors were obtained. Numerical results are plotted for various ring-shaped crack sizes and transversely isotropic materials. It was found that the width of the plastic zone at the inner edge of the crack was greater than the outer one.     

Elastoplastic analysis of nonlinearly hardening variable thickness annular disks under external pressure

Based on von Mises’ yield criterion, deformation theory of plasticity and Swift’s hardening law, elasto-plastic deformation of variable thickness annular disks subjected to external pressure is studied. A nonlinear shooting method using Newton’s iterations with numerically approximated tangent is designed for the solution of the problem. Considering a thickness profile in the form of a general parabolic function, the condition of occurrence of plastic deformation at the inner and outer edges of the annular disk is investigated. A critical disk profile is determined and the corresponding elastic–plastic stresses as well as the residual stress distribution upon removal of the applied pressure are computed and discussed.     

Elastic–plastic stresses in linearly hardening rotating solid disks of variable thickness

The distribution of stress, displacement and plastic strain in a rotating elastic–plastic solid disk of variable thickness in a power function form is investigated. The analysis is based on Tresca's yield condition, its associated flow rule and linear strain hardening material behavior. An analytical solution is obtained and numerical results are presented for different values of the geometric parameters. The validity of the solution is demonstrated by comparing the results with those for a uniform thickness disk available in the literature.     

Elastic–plastic and elastic anti-plane shear of two parallel cracks

Two infinite interacting parallel cracks in an elastic–plastic and in an elastic body under anti-plane strain (mode III) loading conditions are considered. The body is subjected to vanishing remote loading and the cracks are traction free. Closed-form solution is found for the elastic–plastic problem in terms of elementary functions, where the shape of the plastic boundary is obtained. The complete stress distribution is obtained in an inverse form i.e. physical coordinates are functions of stresses.     

Elastic and elastic–plastic singular fields around a crack-tip in particulate-reinforced composites with progressive debonding damage

This paper deals with elastic and elastic–plastic singular fields around a crack-tip in particulate-reinforced composites with debonding damage of particle-matrix interface. Numerical analyses are carried out on a crack-tip field in elastic-matrix and elastic–plastic-matrix composites reinforced with elastic particles, using a finite element method developed based on an incremental damage theory of particulate-reinforced composites. A particle volume fraction and interfacial strength between particles and matrix of the composites are parametrically changed. In the elastic-matrix composites, a unique elastic singular field is created on the complete damage zone in the vicinity of a crack-tip in addition to the conventional elastic singular field on the no damage zone. The macroscopic stress level around a crack-tip is reduced by the debonding damage while the microscopic stress level of the matrix remains unchanged. In the elastic–plastic-matrix composites, the damage zone develops in addition to the plastic zone due to matrix plasticity, and both the macroscopic and microscopic stress revels around a crack-tip are reduced by the debonding damage. It is concluded from the numerical results that the toughening due to damage could be expected in the elastic–plastic-matrix composites, while it is questionable in the elastic-matrix composites.     

Limit Analysis of Structures with Stochastic Strengths by a Static Approach

A procedure for the evaluation of the conditional probability of collapse of elastic–plastic structures with stochastic strengths is presented. The procedure represents an extension to the stochastic framework of the static approach, coupled with the method of redundant unknowns, well established in the classical deterministic limit analysis. In this paper, a kinematic approach for probabilistic limit analysis, provided in the literature, is also recalled in order to show the duality with the proposed static approach.A beam of elastic perfectly plastic material with correlated stochastic strengths is studied and the influence of the degree of correlation between cross-sections is evidenced. When a low correlation is encountered an accurate discretisation is necessary. Furthermore, investigation of the influence of different collapse events on the conditional probability of collapse is conducted for a frame structure.     

Approximate elastic–plastic analysis of a notched bar under tensile and torsional fatigue

An approximate model is presented for estimating elastic–plastic stresses and strains in a notched bar subjected to synchronous non-proportional tensile and torsional cyclic loading. To begin with, it is applied to a multiaxial synchronous proportional loading system. A detailed FE analysis is performed for an axisymmetric bar of circular cross-section with a circumferential notch. The accuracy of the proposed approximate model is first established. The elastic–plastic material property is described by the von Mises yield criterion and a linear kinematic hardening rule. The comparison of the FE results with those estimated by the proposed approximate model shows good agreement for all loading cases investigated.     

Elastoplastic deformations of rotating parabolic solid disks using Tresca''s yield criterion

Analytical solutions for the stress distribution in rotating parabolic solid disks are obtained. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening. It is shown that, the deformation behavior of the convex parabolic disk is similar to that of the uniform thickness disk, but in the case of concave parabolic solid disk, it is different. In the latter, the plastic core consists of three different plastic regions with different mathematical forms of the yield criteria. Accordingly, three different stages of elastic–plastic deformation occur. All these stages of elastic–plastic deformation are studied in detail. It is also shown mathematically that in the limiting case the parabolic disk solution reduces to the solution of rotating uniform thickness solid disk.     

Perfectly plastic stress field at a stationary crack tip

Under the hypothesis that all the perfectly plastic stress components at a orach tip are the functions of θ only, making use of yield conditions and equilibrium equations. we derive the generally analytical expressions of the perfectly plastic stress field at a crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the tips of Mode Ⅰ Mode Ⅱ, Mode Ⅲ and Mixed Mode Ⅰ-Ⅱ cracks are obtained.     

Crack tip singular fields in ductile crystals with taylor power-law hardening. I: Anti-plane shear

Asymptotic singular solutions of the HRR type are presented for anti-plane shear cracks in ductile crystals. These are assumed to undergo Taylor hardening with a power-law relation between stress and strain at sufficiently large strain. Results are given for several crack orientations in fcc and bcc crystals. The neartip region divides into angular sectors which are the maps of successive flat segments and vertices on the yield locus. Analysis is simplified by use of new general integrals of crack tip singular fields of the HRR type. It is conjectured that the single crystal HRR fields are dominant only over part of the plastic region immediately adjacent to the crack tip, even at small scale yielding, and that their domain of validity vanishes as the perfectly plastic limit is approached. This follows from the fact that while in the perfectly plastic limit the HRR stress states approach the correct discontinuous distributions of the complete elasticideally plastic solutions for crystals (Rice and Nikolic, J. Mech. Phys. Solids33, 595 (1985)), the HRR displacement fields in that limit remain continuous. Instead, the complete elastic-ideally plastic solutions have discontinuous displacements along planar plastic regions emanating from the tip in otherwise elastically stressed material. The approach of the HRR stress fields to their discontinuous limiting distributions is illustrated in graphical plots of results. A case examined here of a fcc crystal with a crack along a slip plane is shown to lead to a discontinuous near-tip stress state even in the hardening regime.Through another limiting process, the asymptotic solution for the near-tip field for an isotropic material is also derived from the present single crystal framework.     

Meshless analysis of plasticity with application to crack growth problems

The governing equations for classical rate-independent plasticity are formulated in the framework of meshless method. The special J₂ flow theory for three-dimensional, two-dimensional plane strain and plane stress problems are presented. The numerical procedures, including return mapping algorithm, to obtain the solutions of boundary-value problems in computational plasticity are outlined. For meshless analysis the special treatment of the presence of barriers and mirror symmetries is formulated. The crack growth process in elastic–plastic solid under plane strain and plane stress conditions is analyzed. Numerical results are presented and discussed.     

On the Torsion of a Class of Nonlinear Fiber Composite Cylinders

This paper presents an analysis of the torsion of a solid or annular circular cylinder consisting of nonlinear material in the form of an elastic matrix with embedded unidirectional elastic fibers parallel to the cylinder axis. The specific class of composite considered is one for which nonlinear fiber-matrix interface slip is captured by uniform cohesive zones of vanishing thickness. Previous work on the effective antiplane shear response of this material leads to a stress–strain relation depending on the interface slip together with an integral equation governing its evolution. Here, we obtain an approximate single mode solution to the integral equation and utilize it to solve the torsion problem. Equations governing the radial distributions of shear stress and interface slip are obtained and formulae for torque–twist rate are presented. The existence of singular surfaces, i.e., surfaces across which the slip and the shear stress experience jump discontinuities are analyzed in detail. Specific results are presented for an interface force law that allows for interface failure in shear.     

Interaction of microcracks with a macrocrack yielded in a narrow strip

Plastic zone growth of collinear cracks has had a longstanding interest in ductile fracture. This work further considers yield zone growth in an isotropic, homogeneous elastic–perfectly plastic infinite plate containing a macrocrack with several neighboring microcracks. Normal loading is considered at distances far away from the cracks. The strip yield is adopted where the plastic zone is assumed to be confined to two narrow strips extending from the ends of a finite length crack while the microcracks are assumed to be elastic. The plastic zone length and crack opening displacement are found from asymptotic solution and compared with finite element solution.     

A finite thickness band method for ductile fracture analysis

We present a finite element method with a finite thickness embedded weak discontinuity to analyze ductile fracture problems. The formulation is restricted to small geometry changes. The material response is characterized by a constitutive relation for a progressively cavitating elastic–plastic solid. As voids nucleate, grow and coalesce, the stiffness of the material degrades. An embedded weak discontinuity is introduced when the condition for loss of ellipticity is met. The resulting localized deformation band is given a specified thickness which introduces a length scale thus providing a regularization of the post-localization response. Also since the constitutive relation for a progressively cavitation solid is used inside the band in the post-localization regime, the traction-opening relation across the band depends on the stress triaxiality. The methodology is illustrated through several example problems including mode I crack growth and localization and failure in notched bars. Various finite element meshes and values of the thickness of the localization band are used in the calculations to illustrate the convergence with mesh refinement and the dependence on the value chosen for the localization band thickness.