Unified solution of a borehole problem with size effect

Based on the strain gradient theory and the unified yield criterion, the borehole problem of an elasto-plastic plane strain body containing a traction-free hole subjected to uniform far-field stress is studied. The unified expressions for the plastic region radius and the stress concentration factor considering the size effect are derived on the basis of the unified yield criterion, respectively, which can be reduced to those based on the Tresca, von Mises and twin-shear criteria. The dependences of the plastic region radius and the stress concentration on the yield criteria, the material strain hardening level and the size effect are discussed. As a result, it is concluded that the influences of yield criteria and strain hardening level on the plastic region radius and the stress concentration are significant, and the size effect cannot be ignored when the hole size is in the order of tens of microns.     

Incremental constitutive equation for large strain elasto plasticity

Starting from the standard theory with intermediate configuration for finite deformations of an isotropic elasto-plastic material with isotropic hardening, rate type constitutive equations are obtained. The small elastic strain approximation is then discussed and it is shown that, in this approximation, these equations reduce to Hill's formalism of large strain elasto-plasticity obtained from the classical Prandtl-Reuss relations of infinitesimal plasticity by substituting for the infinitesimal strain rate, stress and stress rate respectively the rate of deformation tensor, the Cauchy stress tensor and the Jaumann stress rate tensor. The limiting case of perfect plasticity is also investigated.     

Asymptotic homogenization algorithm for reinforced metal-matrix elasto-plastic composites

The theory of the two-scale convergence was applied to homogenization of initial flow stresses and hardening constants in some exponential hardening laws for elasto-plastic composites with a periodic microstructure. The theory is based on the fact that both the elastic and the plastic part of the stress field two-scale converge to a limit, which can be factorized by parts, one of which depends only on the macroscopic, and the other one – only on the microscopic characteristics. The first factor is represented in terms of the homogenized stress tensor and the second factor – in terms of stress concentration tensor, that relates to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to a composite, that consists of the metallic elasto-plastic matrix with Ludwik and Hocket–Sherby hardening law and pure elastic silica inclusions. Results were compared with those of averaging based on the self-consistent methods.     

Constitutive model and finite element formulation for large strain elasto-plastic analysis of shells

This contribution presents a refined constitutive and finite element formulation for arbitrary shell structures undergoing large elasto-plastic deformations. An elasto-plastic material model is developed by using the multiplicative decomposition of the deformation gradient and by considering isotropic as well as kinematic hardening phenomena in general form. A plastic anisotropy induced by kinematic hardening is taken into account by modifying the flow direction. The elastic part of deformations is considered by the neo-Hookean type of a material model able to deal with large strains. For an accurate prediction of complex through-thickness stress distributions a multi-layer shell kinematics is used built on the basis of a six-parametric shell theory capable to deal with large strains as well as finite rotations. To avoid membrane locking in bending dominated cases as well as volume locking caused by material incompressibility in the full plastic range the displacement based finite element formulation is improved by means of the enhanced assumed strain concept. The capability of the algorithms proposed is demonstrated by various numerical examples involving large elasto-plastic strains, finite rotations and complex through-thickness stress distributions.     

Discrete element methods for the plastic analysis of structures subjected to cyclic loading

Methods for the analysis of complex, highly redundant structures subjected to intermittent loads causing biaxial membrane stress and stress reversal into the plastic range are presented. The Bauschinger effect in multi-axial stress is taken into account by the use of Ziegler's modification of Pragers kinematic hardening theory. The implementation of this plasticity theory in the discrete element methods involves the application of the loading in small increments. A linear relationship between increments of plastic strain and of stress, arising out of the theory, is used in conjunction with a linear matrix equation that governs the elastic behaviour of the structure. In the latter equation, plastic strains are interpreted as initial strains. A solution to the linear matrix equation, expressed in terms either of stress or of total strain, may be obtained by utilizing one of two alternative procedures. The methods are capable of treating materials which exhibit elastic–plastic behaviour involving ideal plasticity, linear or non-linear strain hardening, or limited strain hardening. Application is made to several representative structures. Comparison of some of the results with existing test data for both monotonic and reversed loading shows good correlation.     

Approximate local elasto-plastic solution for notched plates undergoing cyclic tensile loading

This work presents a simple methodology to estimate the inelastic stress and strain histories at the root of the notch of a thin elasto-plastic plate undergoing any complex non-monotonic tensile loading. It is proposed an extension of the classical Neuber and Linear projection rules to the case of non-monotonic loading. Using such extended projection techniques, the stress–strain curve at the root of the notch is similar to the curve obtained in a non-monotonic tensile test with prescribed strain. In this test, the limit strain at each reversal depends on the loading history and on the adopted projection technique, what is very important mainly if cyclic hardening (or softening) occurs. The basic idea is to replace this experimental curve by a numerically generated one. A very easy to implement algorithm is proposed to automatically generate the stress–strain curve and to perform the projection. A constitutive theory that accounts for the hardening effects induced by cyclic plasticity is considered. A simple procedure to identify all the material constants that arise in this theory from only one cyclic tensile test with controlled deformation is also presented. Examples concerning cyclic loading in notched aluminium plates are presented to show the main features of the proposed methodology.     

Material parameters for pressure-dependent yielding of unidirectional fiber-reinforced polymeric composites

We study elasto-plastic deformations of unidirectional fiber reinforced polymeric composites (UFPCs) with fibers assumed to deform elastically and the matrix elasto-plastically. The matrix’s and hence composite’s plastic deformations are analyzed by using both the pressure-independent von Mises yield surface and the pressure-dependent Drucker–Prager yield surface and the associated flow rules. In both cases the strain hardening of the matrix is considered and values of material parameters for the matrix are obtained by computing the effective stress versus the effective plastic strain curves from experimental uniaxial stress–strain curves. Values of parameters in the yield surface for the UFPC in terms of those of the matrix and the volume fraction of fibers are found by using a micromechanics approach. Wherever possible, the computed results are compared with the corresponding experimental findings available in the literature. Significant contributions of the work include providing a methodology for determining values of elasto-plastic material parameters for a UFPC from those of its constituents and their volume fractions, and giving expressions in terms of volume fractions of fibers for material parameters appearing in the yield surface of the composite.     

A plasticity model for multiaxial cyclic loading and ratchetting

Summary The nonlinear behavior of metals when subjected to monotonic and cyclic non-proportional loading is modeled using the proposed hardening rule. The model is based on the Chaboche [1], [2] and Voyiadjis and Sivakumar [3], [4] models incorporating the bounding surface concept. The evolution of the backstress is governed by the deviatoric stress rate direction, the plastic strain rate, the backstress, and the proximity of the yield surface from the bounding surface. In order to ensure uniqueness of the solution, nesting of the yield surface with the bounding surface is ensured. The prediction of the model in uniaxial cyclic loading is compared with the experimental results obtained by Chaboche [1], [2]. The behavior of the model in multiaxial stress space is tested by comparing it with the experimental results in axial and torsional loadings performed by Shiratori et al. [5] for different stress trajectories. The amount of hardening of the material is tested for different complex stress paths. The model gives a very satisfactory result under uniaxial, cyclic and biaxial non-proportional loadings. Ratchetting is also illustrated using a non-proportional loading history.     

Numerical study on patterning of shear bands in a Cosserat continuum

Summary Numerical simulation of patterns of shear bands in biaxial compression tests using an elasto-plastic Cosserat constitutive equation is presented. Random distribution of the material properties acts as a trigger for the localized deformation. Two types of stress-strain curves, namely strain softening and strain softening followed by strain hardening, are investigated. It is shown that the characteristic of the stress-strain curve is crucial for the patterning of shear bands. While calculations with the stress-strain curve with solely softening yield only one single shear band, a flock of shear bands can be obtained with the stress-strain curve with softening followed by hardening. Benefited from the characteristic length provided by the Cosserat elasto-plastic constitutive equation, the dependence of the calculation on the mesh-size is avoided.     

Mode I plane stress crack-tip constraint for elastic-plastic materials with different strain hardening

Finite element analyses and simulations have been undertaken to investigate the triaxial constraint in the crack-tip regions of a stationary crack and a steady-state growing crack under mode I plane stress for elastic-plastic materials with different strain hardening. The results show that the triaxial constraint in the crack-tip region is independent of specimen geometry, and material strain hardening, both for a stationary and an extending crack quasi-statically. The triaxial constraints for the various configurations examined are in better accordance with those required by the HRR solution for a stationary crack, which defines the low and similar constraints in crack-tip regions for different material strain hardening in the plane stress case. Along the entire ligament ahead of a crack tip, the constraint level transites gradually from that defined by the HRR solution within the near tip zone to that characterized by the stress intensity factor K _I in the far field.     

Elasto-plastic stress analysis and burst strength evaluation of Al-carbon fiber/epoxy composite cylindrical laminates

This paper concentrates on the elastic–plastic stress analysis and damage evolution of the Al-carbon fiber/epoxy composite cylindrical laminates under internal pressure and thermal residual stress. Firstly, the elastic stress analysis of the composite laminates is performed by using the classical laminate theory. Secondly, the elasto-plastic stress analysis of the liner layer is further conducted by employing the power hardening theory and the Hencky equation in the plastic theory. Finally, an universal solution algorithm based on the last-ply failure criterion is proposed to explore the damage evolution and the burst strength of the composite laminates. Effects of the winding angle and number of the composite layers as well as the thermal residual stress are addressed. The calculated burst strengths are also compared with the experimental results.     

Dynamic elasto-plastic response of beams — A new model

A new approximate model to analyze the dynamic elasto-plastic large deformation response of beams is presented. The model beam is composed of two rigid parts interconnected by a gap of zero width which is built of fibers having an imaginary length. The imaginary length governs the strains and stresses in the beam, for equal deflections in the real and model beams, and is found to be almost constant in the elastic and elasto-plastic domains. The deflected shape is triangular because interest is focused on midspan deflection, while advanced existing models [13] can be used to determine the deflected shape. Comparison of predicted final deflections with test results show very good correspondence. Apart from the final deflection, which may easily be measured in tests, the model also calculates the time dependence of the dynamic reactions, bending moment and membrane force, displacement, velocity and acceleration as well as stress and strain distributions at selected times. Calculation of a complete response time history requires only a few seconds. The proposed model is extendable to various boundary conditions and load distributions and might be generalized to include strain hardening and rate effects.     

On the Determination of the Stress State of a Solid Cylinder in Tension with Torsion

The solution of problems on the determination of the stress state within the framework of flow theory with isotropic hardening has been obtained for a solid cylinder made of a material sensitive to the strain rate under conditions of proportional loading in tension combined with torsion. An analysis of the obtained expressions has been performed. The results of tensile tests with simultaneous torsion under conditions of superplasticity were used to plot a stress-strain diagram for titanium alloy VT9 with the use of the expressions for the stress intensity.     

Kinematic hardening rules for modeling uniaxial and multiaxial ratcheting

Ratcheting, which is the strain accumulation observed under the unsymmetrical stress controlled loading and non-proportional loadings, is modeled using the simplified viscoplasticity theory based on overstress (VBO). The influences of kinematic hardening laws on the uniaxial and multiaxial non-proportional ratcheting behavior of CS 1026 carbon steel have been investigated. The following kinematic hardening rules have been considered: the classical kinematic hardening rule, the kinematic hardening rules introduced by Armstrong–Frederick, Burlet–Cailletaud and the modified Burlet–Cailletaud. The investigated loading conditions include uniaxial stress controlled test with non-zero mean stress, and axial strain controlled cyclic test of thin-walled tubular specimen in the presence of constant pressure. Numerical results are compared with the experimental data obtained by Hassan and Kyriakides [Hassan T, Kyriakides S. Ratcheting in cyclic plasticity, part I: uniaxial behavior. Int J Plast 1992;8:91–116] and Hassan et al. [Hassan T, Corona E, Kyriakides S. Ratcheting in cyclic plasticity, part I: multiaxial behavior. Int J Plast 1992;8:117–146]. It is observed that all investigated kinematic hardening rules do not improve ratcheting behavior under multiaxial loading, but over-prediction still exists.     

Elasto-plastic strain analysis by a semi-analytical method

The aim of this paper is to develop a simulation model of large deformation problems following a semi-analytical method, incorporating the complications of geometric and material non-linearity in the formulation. The solution algorithm is based on the method of energy principle in structural mechanics, as applicable for conservative systems. A one-dimensional solid circular bar problem has been solved in post-elastic range assuming linear elastic, linear strain hardening material behaviour. Type of loading includes uniform uniaxial loading and gravity loading due to body force, whereas the geometry of the bar is considered to be non-uniformly taper. Results are validated successfully with benchmark solution and some new results have also been reported. The location of initiation of elasto-plastic front and its growth are found to be functions of geometry of the bar and loading conditions. Some indicative results have been presented for static and dynamic problems and the solution methodology developed for one-dimension has been extended to the elasto-plastic analysis of two-dimensional strain field problems of a rotating disk.     

Elasto-plastic finite element analysis

An elasto-plastic finite element small displacement procedure to predict the behaviour of structures under cyclic loads is presented. The solution combines incremental procedure with correction for equilibrium and iterative scheme. Applications are made for a rectangular strip and a notched bar under cyclic axial loads. Flow theory of plasticity with either isotropic hardening or kinematic linear hardening is used.     

Annealing model of elasto-plastic solids

The paper develops a model of elasto-plastic solids in which the effect of strain hardening is undergoing relaxation or annealing. A system of constitutive equations of this model is formulated and discussed. It is shown that the existence of annealing implies the existence of creep under constant stress and stress relaxation under constant strain.     

Effect of material plasticity on fatigue crack propagation under complex stress state

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. However, this fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, a modification has been made to G _max criterion to implement the consideration of the plastic strain energy. This criterion is extended to study the fatigue crack growth characteristics of mixed mode cracks in steel pipes. To predict crack propagation due to fatigue loads, a new elasto-plastic energy model is presented. This new model includes the effects of material properties like strain hardening exponent n, yield strength σ_y and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.     

Influences of grain size and precracking load on the critical stress intensity factor of mild steel

The effects of grain size and precracking load on the critical stress intensity factor are studied. A plane stress model of elastic-plastic stress distribution which includes the strain hardening effects is used. The effects of residual stresses and strain hardening due to fatigue load are calculated by choosing plastic zone size as fracture criterion. Experimental results are obtained to demonstrate the reliability of theoretical calculations.