A plastic flow rule at a yield vertex

A yield vertex having three distinct facets at the uniaxial stress point is considered. An associated plastic flow rule is constructed using a new 3 × 3 coupled hardening matrix. This has the property that the incipient shear modulus is less than the elastic value. The approach is novel and distinct from previous work having that conclusion. In particular, all the relevant incipient moduli governing fully active in-plane loading can be fitted, if desired, to those values which J₂-deformation theory would require. To that extent the proposed incremental theory therefore legitimizes the use of the latter moduli, for example in certain ‘paradoxical’ buckling problems. When the moduli of the incremental theory are so chosen, the domain of stress-rate vectors enforcing loading is a calculable pyramid which contracts from the Mises half-space as the stress increases beyond yield. This domain becomes wider, at a given stress, as the initial smooth curvature of the stress-strain curve is imagined to become sharper.     

Numerical study of the effects of constitutive models on plastic buckling of plate elements

Compared with experiments, the J₂ deformation theory of plasticity is known to predict plastic buckling with better accuracy than the more accepted incremental J₂ flow theory. This paradox is commonly known as the ‘plastic buckling paradox’. In an attempt to analyse this discrepancy, the two mentioned constitutive models were implemented in a non-linear finite element code, along with a third non-associative J₂ flow theory. The latter model incorporates a vertex-type plastic flow rule. Using these three constitutive models, the buckling behaviour of plate outstand elements was investigated. Comparisons between the buckling strengths derived are presented. The non-linear static buckling simulations show that the instability introduced by the alternative flow rule of the non-associative model has substantial influence on the buckling behaviour. The acceptance of only small departures from normality was shown to reduce the predicted ultimate capacity of the plates. Furthermore, for plates with small plate slendernesses it was found that the imperfection sensitivity was significantly reduced when using the non-associative flow rule.     

Effect of yield surface vertex on crack-tip fields in mode III

An asymptotic crack-tip analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic plastic solid, as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the J₂ flow theory of plasticity.     

On the plastic bifurcation and post-bifurcation of axially compressed beams

The paper presents two new results in the domain of the elastoplastic buckling and post-buckling of beams under axial compression. (i) First, the tangent modulus critical load, the buckling mode and the initial slope of the bifurcated branch are given for a Timoshenko beam (with the transverse shear effects). The result is derived from the 3D J₂ flow plastic bifurcation theory with the von Mises yield criterion and a linear isotropic hardening. (ii) Second, use is made of a specific method in order to provide the asymptotic expansion of the post-critical branch for a Euler-Bernoulli beam, exhibiting one new non-linear fractional term. All the analytical results are validated by finite element computations.     

Effect of initial imperfections in geometry on the elastic-plastic stability of a thin annular plate

In this paper,Neale’S generalized variational principle aboutincremental boundarg-value problems is utilized to study theeffect of initial imperfections in geometry on tbe criticalloads of elasticoplastic buckling of thin annular plates.Thecalculations show that.if the effect of initial imperfectionsin geometrg is taken into account in the solutions by J_2 in-cremental theorg.the results are very close to the bifurca-tional buckling loads of the perfect annular Plates accordingto the plastic deformation theorg.     

Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress,the modified couple stress theory(MCST),and the nonlocal elasticity theories using the differential quadrature method(DQM)is presented.Main advantages of the MCST over the classical theory(CT)are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter.Based on the nonlinear von K′arm′an assumption,the governing equations of equilibrium for the micro-classical plate considering midplane displacements are derived based on the minimum principle of potential energy.Using the DQM,the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained.Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature.A parametric study is conducted to show the effects of the aspect ratio,the side-to-thickness ratio,Eringen’s nonlocal parameter,the material length scale parameter,Young’s modulus of the surface layer,the surface residual stress,the polymer matrix coefficients,and various boundary conditions on the dimensionless uniaxial,biaxial,and shear critical buckling loads.The results indicate that the critical buckling loads are strongly sensitive to Eringen’s nonlocal parameter,the material length scale parameter,and the surface residual stress effects,while the effect of Young’s modulus of the surface layer on the critical buckling load is negligible.Also,considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate.The results show that the critical biaxial buckling load increases with an increase in G12/E2and vice versa for E1/E2.It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude.Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios,it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.     

Buckling of a pre-compressed or pre-stretched membrane in shear flow

Membranes enclosing capsules and biological cells undergo periodic compression and stretching due to an imparted hydrodynamic traction as they rotate in a shear flow. Compression may cause transient or permanent buckling manifested by the onset of wrinkled shapes. To study the effect of pre-compression and pre-stretching on the critical conditions for buckling, the response of an elastic circular plate flush mounted on a plane wall and deforming under the action of a uniform tangential load due to an over-passing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated resulting in a fourth-order partial differential equation with position-dependent coefficients parametrized by the Poisson ratio. Solutions are computed by applying Fourier series expansions to derive an infinite system of coupled ordinary differential equations, and then implementing orthogonal collocation. The solution space is illustrated, critical values for buckling are identified, the associated eigenfunctions representing possible modes of deformation are displayed, and the effect of the Poisson ratio is discussed.     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.     

Multiaxial constitutive model accounting for the strength-differential in Inconel 718

The nickel-base alloy Inconel 718 exhibits a strength-differential, that is, a different plastic flow behavior in uniaxial tension and uniaxial compression. A phenomenological viscoplastic model founded on thermodynamics has been extended for material behavior that deviates from classical metal plasticity by including all three stress invariants in the threshold function. The model can predict plastic flow in isotropic materials with or without a flow stress asymmetry as well as with or without pressure dependence. Viscoplastic material parameters have been fit to pure shear, uniaxial tension, and uniaxial compression experimental results at 650°°C. Threshold function material parameters have been fit to the strength-differential. Four classes of threshold functions have been considered and nonproportional loading of hollow tubes, such as shear strain followed by axial strain, has been used to select the most applicable class of threshold function for the multiaxial model as applied to Inconel 718 at 650 °C. These nonproportional load paths containing corners provide a rigorous test of a plasticity model, whether it is time-dependent or not. A J₂J₃ class model, where J₂ and J₃ are the second and third effective deviatoric stress invariants, was found to agree the best with the experimental results.     

Matched asymptotic expansions in nonlinear fracture mechanics—II. Longitudinal shear of an elastic work-hardening plastic specimen

Earlier analysis given by T.M. Edmunds and J.R. Willis (1976) is extended to deal with cracks in elastic work-hardening plastic specimens subjected to longitudinal shear loads. Solutions are expressed in terms of a set of parameters that are determined from linear elastic solutions alone. It is proved, for any specimen geometry and any loading symmetric about the plane of the crack, that a ‘plastic-zone correction’, obtained by solving a linear elastic problem for a crack which is a length r_y longer than the actual crack, provides a two-term asymptotic expansion for the J-integral, if r_y is defined suitably in terms of the linear elastic stress concentration factor and the initial slope of the work-hardening curve. The general method is applied in detail to a strip of finite width containing an edge crack, for which the effect of the work-hardening on the maximum extent of the plastic zone and on the J-integral is summarized graphically.     

Role of elasticity in elastic–plastic fracture: Analytical bi-linear crack-tip fields and finite element analysis

A solution for Model-I plane strain crack tip fields in a bi-linear elastic–plastic material is presented. The elastic–plastic Poisson's ratio is introduced to characterize the influence of elastic deformation on the near tip constraint. Attention is focused on the distribution of elastic/plastic strain energy in the sensitive region of the forward sector ahead of a crack tip. The present study shows that the elastic strain energy can be higher than the plastic strain energy in this sensitive sector while large amount of the plastic strain energy develops outside this sector around the crack tip. The effect of elastic deformation in this sensitive region on the structure of crack-tip fields is considerable and the assumption in some important solutions for crack-tip fields reported in literature that the elastic deformation is small and can be ignored is therefore not physically reasonable. Besides, finite element analysis is carried out to validate the analytical solution and good agreement between them is found. It is seen that the present solution with T-stress can properly describe the crack-tip fields under various constraints for different specimens and an analytical relation is established between the critical value of J-integral, J_c, and T-stress for elastic–plastic fracture.     

Plastic buckling of cylindrical shells under biaxial loading

The predictions for plastic buckling of shells are significantly affected by the plasticity model employed, in particular in the case of nonproportional loading. A series of experiments on plastic buckling of cylindrical aluminum alloy shells under biaxial loading (external pressure and axial tension), with well-defined loading and boundary conditions, was therefore carried out to provide experimental data for evaluation of the suitability of different, plasticity models. In the experiments, initial imperfections and their growth under load were measured and special attention was paid to buckling detection and load path control. The Southwell plot was applied with success to smooth the results. The results show that axial tension decreases resistance to buckling under external pressure in the plastic region due to softening of the material behavior. Comparison with numerical calculations usingJ ₂ deformation and incremental theories indicate that both theories do not predict correctly plastic buckling under nonproportional loading.Babcock (SEM Member), deceased, was Professor of Aeronautics and Applied Mechanics, California Institute of Technology, Pasadena, CA 91125.     

Bifurcation buckling of shells of revolution including large deflections,plasticity and creep

A summary is first presented of the conceptual difficulties and paradoxes surrounding plastic bifurcation buckling analysis. Briefly discussed are nonconservativeness, loading rate during buckling, and the discrepancy of buckling predictions with use of J₂ flow theory vs J₂ deformation theory. The axisymmetric prebuckling analysis, including large deflections, elastic-plastic material behavior and creep is summarized. Details are given on the analysis of nonsymmetric bifurcation from the deformed axisymmetric state. Both J₂ flow theory and J₂ deformation theory are described. The treatment, based on the finite-difference energy method, applies to layered segmented and branched shells of arbitrary meridional shape composed of a number of different elastic-plastic materials. Numerical results generated with a computer program based on the analysis are presented for an externally pressurized cylinder with conical heads.     

A critical state sand model with elastic–plastic coupling

Soil elastic moduli are highly pressure-dependent. Experimental findings have indicated that the elastic shear modulus of sands depends on p^χ, where p is mean principal effective stress and χ is a non-dimensional parameter. χ practically remains unchanged for shear strains less than 10⁻⁵ where the mechanical behavior is purely elastic. However, experiments have revealed that the emergence of plasticity for shear strains larger than 10⁻⁵ provokes a gradual increase in χ. Technically, this observation is an elastic–plastic coupling effect in which plasticity causes to change the elastic characteristics. Here, this issue is considered in hyper-elasticity framework in conjunction with a critical state compatible bounding surface plasticity platform for granular soils. To this aim, constitutive equations linking χ to a proper kinematic hardening parameter are presented. Then, using the proposed approach, a hyper-elastic theory is modified to consider the mentioned elastic–plastic coupling effect in the whole domain of the elastoplastic behavior. Adopting the improved hyper-elasticity necessitates the modification of a number of basic plasticity platform elements. In this regard, dilatancy and plastic hardening modulus of the bounding surface platform are modified. Successful performance of the modified constitutive model is presented against experimental data of loading/unloading triaxial tests.     

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

Thermoelastic buckling of steel columns with load-dependent supports

The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient α or the stiffening rate β of thermal restraints will increase the critical temperature.     

Constitutive modelling of cyclic plasticity of metal particulate-reinforced brittle matrix composites under compression-compression loading

The Mori-Tanaka approach is used to modelling metal particulate-reinforced brittle matrix composites under cyclic compressive loading. The J₂-flow theory is considered as the relevant physical law of plastic flow in inclusions. Ratchetting of the composite is prevented by the strong constraint exerted by the matrix on the inclusions, even under the assumption of evanescent kinematic hardening. However, the weakening constraint power of the matrix caused by microfracture damage around inclusions is closely coupled with the plasticity of inclusion and leads to ratchetting even when the plastic deformation of inclusions is described by an isotropic hardening rule. A detailed parametric study has revealed that ratchetting is followed by either plastic or elastic shakedown, depending on the load amplitude, composite parameters and the mean length of microcracks.     

弹塑性板壳结构非线性有限元分析

本文根据塑性流动理论的基本公式，由隐式积分导出了与路径无关的变量更新算法和一致切线模量。采用单元广义应力应变直接离散塑性流动定律，构造了杂交应力单元一致切线刚度矩阵的显式表达式，编制了结构有限元程序ＳＡＦＥ，数值算例表明：本文的计算方法和计算程序是正确可靠的，可用于弹塑性板壳结构的非线性分析，计算结果屈曲临界载荷和极限承载能力。     

Strain gradient plasticity analysis of elasto-plastic contact between rough surfaces

From a microscopic point of view, the real contact area between two rough surfaces is the sum of the areas of contact between facing asperities. Since the real contact area is a fraction of the nominal contact area, the real contact pressure is much higher than the nominal contact pressure, which results in plastic deformation of asperities. As plasticity is size dependent at size scales below tens of micrometers, with the general trend of smaller being harder, macroscopic plasticity is not suitable to describe plastic deformation of small asperities and thus fails to capture the real contact area and pressure accurately. Here we adopt conventional mechanism-based strain gradient plasticity (CMSGP) to analyze the contact between a rigid platen and an elasto-plastic solid with a rough surface. Flattening of a single sinusoidal asperity is analyzed first to highlight the difference between CMSGP and J₂ isotropic plasticity. For the rough surface contact, besides CMSGP, pure elastic and J₂ isotropic plasticity analysis is also carried out for comparison. In all cases, the contact area A rises linearly with the applied load, but with a different slope which implies that the mean contact pressures are different. CMSGP produces qualitative changes in the distributions of local contact pressures compared with pure elastic and J₂ isotropic plasticity analysis, furthermore, bounded by the two.