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.     

The torsional buckling of a cruciform column under compressive load with a vertex plasticity model

The torsional buckling of a plastically deforming cruciform column under compressive load is investigated. The problem is solved analytically based on the von Kármán shallow shell theory and the virtual work principle. Solutions found in the literature are extended for path-dependent incremental behaviour as typically found in the presence of the vertex effect that is present in metallic polycrystals.At the critical load for buckling the direction of straining changes by an additional shear component. It is shown that the incremental elastic–plastic moduli are spatially nonuniform for such situations, contrary to the classical J₂ flow and deformation theories. The critical shear modulus that governs the buckling equation is obtained as a weighted average of the incremental elastic–plastic moduli over the cross-section of the cruciform.Using a plasticity model proposed by the authors, that includes the vertex effect, the buckling-critical load is computed for a aluminium column both with the analytical model and a FEM-based eigenvalue buckling analysis. The stable post-buckling path is determined by the energy criterion of path-stability. A comparison with the experimentally obtained classical results by Gerard and Becker (1957) shows good agreement without relying on artificial imperfections as necessary in the classical J₂ flow theory.     

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.     

Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach

The class of generalised standard materials is not relevant to model the non-associative constitutive equations. The possible generalisation of Fenchel's inequality proposed by de Saxcé allows the recovery of flow rule normality for non-associative behaviours. The normality rule is written in the weak form of an implicit relation. This leads to the introduction of the class of implicit standard materials. This formulation is applied to constitutive equations involving non-linear kinematic hardening, indispensable to describe accurately and realistically the cyclic plasticity of metallic materials. For these plastic flow rules shakedown bound theorems can be extended; an analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion is considered and the obtained solution is proved to be exact.     

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.     

Plastic buckling analysis of thick plates using p-Ritz method

This paper is concerned with the application of the p-Ritz method for the plastic buckling analysis of thick plates. In order to allow for the effect of transverse shear deformation in thick plates, the Mindlin plate theory is adopted. The plastic buckling behaviour of the plate is captured by using the incremental and deformation theories of plasticity. The material property of the plate is assumed to obey the Ramberg–Osgood stress–strain relation. The p-Ritz method will be applied to obtain the governing eigenvalue equation for the plastic buckling analysis of uniformly stressed plates with edges defined by polynomial functions. In the p-Ritz method, the displacement functions of the plate are approximately represented by the product of mathematically complete two-dimensional polynomial functions and boundary equations raised to appropriate powers that ensure the satisfaction of the geometric boundary conditions. The validity, convergence and accuracy of the method were demonstrated for various plate shapes such as rectangular, triangular and elliptical shapes. A parametric study was also undertaken to study the plastic buckling behaviour and the effect of transverse shear deformation.     

Buckling and post-buckling of laminated composite plates

This paper addresses the buckling and post-buckling of laminated composite plates using higher order shear deformation theory associated with Green–Lagrange non-linear strain–displacement relationships. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strain conditions at the top and bottom surfaces of the plate in von Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model.     

Roles of plastic spin in shear banding

In this paper, the effects of plastic spin on shear banding and simple shear are examined systematically. Three types of plastic constitutive model with plastic spin are considered: (i) a non-coaxial model in which the direction of the plastic strain rate depends on that of the stress rate; (ii) a strain-softening model based on the J₂ flow theory; and (iii) the pressure-sensitive porous plasticity model. All the constitutive models are formulated in viscoplastic forms and in conjunction with non-local concepts that have been recently focused and discussed. First, behavior in simple shear is examined by numerical analysee with the aforementioned constitutive models. Moreover, some experimental evidences for stress response to simple shear are shown; that is, several large torsion tests of metal tubes and bars are carried out. Next, finite element simulations of shear banding in plane strain tension are performed. A critical effect of plastic spin on shear banding is observed for the noncoaxial model, while an almost negligible effect is observed for the porous model. The identical effects of plastic spin are observed, whether nonlocality exists or not. Finally, we discuss the relationship between the behavior in simple shear and the shear band formation. It is emphasized that this is a critical issue in predicting shear banding in macroscopic grounds.     

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.     

On Generalised and Implicit Normality Hypotheses

The restrictive framework of generalised standard materials can be suitably extended to model non-associative constitutive equations, by exploiting the concept of implicit standard materials, based on the use of a bipotential of dissipation. As presented in this study, it allows one to incorporate, in an easy and elegant way, non-linear kinematic and isotropic hardening in the constitutive equations and to recover useful flow rule normality for these non-associative behaviours. Sommario. Il ristretto ambito dei materiali standard generalizzati è adeguatamente esteso nel lavoro utilizzando il concetto di materiale standard implicito e di bipotenziale di dissipazione. Ciò consente di incorporare nelle equazioni costitutive, in modo semplice ed elegante, leggi non lineari di incrudimento cinematico ed isotropo e di riottenere formalmente, per questi comportamenti di tipo non associato, la condizione di normalità nella legge di flusso.     

Elasto-plastic buckling analysis of laminated plates including interfacial damage

Elasto-plastic buckling of orthotropic laminated plates, which include interfacial damage, is analyzed in detail. Firstly, a novel mixed hardening yield criterion, as an improvement of Hill’s counterpart, is proposed for the orthotropic materials on the basis of the plastic theory. And differing from Hill’s theory, the present yield criterion is related to the spherical tensor of stress. Then, the incremental elasto-plastic constitutive relations of the mixed hardening orthotropic materials are presented. Secondly, the incremental static equilibrium equations for laminated plates including interfacial damage are established based on Von-Karman type theory and the principle of minimum potential energy. Finally, the elasto-plastic buckling of laminated plates are solved by adopting the Galerkin method and iteration scheme. The numerical results show that buckling of the plate occurs easier due to the existence of interfacial damage, and the critical load trends to constant when the interfacial damage approaches a certain degree. Also, the effect of anisotropy on buckling is obvious and the analysis of elasto-plastic buckling is necessary.     

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.     

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.     

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.     

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.     

Optimal design of functionally graded plates with thermo-elastic plastic behaviourConfiguration optimale des plaques fonctionnellement graduées à plasticité thermo-élastique

The thermo-elastic plastic behaviour of functionally graded plates under extremal thermal loading at different boundary conditions is considered. The plates consist of two phases – ZrO₂ ceramics and Ti₆Al₄V alloy. The layers are distributed exponentially through the thickness. The mechanical and thermal properties of both materials strongly depend on temperature. The stress–strain behaviour is investigated by the FEM. To predict the stable state of the structures of interest, several failure criteria are applied. Two cost functions are introduced to optimize the design of the plate. The main results are discussed and graphically illustrated. To cite this article: L. Parashkevova et al., C. R. Mecanique 332 (2004).     

Plastic constraint considerations at the prediction of wide plate failure behaviour from small scale testing, like c.o.d. determinations in s.e.n.b.-specimens

The stress intensity factor, crack opening displacement, J-integral, etc. will depend, besides on load, reduced (physical) cracklength and dimensions also on the plastic constraint situation near the crack tip. Assumedly their critical values are also dependent on this, which can imply an appreciable quantitative consequence for uniaxial gross stresses larger than (2/3) Y, as elaborated. This might interfere when trying to predict large sale behaviour of notched plates from small scale behaviour of specimens with the same thickness, as this presumbly does not guarantee equal plastic constraint. An average plastic constraint factor (p.c.f.) estimate for contained plastic flow can be indicated from simultaneous readings of s.i.f. or c.o.d. and gross stress during loading and its decrease followed. This allows to predict the failure conditions at the end of the experiment, provided the dependence of critical c.o.d. on this p.c.f. was previously determined. For arbitrary notched specimens a priori predictions of failure from the known critical c.o.d./p.c.f.-curve can be envisaged, using analytical and numerical evaluations of p.c.f. development at loading. The transition temperature concept is discussed with respect to the p.c.f. The failure prognosis for wide plate testing from small scale testing is reconsidered for deformation controlled loading: for contained plastic flow an alternative for the design-curve is suggested, using the deformation of the gauge length of the specimen defined as the distance between the localisations of load application. Finally the allowable deformation of plates, indicated for the required load in absence of a defect, is analysed for plates with a defect present.     

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J₂-deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H²(Ω) by using monotone operator theory and Browder-Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H²-norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.