The hybrid-stress model for thin plates

The assumed-stress hybrid finite element model is examined for application to the bending analysis of thin plates. A hybrid-stress functional is defined by using a Mindlin-type displacement assumption and including all components of stress. The Euler equations and matrix formulation corresponding to this functional are examined to assess the effects of plate thickness, and a rationale is presented for the selection of stress assumptions so that locking is avoided in the thin plate limit. To illustrate these concepts, a series of linear displacement quadrilateral elements are derived and tested, and the best of these elements is identified for suggested implementation in general-purpose computer programs.     

Hybrid-stress models for elastic–plastic analysis by the initial-stress approach

Two alternative numerical methods are presented, based on the assumed-stress hybrid finite element model and the initial-stress approach, for the elastic–plastic small-deflection analysis of structures under static loading. The use of the initial-stress approach results in a set of simultaneous linear algebraic incremental equations to be solved at each loading step, with the elastic stiffness matrix remaining unchanged throughout the loading process and the effects of plasticity included as equivalent element loads. The derivation of these alternative methods differs in the assigning of the incremental stress which satisfies equilibrium; in one case it is the actual stress increment while in the other it is a fictitious stress increment. An equilibrium imbalance correction is included in each of these methods to prevent drifting of the solution during the incremental process. Example solutions are presented which demonstrate the accuracy of the two methods and permit comparisons of the relative efficiencies of the two methods.     

Elastic-plastic buckling analysis of rectangular plates subjected to biaxial loads

In order to calculate the buckling load of a rectangular plate, the analytical approach is used in this study. The plate is assumed to be simply supported on four edges and loaded by uniform stresses along the edges. If the plate is slender, the buckling is elastic. However, if the plate is sturdy, it buckles in the plastic range. Then, the instantaneous moduli in the constitutive equations depend on the external loading. In this study, the elastic and plastic buckling equations are derived for rectangular plates under biaxial loading, and the corresponding interaction curves are presented. The influences of aspect ratios, load ratios and hardening factors on the buckling stresses are investigated for rectangular plates. From the plastic buckling analysis, the optimal combination of loads is given for the buckling strength.     

Elastic-plastic buckling analysis of rectangular plates subjected to biaxial loads

In order to calculate the buckling load of a rectangular plate, the analytical approach is used in this study. The plate is assumed to be simply supported on four edges and loaded by uniform stresses along the edges. If the plate is slender, the buckling is elastic. However, if the plate is sturdy, it buckles in the plastic range. Then, the instantaneous moduli in the constitutive equations depend on the external loading. In this study, the elastic and plastic buckling equations are derived for rectangular plates under biaxial loading, and the corresponding interaction curves are presented. The influences of aspect ratios, load ratios and hardening factors on the buckling stresses are investigated for rectangular plates. From the plastic buckling analysis, the optimal combination of loads is given for the buckling strength.     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.     

Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

Elastic-plastic analysis of cracked plates in plane stress: An experimental study

Summary An experimental method is presented for the complete solution of the elastic-plastic plane stress problem of an edge-cracked plate obeying the Mises yield criterion and the Prandtl-Reuss incremental stress-strain flow rule. The material of the plate is assumed as a strain-hardened one with different degrees of hardening. The elastic and plastic components of strain were determined by using the method of birefringent coatings cemented on the surface of the metallic specimens made of the material under study. Normal incidence of circularly polarized light yielded the isolinics and isochromatics of the coating which provided the principal elastic strain differences and strain-directions at the interface. Evaluation of the stress intensity factor at the crack tip, by using the Griffith-Irwin definition, gave the sum of principal stresses at the crack tip. These data were sufficient to separate the components of strain at the coating-plate interface by using the classical shear-difference method.The stress components on the partially plastically deformed cracked plate were determined by using the Prandtl-Reuss stress-strain relationships in a step-by-step process following the whole history of loading of the plate. Thus, a radial distribution law for the equivalent stress and strain in all directions of the plate was established which gave the instantaneous position of the elastic-plastic boundary and its evolution during loading, as well as the distribution of elastic and plastic components of stresses allover the plate.Four cases were solved for various amounts of strain-hardening from a quasi perfectly plastic material to an almost brittle strain hardened one. The values of the characteristic parameters defining each type of material were established.The results derived compare excellently with existing ones based either on experimental or numerical solutions and since they are based on both the theory of elasticity and the incremental theory of plasticity they constitute a sound basis for comparison. Moreover, the algorithm based on this hybrid method is fast and stable requiring a minimum computer time, memory and data preparation.     

Elastic-plastic analysis of Saint-Venant torsion problem by a hybrid stress model

An application of the finite element method to inelastic analysis of the Saint-Venant torsion problem, using a triangular element formulated on the basis of the hybrid stress approach, is described. The stress field in the element is defined by a stress function which is assumed to vary linearly within the element. The warping function on the element boundary is also defined by a linear expression. Numerical results indicate that the method gives an accurate stress solution and is appropriate to the elastic-plastic analysis of bars of arbitrary cross-section which may include multiply connected regions. It is further shown that such phenomena as planar orthogonal plastic anisotropy, strain-hardening and unloading of relevant twisted bar can be treated in a unified manner.     

An invariant eight-node hybrid-stress element for thin and thick multilayer laminated plates

A hybrid-stress eight-node isoparametric element is developed for the analysis of thin or thick multilayer fiber-reinforced composite plates. Transverse shear deformation effects are included by allowing for individual layer cross-section warping for thick laminates, or alternatively, laminate non-normal cross-section rotations for thin to moderately thick laminates. All stress components are included and are interpolated independently within each layer. Interlayer surface traction continuity and appropriate upper/lower surface traction-free conditions are exactly satisfied. The layer stress field is selected on the basis of earlier single-layer element studies so that the resulting element is naturally invariant with respect to co-ordinate translation or rotations, is non-locking in the thin-plate limit, and the element stiffness is of correct rank. An example for which an elasticity solution is available is used to demonstrate the element performance. Schemes for reduction of element stiffness computation time are also presented.     

Elastic-plastic analysis of surface flaws using a simplified line-spring model

The line-spring model has proven to be an effective tool for evaluating fracture parameters in surface-cracked plates and shells. However, application of the model requires detailed numerical computations, necessitating the availability of a specialized computer code. For approximate engineering calculations a version of the model which is more convenient to implement computationally, would be useful.In this paper a simplified line-spring model is presented along with detailed illustration of its application. The simplification is accomplished by replacing the crack front with a crack of constant depth and treating the ligament “spring” as elastic perfectly plastic. Despite its simplicity the model gives reasonably accurate predictions of fracture parameters, such as the J-integral or crack opening displacement (COD) at the root of surface cracks. This will be demonstrated by comparing analytical results for J and COD with previously published experimental data for surface-cracked steel plates.     

Analysis of axisymmetric structures under arbitrary loading using the hybrid-stress model

The analysis of axisymmetric structures in which geometric and material properties are independent of the circumferential co-ordinate, but for which loading may be dependent on this co-ordinate, is considered. The analysis is performed by expressing the field variables in a Fourier series and results in the solutions of a series of two-dimensional problems. The assumed-stress hybrid model is used and the element matrices for each harmonic are derived. To assess the procedure, two-4 node elements, differing in the form of the assumed stress field, are developed; both elements are shown to possess the correct rank. A series of example problems are considered, and the better of the two elements is identified.     

基于微平面弹塑性应力-应变关系的混凝土本构模型

在B.P. Bazant等人提出的混凝土微平面本构模型M2的基础上,将微平面上的应力分解为体、偏、剪三个分量,根据各个分量的物理意义定义了相应的应力-应变关系函数,即理想弹塑性函数。引入了破断应变的概念,当微平面应变达到破断应变后应力减为零。介绍了模型参数的确定方法。最后通过三个算例初步验证了本文所建议模型的合理性和正确性。     

Multilayer hybrid-stress finite element analysis of composite laminate with through-thickness cracks

An assumed hybrid-stress finite element model using a simple composite multilayer element is developed to analyze generally thin or moderately thick composite laminates with through-thickness cracks. The assumed stress field satisfies: (i) equilibrium conditions within each layer, (ii) the traction reciprocity conditions at interlaminar boundaries, and (iii) the traction-free boundary conditions at the top and bottom faces of the laminate. Since the number of nodes and assumed stress parameters are independent of the number of layers, the multilayer element devised is quite effective especially for the laminate with a large number of layers. Several selected composite laminates with a through-thickness edge crack are solved. Many fewer degrees of freedom and only a one-step solution are necessary for the present technique. The variations of mixed-mode stress intensity factors across the thickness of the composite laminate are also computed. Excellent agreements between the present results and referenced solutions are drawn. The technique developed is also applicable to analyze the structural behaviors of the cracked laminate with arbitrary fiber orientation and stacking sequence for which the stress singularity has not yet been found.     

A numerical approach to the analysis of failure modes in anisotropic plates

This study is motivated by the attempt to characterize failure modes of silicon chips commonly used in electronic industries. Previous experimental investigations provided the failure probability of dies made of a single-crystal and produced a large variety of crack patterns, but were not able to elucidate the link between defect distributions and crack initiation and propagation. To get some insight in the fracture activation and propagation mechanisms, we resort to finite element analyses and adopt an explicit methodology for crack tracking, based on the self-adaptive insertion of cohesive elements into a coherent mesh of solid elements. Finite kinematics material models with anisotropic features for both bulk and cohesive surfaces are employed to describe the behavior of single-crystal silicon plates undergoing a particular bending test up to failure. The cohesive model adopted in the calculation is fully anisotropic and newly formulated to accomplish the present study. Numerical simulations considering different material properties were able to ascertain the effects of particular flaws on failure modes of brittle silicon plates.     

A symmetric boundary element model for the analysis of Kirchhoff plates

The paper deals with the formulation and implementation of a new symmetric boundary element model for the analysis of Kirchhoff plates. The transversal displacement and normal slope boundary integral equations, usually adopted in the standard boundary element analysis, are considered together with bending moment, twisting moment and equivalent shear boundary integral equations. These equations are weighted by considering distributed sources related to the kinematic and static variables in the virtual-work sense. Moreover, particular attention is paid to the discretization of the boundary variables by shape functions selected in order to ensure continuity over the boundary and symmetry for the matrix system. The evaluation of the highly singular boundary integrals for overlapped integration domains is performed in closed form using a limit approach which provides self-contributions as limit values of non-singular terms. The corner effects and their treatment in the numerical procedure are also discussed. Various numerical examples for plates having different boundary conditions illustrate the performance of the model.     

Elastic-plastic finite element analysis of dynamic fracture

Recent evidence has pointed to the possible inadequacy of elastodynamic treatments of rapid crack propagation and crack arrest. This paper describes the development of a dynamic elastic-plastic finite element capability designed to address this concern by taking direct account of crack tip plasticity. Comparisons with known dynamic fracture mechanics solutions and with experimental data are made to demonstrate the fidelity of the approach. A comparison with an elastodynamic solution in an impact loaded 4340 steel bend specimen is also made. This result reveals that a significant effect of crack tip plasticity may exist even for high strength materials.     

Elastic-plastic analysis of crack in thermally-loaded structures

The stress or strain field near a crack-tip in a thermally-loaded structure has the HRR singularity and its amplitude is determined by the path-independent $\text{J}\text{?}\text{-}\text{integral}$. The $\text{J}\text{?}\text{-}\text{integral}$ is an extension of Rice's $\text{J-}\text{integral}$ and is related to the energy-release rate in thermal stress field. The finite element calculations have been carried out for a plate and a cylinder subjected to transient thermal loading with the temperature dependence of material properties taken into consideration. The computed $\text{J}\text{?}\text{-}\text{values}$ are almost path-independent as predicted theoretically. In an elastic problem, the stress intensity factor estimated from the $\text{J}\text{?}\text{-}\text{integral}$ agrees well with the analytical one. It is concluded that the $\text{J}\text{?}\text{-}\text{integral}$ has a potential ability to evaluate the integrity of structures such as nuclear pressure vessies that are subjected to thermal loading as well as mechanical one.     

A finite element alternating approach for the bending analysis of thin cracked plates

Based on the classical plate theory, the analytical solution for an infinte thin plate containing a crack subjected to arbitrary symmetric bending moments on the crack surfaces is first derived. Using this solution, an efficient and accurate finite element alternating procedure is then devised to deal with symmetric plate bending problems with single or multiple cracks. The interaction effect among cracks and the influence of the geometric boundaries on the calculation of bending stress intensity factors are also presented in detail. Several numerical examples are solved to demonstrate the validity of the approach.     

Elastic-plastic fracture analysis and design using the finite element method

The purpose of analysis in fracture mechanics is to determine characterising parameters which reflect the influence of loading and geometry on the crack tip environment of flawed bodies. Here practical methods are developed which permit the determination of such parameters in general situations. Extensive use of finite element methods has been made to provide relevant field values which are then manipulated to determine the required parameters. However the choice of method to determine field values is arbitrary and is dictated by the ease with which such field values may be found. It is in the manipulation of these values that the fracture mechanics philosophy is introduced.Contributions are made in three areas. First economic methods for the determination of the linear fracture mechanics parameter in general stiuations are developed which are of direct relevance to design procedures. Detailed discussion of the Dugdale model of fracture behaviour is then given and a general method for determining Dugdale model solutions is provided. This method is used to provide solutions for standard specimen geometries and it is suggested that such solutions will enable a rational evaluation of the general applicability of the model. However the method is such that, should sufficient confidence in the model be established, design calculations on its premises may be performed. Finally, it is demonstrated that materials which allow extensive plastic flow at a flaw tip prior to fracture may be analysed using the basic ideas of fracture mechanics. It is shown that a line integral provides a flaw tip environment parameter for materials deforming according to the physically appropriate Prandtl-Reuss laws of plasticity. It is hoped that these results will indicate a rational approach to correlating fracture behaviour in such situations.