Finite element techniques in plane limit problems

Summary Limit analysis of plane structures is formulated by applying a finite element linear programming technique that was previously proposed by the same author.Three different element models are considered: the constant stress compatible model, the linear stress equilibrium model, the quadratic stress equilibrium model. Some notes are devoted to the construction of the stress fields with reference to the adopted method.Some suitably chosen numerical applications are then proposed in order to compare the different approaches as far as numerical efficiency and result accuracy are concerned. The possibility of improving the linear programming procedure is also discussed, and in particular a method is proposed to find out an initial feasible vector on the basis of mechanical considerations, in order to overcome the necessity of the so-called M-phase of the simplex method.Numerical results are given together with the comments.

---

Sommario Si formula il problema dell'analisi limite di strutture piane secondo un metodo degli elementi finiti già proposto dall'autore.Nel presente lavoro vengono considerati tre diversi modelli di elementi e precisamente: il modello congruente a sforzo costante e quelli equilibrati a campi di sforzo lineare e quadratico. La costruzione dei tre modelli viene presentata in dettaglio nell'ambito del metodo adottato.Vengono quindi presentati alcuni esempi di applicazione opportunamente scelti, con lo scopo di confrontare l'efficienza numerica e l'accuratezza di risultati ottenuti con l'impiego dei tre modelli. Si discute inoltre la possibilità di migliorare l'efficacia dell'algoritmo matematico ed in particolare si suggerisce un metodo che, individuando una soluzione ammissibile di partenza sulla scorta di considerazioni meccaniche, permette di evitare la cosiddetta fase M della programmazione lineare.I commenti sono accompagnati dai risultati numerici.

---

---

Research supported by the C.N.R. (Italian National Research Council).     

Finite element analysis of plane soil cutting

This study develops the finite element method (FEM) of solution to provide a theoretical means for determination of soil performance under the actions of a cutting blade—and the forces required to promote cutting. The developed FEM takes into account the effect of progressive and continuous cutting of the clay soil at the tip of the blade, with possible development of failure zones in the soil whenever the shear strength of the soil is exceeded. The solution provides detailed information on stress and deformation fields in the soil, together with tangential and normal pressures developed at the blade soil interface Correspondence between theoretically computed displacement fields and measured values has been obtained. In addition, the theoretically computed and experimentally measured values for forces developed in blade thrust are seen to be in close agreement.     

Finite element analysis of localization in FCC polycrystalline sheets under plane stress tension

Localization phenomena in thin sheets subjected to plane stress tension are investigated. The sheet is modelled as a polycrystalline aggregate, and a finite element analysis based on rate-dependent crystal plasticity is developed to simulate large strain behaviour. Accordingly, each material point in the specimen is considered to be a polycrystalline aggregate consisting of a large number of FCC grains. The Taylor model of crystal plasticity theory is assumed. This analysis accounts for initial textures as well as texture evolution during large plastic deformations. The numerical analysis incorporates certain parallel computing features. Simulations have been carried out for an aluminum sheet alloy, and the effects of various parameters on the formation and prediction of localized deformation (in the form of necking and/or in-plane shear bands) are examined.     

Numerical plane stress elastic–perfectly plastic crack analysis under Tresca yield condition with comparison to Dugdale plastic strip model

A number of plane stress numerical analyses of the mode I elastoplastic fracture mechanics problem have been performed in the past using the Huber–Mises yield criterion. This study employs instead the Tresca yield condition using an incremental theory of plasticity for a stationary crack. A commercial finite element program is used to solve the opening mode of fracture problem (mode I) for a square plate containing a central crack under generalized plane stress loading conditions. A biaxial uniform tensile traction is applied to the edges of a thin plate composed of a linear elastic non-work hardening material under small strain assumptions. The finite element results are compared with the analytical predictions of the Dugdale plastic strip model for a crack in an infinite plate subject to a biaxial uniform load at infinity.     

基于微面有效应力矢量的各向异性屈服准则

基于微面模型,定义损伤变量为微面上有效承载面积的减少. 将Kachanov的一维有效 应力概念推广到三维,提出微面有效应力矢量的概念. 根据微面的有效应力矢量,将无损材 料的宏观应力张量及不变量与微面应力矢量的积分关系拓展到有损材料,得到了有损材料的 宏观有效应力张量及其不变量与宏观名义应力张量、微面面积损伤组构张量之间的关系. 将 无损材料的以应力张量不变量表示的Drucker-Prager准则推广到有损材料,建立了含缺陷 材料的各向异性屈服准则. 对有损材料,宏观有效应力张量与Murakami的有效应力张量具 有相同的形式,各向异性强度准则与Liu等提出的扩展Hill准则有相同的形式,当不考虑 静水应力对屈服的影响时,它与Hill准则具有相同的形式.     

直杆碰撞刚性壁弹塑性动力后屈曲有限元分析

利用显式动力学有限元方法对直杆弹塑性动力后屈曲进行了分析,模拟了直杆轴向碰撞动力屈曲的变形及发展过程。分析中在直杆碰撞端局部临界屈曲长度范围内引入半正弦波形式的初始缺陷,计算结果与文献中的实验数据获得了很好的一致。分析结果表明,随着碰撞过程中所产生的应力波逐渐向前传播,后屈曲变形过程中所呈现的多个半波形式的高阶屈曲模态由初始具有单个半波形式的简单屈曲模态迅速演变而成。分析结果同时也揭示了直杆动力屈曲变形发展的机理,以及轴向应力波和屈曲变形的相互作用规律。     

不同后继屈服函数在弹塑性有限元中的应用评估

构造了一种在π平面上为椭圆的后续屈服函数,将该屈服函数、“Tresca”及“Misses型”后继屈服函数分别应用于干涉孔及冷胀孔问题的弹塑性有限元分析,并对这三种屈服函数在弹塑性有限元中的应用进行了评估。本文三种屈服函数在弹塑性问题中的应用评估对更有效地解决工程中的弹塑性问题有着实际的应用价值。     

Finite element analysis of axisymmetric elastic body problems

Linear form functions are commonly used in a long time for a toroidal volume element swept by a triangle revolved about the symmetrical axis for general axisymmetrical stress problems. It is difficult to obtain the rigidity matrix by exact integration, and instead, the method of approximate integration is used. As the locations of element close to the symmetrical axis, the accuracy of this approximation deteriorates very rapidly. The exact integration have been suggested by various authors for the calculation of rigidity matrix. However, it is shown in this paper that these exact integrations can only be used for those axisymmetric bodies with central hole. For solid axisymmetric body, it can be proved that the calculation fails due to the divergent property of rigidity matrix integration. In this paper a new form function is suggested. In this new form function, the radial displacementu vanishes as radial coordinatesr approach to zero. The calculated rigidity matrix is convergent everywhere, including these triangular toroidal element closed to the symmetrical axis. This kind of element is useful for the calculation of axisymmetric elastic solid body problems.     

Dual-parameter elastic-plastic fracture criterion of metals and alloys in the plane strain case

In recent years, some investigators discussed the applicability of the HRR theory for engineering materials based on the results of numerical analyses and experimental studies. In the present paper, the finite element method is employed to analyze the crack tip fields of the engineering elastic-plastic material with a variety of geometry configurations of cracked specimens from elastic state to intensely general yielded state in the plane strain case. The results indicate that the HRR theory loses its validity of application for engineering elastic-plastic materials in the plane strain case. The reasons for this are analyzed. A dual-parameter fracture criterion is suggested for this case.     

A plane stress formulation for elastic-plastic deformation of unidirectional composites

A plane stress analysis of the elastic-plastic deformation of unidirectional composites is presented. A continuum model based on the solid-mixture concept is selected as the basis for the analysis. Model parameters, including process-dependent variables, are deduced from experiments performed on unidirectional composites. A computer program MET1MAT has been developed accordingly and tested for a few simple in-plane loading cases. Experimental data for uniaxial tests performed in longitudinal and transverse directions and for a few biaxial tests are presented to substantiate the analysis. And, finally, application of the results to laminated metal matrix composites is discussed.     

Modified Burzynski criterion with non-associated flow rule for anisotropic asymmetric metals in plane stress problems

The Burzynski criterion is developed for anisotropic asymmetric metals with the non-associated flow rule (NAFR) for plane stress problems. The presented pressure depending on the yield criterion can be calibrated with ten experimental data, i.e., the tensile yield stresses at 0°, 45°, and 90°, the compressive yield stresses at 0°, 15°, 30°, 45°, 75°, and 90° from the rolling direction, and the biaxial tensile yield stress. The corresponding pressure independent plastic potential function can be calibrated with six experimental data, i.e., the tensile R-values at 0°, 15°, 45°, 75°, and 90° from the rolling direction and the tensile biaxial R-value. The downhill simplex method is used to solve these ten and six high nonlinear equations for the yield and plastic potential functions, re- spectively. The results show that the presented new criterion is appropriate for anisotropic asymmetric metals.     

An hysteretic quadrilateral plane stress element

In this work, a new plane stress element is proposed for the nonlinear static and dynamic analysis of plane stress/plane strain problems. The four node quadrilateral element formulation for the elastic case is extended by introducing a novel hysteretic constitutive relation, based on the Bouc–Wen model of hysteresis. The hysteretic model introduced is directly derived from the governing equations of classical plasticity based on the flow rule and specific hardening law. The stiffness matrix of the element is formulated using the principle of virtual displacements, where the elastic stress–strain relation is substituted by the hysteretic relation proposed. The derived stiffness matrix is expressed as a smooth function of the internal stress field both in the elastic and inelastic regime. The efficiency of the proposed element in the simulation of the cyclic behavior in plane structures is presented through illustrative examples.     

A new boundary element approach of modeling singular stress fields of plane V-notch problems

In this paper, a new boundary element (BE) approach is proposed to determine the singular stress field in plane V-notch structures. The method is based on an asymptotic expansion of the stresses in a small region around a notch tip and application of the conventional BE in the remaining region of the structure. The evaluation of stress singularities at a notch tip is transformed into an eigenvalue problem of ordinary differential equations that is solved by the interpolating matrix method in order to obtain singularity orders (degrees) and associated eigen-functions of the V-notch. The combination of the eigen-analysis for the small region and the conventional BE analysis for the remaining part of the structure results in both the singular stress field near the notch tip and the notch stress intensity factors (SIFs).Examples are given for V-notch plates made of isotropic materials. Comparisons and parametric studies on stresses and notch SIFs are carried out for various V-notch plates. The studies show that the new approach is accurate and effective in simulating singular stress fields in V-notch/crack structures.     

有限元表面应力计算

用有限元[1]通用程序进行结构计算时,最常用的是位移法,因而计算得到的位移有较高的精度。由位移计算应力时,有限元法应用的是应力-应变关系和应变-位移关系,其中应变-位移是微商关系。在数值计算中,微商只能转化为差商等用插值近似处理。这样,虽然位移精度高,但应力的计算精度就被大打折扣。本文应用弹性力学辛体系理论[2],解析求解了位移和应力的影响函数。利用有限元程序计算得到的位移,由功互等定理,不需要微分插值,就可以得到指定点的应力,应力精度大大提高。工程实际中有许多问题的最大应力往往发生在构件表面。针对表面应力问题,本文给出了半平面表面应力的影响函数,进行了数值算例计算。计算结果表明,用本文提出的影响函数法求解一点的应力,其精度明显提高,并且计算结果有很好的稳定性。用本文的影响函数法编制成子程序,可作为有限元软件应力计算的一个模块,可以更好地发挥有限元程序的功效。     

Finite element simulation of solidification problems

The modelling of liquid-solid phase change phenomena is extremely important in many areas of science and engineering. In particular, the solidification of molten metals during various casting methods in the foundry, provides a source of important practical problems which may be resolved economically with the aid of computational models of the heat transfer processes involved. Experimental design analysis is often prohibitively expensive, and the geometries and complex boundary conditions encountered preclude any analytical solutions to the problems posed. Thus the motivation for numerical simulation and computer aided design (CAD) systems is clear, and several mathematical/computational modelling techniques have been brought to bear in this area during recent years.This paper reports on the application of the finite element method to solidification problems, principally concerning industrial casting processes. Although convective heat transfer has been modelled, the work herein considers only heat conduction, for clarity. The heat transfer model has also been coupled with thermal stress analysis packages to predict mechanical behaviour including cracking and eventual failure, but this is reported elsewhere.Following the introduction, the mathematical and computational modelling tools are described in detail, for completeness. A discussion on the handling of the phase change interface and latent heat effects is then presented. Some aspects of the solution procedures are examined next, together with special techniques for dealing with the mold-metal interface. Finally, some numerical examples are presented which substantiate the capabilities of the finite element model, in both two and three dimensions.Nomenclature c heat capacity - C capacitance matrix - f time function - F loading term - h heat convection coefficient - H specific enthalpy - |J| Jacobian determinant - || patch approximation to |J| - k thermal conductivity - K conductance matrix - L latent heat - unit outward normal - N _i nodal shape function - q known heat flux - R _i nodal heat capacity - S phase change interface - t time - T temperature - known boundary temperature - T vector of nodal temperatures - T _a ambient temperature - T _c solidification temperature - T _L liquidus temperature - T ₀ initial temperature - T _s solidus temperature - x space coordinates - interface heat transfer coefficient - iteration parameter - boundary of domain - T solidification range - t timestep magnitude - vector gradient operator - convergence tolerance - timestepping parameter - t known vector in alternating-direction formulation - Laplace modifying parameter - (, ) local space coordinates - density - time limit - () shape function factor - () shape function factor - domain of interest     

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.     

Solution of elastic-plastic shallow shell problems by the boundary element method

The boundary integral equations for elasto-plastic problems of shallow shells are established by using the fundamental solutions of shallow shells derived previously. The strains and stress-resultants in the plastic region are used as unknown variables. The simultaneous nonlinear equations of these variables and unknown boundary values are established and solved by direct iteration method.     

Diagonalized consistent mass matrix and the dynamical finite element analysis of elastic-plastic impact in axisymmetrical problems

In this paper, the diagonalized consistent mass matrix is found for the triangular ring element in axisymmetrical problems. The results of this work eliminate the feeling of uncertainty and arbitrariness of lumped mass method on the one hand and the difficulty of computation due to non-diagonalized character of consistent mass method on the other. This paper gives also the foundations of the finite element analysis of elastic-plastic axisymmtrical impact problems.