A return mapping algorithm for unified strength theory model

A return mapping algorithm in principal stress space for unified strength theory (UST) model is presented in this paper. In contrast to Mohr–Coulomb and Tresca models, UST model contains two planes and three corners in the sextant of principal stress space, and the apex is formed by the intersection of 12 corners rather than the six corners of Mohr–Coulomb in the whole principal stress space. In order to utilize UST model, the existing return mapping algorithm in principal stress space is modified. The return mapping schemes for one plane, middle corner, and apex of UST model are derived, and corresponding consistent constitutive matrices in principal stress space are constructed. Because of the flexibility of UST, the present model is not only suitable for analysis based on the traditional yield functions, such as Mohr–Coulomb, Tresca, and Mises, but might also be used for analysis based on a series of new failure criteria. The accuracy of the present model is assessed by the iso‐error maps. Three numerical examples are also given to demonstrate the capability of the present algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

A comparative study of stress update algorithms for rate‐independent and rate‐dependent crystal plasticity

The paper presents a comparative discussion of stress update algorithms for single‐crystal plasticity at small strains. The key result is a new unified fully implicit multisurface‐type return algorithm for both the rate‐independent and the rate‐dependent setting, endowed with three alternative approaches to the regularization of possible redundant slip activities. The fundamental problem of the rate‐independent theory is the possible ill condition due to linear‐dependent active slip systems. We discuss three possible algorithmic approaches to deal with this problem. This includes the use of alternative generalized inverses of the Jacobian of the currently active yield criterion functions as well as a new diagonal shift regularization technique, motivated by a limit of the rate‐dependent theory. Analytical investigations and numerical experiments show that all three approaches result in similar physically acceptable predictions of the active slip of rate‐independent single‐crystal plasticity, while the new proposed diagonal shift method is the most simple and efficient concept. Copyright © 2001 John Wiley & Sons, Ltd.     

基于二阶组构张量的各向异性屈服准则

将四个屈服准则:Tresca准则、Mises准则、Mohr-Coulomb准则以及Drucker-Prager准则归类为剪切屈服准则。Tresca准则和Mohr-Coulomb准则是关于最不利截面的剪切屈服准则,而Mises准则和Drucker-Prager准则是关于各方向截面的剪应力和正应力的某种综合度量的八面体剪应力和八面体正应力的剪切屈服准则。从方向函数(ODF)的概念入手,将各方向截面的剪应力和正应力综合度量直接取为所有方向截面上的剪应力和正应力的平均。对各向同性材料,提出了平均剪切屈服度准则:当平均剪应力和平均正应力的组合达到某一极限值时,材料开始屈服。研究表明,平均剪切屈服准则与Drucker-Prager准则具有相同的形式,当不考虑平均正应力对屈服的影响时,它与Mises准则具有相同的形式。针对由各向异性损伤导致的材料各向异性强度问题,定义截面上的有效正应力和有效剪应力则分别为截面上的法向力和切向力与有效承载面积之比,基于截面上的有效应力提出了各向异性材料的平均剪切屈服准则。各向异性损伤引起的截面上有效应力放大系数为方向函数,可以采用二阶组构张量来近似表示,在任意坐标系中,各向异性屈服准则为应力分量的二次齐次式,导出了其中的系数与二阶组构张量之间的显式关系式。在二阶组构张量的主轴坐标系内,各向异性屈服准则与殷有泉的拓展Hill准则形式完全相同,当不考虑正应力对屈服的影响时,它与Hill准则具有相同的形式。     

INTERELEMENT STRESS EVALUATION BY BOUNDARY ELEMENTS

The conventional boundary element method employs piecewise shape functions which lead to stress discontinuity at the interelement boundary. This paper derives formulae for boundary stress and boundary stress gradient based on boundary element solutions, and discusses error propagation in stress evaluation due to errors in the displacement boundary values. The nature of stress discontinuity is investigated. A simple post-processing scheme is presented using continuities of stress and stress gradient along a traction boundary as two extra conditions, so that a high-order shape function can be employed for the evaluation of stress and stress gradient on the interelement boundary. Four numerical examples are used to demonstrate the accuracy of the proposed post-processing scheme. The numerical results show that as compared with the conventional method, the post-processing method can significantly improve the stress and stress gradient on the traction boundaries, especially in the area of stress concentration.     

A general framework for generating convex yield surfaces for anisotropic metals

Summary The aim of this paper is to develop a general procedure to create yield surfaces (both isotropic and anisotropic) for elastic plastic metals with particular reference to sheet metal forming operations. Due to the fact that the forming limit of sheet metals requires a very accurate prediction of the yield limits and their normals, it is desirable to create yield functions that offer independent control of the yield points and the normal to the yield surface at a particular stress state without affecting its global convexity properties and the yield points at other states of stress. We achieve this by creating a class of yield surfaces that are obtained by the intersection of a number of elementary convex surfaces (such as planes and cylinders); the edges and corners that are a result of the intersection are smoothened by aL ^p-norm smoothing technique to give rounded corners. The yield surfaces generated by this procedure are centrosymmetric in nature and are thus limited in their applicability.We show that many of the standard yield surfaces can be recast into this form. We obtain a new yield surface that fits the experiments obtained for the 6022-T4 aluminum alloy as reported by Barlat et al. [1]. The same yield surface is also used to fit the results of the Taylor-Bishop-Hill polycrystal theory applied to a material with copper texture. The data were obtained from Choi et al. [2].     

A method for smoothing multiple yield functions

Many models of plasticity are built using multiple, simple yield surfaces. Examples include geomechanical models and crystal plasticity. This leads to numerical difficulties, most particularly during the stress update procedure, because the combined yield surface is nondifferentiable, and when employing implicit time stepping to solve numerical models, because the Jacobian is often poorly conditioned. A method is presented that produces a single C² differentiable and convex yield function from a plastic model that contains multiple yield surfaces that are individually C² differentiable and convex. C² differentiability ensures quadratic convergence of implicit stress-update procedures; convexity ensures a unique solution to the stress update problem, whereas smoothness means the Jacobian is much better conditioned. The method contains just one free parameter, and the error incurred through the smoothing procedure is quantified in terms of this parameter. The method is illustrated through three different constitutive models. The method's performance is quantified in terms of the number of iterations required during stress update as a function of the smoothing parameter. Two simple finite-element models are also solved to compare this method with existing approaches. The method has been added to the open-source “MOOSE” framework, for perfect, nonperfect, associated, and nonassociated plasticity.     

The uniaxial strain compression of porous and fully dense beryllium

Compressive stress versus uniaxial compressive strain curves obtained using a simple closed die are presented for SP-100-C beryllium of initial fractional porosities varying from 0.05 to 0.26, and for fully dense ingot beryllium. The relationship between porosity and compressive yield stress is found to be linear over the entire porosity range with a discontinuity at 0.065 fractional porosity. It is suggested that the discontinuity is related to the transition from isolated to predominantly interconnected porosity. It is deduced that the pressure required for full compaction varies with density and is estimated to range from 3.65×10³ MN m^?2 at 0.05 to 17.2×10³ MN m^?2 at 0.26 fractional porosity.     

Numerical modelling of fracture initiation and propagation in biaxial tests on rock samples

A two-dimensional boundary element code, based on the displacement discontinuity method is used to simulate a confined compression test. The method takes account of the granular nature of the rock and of the presence of pre-existing defects. Fracture propagation is thought to depend, amongst other factors, on the crack orientation, the residual friction angle, the dilation angle, and the confining pressure. To obtain a more precise understanding of the influence of these properties on the crack growth process, their influence on the normal stress and the excess shear stress on potential fracture planes ahead of the crack tip is investigated for a single crack configuration. The orientation of the potential fracture planes proves to be the most important parameter determining fracture growth. A series of numerical experiments is carried out to determine the influence of the tessellation pattern used to represent the granular nature of the rock. Both the influence of the type of tessellation and the tessellation density are evaluated, and reasons for the differences in behaviour are presented. The results of the simulations with the Delaunay and a Voronoi tessellation with internal fracture paths compare well with the fracture pattern obtained in laboratory tests. The pre-peak non-linearity in the stress-strain response obtained with the Voronoi tessellation and the post-peak strain softening obtained with the Delaunay tessellation are combined in one model. For that purpose, a Voronoi tessellation with internal fracture paths is used, whereby the properties of the elements of the polygons and of the internal fracture paths are assigned different values. The role that is played by shear failure and the influence of dilation on the localisation process is determined by means of some further numerical experiments. It is shown that at the scale, at which the material is modelled, shear failure is required for a shear band to develop.     

接管轴向推力作用下开孔－补强区应力分布规律的试验研究

对接管轴向推力作用下圆筒形压力容器开孔－补强区的应力分布规律进行了试验研究．工作针对３台具有不同ｄ／Ｄ比的接管及标准补强圈补强的试验容器进行．研究结果表明，在补强圈补强的范围内，容器上的应力明显降低。但由于几何尺寸的变化及焊缝的加强作用，在补强圈外边缘特别在横向截面（θ＝９０°）内，容器中将出现一高的不连续应力，从而使该区域成为整个接管最危险的区域．     

Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design

A continuum‐based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non‐linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non‐linear dynamic problems, such as crashworthiness design. The first‐order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path‐dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate‐based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.     

Tensile crack tip fields in elastic-ideally plastic crystals

Crack tip stress and deformation fields are analyzed for tensile-loaded ideally plastic crystals. The specific cases of (0 1 0) cracks growing in the [1 0 1] direction, and (1 0 1) cracks in the [0, 1, 0] direction, are considered for both fcc and bcc crystals which flow according to the critical resolved shear stress criterion. Stationary and quasistatically growing crack fields are considered. The analysis is asymptotic in character; complete elastic-plastic solutions have not been determined. The near-tip stress state is shown to be locally constant within angular sectors that are stressed to yield levels at a stationary crack tip, and to change discontinuously from sector to sector. Near tip deformations are not uniquely determined but fields involving shear displacement discontinuities at sector boundaries are required by the derived stress state. For the growing crack both stress and displacement must be fully continuous near the tip. An asymptotic solution is given that involves angular sectors at the tip that elastically unload from, and then reload to, a plastic state. The associated near-tip velocity field then has discontinuities of slip type at borders of the elastic sectors. The rays, emanating from the crack tip, on which discontinuities occur in the two types of solutions are found to lie either parallel or perpendicular to the family of slip plane traces that are stressed to yield levels by the local stresses. In the latter case the mode of concentrated shear along a ray of discontinuity is of kink type. Some consequences of this are discussed in terms of the dislocation generation and motion necessary to allow the flow predicted macroscopically.     

A 2‐D meshless model for jointed rock structures

According to the characteristic structural features of jointed rock structures, a meshless model is proposed for the mechanics analysis of jointed rock structures based on the moving least‐squares interpolants. In this model, a jointed rock structure is regarded as a system of relatively intact rock blocks connected by joints or planes of discontinuity; these rock blocks are modelled by general shaped anisotropic blocks while these joints and planes of discontinuity are modelled by interfaces. The displacement field of each block is constructed by the moving least‐squares interpolants with an array of points distributed in the block. To deal with the discontinuities of rock structures, the displacement fields are constructed to be discontinuous between blocks. The displacement fields and their gradients are continuous in each block, hence no post processing is required for the output of strains and stresses. The finite element mesh is totally unnecessary, so the time‐consuming mesh generation is avoided. The rate of convergence can exceed that of finite elements significantly, and a high resolution of localized steep gradients can be achieved. Furthermore, the discontinuities of rock structures are also fully taken into consideration. The present method is developed for two‐dimensional linear elastic analysis of jointed rock structures, and can be extended to three‐dimensional and non‐linear analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical modeling of 2-D smooth crack growth

Numerical algorithm to simulate 2-D smooth crack is presented. The stepwise method based on local criteria of propagation is used. Two crack propagation criteria are employed. At the first stage of propagation, the maximum tensile stress criterion is used to take into account the abrupt change in tangent direction. At subsequent stages, the assumption that the stress intensity factor (SIF) K_{_2=} 0 at the current crack tip is exploited. The analytical formulae for calculating SIFs are given. The displacement discontinuities (DD) involved in these formulae are found from the numerical solution of a complex hypersingular integral equation (CHSIE) for a piecewise homogeneous plane with curvilinear cracks. The new mechanism of smooth approximation of the crack path by circular arcs at each propagation stage is suggested. Numerical results are given. They confirm the efficiency of the algorithm suggested. This revised version was published online in August 2006 with corrections to the Cover Date.     

材料三剪屈服准则研究

在分析Tresca屈服准则和双剪屈服准则的基础上,通过考虑十二面体单元主剪面上所有三个主剪应力的共同作用,提出了材料的三剪屈服准则,并对该准则作了极限线分析、应力莫尔圆分析、中间主应力效应曲线分析和滑移面方向分析。结果表明,新准则与统一屈服准则一样表示的是一个屈服准则系列,其中Tresca屈服准则和vonMise屈服准则是该屈服准则的特例。但与双剪统一屈服准则不同,新屈服准则的极限线和中间主应力效应曲线是非线性的,能够描述材料的非线性屈服特性。另外,也克服了双剪统一屈服准则中存在的在某些特定应力状态下会得出双重滑移面并且滑移面方向会发生变化的问题。     

An approximated computational method for fast stress reconstruction in large strain plasticity

This article describes a numerical method to reconstruct the stress field starting from strain data in elastoplasticity. Usually, this reconstruction is performed using the radial return algorithm, commonly implemented also in finite element codes. However, that method requires iterations to converge and can bring to errors if applied to experimental strain data affected by noise. A different solution is proposed here, where an approximated numerical method is used to derive the stress from the strain data with no iterations. The method is general and can be applied to any plasticity model with a convex surface of the yield locus in nonproportional loading. The theoretical basis of the method is described and then it is implemented on two constitutive models of anisotropic plasticity, namely, Hill48 and Yld2000-2D. The accuracy of the proposed method and the advantage in terms of computational time with respect to the classical radial-return algorithm are discussed. The possibility of using such method to reconstruct the stress field in case of few temporal data and noisy strain fields is also investigated.     

Local and global instabilities in nanosize rectangular prismatic gold specimens

We use molecular mechanics simulations with the tight-binding potential to study local and global instabilities in initially defect-free nanosize rectangular prismatic specimens of gold deformed in tension/compression and simple tension/compression. Whereas in simple tension/compression atoms on end faces are constrained to move axially but are free to move laterally and the cross-sectional dimensions of end faces can change, in tension/compression all three components of displacements of atoms on end faces are prescribed and the cross-section of an end face does not change. The three criteria used to delineate local instabilities in a specimen are: (i) a component of second-order spatial partial derivatives of the displacement field has large value relative to its average value in the body, (ii) the minimum eigenvalue of the Hessian of the potential energy of an atom is negative, (iii) a relatively high value of the common neighborhood parameter. A specimen becomes globally unstable when its potential energy decreases noticeably with a small increase in its deformations. It is found that the three criteria for local instability are met essentially simultaneously at the same atomic position. Deformations of interior points of a specimen are different when it is deformed in simple tension/compression from those in tension/compression. It is found that the initial unloaded configuration (or the reference configuration) of the minimum potential energy has significant in-plane stresses on the bounding surfaces and non-zero normal stresses at interior points. This initial stress distribution satisfies Cauchy’s equilibrium equations for a continuum. In deformations of a nanobar studied here, the yield stress defined as the average axial stress when the average axial stress vs. the average axial strain curve exhibits a sharp discontinuity depends upon the specimen size. It is shown possibly for the first time that deformations of the specimen are reversible if it is unloaded prior to yielding but have a permanent strain if unloaded after it has yielded. Because of residual stresses in the reference configuration, the average axial stress at yield in compression is nearly one-half of that in tension. The slope of the average axial stress vs. the average axial strain curve during unloading after it has yielded is the same as that during initial loading up to the yield point.     

ESTIMATED AND EXPERIMENTAL FATIGUE LIVES OF 30CrNiMo8 STEEL UNDER IN-AND OUT-OF-PHASE COMBINED BENDING AND TORSION WITH VARIABLE AMPLITUDES

Abstract— Calculated fatigue lives, based on three criteria for multiaxial random fatigue, were compared with lives obtained from tests on cylindrical specimens of 30CrNiMo8 steel subjected to in- and out-of-phase bending and torsion at variable amplitudes. In the chosen fatigue criteria the expected position of the fracture plane, determined from a variance method for the equivalent stress, were taken into account. The equivalent stress history was related to the rain flow method and fatigue damage was evaluated from the Palmgren–Miner hypothesis. It has been shown that the expected fatigue fracture planes agree with those determined by experiments. The most realistic estimations of fatigue life were obtained by the criterion of maximum shear and normal stresses on the fracture plane using a modified shear stress.     

Integration schemes for von‐Mises plasticity models based on exponential maps: numerical investigations and theoretical considerations

We consider three different exponential map algorithms for associative von‐Mises plasticity with linear isotropic and kinematic hardening. The first scheme is based on a different formulation of the time continuous plasticity model, which automatically grants the yield consistency of the method in the numerical solution. The second one is the quadratically accurate but non‐yield consistent method already proposed in Auricchio and Beirão da Veiga (Int. J. Numer. Meth. Engng 2003; 56 : 1375–1396). The third method is an improved version of the second one, in which the yield consistency condition is enforced a posteriori. We also compare the performance of the three methods with the classical radial return map algorithm. We develop extensive numerical tests which clearly show the main advantages and disadvantages of the three methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Two bonded half planes with a crack going through the interface

The plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case. The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements.     

Local and overall flow in composites predicted by micromechanics

Rate-dependent inelastic flow in metal matrix composites subjected to multiaxial stress states is quantified by flow surfaces, which are geometrically analogous to yield surfaces. The definition of flow is important because the most meaningful definition from a theoretical viewpoint, dissipation, is not measurable in the laboratory. Inelastic power is measurable, but differs from the dissipation due to residual stresses and evolution of the material state. Since experiments are necessary for development and validation of models, both definitions are important and considered here. The relationship between local flow in the matrix and overall flow of the composite is explored using finite element and generalized method of cells micromechanical analyses. The loci of flow surfaces in the axial–transverse and transverse–transverse stress planes are plotted. At the threshold, the overall flow surface is the intersection of all the local flow surfaces. Beyond the threshold, the intersection of all the local flow surfaces is smaller than the overall flow surface and differences between the dissipation and inelastic power are notable. Most importantly, the directions of the overall inelastic strain rate vectors are generally not normal to the overall surface of constant dissipation after the material state has begun to evolve. Thus, an associative macroscale continuum model will be, at best, approximate. Interestingly, local flow surfaces beyond the threshold are not necessarily convex when plotted in the overall stress plane. This is due to the existence of residual stresses. In addition, the generalized method of cells was found to accurately estimate the inner and outer envelopes of the local flow surface cluster with a surprisingly small number of subcells.