Non-associated Reuleaux plasticity: Analytical stress integration and consistent tangent for finite deformation mechanics

Analytical backward Euler stress integration is presented for a volumetrically non-associated pressure-sensitive yield criterion based on a modified Reuleaux triangle. This advances previous work on associated Reuleaux plasticity using energy-mapped stress space. The analytical solution is 2–4 times faster than a standard numerical backward Euler algorithm. The merit in transforming to (and operating in) this space is that the stress return is truly the closest point on the surface to the elastic trial state. The paper includes a tension cut-off (formed by a second cone) and describes the steps necessary to allow the model’s incorporation within a finite deformation framework. Finite-element results show a 59% runtime saving for a modified Reuleaux model over a Willam–Warnke cone giving comparable accuracy in a thick-walled cylinder expansion problem. The consistent tangent provides asymptotically quadratic convergence in the Newton–Raphson scheme under both (i) small strain, infinitesimal deformation and (ii) large strain, finite deformation finite-element simulations. It is shown that the introduction of non-associated flow changes the plastic deformation field and reduces the heave predicted in a plane strain rigid strip-footing problem. The proposed model offers a significant improvement over the Drucker–Prager and Mohr–Coulomb formulations by better reproducing the material dependence on the Lode angle and intermediate principal stress, at little extra computational effort.     

Reuleaux plasticity: Analytical backward Euler stress integration and consistent tangent

Analytical backward Euler stress integration is presented for a deviatoric yielding criterion based on a modified Reuleaux triangle. The criterion is applied to a cone model which allows control over the shape of the deviatoric section, independent of the internal friction angle on the compression meridian. The return strategy and consistent tangent are fully defined for all three regions of principal stress space in which elastic trial states may lie. Errors associated with the integration scheme are reported. These are shown to be less than 3% for the case examined. Run time analysis reveals a 2.5–5.0 times speed-up (at a material point) over the iterative Newton–Raphson backward Euler stress return scheme. Two finite-element analyses are presented demonstrating the speed benefits of adopting this new formulation in larger boundary value problems. The simple modified Reuleaux surface provides an advance over Mohr–Coulomb and Drucker–Prager yield envelopes in that it incorporates dependencies on both the Lode angle and intermediate principal stress, without incurring the run time penalties of more sophisticated models.     

On the superlinear convergence in computational elasto-plasticity

We consider the convergence properties of return algorithms for a large class of rate-independent plasticity models. Based on recent results for semismooth functions, we can analyze these algorithms in the context of semismooth Newton methods guaranteeing local superlinear convergence. This recovers results for classical models but also extends to general hardening laws, multi-yield plasticity, and to several non-associated models. The superlinear convergence is also numerically shown for a large-scale parallel simulation of Drucker–Prager elasto-plasticity and an example for the modified Cam-clay model.     

Implicit Numerical Integration of Nonsmooth Multisurface Yield Criteria in the Principal Stress Space

Plasticity models employing multiple yield surfaces are frequently met in plasticity theory and engineering practice. The multiple yield surfaces may or may not intersect in a smooth manner, with the latter case being a superset of the former, typically encountered more often in engineering problems and covered within this work. Prominent members of this class of plasticity models can be found in a wide range of applications in Soil Mechanics, Rock Mechanics, Damage Mechanics, metal plasticity, concrete modeling or in the modeling of brittle or cohesive/frictional materials. While the importance of these models is widely acknowledged, difficulties arising from the singularities induced by the nonsmooth intersections of the yield surfaces, introduce severe algorithmic and numerical complexities that usually require specific treatment of each model. Three main problems can be identified, originating mainly from the application of the normality hypothesis, namely that (a) the elastic domain is subdifferential with respect to the stress vector at the intersections of the yield surfaces, (b) severe numerical errors are present in the vicinity of intersections since the derivatives of the yield surfaces are not always defined in these areas and (c) the set of the yield surfaces considered to be ultimately active is not known a priori. The objectives of this article are (a) to provide a comprehensive review of the related literature and an extensive overview of the solution techniques proposed by different researchers, (b) to present the formulation and propose the algorithmic treatment for the problem of nonsmooth multisurface plasticity models and finally (c) to give implementation details for some of the most widely used nonsmooth multisurface plasticity models. The proposed algorithm is based on a spectral representation of stresses and strains for the case of infinitesimal deformation plasticity and the reformulation of the return mapping scheme in principal stress directions. The determination of the set of the yield surfaces that will remain ultimately active is identified by involving a systematic enforcement of the Karush–Kuhn–Tucker conditions, providing in this way a surface agnostic implementation. Four representative examples of nonsmooth, multisurface plasticity models, are extensively presented and examined within the framework of the proposed algorithm. These are the Tresca, the Mohr–Coulomb, the Hoek–Brown and the Drucker–Prager (in the case that it is accompanied with a tension cut–off type surface) yield criteria, all of which are well established in the related literature and engineering practice. The efficiency, robustness and accuracy of the proposed algorithm is demonstrated through a series of numerical examples with excellent results.     

Cap models for clay strata to footing loads

Plasticity cap models are applied to obtain progressive failure solutions of flexible and smooth strip footings as well as rigid and rough strip footings on an overconsolidated stratum of clay. The comparative study of the elastic-plastic small deformation response of clay to footing loads is made within the framework of finite element analysis with different plasticity models based on a non-associated and the associated flow rule. More specifically, the analyses include the following features: (i) Drucker-Prager perfect plastic model with different methods in determining the material constants using the associated flow rule, (ii) Drucker-Prager perfect plastic model with the associated flow rule and a non-associated flow rule, and (iii) Drucker-Prager perfect plastic yield surface with a work-hardening plane cap and a work-hardening elliptic cap and their associated flow rules.     

The use of the Hoffman yield criterion in finite element analysis of anisotropic composites

A return mapping algorithm has been developed for the Hoffman yield function of anisotropic plasticity. The accuracy of the algorithm has been assessed by means of iso-error maps for trial stress increments in the deviatoric and volumetric plane. A tangent operator that is consistent with the developed integration algorithm has been formulated. The Hoffman model has been applied to a plate structure and to two shell structures.     

A finite element perturbation method for the prediction of rockburst

A finite element model is proposed to predict the potential occurrence of rockburst in underground openings. It is assumed that rockburst results due to instability of rock mass and the analysis is based on a perturbation technique. The rock mass is considered to be elastic-brittle-plastic. A novel method of elastic supports is used to simulate the effects of unbounded rock mass. Mohr-Coulomb and Hoek–Brown failure criteria are used and both associated and non-associated plasticity are considered. The effectiveness and efficiency of the proposed numerical technique are demonstrated by comparing finite element results with analytical solutions for ‘deep’ circular openings in rock mass subject to hydrostatic in situ stress. Analyses of example problems are presented to demonstrate the potential occurrence of rockburst predicted by using the proposed model.     

Finite elements in non-associated plasticity—axisymmetric necking in void-containing materials

Numerical procedures for the solution of large strain non-associated elasto-plasticity problems by the finite element method are presented. Both the Cartesian and natural finite element formulations are discussed in detail. As an example of the theory of plasticity without normality rule, a plasticity of void-containing metals is considered. The paper closes with a numerical study of necking in an axisymmetric specimen with initial imperfections resulting from a non-uniform distribution of voids.     

An interpretation of Mahalanobis distance in the dual space

Using the concept of a dual space, nk-dimensional vectors can be viewed as k points in an n-dimensional co-ordinate system. The relationships between the basic statistical properties of a k-variate sample and the geometrical properties of such a space are developed and the concept extended to two samples drawn from different populations, with derivation of the geometrical meaning of Mahalanobis distance. This geometrical approach provides valuable insight into why different feature subsets may or may not have high discriminatory potential, and shows that clustering in the dual space, or its subspaces, does not necessarily yield an effective feature selection technique.     

Elasto-plastic analysis of a plane stress problem using linearized yield surface and mixed hardening rule

The objective of this paper is to present the mathematical formulations in the incremental theory of plasticity, which is based on the mixed hardening rule and a linear yield surface. A three-parameter, uniaxial symmetric, linear yield surface suitable for tension-weak as well as equal tension and compression yield stress material is presented. This yield condition, along with the mixed hardening and associated flow rules is used to formulate the constitutive laws for sides and corners of the yield surface. The formulation is based on incremental plasticity with the assumption of small displacements and is suitable for plane stress problems under monotonie and cyclic loading.

The mixed hardening rule, which is mathematically modeled, could be changed to either kinematic or isotropic hardening by a simple change in the model. This hardening rule could handle different degrees of Bauschinger effect, as opposed to kinematic hardening, which assumes only an ideal Bauschinger effect, or isotropic hardening, which does not account for the effect at all. The theory is applied to a ductile material using the finite element method and cyclic loading.     

Cyclic plasticity models and application in fatigue analysis

An analytical procedure for prediction of the cyclic plasticity effects on both the structural fatigue life to crack initiation and the rate of crack growth is presented. The crack initiation criterion is based on the Coffin-Manson formulae extended for multiaxial stress state and for inclusion of the mean stress effect. This criterion is also applied for the accumulated damage ahead of the existing crack tip which is assumed to be related to the crack growth rate. Three cyclic plasticity models, based on the concept of combination of several yield surfaces, are employed for computing the crack growth rate of a cracked plane stress panel under several cyclic loading conditions.     

Robot location determination in a complex environment by multiple marks

A new approach of using multiple marks for determining three-dimensional robot location in a complex environment, which may be a composed space or a space with rough terrain, is proposed. One principal idea is to pre-locate a large enough number of marks with different labels at different planar places that at least one mark can be seen by the camera, wherever the robot is in the working space. The other concept is that by integrating the pattern recognition technique with the three-dimensional geometrical transformation method, the system can determine the robot location without the need of fitting any prerequisite alignment. This novel property enables the robot to work in a space with rough terrain. Experimental results from using three marks in a space composed of two warehouses have shown that the location error is less than 3% on average.     

A compact and grounded comb‐shaped microstrip antenna: A key to realize enhanced radiation performance

A compact grounded comb‐shaped single‐element microstrip antenna is proposed with wide beam width and symmetrical radiation pattern in both principal and diagonal planes. The proposed antenna exhibits excellent isolation of around 35 dB between co‐polarized and cross‐polarized radiation while the same for a conventional patch is only 15 dB. Around 105°‐110° of 3 dB beamwidth is achieved in both principal and diagonal (skew) planes at the center frequency. Around 1.5 dB of front‐to‐back radiation isolation is found from the proposed patch. Most importantly, the proposed antenna produces 65% and 35% broader beam in H‐ and E‐planes, respectively, than that of a classical microstrip antenna. Furthermore, in all principal and diagonal planes, the radiation pattern is found to be symmetrical in a wide angular region (?125° to +125°).     

一类三次隐式代数曲面参数化研究

在CAGD中隐式曲面和参数曲面作为曲面的两种表示形式各有其内在的优点 ,多年来如何有效地实现二者的相互转换一直是CAGD的一个热点问题 对一类GC1拼接两个二次曲面的三次混合代数曲面进行了研究 ,提出一种基于同轴平面束与代数曲面相交的几何化参数化方法 与传统参数化方法相比 ,该方法结构直观且具有可使三次代数曲面位于 [0 ,1]× [0 ,1]参数区间内 ,以及曲面的边界位于等参数线上等特点 ,利用这种参数曲面可以方便地实现机器作图和各种操作 实验结果验证了方法的有效性     

Interaction problems in soil-structure mechanics with material nonlinearity

Finite element analysis of soil-structure stress interaction and sand-clay flow interaction is discussed in the paper. The soil may be a one- or two-phase medium, depending on if fully drained, partly drained (consolidation) or undrained condition is considered. Transient behaviour is investigated.Yield hinge theory is adopted for the structural members while visco-plasticity, plasticity or a combination of visco-plasticity and plasticity are adopted for the soil skeleton to account for inviscid yield and creep. Adopted yield criterions are of cone type and according to the Critical State Theory.Finite element calculations for the stability analysis of a natural slope in clay and the interaction of a frame resting on a consolidating elastic-plastic subsoil are presented.     

Simulation Studies For The Performance Analysis Of The Reconstruction Of A Line In 3-D From Two Arbitrary Perspective Views Using Two Plane Intersection Method

Objects in nature as well as man made are bound by planar as well as curvilinear surfaces. Hence the reconstruction process of three dimensional objects involves obtaining the geometrical attributes of these planar and curvilinear surfaces, lines, points etc. If a line in 3-D space is viewed from two arbitrary positions, two different images of the same line are obtained. The correspondence between this pair of projections of the line is assumed to be established in this work. The work presented in this paper describes a method of reconstruction of a line in 3-D as an intersection of two planes containing the respective projected line images. The parameters of the line in 3-D space are obtained from the parameters of projected images of the line. Relevant mathematical formulations and analytical solutions for obtaining the parameters of the reconstructed line are given. The effect of noise in the reconstruction and the efficiency of the described reconstruction methodology are studied by simulation studies.     

Elasto-plastic response of uniformly loaded sector plates: full-section yield model predictions and spread of plasticity

The governing equations for an elasto-plastic large deflection analysis of pressure loaded sector plates, based on the Ilyushin full-section yield criterion and the flow theory of plasticity, are presented. An outline of the solution of these equations using a finite difference implementation of the dynamic relaxation algorithm is given. Numerical results are presented for uniformly loaded slender and stocky sector plates with a 60° sector angle. In particular, first yield data are presented as well as the results of two parameter studies. The first study shows the effects of flexural and in-plane edge conditions on the deflections and stress resultants/couples at the centre of the sector plate and the second contrasts the spread of plasticity in slender sector plates with simply supported and clamped in-plane fixed edges.     

圆柱型空间缩放机构用于步行机构时的优化设计

空间缩放机构由于在其运动主平面内的运动解耦性,被用作步行机构.本文作者认为圆柱型空间缩放机构用作步行机构有许多有利的方面.文中讨论了将该机构用于步行机构时机构运动的约束条件,对机构在有约束条件下进行了优化处理.设计出旨在具有最大运动空间的有约束的优化机构,并将此优化结果用于实际模型的设计.     

A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations

By assuming from the outset hyperelastic constitutive behavior, an alternative approach to finite deformation plasticity and viscoplasticity is proposed whereby the need for integration of spatial rate constitutive equations is entirely bypassed. To enhance the applicability of the method, reference is made to a general formulation of plasticity and viscoplasticity which embodies both the multiplicative and additive theories. A new return mapping algorithm capable of accommodating general yield conditions, arbitrary flow and hardening rules and non-constant tangent elasticities is proposed. Finally, a numerical example is presented which illustrates the excellent performance of the method for very large time steps.