An implicit three-dimensional model for describing the inelastic response of solids undergoing finite deformation

The aim of this paper is to develop a new unified class of 3D nonlinear anisotropic finite deformation inelasticity model that (1) exhibits rate-independent or dependent hysteretic response (i.e., response wherein reversal of the external stimuli does not cause reversal of the path in state space) with or without yield surfaces. The hysteresis persists with quasistatic loading. (2) Encompasses a wide range of different types of inelasticity models (such as Mullins effect in rubber, rock and soil mechanics, traditional metal plasticity, hysteretic behavior of shape memory materials) into a simple unified framework that is relatively easy to implement in computational schemes and (3) does not require any a priori particular notion of plastic strain or yield function. The core idea behind the approach is the development of an system of implicit rate equations that allow for the continuity of the response but with different rates along different directions. The theory, which is in purely mechanical setting, subsumes and generalizes many commonly used approaches for hypoelasticity and rate-independent plasticity. We illustrate its capability by modeling the Mullins effect which is the inelastic behavior of certain rubbery materials. We are able to simulate the entire cyclic response without the use of additional internal variables, i.e., the entire response is modeled by using an implicit function of stress and strain measures and their rates.     

A Study on Finite Orthotropic Plasticity using Generalized Measures

A numerical study of anisotropic plasticity using generalized strain and stress measures [1] according to [2] is presented. The constitutive frame is described and an algorithmic implementation is proposed. The coaxiality of generalized and standard deformation measures is exploited for the numerical treatment of the projection tensors.     

考虑主应力轴旋转的土体本构关系研究进展

对目前国内外考虑主应力轴旋转的试验研究及本构模型研究进行了总结分析,并对进一步研究提出了相应的建议．基于不同的加载条件,从纯主应力轴旋转和耦合主应力轴旋转两个方面,较全面的描述了主应力轴旋转情况下土体的基本变形特性,并对考虑主应力轴旋转的土体变形试验提出了进一步研究的建议．较为系统地评述了当前较有代表性的考虑主应力轴旋转的土体本构模型（边界面模型、多机构模型、运动硬化模型和广义塑性模型）,得出了广义塑性模型更适合用来描述考虑主应力轴旋转的土体变形特性的结论．总结未来考虑主应力轴旋转的土体本构关系研究的主要方向是:把握主应力轴旋转情况下土体变形的本质特性,建立推理严密、形式简单、适用方便的本构模型,并用来指导工程实践．     

A finite elasto-plasticity model for thick adhesive layers based on generalized stress-strain measure

The collective term adhesives includes a wide field of materials with a diversity of different material properties. Regarding high-strength adhesives, the assumption of small strains often holds according to their brittle behavior. The experience with plasticity models based on the additive decomposition into elastic and inelastic strains indicates an appropriate approach to characterize such materials. In some cases, due to a more ductile material response, the assumption of infinitesimal strains is not valid anymore. In particular this is the case for high-strength adhesives with additives like rubber. But ductile behavior is also observed for specific stress states in one adhesive, e.g. when the behavior for tensile is quite brittle while large shear-strains could appear. The objective of this work is to overcome the theoretical restriction of small strains and to archive the practical experiences. For the failure criterion two stress invariants are used, which involves the hydrostatic pressure as well as the deviator stress state. The flowrule is introduced for the evolution of the inelastic variables. Herein the flow rule has to be of non-associated type to ensure the thermodynamical consistency of the model. The plasticity model also includes hardening as well as softening. The presented finite strain model makes use of the fact that the eigenvalues for Green-Lagrange strains and generalized strains are the same. Thus the limit of applicability is extended to finite strains. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Interface Models With Damage in Gradient Plasticity

Gradient plasticity can account for experimentally observed size effects. Here, a previously developed gradient plasticity model is extended to account for interface delamination processes. The crystal plasticity model is based on the gradient of an equivalent plastic strain measure. A modification of the related boundary conditions allows for the formulation of a generalized cohezive zone model which can take into account the effect of interface delamination on the gradient plasticity solution. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

简单剪切有限塑性内时理论分析

在有限塑性内时理论中引入Jaumann率、广义Jaumann率、扶率及Wu率,并以此分析了简单剪切大变形问题．结果验证了简单剪切变形中,采用次弹性或内时刚塑性材料的Jaumann率客观模型,随单调递增的剪切变形剪切应力和法向应力都会出现振荡现象．这说明振荡现象的出现不取决于弹塑性模型,而与选取不同的客观率有很大的关系．同时指出在简单剪切大变形时,法向应力并不为零．     

Dynamic multi-dimensional bin packing

A natural generalization of the classical online bin packing problem is the dynamic bin packing problem introduced by Coffman et al. (1983) [7]. In this formulation, items arrive and depart and the objective is to minimize the maximal number of bins ever used over all times. We study the oriented multi-dimensional dynamic bin packing problem for two dimensions, three dimensions and multiple dimensions. Specifically, we consider dynamic packing of squares and rectangles into unit squares and dynamic packing of three-dimensional cubes and boxes into unit cubes. We also study dynamic d-dimensional hypercube and hyperbox packing. For dynamic d-dimensional box packing we define and analyze the algorithm NFDH for the offline problem and present a dynamic version. This algorithm was studied before for rectangle packing and for square packing and was generalized only for multi-dimensional cubes. We present upper and lower bounds for each of these cases.     

Instability prevention at the modeling of large elasto-plastic strains under cyclic loading

The cyclic instability phenomenon is investigated at the modeling of large elasto-plastic strains. The possible causes of the cyclic instability and conditions ensuring cyclical stability of elasto-plastic models are analyzed for the case of large strains. Among the possible causes of the cyclic instability the following are considered: the method of strain decomposition on elastic and plastic parts; the constitutive law for the elastic deformation (hypo- and hyper-elasticity); the constitutive equation for the plastic deformation; the constitutive relation for the plastic spin; kinematic hardening law, in particular, the type of the objective rate in the generalized Prager's law. Predictions of 50 various models of the elasto-plastic material have been compared in order to find the causes of the cyclic instability. Two test problems are considered: cyclic simple shear, combined cyclic simple shear and tension-compression. Results of numerical experiments for the various material models are presented and discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the absence of a preferred configuration in constitutive relations of finite strain elasto-plasticity

There exists a large variety of different techniques used to generalize the Prandtl-Reuss relations of small strain elasto-plasticity to finite strains. Obviously, any systematic study of such generalization techniques should be based on some sound principles. In this work, one aspect is taken into account, concerning arbitrary isochoric change of the local reference configuration. We analyze some models of finite strain elasto-plasticity in order to find out if the specific choice of the reference configuration can affect the predicted stress response. For material models which remain invariant under the reference change, no preferred reference configuration exists. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cyclic subsemigroups of symmetric inverse semigroups

A generalization of the cycle notation for permutations is introduced for partial one-one transformations (charts). Notational representation theorems for charts that generalize those of permutations are given. Notational multiplication of charts is developed and then applied to yield a transparent proof of Frobenius' result which bounds the idempotent in the cyclic subsemigroup. Lastly, the well known result that the structure of the cyclic subgroups of the finite symmetric groups is determined from combinations of disjoint cycles is generalized to the cyclic subsemigroups of the finite symmetric inverse semigroups.     

Experimental and Numerical Investigation of Nonlinear Piezoelectric Effects Including Minor Hystereses

Piezoelectric ceramics are often used in active structures for shape and vibration control. Since the operation range is not limited to small signals the nonlinear behaviour of the actuator under high electric loads has to be known. There are several approaches in literature to model the hysteretic effects, each having its assets and drawbacks. When a model is able to reproduce the minor loops of the strain - electric field hysteresis, it often lacks the consideration of stress dependence which is fundamental for actuators attached to elastic structures. On the other hand constitutive models which take into account all ferroelectric and ferroelastic effects are not capable of representing the minor hystereses in acceptable calculation times. In this work a phenomenological constitutive model is verified using the experimental data of an active plate structure. Therefore, the ceramic is characterised under mechanically unconstrained conditions and afterwards attached onto a steel plate. The bonding to the substructure leads to a mechanical stress depending on the actuation state of the ceramic. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Microplane modeling of cyclic behavior of concrete: a gradient plasticity-damage formulation

A microplane model is developed to simulate the behavior of concrete under cyclic loading conditions. Pure damage mechanics or pure plasticity models yield satisfactory results for concrete under monotonic loading but cannot capture correctly the unloading and reloading response. Therefore, coupling damage and plasticity is necessary for accurate constitutive modeling of concrete. The microplane model offers a straightforward approach to simulate induced anisotropy by formulating the material laws on many randomly oriented planes. Distinguishing between compression and tension response using the proper plastic yield function and damage laws is considered. Furthermore, gradient enhancement is employed to handle the pathological mesh sensitivity related to strain softening. The new formulation is implemented within a 3D finite element code and a numerical example is simulated and compared to experiments in order to evaluate the capabilities of the model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards variational constitutive updates for finite strain plasticity models based on non-associative evolution equations

In this contribution, first steps towards variational constitutive updates for finite strain plasticity theory based on non-associative evolution equations are presented. These schemes allow to compute the unknown state variables such as the plastic part of the deformation gradient, together with the deformation mapping, by means of a fully variational minimization principle. Therefore, standard optimization algorithms can be applied to the numerical implementation leading to a very robust and efficient numerical implementation. Particularly, for highly non-linear, singular or nearly ill-posed physical models like that corresponding to crystal plasticity showing a large number of possible active slip planes, this is a significant advantage compared to standard constitutive updates such as the by now classical return-mapping algorithm. While variational constitutive updates have been successfully derived for associative plasticity models, their extension to more complex constitutive laws, particularly to those featuring non-associative evolution equations, is highly challenging. In the present contribution, a certain class of non-associative finite strain plasticity models is discussed and recast into a variationally consistent format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized boundary conditions on representative volume elements and their use in determining the effective material properties

We discuss generalized boundary conditions for representative volume elements (RVE), which include the classical boundary conditions as special cases. From the generalization, stochastic boundary conditions are derived. These allows to adjust the the stiffness of the boundary conditions smoothly between the extremal cases of homogeneous strain and homogeneous stress boundary conditions. We found that it needs to be distinguished between the resistance of the boundary conditions against homogeneous and inhomogeneous RVE deformation. The stochastic BC can combine the moderate stiffness of the well known periodic boundary conditions with the high resistance against localization of the homogeneous strain boundary conditions. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Method to Estimate the Inelastic Response in Notched Structures of Anisotropic Materials with Elastic ‐ Perfectly Plastic Behavior

The fatigue strength of engineering structures is often limited by notches which arise from constructive features or manufacturing defects. The constitutive behavior in notch regions is then characterized by small plastic zones which are contained in an elastic region. To avoid costly plastic calculations, approximate methods have been developed to estimate the inelastic stress‐strain response at the notch tip. One of the first and best known approximate models is that of Neuber which, over the last 40 years, has received considerable attention particularly in connection with fatigue life prediction. Numerous studies have been conducted to the verification and the generalization of the Neuber approach which respect to multiaxiality, cyclic loading and creep conditions. Recently an extension of the Neuber method to anisotropic materials has been proposed in [1] and applied to directionally solidified and single crystal Nickel based superalloys as they are used in high temperature material applications. In this short notice we modify the approach in [1] for the special case of an elastic ‐ perfectly plastic anisotropic material.     

Analytic solutions of a reducible strain gradient elasticity model for solid cylinder with a cavity and its application in zonal failure

In this work, the reducibility condition of the fourth-order equilibrium equation in the strain gradient elasticity (SGE) model for solid cylinder with a cavity is obtained. When the reducibility condition is satisfied, the analytic displacement, generalized radial stress, and generalized angular stress can be solved out, and according to the higher-order coefficients, internal length scale, and Lamé constants, the displacement, generalized radial stress, and generalized angular stress are classified into four types: (1) conventional elasticity solution, (2) quasiperiodic SGE solution, (3) monotonous SGE solution, and (4) non-real-number solution. Quasiperiodic generalized radial stress and generalized angular stress are used to explain the occurrence of zonal failure of surrounding rock of a circular roadway. Numerical analysis with MATLAB is applied to study the influence of loading on zonal failure of surrounding rock of a circular raodway.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

Exponentiated generalized linear exponential distribution

A new generalization of the linear exponential distribution is recently proposed by Mahmoud and Alam [1], called as the generalized linear exponential distribution. Another generalization of the linear exponential was introduced by Sarhan and Kundu  and , named as the generalized linear failure rate distribution. This paper proposes a more generalization of the linear exponential distribution which generalizes the two. We refer to this new generalization as the exponentiated generalized linear exponential distribution. The new distribution is important since it contains as special sub-models some widely well known distributions in addition to the above two models, such as the exponentiated Weibull distribution among many others. It also provides more flexibility to analyze complex real data sets. We study some statistical properties for the new distribution. We discuss maximum likelihood estimation of the distribution parameters. Three real data sets are analyzed using the new distribution, which show that the exponentiated generalized linear exponential distribution can be used quite effectively in analyzing real lifetime data.     

Simulation of thick adhesive layers at finite strains utilizing an associated flow rule in a thermodynamically consistent framework

The application of elastoplastic material models is commonly used for the modelling of adhesive layers with high strength adhesives as realized with polyurethane or epoxy resin. To fulfill thermodynamic consistency often restrictions on the choice of material parameters are requested. One of them is the introduction of a non-associated flow rule, which always ensures positive dissipation. Nevertheless, this assumption is a non-essential criterion, which will be addressed in this work. Continuing along this argumentation, the constitutive relations for the material is modified based on an associated flow rule. The applied model for the simulation of the adhesives is based on a small strain theory. A yield surface including two stress invariants, the hydrostatic pressure as well as the deviator stress state, set the elastic limit of the material response. Linear as well as exponential hardening is incorporated and material softening that arises subsequently is also included by substituting effective invariants in the yield function. This material model as proposed from literature was extended to finite strain application with the concept of generalized stress-strain-measure, which was realized in a previous work. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)