Bending of an elastoplastic Hencky bar-chain: from discrete to nonlocal continuous beam models

The static behavior of an elastoplastic one-dimensional lattice system in bending, also called a microstructured elastoplastic beam or elastoplastic Hencky bar-chain (HBC) system, is investigated. The lattice beam is loaded by concentrated or distributed transverse monotonic forces up to the complete collapse. The phenomenon of softening localization is also included. The lattice system is composed of piecewise linear hardening–softening elastoplastic hinges connected via rigid elements. This physical system can be viewed as the generalization of the elastic HBC model to the nonlinear elastoplasticity range. This lattice problem is demonstrated to be equivalent to the finite difference formulation of a continuous elastoplastic beam in bending. Solutions to the lattice problem may be obtained from the resolution of piecewise linear difference equations. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process. The new nonlocal elastoplastic theory associated with both a distributed nonlocal elastoplastic law coupled to a cohesive elastoplastic model depends on length scales calibrated from the spacing of the lattice model. Differential equations of the nonlocal engineering model are solved for the structural configurations investigated in the lattice problem. It is shown that the new micromechanics-based nonlocal elastoplastic beam model efficiently captures the scale effects of the elastoplastic lattice model, used as the reference. The hardening–softening localization process of the nonlocal continuous model strongly depends on the lattice spacing which controls the size of the nonlocal length scales.     

Novel variational formulations for nonlocal plasticity

A nonlocal structural model of softening plasticity is considered in the framework of the internal variable theories of inelastic behaviours of associative type. The finite-step nonlocal structural problem in a geometrically linear range is formulated according to a backward difference scheme for time integration of the flow rule. The related finite-step variational formulation in the complete set of local and nonlocal state variables is recovered. A family of mixed nonlocal variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. The specialization of a mixed variational formulation to existing nonlocal models of softening plasticity, assuming both linear and nonlinear constitutive behaviour, is provided to show the effectiveness of the theory.     

Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage

The paper shows that spectral wave propagation analysis reveals in a simple and clear manner the effectiveness of various regularization techniques for softening materials, i.e., materials for which the yield limits soften as a function of the total strain. Both plasticity and damage models are considered. It is verified analytically in a simple way that the nonlocal integral-type model with degrading yield limit depending on the total strain works correctly if and only one adopts an unconventional nonlocal formulation introduced in 1994 by Vermeer and Brinkgreve (and in 1996 by Planas, and by Strömberg and Ristinmaa), which is here called, for the sake of brevity, ‘over-nonlocal’ because it uses a linear combination of local and nonlocal variables in which a negative weight imposed on the local variable is compensated by assigning to the nonlocal variable weight greater than 1 (this is equivalent to a nonlocal variable with a smooth positive weight function of total weight greater than 1, normalized by superposing a negative delta-function spike at the center). The spectral approach readily confirms that the nonlocal integral-type generalization of softening plasticity with an additive format gives correct localization properties only if an over-nonlocal formulation is adopted. By contrast, the nonlocal integral-type generalization of softening plasticity with a multiplicative format provides realistic localization behavior, just like the nonlocal integral-type damage model, and thus does not necessitate an over-nonlocal formulation. The localization behavior of explicit and implicit gradient-type models is also analyzed. A simple analysis shows that plasticity and damage models with gradient-type localization limiter, whether explicit or implicit, have very different localization behaviors.     

Nonlocal and gradient rate plasticity

This paper deals with a formulation of nonlocal and gradient plasticity with internal variables. The constitutive model complies with local internal variables which govern kinematic hardening and isotropic softening and with a nonlocal corrective internal variable defined either as the sum between a new internal variable and its spatial weighted average or as the gradient of a measure of plastic strain. The rate constitutive problem is cast in the framework provided by the convex analysis and the potential theory for monotone multivalued operators which provide the suitable tools to perform a theoretical analysis of such nonlocal and gradient problems. The validity of the maximum dissipation theorem is assessed and constitutive variational formulations of the rate model are provided. The structural rate problem for an assigned load rate is then formulated. The related variational formulation in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular the generalization to the present nonlocal and gradient model of the principles of Prager–Hodge, Greenberg and Capurso–Maier is presented. Finally nonlocal variational formulations provided in the literature are derived as special cases of the proposed model.     

Reconcile the perfectly elastoplastic model to simulate the cyclic behavior and ratcheting

Perfectly elastoplastic constitutive model is modified through a smoothing factor introduced by Liu [Liu, C.-S., 2003. Smoothing elastoplastic stress–strain curves obtained by a critical modification of conventional models. Int. J. Solids Struct. 40, 2121–2145]. The new model allows plasticity to happen in a non-zero-measure yield volume in stress space, rather than that of conventional zero-measure yield surface, and within the yield volume the plastic modulus is varying continuously. It endows a specific strain-hardening rule of flow stress and is able to describe the phenomena of strain hardening, cyclic hardening, the Bauschinger effect, mean-stress relaxation, strain ratcheting, out-of-phase hardening, as well as erasure-of-memory. In order to suppress the over prediction of ratcheting we consider a scalar function of smoothing factor, which can simulate the saturation behavior of uniaxial/multiaxial strain ratcheting. These effects are demonstrated through numerical examples. The existence of stress equilibrium point and limiting surface is a natural result without requiring an extra design. Moreover, the non-linear constitutive equations can be converted into a linear system for augmented stress in the Minkowski space, of which the symmetry group is a proper orthochronous Lorentz group SO_o(5, 1). The augmented stress is a time-like vector, moving on hyperboloids inside the cone. When taking the Prager kinematic hardening rule into account we can simulate some cyclic behaviors of SAE 4340 and grade 60 steels within a certain accuracy through the use of only three material constants and a fixed smoothing factor. To simulate the ratcheting behaviors of SS304 stainless steel we allow the smoothing factor to be an exponential decaying function of λ.     

Nonlinear softening and hardening nonlocal bending stiffness of an initially curved monolayer graphene

In the present article, the governing nonlinear nonlocal elastic equations are obtained for a monolayer graphene with an initial curvature and the related softening and hardening bending stiffness is analytically calculated. The effects of large deformation, initial curvature, discreteness and direction of chiral vector on the bending stiffness of the monolayer graphene are discussed in detail. A behavior more complex than previously reported in the literature emerges. It is found that the bending stiffness of graphene strongly depends on the initial configuration, showing not obvious maxima and minima, and suggesting the possibility of a smart tuning.     

Mechanical behaviors of electrostatic microresonators with initial offset imperfection: qualitative analysis via time-varying capacitors

This paper investigates analytically and numerically the effect of initial offset imperfection on the mechanical behaviors of microbeam-based resonators. Symmetry breaking of DC actuation, due to different initial offset distances of microbeam to lower and upper electrodes, is concerned. For qualitative analysis, time-varying capacitors are introduced and a lumped parameter model, considering nonlinear electrostatic force and midplane stretching of microbeam, is adopted to examine the system statics and dynamics. The Method of Multiple Scales (MMS) is applied to determine the primary resonance solution under small vibration assumption. Meanwhile, the Finite Difference Method (FDM) combined with Floquet theory is utilized to generate frequency response curves for medium- and large-amplitude vibration simulations. Static bifurcation, phase portrait and Hamiltonian function are firstly investigated to examine the system inherent behaviors. Besides, basins of attraction are briefly depicted to grasp the effects of initial offset and AC excitation on the system global dynamics. Then, variation of equivalent natural frequency versus DC voltage is analyzed. Results show that initial offset may induce complex frequency rebound phenomenon as well as a separate frequency branch under secondary pull-in condition. In what follows, emergences of softening, linear and hardening vibration are classified through discussing a key parameter obtained from the frequency response equation. New linear behavior induced by initial offset imperfection is found, which exhibits much higher sensitivity to DC voltage. Medium- and large-amplitude in-well motions are also investigated, indicating the existence of alternations of softening and hardening behaviors. Finally, lumped parameters are deduced via Galerkin procedure, and case studies are provided to illustrate the effectiveness of the whole analysis.     

Variational formulations,convergence and stability properties in nonlocal elastoplasticity

A thermodynamically consistent formulation of nonlocal plasticity in the framework of the internal variable theories of inelastic behaviors of associative type is presented. A family of mixed variational formulations, with different combinations of state variables, is provided starting from the finite-step nonlocal elastoplastic structural problem. It is shown that a suitable minimum principles provides a rational basis to exploit the iterative elastic predictor-plastic corrector algorithm in terms of the dissipation functional. A sufficient condition is proved for the convergence of the iterative elastic predictor-plastic corrector algorithm based on a suitable choice of the elastic operator in the prediction phase and a necessary and sufficient condition for the existence of a unique solution (if any) of the nonlocal problem at hand is then provided. The nonlinear stability analysis of the nonlocal problem is carried out following the concept of nonexpansivity proposed in local plasticity.     

基于能量等效原理的应变局部化分析:Ⅰ.一维解析解

基于热力学第一定律和非局部塑性理论,提出了一种求解应变局部化问题的非局部方法.对材料的每一点定义了局部和非局部两种状态空间,局部状态空间的内变量通过非局部权函数映射到非局部空间,成为非局部内变量.在应变软化过程中,局部状态空间中的塑性变形服从正交流动法则,材料的软化律在非局部状态空间中被引入.通过两个状态空间的塑性应变能耗散率的等效,得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布.应用本方法导出了一维应变局部化问题的解析解.解析解表明,应变局部化区域的尺寸只与材料内尺度有关;对于高斯型非局部权函数,局部化区域的尺寸大约是材料内尺度的6倍.一维算例表明,局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关,变形局部化区域的尺寸随着材料内尺度的减小而减小,同时塑性应变也随着材料内尺度的减小变得更加集中.当内尺度趋近于零时,应用本文方法得到的解与采用传统的局部塑性理论得到的解相同.     

基于统一强度理论的岩石弹塑性本构模型及其数值实现

基于弹塑性力学理论,以统一强度准则为屈服准则,建立了考虑硬化/软化行为和应变率效应的岩石弹塑性本构模型;采用Fortran语言通过LS-DYNA的用户自定义材料接口(Umat)对该弹塑性本构模型进行编程,并把该程序生成求解器以达到对该模型进行应用的目的;通过岩石的单轴压缩实验和SHPB实验对所建的弹塑性本构模型进行验证,结果表明,该弹塑性本构模型能够反映岩石在准静态和动态下的力学行为。     

Prediction of the initial thickness of shear band localization based on a reduced strain gradient theory

Plastic flow localization in ductile materials subjected to pure shear loading and uniaxial tension is investigated respectively in this paper using a reduced strain gradient theory, which consists of the couple-stress (CS) strain gradient theory proposed by Fleck and Hutchinson (1993) and the strain gradient hardening (softening) law (C–W) proposed by Chen and Wang (2000). Unlike the classical plasticity framework, the initial thickness of the shear band and the strain rate distribution in both cases are predicted analytically using a bifurcation analysis. It shows that the strain rate is obviously non-uniform inside the shear band and reaches a maximum at the center of the shear band. The initial thickness of the shear band depends on not only the material intrinsic length l_cs but also the material constants, such as the yield strength, ultimate tension strength, the linear hardening and softening shear moduli. Specially, in the uniaxial tension case, the most possible tilt angle of shear band localization is consistent qualitatively with the existing experimental observations. The results in this paper should be useful for engineers to predict the details of material failures due to plastic flow localization.     

Fractional visco-elastic Euler–Bernoulli beam

Aim of this paper is the response evaluation of fractional visco-elastic Euler–Bernoulli beam under quasi-static and dynamic loads. Starting from the local fractional visco-elastic relationship between axial stress and axial strain, it is shown that bending moment, curvature, shear, and the gradient of curvature involve fractional operators. Solution of particular example problems are studied in detail providing a correct position of mechanical boundary conditions. Moreover, it is shown that, for homogeneous beam both correspondence principles also hold in the case of Euler–Bernoulli beam with fractional constitutive law. Virtual work principle is also derived and applied to some case studies.     

Nonlocal microplane model with strain-softening yield limits

The paper deals with the problem of nonlocal generalization of constitutive models such as microplane model M4 for concrete, in which the yield limits, called stress–strain boundaries, are softening functions of the total strain. Such constitutive models call for a different nonlocal generalization than those for continuum damage mechanics, in which the total strain is reversible, or for plasticity, in which there is no memory of the initial state. In the proposed nonlocal formulation, the softening yield limit is a function of the spatially averaged nonlocal strains rather than the local strains, while the elastic strains are local. It is demonstrated analytically as well numerically that, with the proposed nonlocal model, the tensile stress across the strain localization band at very large strain does soften to zero and the cracking band retains a finite width even at very large tensile strain across the band only if one adopts an “over-nonlocal” generalization of the type proposed by Vermeer and Brinkgreve [In: Chambon, R., Desrues, J., Vardoulakis, I. (Eds.), Localisation and Bifurcation Theory for Soils and Rocks, Balkema, Rotterdam, 1994, p. 89] (and also used by Planas et al. [Basic issue of nonlocal models: uniaxial modeling, Tecnical Report 96-jp03, Departamento de Ciencia de Materiales, Universidad Politecnica de Madrid, Madrid, Spain, 1996], and by Strömberg and Ristinmaa [Comput. Meth. Appl. Mech. Eng. 136 (1996) 127]). Numerical finite element studies document the avoidance of spurious mesh sensitivity and mesh orientation bias, and demonstrate objectivity and size effect.     

On the plastic bifurcation and post-bifurcation of axially compressed beams

The paper presents two new results in the domain of the elastoplastic buckling and post-buckling of beams under axial compression. (i) First, the tangent modulus critical load, the buckling mode and the initial slope of the bifurcated branch are given for a Timoshenko beam (with the transverse shear effects). The result is derived from the 3D J₂ flow plastic bifurcation theory with the von Mises yield criterion and a linear isotropic hardening. (ii) Second, use is made of a specific method in order to provide the asymptotic expansion of the post-critical branch for a Euler-Bernoulli beam, exhibiting one new non-linear fractional term. All the analytical results are validated by finite element computations.     

Post-buckling behaviour of slender structures with a bi-linear bending moment-curvature relationship

Certain classes of slender structures of complex cross-section or fabricated from specialised materials can exhibit a bi-linear bending moment-curvature relationship that has a strong influence on their global structural behaviour. This condition may be encountered, for instance, in (a) non-linear elastic or inelastic post-buckling problems if the cross-section stiffness may be well approximated by a bi-linear model; (b) multi-layered structures such as stranded cables, power transmission lines, umbilical cables and flexible pipes where the drop in the bending stiffness is associated with an internal friction mechanism. This paper presents a mathematical formulation and an analytical solution for such slender structures with a bi-linear bending moment versus curvature constitutive behaviour and subjected to axial terminal forces. A set of five first-order non-linear ordinary differential equations are derived from considering geometrical compatibility, equilibrium of forces and moments and constitutive equations, with hinged boundary conditions prescribed at both ends, resulting a complex two-point boundary value problem. The variables are non-dimensionalised and solutions are developed for monotonic and unloading conditions. The results are presented in non-dimensional graphs for a range of critical curvatures and reductions in bending stiffness, and it is shown how these parameters affect the structure's post-buckling behaviour.     

一般加载规律的弹塑性本构关系

将有关文献给出一般加载规律一维全量理论的简单模型推广到一般加载规律的一维增量理论,进而推广到一般加载规律的多维增量理论,在此基础上,建立了推导一般加载规律的多维增量理论的本构关系的一种途径。应用这种途径,从应力空间的加载函数和应变空间的加载函数出发,推导了等向强化材料和被加热的等向强化材料的一般加载规律的弹塑性本构关系的两种表示形式。理论和实例均表明,这种途径对等向强化材料、随动强化材料和理想弹塑性材料均适用。     

Nonlinear interaction of geometrical and material properties in sandwich beam instabilities

The first part of this paper is dedicated to the analytical and numerical characterization of local and global sandwich beam instabilities in a perfect linear framework. Analytical loads are extracted from an original unified model and used to understand in depth, through a parametric study, the role played by each geometrical and material parameter in the development of global as well as local instabilities. Also, the effects of the combinations of these characteristics is used to draw precious design indications. A low CPU time-consuming simplified model is then built and assessed. Critical loads and wavelengths computed from this model are shown to correlate very well with analytical predictions. It is established that this first approach is essential in order to lead to more detailed investigations in a numerical nonlinear framework which is the aim of the second part. The first geometrical nonlinear investigations in which linear elastic materials are considered permit to isolate sandwich configurations developing super- or sub-critical post-buckling behaviours. As a general trend, unstable behaviours are rather related to the occurrence of geometrical localizations along the beam. This is illustrated by the drastic effects of the so-called interactive buckling onto the whole stiffness of the sandwich beam. Moreover, it is shown that sandwiches are very sensitive towards imperfection sizes and forms. Eventually, an elastoplastic constitutive law is introduced for the core. It is demonstrated that plastic flow and strain localization in the core, combined with the occurrence of instabilities, are associated with a drastic drop in the global beam stiffness and with a strong decrease of the maximum limit load for some cases. The phenomenon of shear crimping is also observed which can be assimilated to a post-bifurcated development of the global buckling mode.     

Modeling of intra-crystalline hardening of materials with particles

A two-level homogenization approach is developed for the micromechanical modeling of the elastoplastic behavior of polycrystals containing intracrystalline non-shearable particles. First, a micro-meso transition is employed to establish a constitutive relation for a single crystal containing particles. The behavior of an equivalent single crystal with particles is derived from the classical formulation of plasticity of the single crystal based on the Schmid's law and crystallographic gliding. Then, the transition to the macroscopic scale is performed with a self-consistent scheme to determine the elastoplastic behavior of the macro homogeneous material. The obtained global behavior is characterized by a mixed anisotropic and kinematic hardening related to an evolution of inter- and intra-granular material microstructure. Results have been analyzed in light of second and third order internal stresses developed during the plastic flow. Especially, yield surfaces have been determined for various preloadings and particle volume fractions.     

Analysis of the localization of deformation and the complete stress–strain relation for mesoscopic heterogeneous brittle rock under dynamic uniaxial tensile loading

Stress redistribution induced by excavation results in the tensile zone in parts of the surrounding rock mass. It is significant to analyze the localization of deformation and damage, and to study the complete stress–strain relation for mesoscopic heterogeneous rock under dynamic uniaxial tensile loading. On the basis of micromechanics, the complete stress–strain relation including linear elasticity, nonlinear hardening, rapid stress drop and strain softening is obtained. The behaviors of rapid stress drop and strain softening are due to localization of deformation and damage. The constitutive model, which analyze localization of deformation and damage, is distinct from the conventional model. Theoretical predictions have shown to consistent with the experimental results.     

On geometrically exact post-buckling of composite columns with interlayer slip—The partially composite elastica

This paper is focused on the geometrically exact elastic stability analysis of two interacting kinematically constrained, flexible columns. Possible applications are to partially composite or sandwich columns. A partially composite column composed of two inextensible elastically connected sub-columns is considered. Each sub-column is modeled by the Euler–Bernoulli beam theory and connected to each other via a linear constitutive law for the interlayer slip. The paper discusses the validity of parallel and translational kinematics beam assumptions with respect to the interlayer constraint. Buckling and post-buckling behavior of this structural system are studied for cantilever columns (clamped-free boundary conditions). A variational formulation is presented in order to derive relevant boundary conditions in a geometrically exact framework. The exact post-buckling behavior of this partially composite beam-column is investigated analytically and numerically. The Euler elastica problem is obtained in the case of non-composite action. The “partially composite elastica” is then treated analytically and numerically, for various values of the interaction connection parameter. An asymptotic expansion is performed to evaluate the symmetrical pitchfork bifurcation, and comparisons are made with some exact numerical results based on the numerical treatment of the non-linear boundary value problem. A boundary layer phenomenon, similar to that also observed for the linearized bending analysis of partially composite beams, is observed for large values of the connection parameter. This boundary layer phenomenon is investigated with a straightforward asymptotic expansion, that also is valid for large rotations. Finally, the paper analyses the effect of some additional imperfection eccentricities in the loading mode, that lead to some pre-bending phenomena.