Numerical simulation of shear band formation in elastic material by energy relaxation

Following up the previous work, [1], a new approach to the problem of shear localization is proposed, based on energy minimization principles associated with micro-structure developments. In this approach, two different material models are included. They represent the behaviour of material at very small strain and very large strain, respectively. Herein, shear bands are treated as the micro-shearing of rank-one laminates. The thickness of a shear band represented by its volume fraction is assumed to tend to zero while the strain inside the shear band tends to infinity. The existence of shear bands in the structure leads to an ill-posed problem which can be solved by means of energy relaxation. The performance of the proposed energy relaxation is demonstrated through numerical simulation of a tension test under plane strain conditions. The presented numerical simulation shows that there is no mesh-dependence. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of shear localization in elastic material by means of energy relaxation

A new approach to the problem of shear localization is proposed. It is based on the energy minimization principle associated with micro–structure developments and the micro–shearing of a rank–one laminate which is aligned to shear band. The thickness of the shear band represented by the volume fraction is assumed to tend to zero. The non–convex energy due to the formation of the shear band is solved by energy relaxation in order to ensure that the problem is well–posed. An application of the proposed formulation to isotropic linear elastic material is presented. The capability of the proposed energy relaxation is demonstrated through numerical simulation of a plane strain tension. The numerical results demonstrate that there is no mesh–dependence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plane harmonic wave propagation in three-dimensional composite media

An important characteristic of waves propagating through periodic materials is the existence of stop bands. A stop band implies the range of frequencies over which a medium completely reflects all incident waves and there is no transmission. Predicting stop band phenomena in periodic materials is regarded as the first step toward designing composite microstructures capable of propagating energy in a predetermined manner. In this paper a global–local modeling methodology previously proposed by the authors is used to successively predict stop bands in three-dimensional composite media. Numerical results reveal that the first stop band of the considered microstructures occurs where an acoustic shear mode veers with the lowest optical branch of the same symmetry class.     

On the variationally consistent modeling of material failure

Material failure associated with cracks or shear bands is frequently analyzed by utilizing so-called cohesive models. Such models are based on traction-separation laws. Within such approaches, the stress vector of the considered crack or shear band is related to its conjugate variable being the respective displacement jump (such as the material separation or the crack opening). In the present work, a framework suitable for the analysis of shear bands is discussed. All models belonging to that framework are consistently derived from thermodynamical principles. Hence, the second law of thermodynamics is automatically fulfilled. Furthermore, a variational principle strongly relying on the postulate of maximum dissipation is elaborated leading finally to a variationally consistent implementation. More precisely, all state variables, together with the unknown deformation mapping, follow naturally from minimizing an incrementally defined potential within the presented algorithmic formulation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of shear bands in solids by means of computational homogenization

A novel fully three–dimensional framework for the numerical analysis of shear bands in solids undergoing large deformations is presented. The effect of micro shear bands on the macroscopic material response is computed by means of a homogenization strategy. More precisely, a strain–driven approach which complies well with displacement–driven finite element formulations is adopted. The proposed implementation is based on periodic boundary conditions for the micro–scale. Details about the implementation of the resulting constraints into a three–dimensional framework are discussed. The shear bands occurring at the micro–scale are modeled by a cohesive zone law, i.e., the tangential component of the traction vector governs the relative shear sliding displacement. This law is embedded into a Strong Discontinuity Approach (SDA). To account for realistic sliding modes, multiple shear bands are allowed to form and propagate in each finite element. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topology optimization of periodic Mindlin plates

Periodic structures exhibit unique dynamic characteristics that make them act as tunable mechanical filters for wave propagation. As a result, waves can propagate along the periodic structures only within specific frequency bands called the ‘pass bands’ and wave propagation is completely blocked within other frequency bands called the ‘stop bands’ or ‘band gaps’. The spectral width of these bands can be optimized using topology optimization. In this paper, topology optimization is used to maximize the fundamental natural frequency of Mindlin plates while enforcing periodicity. A finite element model for Mindlin plates is presented and used along with an optimization algorithm that accounts for the periodicity constraint in order to determine the optimal topologies of plates with various periodic configurations. The obtained results demonstrate the effectiveness of the proposed design optimization approach in generating periodic plates with optimal natural frequency and wide stop bands. The presented approach can be invaluable design tool for many structures in order to control the wave propagation in an attempt to stop/confine the propagation of undesirable disturbances.     

爆炸和冲击载荷下金属材料及结构的动态失效仿真

通过数值模拟研究爆炸冲击载荷下金属材料和结构的动态失效规律,对表征爆炸冲击毁伤效应及设计新型抗冲击结构有重要意义.强动载下金属材料失效涉及材料大变形、热力耦合、材料状态变化等多个复杂物理过程,给数值仿真带来了极大挑战,其中包括裂纹、剪切带等复杂失效模式的几何描述、动态失效准则的确定、塑性与损伤耦合演化的描述等问题.针对...     

Identification of the Material Parameters of a Micropolar Model for Granular Materials

Granular frictional materials show a complex stress‐strain behaviour depending on the stress state and the load history. Furthermore, biaxial experiments exhibit the occurrence of shear band phenomena as the result of the localization of plastic strains. It is well known that the onset of shear bands is associated with microrotations of the granular microstructure, which has a significant influence on the macroscopic behaviour. Consequently, the macroscopic material must result in a micropolar model, which incorporates rotational degrees of freedom. After the formulation of the constitutive equations and the numerical implementation, it is necessary to determine all required material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic stress-softening model for compressible solids

A general constitutive theory for anisotropic stress softening in compressible solids is presented. The constitutive equation describes anisotropic strain induced behaviour of an initially “isotropic” virgin material. Parameters which characterise damage are proposed together with a concept of damage function. In order to develop an anisotropic stress-softening theory for compressible materials in close parallel to a recent incompressible anisotropic theory, the right stretch tensor is decomposed into its isochoric and dilatational parts. The ’free’ energy is expressed as a function of the dilatation, modified principal stretches, a volume change parameter and invariants of the dyadic products of the principal directions of the right stretch tensor and two structural tensors. A class of free energy functions is discussed and a special form of this class which satisfies the Clausius–Duhem inequality is proposed. Results of the theory applied to uniaxial tension, bulk compression and simple shear deformations are given. A sequence of deformations involving shear, hydrostatic-compression and hydrostatic-tension deformations is also investigated. In the case of hydrostatic-tension deformation, the stress softening is due to cavitation damage. The theoretical results obtained are consistent with expected behaviour and compare well with experimental data.     

Scalar Conservation Laws with Nonconstant Coefficients with Application to Particle Size Segregation in Granular Flow

Granular materials will segregate by particle size when subjected to shear, as occurs, for example, in avalanches. The evolution of a bidisperse mixture of particles can be modeled by a nonlinear first order partial differential equation, provided the shear (or velocity) is a known function of position. While avalanche-driven shear is approximately uniform in depth, boundary-driven shear typically creates a shear band with a nonlinear velocity profile. In this paper, we measure a velocity profile from experimental data and solve initial value problems that mimic the segregation observed in the experiment, thereby verifying the value of the continuum model. To simplify the analysis, we consider only one-dimensional configurations, in which a layer of small particles is placed above a layer of large particles within an annular shear cell and is sheared for arbitrarily long times. We fit the measured velocity profile to both an exponential function of depth and a piecewise linear function which separates the shear band from the rest of the material. Each solution of the initial value problem is nonstandard, involving curved characteristics in the exponential case, and a material interface with a jump in characteristic speed in the piecewise linear case.     

Simultaneous confidence bands for the integrated hazard function

The construction of the simultaneous confidence bands for the integrated hazard function is considered. The Nelson–Aalen estimator is used. The simultaneous confidence bands based on bootstrap methods are presented. Four methods of construction of such confidence bands are proposed. The weird and conditional bootstrap methods are used for resampling. Simulations are made to compare the actual coverage probability of the bootstrap and the asymptotic simultaneous confidence bands. It is shown that the equal-tailed bootstrap confidence band has the coverage probability closest to the nominal one. We also present application of our confidence bands to the data regarding survival after heart transplant. This research was partly supported by AGH grant No. 10.420.03.     

Kink banding in laminated composite structures

An analytical model for kink band deformations occurring in laminated composites is proposed based on geometric and potential energy principles. Whilst nonlinear material behaviour and imperfections are included the model considers a symmetric, multi-directional laminate lay-up where the kink band is assumed to occur in the central 0° laminae. First results compared to data in the literature are presented offering encouragement. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory

Buckling analysis of functionally graded micro beams based on modified couple stress theory is presented. Three different beam theories, i.e. classical, first and third order shear deformation beam theories, are considered to study the effect of shear deformations. To present a profound insight on the effect of boundary conditions, beams with hinged-hinged, clamped–clamped and clamped–hinged ends are studied. Governing equations and boundary conditions are derived using principle of minimum potential energy. Afterwards, generalized differential quadrature (GDQ) method is applied to solve the obtained differential equations. Some numerical results are presented to study the effects of material length scale parameter, beam thickness, Poisson ratio and power index of material distribution on size dependent buckling load. It is observed that buckling loads predicted by modified couple stress theory deviates significantly from classical ones, especially for thin beams. It is shown that size dependency of FG micro beams differs from isotropic homogeneous micro beams as it is a function of power index of material distribution. In addition, the general trend of buckling load with respect to Poisson ratio predicted by the present model differs from classical one.     

A constitutive model for granular materials with microstructures using the concept of energy relaxation

In this paper, we present a constitutive model for granular materials exhibiting microstructures using the concept of energy relaxation. Within the framework of Cosserat continuum theory the free energy of the material is enriched with an interaction energy potential taking into account the counter rotations of the particles. The enhanced energy potential fails to be quasiconvex. Energy relaxation theory is employed to compute the relaxed energy which yields all possible displacement and micro-rotations field fluctuations as minimizers. Based on a two-field variational principle the constitutive response of the material is derived. The developed constitutive model is then implemented in a finite element analysis program using the finite element method. Numerical simulations are presented to observe the localized deformation phenomenon in a granular medium. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

剪切变形下非晶态高聚物的力学行为

基于非平衡态热力学理论，提出了一个适用于不可压材料的新的热粘弹性本构模型．该模型将橡胶弹性理论中的非高斯分子网络模型推广到计及粘性和热效应的情形．通过引入一组二阶张量形式的内变量，建议了一个新的Helmholtz自由能表达式，从而可以用来合理描述内变量的演化规律．根据以上模型，重点研究了热粘弹性材料在简单剪切变形下的力学行为，考察了由于分子链取向分布的变化而产生的“粘性耗散诱导”各向异性，讨论了应变率效应和由于粘性耗散而导致的热软化效应对剪应力的影响．理论预测结果与G’Sell等人的实验数据的定性比较表明了新的本构模型的有效性。     

Modeling and simulation of chip formation in high speed cutting

The purpose of this short work is the thermomechanical modeling of shear band and chip formation during high-speed cutting. Shear bands develop in areas of maximal mechanical dissipation in which temperature-dependent softening dominates strainand strain-rate-dependent hardening. In the simulations, the well-known problem of the mesh-dependence of the shear-band development is addressed, involving both mesh size and mesh orientation. An example simulation is presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptivity in the context of high speed cutting

The purpose of this short work is the thermomechanical modeling of shear band and chip formation during high-speed cutting. Shear bands develop in areas of maximal mechanical dissipation in which temperature-dependent softening dominates strainand strain-rate-dependent hardening. In the simulations, the well-known problem of the mesh-dependence of the shear-band development is addressed, involving both mesh size and mesh orientation. An example simulation is presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Closed Unstretchable Knotless Ribbons and the Wunderlich Functional

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

     

Band structure of the spectra of Hamiltonians of regular polynucleotide duplexes

We calculate the band structure of the spectra of Hamiltonians of regular DNA duplexes and show that in single-stranded periodic polynucleotides whose period is determined by the number m of nucleotides in an elementary cell, the spectrum consists of m nonintersecting energy bands. In DNA duplexes, the number of energy bands is equal to 2m, and the bands can intersect. Discrete energy levels can be present in forbidden bands in the case of (semi)bounded chains or duplexes.     

Simulation-Based Simultaneous Confidence Bands in Multiple Linear Regression With Predictor Variables Constrained in Intervals

This article presents a method for the construction of a simultaneous confidence band for the normal-error multiple linear regression model. The confidence bands considered have their width proportional to the standard error of the estimated regression function, and the predictor variables are allowed to be constrained in intervals. Past articles in this area gave exact bands only for the simple regression model. When there is more than one predictor variable, only conservative bands are proposed in the statistics literature. This article advances this methodology by providing simulation-based confidence bands for regression models with any number of predictor variables. Additionally, a criterion is proposed to assess the sensitivity of a simultaneous confidence band. This criterion is defined to be the probability that a false linear regression model is excluded from the band at least at one point and hence this false linear regression model is correctly declared as a false model by the band. Finally, the article considers and compares several computational algorithms for obtaining the confidence band.