A finite element method on the plane elastic material analysis

Based on the finite element displacement method, a finite element method on the analysis of mechanical behaviour of plane elastic materials is proposed in this paper. By using this method and the corresponding computational program, the material behaviour of any unknown plane elastic material can be determined and all the elastic constants can be calculated.     

Shear stress analysis in engineering beams using deplanation field of special 2-D finite elements

This paper deals with the 2-D finite element shear stress analysis in beams, loaded by bending with shear and St. Venant’s torsion. The properties of these finite elements, like stiffness matrices as well as load vectors, are derived on the basis of their axial nodal displacements, e.g. by warping field. Proposed finite elements enable stress analysis independently of both cross-sectional member shape and material properties. Stiffness matrices and load vectors are derived for several finite element types. Material is assumed to be isotropic and linear elastic. For justification of the proposed stress analysis procedure, some examples are presented.     

A variable stiffness joint with superelastic material

The aim of this paper is to provide the design of a variable stiffness joint able to perform large rotations without undergoing plastic strain. The joint comprises two leaf springs of variable length, allowing to adjust the stiffness of the joint according to the desired performance. Large rotations are achieved by employing superelastic material for the leaf springs, instead of common linear elastic material. The behaviour of the joint has been investigated by means of finite element analyses; in addition, the first attempt to develop a simple mathematical model for the superelastic material as alternative to finite element method is described.     

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.     

Finite element solutions for nonhomogeneous nonlocal elastic problems

A finite element procedure for analysing nonhomogeneous nonlocal elastic 2D problems is presented and discussed. The procedure grounds on a variationally consistent approach known, in the relevant literature, as Nonlocal Finite Element Method. The latter is recast making use of a recently theorized phenomenological strain-difference-based nonhomogeneous nonlocal elastic model. The peculiarities of the numerical procedure together with the pertinent nonlocal operators are expounded and discussed. Two simple numerical 2D examples close the paper.     

Effective thermoelastic properties of graded doublyperiodic particulate matrix composites in varying externalstress fields

We consider a linear elastic composite medium, which consists of a homogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in a doubly periodicarray and subjected to inhomogeneous boundary conditions. The hypothesis of effective fieldhomogeneity near the inclusions is used. The general integral equation obtained reduces theanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusions insome representative volume element (RVE) . The integral equation is solved by a modifiedversion of the Neumann series; the fast convergence of this method is demonstrated for concreteexamples. The nonlocal macroscopic constitutive equation relating the cell averages of stress andstrain is derived in explicit iterative form of an integral equation. A doubly periodic inclusion fieldin a finite ply subjected to a stress gradient along the functionally graded direction is considered.The stresses averaged over the cell are explicitly represented as functions of the boundaryconditions. Finally, the employed of proposed explicit relations for numerical simulations oftensors describing the local and nonlocal effective elastic properties of finite inclusion pliescontaining a simple cubic lattice of rigid inclusions and voids are considered. The local andnonlocal parts of average strains are estimated for inclusion plies of different thickness. Theboundary layers and scale effects for effective local and nonlocal effective properties as well as foraverage stresses will be revealed.     

Size-dependent thermoelasticity of a finite bi-layered nanoscale plate based on nonlocal dual-phase-lag heat conduction and Eringen's nonlocal elasticity

The size effects on heat conduction and elastic deformation are becoming significant along with the miniaturization of the device and wide application of ultrafast lasers.In this work,to better describe the transient responses of nanostructures,a size-dependent thermoelastic model is established based on nonlocal dual-phase-lag(N-DPL)heat conduction and Eringen's nonlocal elasticity,which is applied to the one-dimensional analysis of a finite bi-layered nanoscale plate under a sudden thermal shock.In the numerical part,a semi-analytical solution is obtained by using the Laplace transform method,upon which the effects of size-dependent characteristic lengths and material properties of each layer on the transient responses are discussed systematically.The results show that the introduction of the elastic nonlocal parameter of Medium 1 reduces the displacement and compressive stress,while the thermal nonlocal parameter of Medium 1 increases the deformation and compressive stress.These findings may be beneficial to the design of nano-sized and multi-layered devices.     

A nonlocal model with strain-based damage

A thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented. The damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena. An important advantage of models with strain-based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations. A mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two-field variational formulation. Hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two-field variational formulation. Two examples concerning a one-dimensional bar in simple tension and a two-dimensional notched plate are addressed. No mesh dependence or boundary effects are apparent.     

A nonlocal constitutive model generated by matrix functions for polyatomic periodic linear chains

We establish a discrete lattice dynamics model and its continuum limits for nonlocal constitutive behavior of polyatomic cyclically closed linear chains being formed by periodically repeated unit cells (molecules), each consisting of \({n \geq 1}\) atoms which all are of different species, e.g., distinguished by their masses. Nonlocality is introduced by elastic potentials which are quadratic forms of finite differences of orders \({m \in \mathbf{N}}\) of the displacement field leading by application of Hamilton’s variational principle to nondiagonal and hence nonlocal Laplacian matrices. These Laplacian matrices are obtained as matrix power functions of even orders 2m of the local discrete Laplacian of the next neighbor Born-von-Karman linear chain. The present paper is a generalization of a recent model that we proposed for the monoatomic chain. We analyze the vibrational dispersion relation and continuum limits of our nonlocal approach. “Anomalous” dispersion relation characteristics due to strong nonlocality which cannot be captured by classical lattice models is found and discussed. The requirement of finiteness of the elastic energies and total masses in the continuum limits requires a certain scaling behavior of the material constants. In this way, we deduce rigorously the continuum limit kernels of the Laplacian matrices of our nonlocal lattice model. The approach guarantees that these kernels correspond to physically admissible, elastically stable chains. The present approach has the potential to be extended to 2D and 3D lattices.     

Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

The homogenization results obtained by Bacca et al. (2013a), to identify the effective second-gradient elastic materials from heterogeneous Cauchy elastic solids, are extended here to the case of phases having non-isotropic tensors of inertia. It is shown that the nonlocal constitutive tensor for the homogenized material depends on both the inertia properties of the RVE and the difference between the effective and the matrix local elastic tensors. Results show that: (i) orthotropic nonlocal effects follow from homogenization of a dilute distribution of aligned elliptical holes and, in the limit case, of cracks; (ii) even under the dilute assumption and isotropic local effective behaviour, homogenization may lead to effective nonlocal orthotropic properties.     

Non-local criteria for the borehole problem: Gradient Elasticity versus Finite Fracture Mechanics

Two nonlocal approaches are applied to the borehole geometry, herein simply modelled as a circular hole in an infinite elastic medium, subjected to remote biaxial loading and/or internal pressure. The former approach lies within the framework of Gradient Elasticity (GE). Its characteristic is nonlocal in the elastic material behaviour and local in the failure criterion, hence simply related to the stress concentration factor. The latter approach is the Finite Fracture Mechanics (FFM), a well-consolidated model within the framework of brittle fracture. Its characteristic is local in the elastic material behaviour and non-local in the fracture criterion, since crack onset occurs when two (stress and energy) conditions in front of the stress concentration point are simultaneously met. Although the two approaches have a completely different origin, they present some similarities, both involving a characteristic length. Notably, they lead to almost identical critical load predictions as far as the two internal lengths are properly related. A comparison with experimental data available in the literature is also provided.

     

Finite element modelling of rubber-like polymers based on chain statistics

In this work finite element simulations are conducted based on the micro structure of polymers in order to transfer the information of the micro level to the macro level. The micro structure of polymers is characterized by chain-like macromolecules linked together at certain points. In this way an irregular three-dimensional network is formed. Many authors use the tool of statistical mechanics to describe the deformation behaviour of the entire network. Most of these concepts can be reformulated as traditional continuum mechanical formulations. They are, however, restricted to affine deformation, regular chain arrangements and purely elastic material behaviour. For this reason, in the present contribution, we propose a new finite element-based simulation method for polymer networks which enables us to include non-affinity and arbitrary chain configurations. It can be easily extended to include chain breakage and reconnection.The polymer structure to be investigated, e.g. a rubber boot or a seal, is discretized by means of tetrahedral elements. To each edge of a tetrahedral element one truss element is attached which models the force–stretch behaviour of a bundle of polymer chains. Each of these tetrahedral unit cells represents the micro mechanical material behaviour in a certain point of the network. The proposed method provides the possibility to observe how changes at the microscopic level influence the macroscopic material behaviour. Such information is especially valuable for the polymer industry.     

Triply periodical particulate matrix compositesinvarying external stress fields

We consider a linear elastic composite medium, which consists of ahomogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in aperiodic arrayand subjected to inhomogeneous boundary conditions. The hypothesis of effectivefieldhomogeneity near the inclusions is used. The general integral equation obtained reducestheanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusionsinsome representative volume element (RVE) . The integral equation is solved by theFouriertransform method as well as by the iteration method of the Neumann series ( first-orderapproximation) . The nonlocal macroscopic constitutive equation relating the unit cellaverages ofstress and strain is derived in explicit closed forms either of a differential equation ofasecond-order or of an integral equation. The employed of explicit relations fornumericalestimations of tensors describing the local and nonlocal effective elastic properties aswell asaverage stresses in the composites containing simple cubic lattices of rigid inclusions andvoids areconsidered.     

基于应变梯度有限元的单层石墨烯振动研究

建立了单层石墨烯等效非局部薄板的一种新的有限元模型,并运用有限元法分析不同边界条件下单层石墨烯振动的小尺度效应。给出了基于弹性应变梯度理论下Kirchhoff板的振动方程。发展了一种4节点24自由度的板单元,用于离散化求解考虑微纳结构尺度效应的高阶微分方程。在研究四边简支板振动时,考虑应变梯度的非局部弹性有限元数值计算结果与理论分析结果相一致。用有限元方法研究了不同尺寸、振动波长、振动模态阶数、边界条件类型以及非局部参数的单层石墨烯振动。     

A 2D finite element based on a nonlocal constitutive model describing localization and propagation of phase transformation in shape memory alloy thin structures

Here, the effects of localization and propagation of martensitic phase transformation on the response of SMA thin structures subjected to thermo-mechanical loadings are investigated using nonlocal constitutive model in conjunction with finite element method. The governing equations are derived based on variational principle considering thermo-mechanical equilibrium and the spatial distribution of the nonlocal volume fraction of martensite during transformation. The nonlocal volume fraction of martensite is defined as a weighted average of the local volume fraction of martensite over a domain characterized by an internal length parameter. The local version of the thermo-mechanical behavior model derived from micromechanics considers the local volume fraction of martensite and the mean transformation strain. A 4-noded quadrilateral plane stress element with three degrees of freedom per node accounting for in-plane displacements and the nonlocal volume fraction of martensite is developed. Numerical simulations are conducted to bring out the influence of material and geometrical heterogeneities (perturbations/defects) on the localization and propagation of phase transformation in SMA thin structures. Also, a sensitivity analysis of the material response due to the localization and the other related model parameters is carried out. The detailed investigation done here clearly shows that the localization of phase transformation has significant effect on the response of shape memory alloys.     

SBFEM与非局部宏微观损伤模型相结合的准脆性材料开裂模拟

混凝土是一种被广泛应用于土木和水利工程中的准脆性材料, 在各种内外部因素的作用下, 开裂是混凝土结构最为普遍的破坏形式, 准确模拟结构的开裂过程, 对于结构的安全评估至关重要. 将比例边界有限元与非局部宏微观损伤模型相结合提出一种准脆性材料开裂模拟新方法. 以比例边界有限元子域的比例中心作为物质点, 通过两比例中心(物质点对)之间的物质键的正伸长率来定义微细观损伤, 将某点影响域内物质键的微细观损伤加权平均得到该点的宏观拓扑损伤. 再引入能量退化函数, 将宏观拓扑损伤嵌入到比例边界有限元的基本框架中. 充分利用比例边界有限元网格允许存在悬挂节点的优势, 采用四叉树网格离散技术进行快速、高质量的多级网格划分与过渡. 通过一个I型开裂与一个混合型开裂的两个典型算例, 验证了该方法可捕获结构裂纹扩展路径与荷载变形曲线. 与现有的方法相比, 本文的损伤模型可得到更准确的局部开裂损伤带, 结果更为合理, 且具有更高的计算精度和计算效率. 当损伤过程区网格尺寸小于影响域半径的1/5时, 计算结果不存在网格敏感性问题.      

Effects of structural and material length scales on stress-induced martensite macro-domain patterns in tube configurations

This paper studies the effects of structural and material length scales on the equilibrium domain patterns in thin-walled long tube configurations during stress-induced elastic phase transition under displacement-controlled quasi-static isothermal stretching. A nonconvex and nonlocal continuum model is developed and implemented into a finite element code to simulate the domain formation and evolution during the phase transition. The morphology and evolution of the macro-domains in different tube geometries are investigated by both analytical (energy analysis) and numerical (nonlocal finite element) methods. Energy minimization is used as the principle to explain the experimentally observed macroscopic domain patterns in a NiTi polycrystal tube. It is found that the domain pattern, as the minimizer of the system energy, is governed by the relative values of the material length scale g, tube-wall thickness h and tube radius R through two nondimensional factors: h/R and g/R. Physically, h/R and g/R serve as the weighting factors of bending energy and domain-wall energy over the membrane energy in the minimization of the total energy of the tube system. Theoretical predictions of the effects of these length scales on the domain pattern are quantified and confirmed by the computational parametric study. They all agree qualitatively well with the available experimental observations.     

一种计算复合材料等效弹性性能的有限元方法

在最小二乘意义下提出了一种计算复合材料等效弹性性能的有限元方法.这种方法由于考虑了等效弹性张量各分量之间的耦合关系,所求得的等效弹性常数比传统方法更可靠,可适用于求解含任意形状的夹杂和夹杂物问题.通过算例计算了在不同弹性模量对比度下两相复合材料的等效弹性性能,并与相关的理论及数值结果进行了比较,结果表明,利用该方法计算含夹杂复合材料等效弹性常数是可行的.     

基于多层模拟方法的双层梁断裂有限元分析

本文提出一种用于含分层的双层梁线弹性断裂分析的有限元方法．将上下子梁均模拟为多个子层,采用只有平动位移自由度的新型梁单元,假设单元内的位移沿纵向和横向均线性变化,推导了该单元的单元刚度矩阵．将开裂部分和未开裂部分的子梁进行单元刚度矩阵组装,施加相应的等效结点力,得到整体平衡方程,并结合边界条件进行求解．为验证该方法的有效性和精度,开展非对称双悬臂梁(Asymmetric Double Cantilever Beam, ADCB)和单臂弯曲梁(Single Leg Bending, SLB)试样的断裂分析,利用虚拟裂纹闭合技术(Virtual Crack Closure Technique, VCCT)得到了试样的能量释放率及其分量,并将求得的结果与解析解和二维有限元解进行对比．计算结果表明,相对于传统双层模拟方法,该多层模拟方法能够精确、高效地计算各类梁试样的能量释放率及其分量,并且无需引入界面连续条件．     

Nonlocal elasticity: an approach based on fractional calculus

Fractional calculus is the mathematical subject dealing with integrals and derivatives of non-integer order. Although its age approaches that of classical calculus, its applications in mechanics are relatively recent and mainly related to fractional damping. Investigations using fractional spatial derivatives are even newer. In the present paper spatial fractional calculus is exploited to investigate a material whose nonlocal stress is defined as the fractional integral of the strain field. The developed fractional nonlocal elastic model is compared with standard integral nonlocal elasticity, which dates back to Eringen’s works. Analogies and differences are highlighted. The long tails of the power law kernel of fractional integrals make the mechanical behaviour of fractional nonlocal elastic materials peculiar. Peculiar are also the power law size effects yielded by the anomalous physical dimension of fractional operators. Furthermore we prove that the fractional nonlocal elastic medium can be seen as the continuum limit of a lattice model whose points are connected by three levels of springs with stiffness decaying with the power law of the distance between the connected points. Interestingly, interactions between bulk and surface material points are taken distinctly into account by the fractional model. Finally, the fractional differential equation in terms of the displacement function along with the proper static and kinematic boundary conditions are derived and solved implementing a suitable numerical algorithm. Applications to some example problems conclude the paper.