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Highlights • Residual vibration reduction in translation unconstrained structures is considered. • Two methods to solve the problem arising from rigid body responses are proposed. • A new transient dynamic responses-based optimization method is built. • Model reduction is used to reduce computational burden. • The effectiveness of proposed approaches is verified by various examples. Abstract This paper studies residual vibration reduction in translation unconstrained or partially unconstrained structures by structural optimization. Suppression of elastic residual vibration is of great concern in many fields, and several approaches have been proposed to deal with it through structural optimization. However, these approaches are suitable for completely constrained structures, while unconstrained and partially unconstrained structures are not considered in particular. The dynamic responses of the latter include not only elastic responses but also rigid body responses. The rigid body responses make several effective structural optimization approaches inapplicable. This paper proposes two methods to solve the problem arising from the rigid body responses. Then the abovementioned effective approaches based on completely constrained models are applied to study the optimization of translation unconstrained or partially unconstrained structures to minimize the residual vibration. To reduce the computational burden of optimization, model reduction is further applied in this research. Finally, several numerical examples are presented to illustrate the effectiveness of the proposed approaches. The optimized results for residual vibration minimization show superiority over those obtained with classical objective functions. [ABSTRACT FROM AUTHOR]

• An energy-based homogenization method (EBHM) for the lattice structures with generalized periodicity is developed. • The limitation of classical EBHM, which is only applicable to unit cells with a square shape is overcome. • The closed-form expressions are derived for equivalent properties in terms of the elastic strain energy of the unit cell. • The effect of temperature rise operating conditions on the accuracy of calculations is highlighted. • The computational accuracy of the EBHM, classical homogenization method and direct finite element analysis are investigated. This paper presents an energy-based homogenization method (EBHM) to calculate the equivalent elastic properties of lattice structures with generalized periodicity. Unlike the traditional implementation of the homogenization method, expressions of closed-form are derived for the equivalent elastic matrix, equivalent coefficients of thermal stress and thermal expansion in terms of the elastic strain energy of the unit cell so that the tedious numerical solution and programming are avoided. It is shown that the elastic strain energy can easily be calculated by mapping the unit cell with the imposition of specific periodic boundary conditions. The implementation can resort to any available finite element tools. Numerical examples are used to compare the EBHM with the homogenization mapping method, classical homogenization method and direct finite element analysis (FEA). The computational accuracy is investigated to show the effectiveness of the EBHM. [ABSTRACT FROM AUTHOR]

• Topology optimization using analytic Cosserat continuum for periodic lattices. • Quantitative agreement between Euler–Bernoulli beam and Cosserat continuum results. • Cosserat elasticity is sufficient and necessary to optimize beam lattices. • Local volume fraction affects most the optimal topology and maximum deflections. • Structures made of reinforced honeycomb lattices were the most ridged. This paper presents a novel method based on the Cosserat theory to optimize the topology of metamaterials made of slender beams. First, we filled the gap in the literature and compared the optimal topology of discrete Euler–Bernoulli beam lattices with counterparts obtained using the homogenized Cosserat theory. We investigated the effect of material and numerical parameters on the optimization results and the global stiffness. Finally, the paper highlights the importance of second-order models for slender lattice structures through different macroscopic geometries. For the first time, we presented an excellent quantitative agreement between continuum Cosserat and discrete beam results. We demonstrated that the Cosserat theory is necessary and sufficient to optimize slender, lightweight designs with lattice-based microstructures. Furthermore, the results showed that the locally allowed volume fraction was the most critical limiting parameter when maximizing global stiffness. Finally, we found that the reinforced honeycomb lattice is the stiffest microstructure for a given mass among the investigated forms. [ABSTRACT FROM AUTHOR]

This paper discusses instabilities occurring in thin pressurized membranes, important in biological as well as in engineering contexts. The membranes are represented by only their in-plane stress components, for which an incompressible isotropic hyper-elastic behavior can be assumed. A hydro-static pressurization can give instabilities in the form of limit points with respect to a loading parameter, but also bifurcations and wrinkling. The hyper-elastic material model itself can also, under some circumstances, lead to a bifurcation situation. The instability situations can be included as constraints in a structural optimization. The paper discusses the formulation, the solution methods and some relevant instability situations. Numerical examples considering the pressurization of a flat and a cylindrical pre-stressed membrane illustrate some aspects of instability. [ABSTRACT FROM AUTHOR]

The paper reports the results of a numerical investigation on the relevance of web-triggered local–distortional interaction on the ultimate strength of a large number of cold-formed steel fixed-ended (plain) lipped channel, zed, hat and rack-section columns. These results concern columns with various geometries (cross-section dimensions and unrestrained length) and yield stresses, ensuring a wide variety of combined ratios between the (i) distortional and local critical buckling stresses ( R DL ) and (ii) yield and non-critical buckling stresses ( R y ). To avoid interaction with global buckling, all columns have global critical buckling stresses much higher than their local, distortional and yield counterparts. The aim of this study is to identify combinations of these ratios for which L–D interaction is relevant, in the sense that it affects visibly the column elastic and elastic–plastic post-buckling behaviours, ultimate strength and failure mechanism – special attention is devoted to the ultimate strength erosion. The numerical ultimate strength data obtained are then compared with the predictions of (i) the currently codified DSM (Direct Strength Method) strength curves for the design of columns failing in local and distortional modes and (ii) available DSM-based design approaches specifically developed to handle local–distortional interactive failures. Then, experimental results available in the literature concerning lipped channel and rack-section columns failing in local–distortional interactive modes are used to assess the quality of the estimates provided by the DSM-based design approaches. Finally, the paper closes with some considerations on the design of cold-formed steel columns undergoing L–D interaction. [ABSTRACT FROM AUTHOR]

• Grid point density is introduced to achieve smooth topological boundaries. • Boundary error is proposed to control the distribution of intermediate elements. • Different levels of anisotropic elastic tensor of cancellous bone are considered. • The optimized cancellous bone scaffolds have good printability. In this paper, a topology optimization method is proposed to design bone scaffold structures that match the anisotropic elasticity tensors of cancellous bone. Grid point density and boundary error are introduced in the optimization model to achieve smooth boundaries of scaffold topology and control the distribution of intermediate elements. Combined with finite element analysis, the effects of scaffolds with different levels of elasticity tensors on the stress transfer and distribution of the scaffold-cancellous bone structure are investigated. It is found that scaffolds with the same or slightly lower elastic properties (0.9 times) than cancellous bone transmitted more uniform stresses into the cancellous bone. Finally, we demonstrate that optimized scaffolds can be fabricated by Stereolithography (SLA). [ABSTRACT FROM AUTHOR]

Abstract: The use of negative penalty parameters such as elastic stiffness or mass to enforce constraints has been the subject of several recent papers. These developments began with an article in which rigid supports and connections were modelled by springs of large negative stiffness. This brief review paper discusses the history behind these developments and some emerging challenges and opportunities. [Copyright &y& Elsevier]

• First time a NEFEM formulation is applied in solid mechanics. • HDG-NEFEM enables super-convergence in domains with curved boundaries for linear elements. • HDG-NEFEM passes low and high-order patch tests in domains with curved boundaries. • Reliable error indicators for degree adaptive process driven by both displacement and stress. This paper proposes a new methodology for the solution of two dimensional linear elastic problems in domains with curved boundaries. The proposed method exploits the advantages of the hybridisable discontinuous Galerkin method to obtain an accurate approximation of both the displacement and the stress fields by solving a global problem that only involves the displacement field on the element edges as unknown. In addition, the methodology incorporates the exact boundary representation of the domain by means of the so-called NURBS-enhanced finite element method. Numerical examples are used to illustrate the three main advantages of the proposed method, namely the reproducibility of polynomials in domains with curved boundaries, the super-convergence of the solution even for linear approximation and the effectiveness and reliability of degree adaptive processes driven by displacement or stresses. [ABSTRACT FROM AUTHOR]

Highlights • We assess some 8-node hexahedral three-dimensional solid elements using inf-sup testing. • The test is for coercivity using displacement degrees of freedom only. • Insight into the element capacity to not lock in volumetric, shear, membrane, and pinching actions is obtained. • The novel 3D-MITC8 element is tested and found to perform well. Abstract We assess some 8-node hexahedral three-dimensional (3D) solid elements using inf-sup tests focusing on locking behavior in volumetric, shear, membrane, and pinching actions. Inf-sup tests have been widely used to see whether a finite element discretization is effective. In this paper we use a rather simple approach to inf-sup testing for the coercivity of the discretization to obtain insight into the convergence behavior of the standard displacement-based element, the 8-node element with incompatible modes, and the 3D-MITC8 element, and include geometric nonlinear analysis. We see that the 3D-MITC8 element performs better than the incompatible modes element in plate bending problems and can also be used with a small modification in the solution of thick or moderately thick curved shell structures. [ABSTRACT FROM AUTHOR]

• Combined rh -adaptivity is applied to the local maximum entropy point collocation method for the first time. • Collocation points are relocated driven by material force residuals in r -adaptivity. • Collocation and source points are added in local regions with relatively high errors determined by physical equilibrium residuals in h -adaptivity. • Combined rh -adaptivity exhibits greater efficiency than pure adaptive strategies for linear elasticity problems. Point collocation methods are strong form approaches that can be applied to continuum mechanics problems and possess attractive features over weak form-based methods due to the absence of a mesh. While various adaptive strategies have been proposed to improve the accuracy of weak form-based methods, such techniques have received little attention for strong form-based methods. In this paper, combined rh -adaptivity, in which r - and h -adaptivities are adopted iteratively, is applied to the local maximum entropy point collocation method for the first time to solve linear elasticity problems. Material force residuals act as driving forces in r -adaptivity to relocate collocation points, reducing the error associated with a given point distribution. Physical equilibrium residuals are used as the error estimator in h -adaptivity to determine the insertion locations for new points, diminishing the error caused by inadequate degrees of freedom. Issues arising in mesh-based methods, such as mesh distortion and hanging nodes, are entirely absent from the proposed method. The paper introduces the approach for the first time and the study is therefore confined to 2D domains. Numerical examples are presented to demonstrate the performance of the proposed adaptive strategies, comparing convergence rates and computational costs using uniform refinement, pure r -, h - and combined rh -adaptivities. [ABSTRACT FROM AUTHOR]

Abstract: This paper brings together the principles, equations, and applications of damage modelling and elastic waves propagation, both traditional and state-of-the-art in a review form. It begins with the relevant fundamentals of damage modelling, derives the basic equations of fracture mechanics and elastic wave propagations, and covers advanced topics and applications of Lamb waves that are at the forefront of today’s research. The results obtained indicate that the current approach is capable of detecting cracks and delaminations of very small size, even in the presence of considerable measurement errors. The sections are filled with case studies, worked examples and exercises that make this paper an outstanding resource. [Copyright &y& Elsevier]

Abstract: In this paper we introduce an enhanced implicit algorithm for phenomenological plasticity based on nested surfaces for the description of multiaxial nonlinear hardening. The algorithm uses an implicit version of the Mróz translation rule. This rule is computed explicitly from the trial state and, as a result, the final stress state is obtained solving a nonlinear scalar equation. With the algorithm and the formulation introduced in this paper, there is no need for performing iterations in order to obtain the final active surface. This surface is determined in advance directly from the trial stress state and therefore the efficiency of the algorithm is greatly improved, specially when the number of surfaces describing the stress–strain curve is moderate to large. The algorithm has been implemented in the commercial code ADINA as a user subroutine. [Copyright &y& Elsevier]

A topology optimization technique for systematically designing contact-aided compliant mechanisms (CCM) is presented in this paper. A CCM is a single piece elastic body that uses intermittent contacts in addition to elastic deformation to transmit force and motion. Contact interactions give rise to interesting nonlinear and nonsmooth behaviors even under the small deflection assumption made in this work. The difficulties associated with the nondifferentiability inherent in the CCM systematic synthesis problem are circumvented by using a regularized contact model. This model uses a smooth approximation of the unilateral displacement constraints that are used to model contact interactions. The use of a regularized contact model in the underlying state problem makes it possible to use efficient smooth optimization algorithms for the systematic synthesis of CCMs. The formulation of the design problem for CCMs, sensitivity analysis, and solution methodology are presented. The paper includes CCM designs that exhibit nonsmooth motion and force transmission characteristics, which are not possible or practical with compliant mechanisms that do not use intermittent contact interactions. [Copyright &y& Elsevier]

The paper describes the attributes that should be possessed by a benchmark example for verifying the beam elements used to carry out 3D linear buckling analysis and 3D second-order elastic analysis of steel frames. Based on the attributes described, the paper proposes a suite of benchmark examples selected from the literature. The necessary features of a beam element required to pass the proposed benchmark problems are given, and beam elements that possess these features are cited. The paper also explains the merits of linear buckling analysis examples, and provides a commentary on two well-known examples. [Copyright &y& Elsevier]

In a previous paper [Comput. Methods Appl. Mech. Eng. 190 (2001) 6053], the framework for the mechanics of solids, deformable over fractal subsets, was outlined. Anomalous mechanical quantities with fractal dimensions were introduced, i.e., the fractal stress [σ*], the fractal strain [ϵ*] and the fractal work of deformation W*. By means of the local fractional operators, the static and kinematic equations were obtained, and the Principle of Virtual Work for fractal media was demonstrated. In this paper, the constitutive equations of fractal elasticity are put forward. From the definition of the fractal elastic potential φ*, the linear elastic constitutive relation is derived. The physical dimensions of the second derivatives of the elastic potential depend on the fractal dimensions of both stress and strain. Thereby, the elastic constants undergo positive or negative scaling, depending on the topological character of deformation patterns and stress flux. The direct formulation of elastic equilibrium is derived in terms of the fractional Lame´ operators and of the equivalence equations at the boundary. The variational form of the elastic problem is also obtained, through minimization of the total potential energy. Finally, discretization of the fractal medium is proposed, in the spirit of the Ritz–Galerkin approach, and a finite element formulation is obtained by means of devil’s staircase interpolating splines. [Copyright &y& Elsevier]

Recently, a global functional set has been proposed for the construction of large finite elements, here called “Coons-patch macroelements”, with degrees of freedom at the boundaries only. The background of the method is “Coons interpolation formula”; a mathematical formula capable of describing CAD surfaces initially applied to automobile applications. So far, these macroelements were found to be accurate in potential and plane elasticity problems. Based on these encouraging results, this paper continues the investigation on the universality which Coons interpolation formula offers in the development of a class of macroelements including members of the well-known serendipity family. In this context, it is shown in a systematic way that it is not necessary to use Lagrange polynomials to interpolate the potential along each side of the macroelement but one may choose piecewise-linear and/or cubic B-splines interpolants. Moreover, the paper investigates the performance of Coons-patch macroelements in eigenvalue and scalar wave propagation problems by implementing standard time-integration schemes. [Copyright &y& Elsevier]

• A fail-safe topology optimization is presented for multiscale structures. • The damage in both the macroscale and microscale is considered. • Simplified damage models for truss-like microstructures are utilized. • The use of Heaviside projection speeds up convergence and obtains better results. • Multiscale fail-safe structures are more damage-resistant and self-supporting. This paper presents a novel fail-safe topology optimization method for multiscale structures. The partial damage of both macroscopic and microscopic scales is considered for structural design. To ensure precision, the effective elasticity tensor obtained by the homogenization method is fitted as a high-order polynomial function. Meanwhile, the simplified models of partially damaged truss-like microstructure are employed to reduce the computational cost and the difficulty of fitting. Moreover, Heaviside projection is applied to speed up the convergence and yield a relatively clear configuration. Three numerical examples are tested to demonstrate that the optimized multiscale structures successfully obtain comprehensive performances than optimized solid structures when appropriate microstructure configurations are chosen. Besides, multiscale structures are more self-supporting than solid structures and thus more suitable for additive manufacturing due to the large number of gray elements diffused. [ABSTRACT FROM AUTHOR]

Highlights • Post-buckling analysis of annular sector plate subjected to in-plane shear load at upper surface. • Governing equations are based on three dimensional theory of elasticity. • Non-linear Green strain tensor is used. • Due to weak coupling between shear loads and resulting twisting moments, post-buckling of simply supported FGM plate is of bifurcation-type buckling. Abstract In this paper, shear post-buckling analysis of functionally graded annular sector plates is considered. In-plane shear loads have been applied to either radial, circumferential, or all edges of annular sector plates. Moreover, post-buckling of annular sector plates subjected to in-plane shear load at upper/lower surfaces is investigated for the first time. Material properties are graded asymmetrically in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while the Poisson's ratio is assumed to be constant. The governing equations are based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor instead of the approximate plate theories and Von Karman assumptions. The governing equations are developed based on the principle of virtual work and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio and four different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to weak coupling between applied shear loads and the resulting twisting moments, shear post-buckling behavior of simply supported FGM plates is of the bifurcation-type buckling following a stable post-buckling path. [ABSTRACT FROM AUTHOR]

Conventionally, the seismic response of primary structures such as buildings and secondary systems such as piping is evaluated through uncoupled analyses. Many studies have illustrated that the two systems interact in many different ways (mass interaction, non-classical damping, phasing, etc.). An analysis of the coupled system is not only rational but also eliminates the excessive conservatism that exists in an uncoupled analysis. Consequently, fragility assessments based on uncoupled analysis are also incorrect and a coupled analysis must be conducted in such evaluations. However, nonlinear analyses of such complex systems particularly in the context of fragility assessment, which requires a large number of nonlinear analyses, becomes computationally prohibitive. Tadinada and Gupta (2017) presented an equivalent elastic limit state concept with an intent to reduce the computational effort needed in these assessments and yet evaluate the seismic fragility with sufficient accuracy. This paper outlines some of the limitations that have been experienced in the use of originally proposed equivalent limit-state formulation and presents valuable enhancements. The novel contribution of this study is focused on accounting for the effect of uncertainty in nonlinear characteristics and the effect of non-classical damping. Unlike the originally proposed formulation, the proposed formulation also considers the asymmetric variation of the equivalent limit state with respect to tuning ratio. Furthermore, a Bayesian approach is incorporated into the proposed methodology for increasing the accuracy of seismic fragilities in the case of tuned or nearly tuned primary-secondary systems. Numerical examples are used to illustrate that the modified form improves the accuracy for both the tuned and the detuned systems. In summary, the proposed approach provides an efficient framework of seismic fragility assessment and risk evaluation for coupled systems. [ABSTRACT FROM AUTHOR]

This paper proposes a probabilistic approach for the solution of elastic buckling of columns, involving uncertainties, using stochastic element free Galerkin method. In the present work, modulus of elasticity is modeled as a homogeneous random field. Karhunen-Loeve expansion and shape function method are used to represent random field and their effectiveness is compared in modeling the same in a computationally viable manner. Both Gaussian and non-Gaussian field are considered for the present study. The stochastic eigenvalue problem is solved for first and second moment characteristics of buckling load, using perturbation analysis. Numerical examples of columns with different boundary conditions are solved. Monte Carlo simulation is used as a validation tool. The obtained results are found in good agreement with those obtained by Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

In the present paper two constitutive models, able to reproduce the mechanical behavior of shape memory alloy (SMA), are proposed. The constitutive laws allow considering the simultaneous presence of normal and shear stresses, accounting for its coupling without developing a full three-dimensional (3D) model. In such a way, two different models suitable for deriving the response of SMA devices, both characterized by the presence of only normal and shear stresses, are presented. In particular, the proposed SMA models reproduce the pseudoelastic behavior, the shape memory effect, the reorientation process and account for the different elastic properties of austenite and martensite. The phase transformations are governed by an equivalent stress defined as function of the normal and shear stresses. A robust numerical algorithm, based on a backward Euler time integration within a predictor–corrector technique, is developed for each model. Numerical simulations of experimental evidences, available in literature, are performed to validate the proposed models and computational strategies. In particular, comparisons of the results obtained by the proposed models with experimental data, are performed. A comparison with a 3D model is also carried out. The ability of the proposed models in satisfactorily reproducing the SMA response with reduced computational cost is verified even for complex loading–unloading histories. [ABSTRACT FROM AUTHOR]

This paper presents an effective computational approach comprised of forward and inverse analyses for detection of multiple complex flaw clusters in elastic solids. A three-step detection strategy is introduced for inverse analysis, whereas extended finite element method (XFEM) is adopted for forward analysis. The use of XFEM is to avoid re-meshing during the change of flaw geometries. The three-step detection strategy involves: firstly, an optimization method that couples an improved discrete artificial bee colony algorithm and hierarchical clustering analysis (IDABC-HCA) is used to capture subdomains containing flaws with limited measure points in the global domain; secondly, additional measure points are introduced locally within each captured subdomain, where the number of flaws and the rough geometry of each flaw are quickly determined with the IDABC-HCA; finally, true geometries of flaws are obtained on the basis of the rough geometries by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. To save computational time, “Queue and Kill” method is proposed to actively identify and eliminate the improper candidate flaws and/or flaw clusters. Three numerical examples of multiple flaw detection that include simple and complex flaw geometries are analyzed. The results demonstrate that the proposed approach can effectively detect multiple complex flaw clusters without prior information of the flaw number. [ABSTRACT FROM AUTHOR]

This paper presents a robust numerical model, which takes into account both size-dependent and shear deformation effects, for the bending, buckling and free vibration analyses of functionally graded microplates. The size-dependent effect is captured by using the modified strain gradient elasticity theory with three length scale parameters, whilst the shear deformation effect is accounted by using the third-order shear deformation theory. The rule of mixture is employed to describe the distributions of material phrases through the plate thickness. By using Hamilton’s principle, the governing equations are derived and then discretized by employing an Isogeometric Analysis (IGA) approach, where the Non-Uniform Rational B-Splines (NURBS) basis functions are adopted to meet the C 2 -continuity requirement. Physical mesh convergence and verification studies are performed to prove the accuracy and reliability of the present model. In addition, parametric studies are also carried out to investigate the size effect in conjunction with the influences of gradient index, shear deformation effect and boundary conditions on the responses of microplates. [ABSTRACT FROM AUTHOR]

This article presents an energy-momentum integration scheme for the nonlinear dynamic analysis of planar Euler-Bernoulli beams. The co-rotational approach is adopted to describe the kinematics of the beam and Hermitian functions are used to interpolate the local transverse displacements. In this paper, the same kinematic description is used to derive both the elastic and the inertia terms. The classical midpoint rule is used to integrate the dynamic equations. The central idea, to ensure energy and momenta conservation, is to apply the classical midpoint rule to both the kinematic and the strain quantities. This idea, developed by one of the authors in previous work, is applied here in the context of the co-rotational formulation to the first time. By doing so, we circumvent the nonlinear geometric equations relating the displacement to the strain which is the origin of many numerical difficulties. It is rigorously shown that the proposed method conserves the total energy of the system and, in absence of external loads, the linear and angular momenta remain constant. The accuracy and stability of the proposed algorithm, especially in long term dynamics with a very large number of time steps, is assessed through four numerical examples. [ABSTRACT FROM AUTHOR]

This paper proposes a general finite-element procedure for the nonlinear analysis of cables based on a mixed variational formulation in curvilinear coordinates with finite deformations. The formulation accounts for nonlinear elasticity and inelasticity, overcoming the limitation of recent numerical approaches which integrate explicitly the global balance of linear momentum for a linear elastic material with infinitesimal deformations. The formulation uses a weak form of the catenary problem and of the strain-displacement relation to derive a new family of cable finite elements with a continuous or discontinuous axial force field. Several examples from the literature on nonlinear cable analysis are used to validate the proposed formulation for St. Venant-Kirchhoff elastic materials and neo-Hookean materials. These studies show that the proposed formulation captures the displacements and the axial force distribution with high accuracy using a small number of finite elements. [ABSTRACT FROM AUTHOR]

In this paper, a general elasticity solution for the axisymmetric bending of a linearly elastic annular plate is used to derive an exact finite element for such a plate. We start by formulating an interior plate problem by employing Saint Venant’s principle so that edge effects do not appear in the plate. Then the elasticity solution to the formulated interior problem is presented in terms of mid-surface variables so that it takes a form similar to conventional engineering plate theories. By using the mid-surface variables, the exact finite element is developed both by force- and energy-based approaches. The central, non-standard feature of the interior solution, and the finite element based on it, is that the interior stresses of the plate act as surface tractions on the plate edges and contribute to the total potential energy of the plate. Finally, analytical and numerical examples are presented using the elasticity solution and the derived finite element. [ABSTRACT FROM AUTHOR]

• Use mesoscale elements to statistically characterize elastic and fracture behavior. • Establish convergence metrics appropriate to define isotropic and heterogeneous limits. • Compare representative volume sizes with respect to heterogeneity and anisotropy. • Develop methods to decrease inaccuracy introduced by mesoscale partitioning scheme. • Relate volume element homogenization to stochastic partial differential equations. In this paper, a method is presented to describe the statistical behavior of mesoscale material property fields, which are useful in macroscopic fracture simulation. The use of representative volume element-based properties can result in unsatisfactory fracture patterns because this approach eliminates random variations. In contrast, statistical volume elements can approximate random and inhomogeneous material properties. We compare two material property convergence metrics based on 1) vanishing variations of a property and 2) the convergence of property mean value versus volume element size. We examine the trends with which the properties of a two-phase composite tend toward homogeneous and isotropic limits. Compared to fracture properties, elastic properties reach their homogeneous and isotropic limit at smaller sizes. Accordingly, a volume element size can be chosen such that only the apparent fracture strength remains random and inhomogeneous. This reduces the computational cost and has proven useful in capturing realistic fracture results. In addition, the geometry of the mesoscale partitioning scheme has an important effect on homogenized properties. The intersection of straight edges with inclusions results in nonphysical size effects, increases anisotropy of properties, and results in 30 to 60 times larger representative volume elements compared to a partitioning scheme based on Voronoi homogenization. [ABSTRACT FROM AUTHOR]

The objective of this paper is to assess the reliability and effectiveness of the method of finite spheres, a truly meshless overlapping finite element method, for the solution of practical three-dimensional linear elasticity problems. Advantages include simplified discretization and the elimination of element distortion. The method is implemented in the ADINA finite element program through a user-supplied element subroutine. The solutions of three increasingly complex three-dimensional problems are studied (1) to establish the reliability of the method for practical linear elasticity problems and (2) to assess the effectiveness of the method as compared to the standard finite element method. The solutions indicate that the method of finite spheres is between one and two orders of magnitude more expensive in computational time than the standard finite element method. This is still a promising result since there are significant time savings for the method of finite spheres during the pre-processing phase, particularly in the discretization of complicated three-dimensional geometries and because the overlapping sphere elements can be directly coupled to traditional finite elements. [ABSTRACT FROM AUTHOR]

In this paper, a non-stationary random vibration problem of an elastic half-space under a moving stochastic load modelling the dynamic characteristic of the train load is studied. An analytical method combining the pseudo-excitation method (PEM) and Fourier transformation method (FTM) is proposed. Because the load is moving and random, the responses at fixed locations in the half-space have characteristics of evolutionary non-stationary randomness. By means of the FEM, the non-stationary random vibration analysis is transformed into a conventional moving pseudo harmonic load problem. A closed-form solution of random vibration responses is derived in an integral form, which avoids step-by-step integration in time domain and thus leads to great saving in computational time. An adaptive quadrature method is adopted in order to obtain the desired numerical results of the singular integrand with non-stationary power spectral density. Through numerical examples, the proposed method is first validated, and then the dynamics characteristics of the non-stationary random responses of the half-space in all subsonic, transonic and supersonic cases are analysed. [ABSTRACT FROM AUTHOR]

In this paper a unified finite element methodology based on gradient-elasticity is proposed for both two- and three-dimensional problems, along with some considerations about the best integration rules to be used and a comprehensive convergence study. From the convergence study it has emerged that for both two and three-dimensional problems, the implemented elements show a convergence rate virtually equal to the corresponding theoretical values. Recommendations on optimal element size are also provided. Furthermore, the ability of the proposed methodology to remove singularities in statics has been demonstrated through a couple of examples, in both two and three dimensions. [ABSTRACT FROM AUTHOR]

• A high-order FVM for linear elasticity on unstructured meshes is presented. • High-order interpolation using Moving Least Squares and Local Regression Estimators. • Verification and validation of the proposed method are done using canonical tests. • The LRE method provides accurate results using fewer computational resources. • The Local Regression Estimators method is strongly recommended. This paper presents a high-order finite volume method for solving linear elasticity problems on two-dimensional unstructured meshes. The method is designed to increase the effectiveness of finite volume methods in solving structural problems affected by shear locking. The particular feature of the proposed method is the use of Moving Least Squares (MLS) and Local Regression Estimators (LRE). Unlike other approaches proposed before, these interpolation schemes lead to a natural and simple extension of the classical finite volume method to arbitrary order. The unknowns of the problem are still the nodal values of the displacement which are obtained implicitly in a direct solution strategy. Some canonical tests are performed to demonstrate the accuracy of the method. An analytical example is considered to evaluate the sensitivity of the solution concerning the parameters of the algorithm. A thin curved beam and a crack problem are considered to show that the method can deal with the shear locking effect, stress concentrations, and geometries where unstructured meshes are required. An overall better behavior of the LRE is observed. A comparison between low and high-order schemes is presented, and a set of parameters for the interpolation method is found, delivering good results for the proposed cases. [ABSTRACT FROM AUTHOR]

This paper presents topology optimization of structures with inclusion of microstructural effects. To this end, Mindlin’s elasticity with microstructure theory is used for investigating the scale effects. For implementation of finite element based topology optimization, a separate interpolation scheme for the displacement and micro-deformation field is adopted based on 8-node quadrilateral element. Numerical results show that although the continuum is associated with material length-scales, the minimal topological length-scale cannot be controlled without using a regularization technique. Results demonstrate that the length-scale of the microstructure may have a significant influence on the optimal topologies. [ABSTRACT FROM AUTHOR]

In this paper, a mixed formulation based on the scaled boundary finite element method (SBFEM) is proposed for nearly incompressible linear elasticity. The problems with strong discontinuities in nearly incompressible linear multi-materials are analyzed. The proposed technique avoids volumetric locking and non-physical pressure oscillations at near incompressible limit without requiring any mesh-dependent or stabilization parameters. The pressure field is introduced but it is not independently unknown in the proposed technique. The pressure and displacement are interpolated identically on the boundary. Standard polynomial shape functions and high order line-elements are used. The governing equation of mixed SBFEM in displacement is derived based on the virtual work principle. The displacement, stress and pressure fields along the radial direction are represented analytically, which permits the stress intensity factors to be directly evaluated by definition. [ABSTRACT FROM AUTHOR]

In this paper, an a posteriori error estimator of the recovery type is developed for the gradient elasticity theory of Aifantis. This version of gradient elasticity can be implemented in a staggered way, whereby solution of the classical equations of elasticity is followed by solving a reaction–diffusion equation that introduces the gradient enrichment and removes the singularities. With gradient elasticity, singularities in the stress field can be avoided, which simplifies error estimation. Thus, we develop an error estimator associated with the second step of the staggered algorithm. Stress-gradients are recovered based on the methodology of Zienkiewicz and Zhu, after which a suitable energy norm is discussed. The approach is illustrated with a number of examples that demonstrate its effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, a temporal stabilized nodal integration method (sNIM) using 3-node triangular elements is formulated for elastic-static, free vibration and buckling analyses of Reissner–Mindlin plates. Two stabilization terms are added into the smoothed potential energy functional of the original nodal integration, consisting of squared-residual of equilibrium equations. A gradient smoothing technique (GST) is used to relax the continuity requirement of shape function. The smoothed Galerkin weak form is employed to create discretized system equations, and the node-based smoothing domains are formed to perform the smoothing operation and the numerical integration. A stabilization parameter is finally introduced to the modified system for the sake of curing temporal instability. Numerical tests provide an empirical value range of stabilization parameter, within which very accurate and stable results can be obtained for both static and eigenvalue problems. [ABSTRACT FROM AUTHOR]

The commonly used two-parameter Mooney–Rivlin incompressible hyper-elastic material model can show non-intuitive responses under certain conditions. This paper shows that critical states with non-unique responses occur at least at very specific bi-axial stress states. This can happen for cases where the constant related to the second invariant of strain is positive, but not for the case with this constant equal to zero (the Neo-Hookean case). The dependence of the instabilities on the ratio between the two constitutive constants is shown by evaluated fold lines. The instability is shown to be related to the imposed boundary conditions. An analytical treatment of the problem shows that dynamic edge effects correspond to the static instability. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we extend the Sussman–Bathe spline-based hyperelastic isotropic model to predict the behavior of transversely isotropic isochoric materials. The model is based on an uncoupled decomposition of the stored energy function and a generalization of the inversion formula used by Sussman and Bathe. The present extension introduces some approximations that, in all studied cases, do not yield relevant deviations from the experimental data. The isotropic model results in a particular case of the present formulation. Several possibilities of user-prescribed experimental data are addressed. The model is used to predict experiments of calendered rubber and of biological tissues. [Copyright &y& Elsevier]

Abstract: This paper describes the development of a two-step homogenization approach for evaluating the elastic properties of fiber reinforced concrete (FRC). For that purpose, a finite element model of an FRC unit cell was generated. Prior to the generation of the unit cell finite element model, the interface transition zones (ITZ) and the aggregates were homogenized using an analytical approach. In the first step, the properties of a spherical aggregate and its concentric ITZ layer were homogenized via an analytical procedure. Then a numerical homogenization procedure was applied to the homogenized aggregate, the mortar, and the fibers to obtain the macroscopic properties of the FRC. The suggested framework executes the multi-scale analysis of FRC structures by incorporating an original concrete unit cell generator into a commercial finite element software package intended for simulating nonlinear solid mechanical problems. The results, obtained using the presented algorithm, are in very good agreement with experimental results. [Copyright &y& Elsevier]

Abstract: The main issue of this paper is the stability analysis of elastic systems with random parameters using the Generalized Stochastic Finite Element Method. The Taylor expansion with random coefficients of nth order is used to express all random functions and to determine up to fourth order probabilistic moments of the critical force or critical pressure. The response function method assists to determine higher order partial derivatives of the structural response instead of the Direct Differentiation Method employed widely before. This approach is examined on the classical Euler problem, 2D and 3D steel frames as well as in addition to the cylindrical shell with some geometrical parameters defined as the Gaussian variables. The comparison of the GSFEM versus the Monte-Carlo simulation on the Euler problem proves the probabilistic convergence of this new technique. [Copyright &y& Elsevier]

Abstract: The present work deals with the use of asymptotic numerical methods (ANM) to manage crack onset and crack growth in the framework of continuum damage mechanics (CDM). More specifically, an application of regularization techniques to a 1D cohesive model is proposed. The standard “triangle” damageable elastic model, which is often used in finite element codes to describe fracture of brittle materials, was chosen. Results associated with the load–unload cycle showed that ANM is convenient for numerically taking this specific irregular behavior into account. Moreover, the present paper also shows that the chosen damageable interface model can be introduced in the generalized standard material formalism, thus unabling us to define a complete energy balance associated with the damage process. In such a framework, the damage state is described by a new displacement variable. Finally, a 1D finite element application to a simple elastic damageable structure is shown to highlight the potential of this approach. [Copyright &y& Elsevier]

Abstract: This paper discusses the effect of the structure and packing conditions of centre displacement of the membrane in a circular piezoelectric actuator. A mathematical model was constructed using classical laminated plate theory (CLPT) to describe the transverse displacement of a circular actuator. In the applied loading simulation, the effect on static displacement from varying the thickness in each layer of the structure as well as the packaging boundary constraint methods was studied. Results show the radius and thickness of the thin film has little effect on the maximum displacement, however an increase in elastic modulus decreased the maximum transverse displacement. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an investigation on the elastic lateral-torsional buckling behaviour of discretely restrained tapered beams. Through a model problem consisting of a doubly symmetric web-tapered I-section cantilever acted by a tip load, it is shown how the effects of linearly elastic or rigid restraints can be included in the one-dimensional model previously developed by the authors to characterize the elastic buckling behaviour of thin-walled tapered beams. The restraints may have a translational, torsional, minor axis bending and/or warping character. The resulting self-adjoint eigenproblem is cast in non-dimensional form over a fixed reference domain, a process that leads to the identification of a complete set of independent non-dimensional parameters. It is then reformulated as a standard inhomogeneous boundary value problem and solved numerically with the collocation package colnew. A parametric study is carried out in order to examine in some detail (i) the effectiveness of different types of restraint, (ii) the influence of the restraint stiffness and (iii) the interplay between these two aspects and the degree of web tapering. This study provides insight into the peculiarities of the lateral-torsional buckling behaviour of tapered beams and some seemingly paradoxical results are given a physical explanation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, it is shown that, for planar systems formed from linear-elastic bodies in non-penetrative contact with Coulomb friction, the limit load can be approximated efficiently by the limit loads corresponding to a sequence of contact problems with given friction, via a finite element fixed-point approach. For the auxiliary problem with given friction, a splitting of this problem into two subproblems, one with prescribed normal contact stresses and given friction and one with prescribed tangential contact stresses, is employed. Numerical results for masonry block structures subjected to in-plane loading illustrate the predictive capabilities of this approach. [Copyright &y& Elsevier]

Abstract: Many structural mechanics problems, such as post-buckling of elastica, involve determining multi-equilibrium states of a nonlinear system. Typically, the stable equilibrium states are found by searching for minima of the potential energy function subjected to some constraints. In this paper, the method of genetic algorithm combined with the quasi-Newton method is applied to search for the multiple minima of the strain energy of various elastica problems. The proposed hybrid methodology is relatively straightforward to implement and is adopted to study post-buckling behaviour of planar elastica modelled as multi-link systems for various boundary conditions and sidewall constraints. [Copyright &y& Elsevier]

Abstract: In this paper, known analytical solutions are used to compare various finite element discretizations for the numerical solution of large deformation problems involving hyperelastic materials. This approach is also known as the method of manufactured solutions (MMS). The performance of these different discretizations is analyzed in terms of convergence in a suitable norm with respect to mesh size, and also in terms of computational cost and memory consumption. Some interesting features of mixed finite element discretizations are presented and discussed. The test cases presented can also serve as nontrivial benchmark problems and as a basis for code verification. [Copyright &y& Elsevier]

Abstract: The numerical modelling of interacting elastodynamic solids and acoustic fluids by time-domain finite element procedures is discussed here. In this study, the global model is divided into solid and fluid sub-domains and each sub-domain is analyzed independently, as an uncoupled model. The interaction between the different sub-domains is accomplished by interface forces. In order to improve the efficiency, accuracy and flexibility of the methodology, a multi-time-step method is also considered, allowing optimal time-steps to be selected for each sub-domain of the model. At the end of this paper, numerical applications illustrate the performance and potentialities of the proposed techniques. [Copyright &y& Elsevier]

Abstract: In the present paper, the process of the formulation of the equations of dynamic equilibrium and of the respective equations of natural vibration of Timoshenko beams on three-parameter elastic foundation within the framework of the second order theory is presented. The corresponding mass matrices are also formulated. In order to define the values of the parameters of the three-parameter model, an analytical method is applied. With the aid of two numerical examples, it is proved that the three-parameter model is capable of yielding natural period values that sufficiently approximate those derived by the solution using 2D finite element soil models. [Copyright &y& Elsevier]

Abstract: This paper addresses accuracy issues of nodal stresses in both finite elements and finite volumes. Strategies to evaluate nodal stresses for finite elements are also briefly discussed. Emphasis is placed on the application of finite volumes to solid mechanics and an overview of recent advancements is presented. Numerical simulation of test problems shows that accuracy of nodal stresses evaluated using finite elements is strongly dependent on the recovery strategy. In all cases, stress recovery at boundary nodal points renders large errors for all non-patch schemes. Contrastingly, finite volumes provide an accurate computation of nodal stresses, including at domain boundaries. [Copyright &y& Elsevier]

Abstract: The paper is concerned with dynamic problems of the linear theory of elasticity for anisotropic porous materials according to the Cowin–Nunziato model. In order to take into account the structural damping in elastic media, a new attenuation model which extends the Rayleigh models is used. A set of new finite element schemes is proposed for finding numerical solutions of transient, harmonic, modal and static problems in the context of elastic media with voids. Three-dimensional, plane (plane stress and plane strain) and axisymmetric problems are considered. [Copyright &y& Elsevier]

Abstract: A three-dimensional NMM based on tetrahedral meshes is developed in this paper. First, the displacement functions of manifold elements are formulated by the partition of unity function. Then, the global equilibrium equations for three-dimensional elasto-statics problem are established by minimizing the total potential energy. The stiffness matrix, loading matrix, displacement resistance matrix are derived for program coding. Thereafter, the computational cost of the global stiffness matrix in the NMM is discussed and compared with that in the classical finite element techniques. Finally, two typical examples are investigated to demonstrate the validity of the proposed method. The results indicate that an enough accuracy of approximation will be obtained with decreasing size of the mathematical mesh. The proposed method can be used as potentially powerful tool for three-dimensional structural analysis. [Copyright &y& Elsevier]