Finite elements for elasticity with microstructure and gradient elasticity

We present a general finite element discretization of Mindlin's elasticity with microstructure. A total of 12 isoparametric elements are developed and presented, six for plane strain conditions and six for the general case of three‐dimensional deformation. All elements interpolate both the displacement and microdeformation fields. The minimum order of integration is determined for each element, and they are all shown to pass the single‐element test and the patch test. Numerical results for the benchmark problem of one‐dimensional deformation show good convergence to the closed‐form solution. The behaviour of all elements is also examined at the limiting case of vanishing relative deformation, where elasticity with microstructure degenerates to gradient elasticity. An appropriate parameter selection that enforces this degeneration in an approximate manner is presented, and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid. Copyright © 2007 John Wiley & Sons, Ltd.     

Variational principles and membrane finite elements with drilling rotations for geometrically non-linear elasticity

Variational principles for geometrically non-linear continuum with independent rotation field are constructed based on the polar factorization theorem. Their regularized forms are next discussed suitable for the finite element implementation. The considerations are specialized to a two-dimensional membrane problem, and standard isoparametric interpolations are used in order to construct membrane elements with drilling rotations. The elements are evaluated on a set of problems in geometrically non-linear elastostatics.     

An efficient methodology for stress-based finite element approximations in two-dimensional elasticity

We present an efficient approach to compute the matrices used in finite element formulations where self-equilibrated (zero divergence) approximations of the stress field are used. The fundamental aspect of this approach is that it is applicable to polynomial approximations of high degree (it was tested up to degree 10), providing closed-form expressions for the elementary matrices involved. The components of the stress field are defined as a function of the coordinates in the master element, facilitating the postprocessing of the finite element results and allowing for the consideration of geometric parameters in reduced order models.     

Wrinkled and slack membranes: nonlinear 3D elasticity solutions via smooth DMS‐FEM and experiment

The smooth DMS‐FEM, recently proposed by the authors, is extended and applied to the geometrically nonlinear and ill‐posed problem of a deformed and wrinkled/slack membrane. A key feature of this work is that three‐dimensional nonlinear elasticity equations corresponding to linear momentum balance, without any dimensional reduction and the associated approximations, directly serve as the membrane governing equations. Domain discretization is performed with triangular prism elements and the higher order (C¹ or more) interelement continuity of the shape functions ensures that the errors arising from possible jumps in the first derivatives of the conventional C⁰ shape functions do not propagate because the ill‐conditioned tangent stiffness matrices are iteratively inverted. The present scheme employs no regularization and exhibits little sensitivity to h‐refinement. Although the numerically computed deformed membrane profiles do show some sensitivity to initial imperfections (nonplanarity) in the membrane profile needed to initiate transverse deformations, the overall patterns of the wrinkles and the deformed shapes appear to be less so. Finally, the deformed profiles, computed through the DMS FEM‐based weak formulation, are compared with those obtained through an experiment on an ultrathin Kapton membrane, wherein wrinkles form because of the applied boundary displacement conditions. Comparisons with a reported experiment on a rectangular membrane are also provided. These exercises lend credence to the feasibility of the DMS FEM‐based numerical route to computing post‐wrinkled membrane shapes. Copyright © 2012 John Wiley & Sons, Ltd.     

On the C1 continuous discretization of non‐linear gradient elasticity: A comparison of NEM and FEM based on Bernstein–Bézier patches

In gradient elasticity, the appearance of strain gradients in the free energy density leads to the need of C¹ continuous discretization methods. In the present work, the performances of C¹ finite elements and the C¹ Natural Element Method (NEM) are compared. The triangular Argyris and Hsieh–Clough–Tocher finite elements are reparametrized in terms of the Bernstein polynomials. The quadrilateral Bogner–Fox–Schmidt element is used in an isoparametric framework, for which a preprocessing algorithm is presented. Additionally, the C¹‐NEM is applied to non‐linear gradient elasticity. Several numerical examples are analyzed to compare the convergence behavior of the different methods. It will be illustrated that the isoparametric elements and the NEM show a significantly better performance than the triangular elements. Copyright © 2009 John Wiley & Sons, Ltd.     

GENERALIZED 3-D PAVING: AN AUTOMATED QUADRILATERAL SURFACE MESH GENERATION ALGORITHM

This paper discusses the extension of the paving algorithm for all quadrilateral mesh generation to arbitrary three-dimensional trimmed surfaces. Methods of calculating angles, projecting elements, and detecting collisions between paving boundaries, for general surfaces are presented. Extensions of the smoothing algorithms for three dimensions are set forth. Advances in the use of scalar sizing functions are presented. These functions can be used to better approximate internal mesh density from boundary densities and surface characteristics.     

Analysis of shell-like structures by the Boundary Element Method based on 3-D elasticity: formulation and verification

In this paper, the Boundary Element Method (BEM) for 3-D elastostatic problems is studied for the analysis of shell or shell-like structures. It is shown that the conventional boundary integral equation (CBIE) for 3-D elasticity does not degenerate when applied to shell-like structures, contrary to the case when it is applied to crack-like problems where it does degenerate due to the closeness of the two crack surfaces. The treatment of the nearly singular integrals, which is a crucial step in the applications of BIEs to thin shapes, is presented in detail. To verify the theory, numerical examples of spherical and ellipsoidal vessels are presented using the BEM approach developed in this paper. It is found that the system of equations using the CBIE is well conditioned for all the thickness studied for the vessels. The advantages, disadvantages and potential applications of the proposed BEM approach to shell-like structures, as compared with the FEM regarding modelling and accuracy, are discussed in the last section. Applications of this BEM approach to shell-like structures with non-uniform thickness, stiffeners and layers will be reported in a subsequent paper. © 1998 John Wiley & Sons, Ltd.     

Numerical analysis of edge singularities in three-dimensional elasticity

A numerical method is described for the computation of eigenpairs which characterize the exact solution of linear elastostatic problems in three-dimensions in the vicinity of edge singularities. These may be caused by re-entrant corners, abrupt changes in boundary conditions or material properties. Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes. This paper addresses the problem of determining the edge eigenpairs numerically on the basis of the modified Steklov formulation (presented in Reference 1 in a 2-D framework) in conjunction with the p-version of the finite element method. Numerical results are presented for several cases including isotropic as well as anisotropic multi-material interfaces. © 1997 John Wiley & Sons, Ltd.     

Computational aspects of a new multi‐scale dispersive gradient elasticity model with micro‐inertia

Computational aspects of a recently developed gradient elasticity model are discussed in this paper. This model includes the (Aifantis) strain gradient term along with two higher‐order acceleration terms (micro‐inertia contributions). It has been demonstrated that the presence of these three gradient terms enables one to capture the dispersive wave propagation with great accuracy. In this paper, the discretisation details of this model are thoroughly investigated, including both discretisation in time and in space. Firstly, the critical time step is derived that is relevant for conditionally stable time integrators. Secondly, recommendations on how to choose the numerical parameters, primarily the element size and time step, are given by comparing the dispersion behaviour of the original higher‐order continuum with that of the discretised medium. In so doing, the accuracy of the discretised model can be assessed a priori depending on the selected discretisation parameters for given length‐scales. A set of guidelines can therefore be established to select optimal discretisation parameters that balance computational efficiency and numerical accuracy. These guidelines are then verified numerically by examining the wave propagation in a one‐dimensional bar as well as in a two‐dimensional example. Copyright © 2016 John Wiley & Sons, Ltd.     

三维空间Stokes问题的两个各向异性非协调混合有限元逼近

构造了两个可用于求解三维Stokes问题的各向异性非协调混合有限元格式,在不需要通常的辅助空间的情况下给出相应的收敛性分析和最优的误差估计。这两种单元具有构造简单,整体自由度少的特点,是目前较为理想的单元。该方法对进一步设计相关的自适应算法有潜在的应用价值。     

Symmetric coupling of multi‐zone curved Galerkin boundary elements with finite elements in elasticity

The coupling of Finite Element Method (FEM) with a Boundary Element Method (BEM) is a desirable result that exploits the advantages of each. This paper examines the efficient symmetric coupling of a Symmetric Galerkin Multi‐zone Curved Boundary Element Analysis method with a Finite Element Method for 2‐D elastic problems. Existing collocation based multi‐zone boundary element methods are not symmetric. Thus, when they are coupled with FEM, it is very difficult to achieve symmetry, increasing the computational work to solve the problem. This paper uses a fully Symmetric curved Multi‐zone Galerkin Boundary Element Approach that is coupled to an FEM in a completely symmetric fashion. The symmetry is achieved by symmetrically converting the boundary zones into equivalent ‘macro finite elements’, that are symmetric, so that symmetry in the coupling is retained. This computationally efficient and fast approach can be used to solve a wide range of problems, although only 2‐D elastic problems are shown. Three elasticity problems, including one from the FEM‐BEM literature that explore the efficacy of the approach are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite elements with embedded strong discontinuities for the modeling of failure in solids

This paper presents new finite elements that incorporate strong discontinuities with linear interpolations of the displacement jumps for the modeling of failure in solids. The cases of interest are characterized by a localized cohesive law along a propagating discontinuity (e.g. a crack), with this propagation occurring in a general finite element mesh without remeshing. Plane problems are considered in the infinitesimal deformation range. The new elements are constructed by enhancing the strains of existing finite elements (including general displacement based, mixed, assumed and enhanced strain elements) with a series of strain modes that depend on the proper enhanced parameters local to the element. These strain modes are designed by identifying the strain fields to be captured exactly, including the rigid body motions of the two parts of a splitting element for a fully softened discontinuity, and the relative stretching of these parts for a linear tangential sliding of the discontinuity. This procedure accounts for the discrete kinematics of the underlying finite element and assures the lack of stress locking in general quadrilateral elements for linearly separating discontinuities, that is, spurious transfers of stresses through the discontinuity are avoided. The equations for the enhanced parameters are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the aforementioned cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. Given the locality of all these considerations, the enhanced parameters can be eliminated by their static condensation at the element level, resulting in an efficient implementation of the resulting methods and involving minor modifications of an existing finite element code. A series of numerical tests and more general representative numerical simulations are presented to illustrate the performance of the new elements. Copyright © 2007 John Wiley & Sons, Ltd.     

Asymptotic homogenisation in linear elasticity. Part II: Finite element procedures and multiscale applications

The asymptotic expansion homogenisation (AEH) method can be used to solve problems involving physical phenomena on continuous media with periodic microstructures. In particular, the AEH is a useful technique to study of the behaviour of structural components built with composite materials. The main advantages of this approach lie on the fact that (i) it allows a significant reduction of the problem size and (ii) it has the capability to characterise stress and deformation microfields. In fact, specific equations can be developed to define these fields, in a process designated by localisation and not found on typical homogenisation methods. In the AEH methodology, overall material properties can be derived from the mechanical behaviour of selected periodic microscale representative volumes (also known as representative unit-cells, RUC). Nevertheless, unit-cell based modelling requires the control of some parameters, such as reinforcement volume fraction, geometry and distribution within the matrix material. The need for variety and flexibility leads to the development of automatic geometry generation algorithms. Additionally, the unstructured finite element meshes required by these RUC are usually non-periodic and involve the control of specific periodic boundary conditions. This work presents some numerical procedures developed in order to support finite element AEH implementations, rendering them more efficient and less user-dependent. The authors also present a numerical study of the influence of the reinforcement volume fraction on the overall material properties for a metal matrix composite (MMC) reinforced with spherical ceramic particles. A general multiscale application is shown, with both the homogenisation and localisation procedures.     

Mixed finite element formulation for the general anti-plane shear problem,including mode III crack computations,in the framework of dipolar linear gradient elasticity

A mixed formulation is developed and numerically validated for the general 2D anti-plane shear problem in micro-structured solids governed by dipolar strain gradient elasticity. The current mixed formulation employs the form II statement of the gradient elasticity theory and uses the double stress components and the displacement field as main variables. High order, C ⁰-continuous, conforming basis functions are employed in the finite element approximations (p-version). The results for the mode III crack problem reveal that, with proper mesh refinement at the areas of high solution gradients, the current approximation method captures the exact solution behaviour at different length scales, which depend on the size of material micro-structure. The latter is of vital importance because, near the crack tip, the nature of the exact solution, changes radically as we proceed from the macro- to micro-scale.     

Exploitation of symmetry in graphs with applications to finite and boundary elements analysis

We present explicit and parametric forms of transformation matrices for three well‐known and widely used symmetry groups: S₂, C_2v and C_4v. Group representation theory is the most powerful method for exploiting symmetry. We propose an efficient algorithm for systematic generation of reducible representations that can be combined linearly to obtain the projection operators. The exact column spaces of these projection operators are calculated and integrated through special orderings, leading to exact explicit and parametric forms of transformation matrices. The transformation matrices could be used directly for block diagonalization of single‐variable scalar field problems. Another algorithm is proposed to extend the application of the method to nonscalar and multivariable field problems. Finally, the generality and efficiency of the proposed method in relation to computation times and the accuracy of results are illustrated through examples from spectral decomposition, free vibration, buckling of FEMs and boundary element analysis of a symmetric field. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element modelling of cracked inelastic shells with large deflections: two‐dimensional and three‐dimensional approaches

Higher utilization of structural materials leads to a need for accurate numerical tools for reliable predictions of structural response. In some instances, both material and geometrical non‐linearities are allowed for, typically in assessments of structural collapse or residual strength in damaged conditions. The present study addresses the performance of surface‐cracked inelastic shells with out‐of‐plane displacements not negligible compared to shell thickness. This situation leads to non‐linear membrane force effects in the shell. Hence, a cracked part of the shell will be subjected to a non‐proportional history of bending moment and membrane force. An important point in the discretization of the problem is whether a two‐dimensional model describes the structural performance sufficiently, or a three‐dimensional model is required. Herein, the two‐dimensional modelling is performed by means of a Mindlin shell finite element. The cracked parts are accounted for by means of inelastic line spring elements. The three‐dimensional models employ eight‐noded solid elements. These models also account for ductile crack growth due to void coalescence by means of a modified Gurson–Tvergaard constitutive model, hence providing detailed solutions that the two‐dimensional simulations can be tested against. Using this, the accuracy of the two‐dimensional approach is checked thoroughly. The analyses show that the two‐dimensional modelling is sufficient as long as the cracks do not grow. Hence, using fracture initiation as a capacity criterion, shell elements and line springs provide acceptable predictions. If significant ductile tearing occurs before final failure, the line spring ligaments have to be updated due to crack growth.     

3-D Simulations of Diffusivity Measurements in Liquids with an Applied Magnetic Field

The effect of convective contamination in self-diffusivity experiments of liquid metals is predicted via a three-dimensional (3-D) model that includes an applied magnetic field. A uniform heat flux is applied at the sidewall of the cylindrical ampoule, and heat losses are allowed at the top and bottom walls of the ampoule. A wide range of a uniform, steady, axial magnetic field (from moderate to very strong) is considered in the model. Since the thermal Peclet number, Pe, is very small for the parameters of interest, convective heat transfer is neglected. A large interaction parameter, N, suggests that the flow is inertialess. The temperature and flow problems are solved at steady state while the time-dependent concentration problem is determined for various mass Peclet numbers, P e. In all cases, the output D (i.e., with convective contamination) increases with an increase in the temperature non-uniformity T. The radial and azimuthal velocities are much smaller than the axial velocity in each case. A stronger magnetic field can tolerate a higher temperature non-uniformity T, but T is still less than 0.025 K with a 5 T magnetic field for convective contaminations to be less than 5 of the total mass flux.     

A simple solution method to 3D integral nonlocal elasticity: Isotropic-BEM coupled with strong form local radial point interpolation

Nonlocal theories are of growing interest as they can address problems that lead to unphysical results in the framework of classical models. In this work, a solution procedure for three-dimensional integral nonlocal elastic solid is presented. The approach is based on the partition of the displacement field into complementary and particular parts. The complementary displacement is the solution of a Navier type equation and is obtained by the boundary element method, while the particular displacement is obtained using a local radial point interpolation method. The method is illustrated by comparing the responses to some simple loadings of a solid of finite extent with the original nonlocal model of Eringen and the enhanced model of Polizzotto.     

Ductile tearing in thin aluminum panels: experiments and analyses using large-displacement, 3-D surface cohesive elements

This paper describes a large-displacement formulation for a 3-D, interface-cohesive finite element model and its application to predict ductile tearing in thin aluminum panels. A nonlinear traction-separation relationship defines the constitutive response of the initially zero thickness interface elements. Applications of the model simulate crack extension in C(T) and M(T) panels made of a 2.3 mm thick, Al 2024-T3 alloy tested as part of the NASA-Langley Aging Aircraft program. Tests of the M(T) specimens without guide plates exhibit significant out-of-plane (buckling) displacements during crack growth which necessitates the large-displacement, cohesive formulation. The measured load vs. outside surface crack extension behavior of high constraint (T-stress>0) C(T) specimen drives the calibration process of the cohesive fracture model. Analyses of low constraint M(T) specimens, having widths of 300 and 600 mm and various a/W ratios, demonstrate the capabilities of the calibrated model to predict measured loads and measured outside surface crack extensions. The models capture accurately the strong 3-D effects leading to out-of-plane buckling and various degrees of crack front tunneling in the C(T) and M(T) specimens. Previous analyses of these specimens using a crack tip opening angle (CTOA) criterion for growth show good agreement with measured peak loads. However, without the ability of the interface-cohesive model to predict tunneling behavior, the CTOA approach overestimates crack extensions early in the loading when tunneling behavior dominates the response.