A circular inclusion in a finite domain II. The Neumann-Eshelby problem

Summary This is the second paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. In this part, an exact and closed form solution is obtained for the elastic fields of a circular inclusion embedded in a finite circular representative volume element (RVE), which is subjected to the traction (Neumann) boundary condition. The disturbance strain field due to the presence of an inclusion is related to the uniform eigenstrain field inside the inclusion by the so-called Neumann-Eshelby tensor. Remarkably, an elementary, closed form expression for the Neumann-Eshelby tensor of a circular RVE is obtained in terms of the volume fraction of the inclusion. The newly derived Neumann-Eshelby tensor is complementary to the Dirichlet-Eshelby tensor obtained in the first part of this work. Applications of the Neumann-Eshelby tensor are discussed briefly.     

Inclusion problem of a two-dimensional finite domain: The shape effect of matrix

With the advance in composite mechanics and micromechanics, there are increasing demands for analytical solutions of inclusion problems in a bounded domain. To echo this need, this study is focused on establishing explicit expressions of elastic fields for a 2D elastic domain containing a circular inclusion at center. Unlike the configuration in the classical Eshelby formulation, the elastic domain in this study is bounded and has shapes other than a circle. To circumvent the mathematical difficulty in solving Green’s function in a finite domain, an approach powered by complex potential method, which has been successfully employed to formulate the elastic fields for inclusion problems where matrix is unbounded or bounded by a circle, is extended to finite domains displaying complicated shapes, particularly, a Pascal’s limaçon and a curved square (an approximation of perfect square) in this study. In order to take advantage of the mathematical simplicity inherent in expressing a circular geometry, conformal mapping is used to transform the complex geometry of the finite domain of interest to a unit circle. The governing complex potentials, which capture the discontinuity on the inclusion–matrix interface due to the uniform eigenstrain within the inclusion, are formulated with the aid of Cauchy integral and then explicitly identified by satisfying the prescribed boundary conditions. In this study, the displacement fields for finite domains bounded by a Pascal’s limaçon and a curved square are obtained based on Dirichlet (displacement) boundary conditions imposed by the far field strain. In addition to asymptotical behaviors, firm agreement is also achieved when the analytical solutions based on complex potentials are compared with the FEM results. Furthermore, inverse of the conformal mapping is discussed here in order to get the explicit expression for elastic fields.     

Linear and elastic–plastic fracture mechanics revisited by use of Fourier transforms – theory and application

This paper investigates whether and how discrete Fourier transforms (DFT) can be used to compute the local stress/strain distribution around holes in externally loaded two-dimensional representative volume elements (RVEs). To this end, the properties of DFT are first reviewed and then applied to the solution of linear elastic and time-dependent elastic plastic material response. The equivalent inclusion method is used to derive a functional equation which allows for the numerical computation of stresses and strains within an RVE with heterogeneities of arbitrary shape and stiffness. This functional equation is then specialized to the case of circular and elliptical holes of different minor axes which eventually degenerate into Griffith cracks. The numerically predicted stresses and strains are compared to the corresponding analytical solutions for a single circular as well as an elliptical hole in an infinitely large plate under tension as well as to finite element calculations (for time-independent elastic/plastic material response).     

Micromechanics as a basis of random elastic continuum approximations

The problem of the determination of stochastic constitutive laws for input to continuum-type boundary value problems is analyzed from the standpoint of the micromechanics of polycrystals and matrix-inclusion composites. Passage to a sought-for random continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that an elastic microstructure with piecewise continuous realizations of random tensor fields of stiffness cannot be uniquely approximated by a random field of stiffness with continuous realizations. Rather, two random continuum fields may be introduced to bound the material response from above and from below. As the size of the RVE relative to the crystal size increases to infinity, both fields converge to a deterministic continuum with a progressively decreasing strength of fluctuations. Since the RVE corresponds to a single finite element cell not infinitely larger than the crystal size, two random fields are to be used to bound the solution of a given boundary value problem at a given scale of resolution. The method applies to a number of other elastic microstructures, and provides the basis for stochastic finite differences and elements. The latter point is illustrated by an example of a stochastic boundary value problem of a heterogeneous membrane.     

Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.     

A micromechanics based analysis of hollow fiber composites using DQEM

In this study, a generalized plane strain micromechanical model is presented to obtain micro-stress/strain fields within the unidirectional (UD) hollow fiber reinforced composites. In addition, the thermally induced residual stresses during cooling down process, overall elastic properties and energy absorption capability of hollow reinforced composite are studied. The representative volume element (RVE) of the composite consists of a quarter of the fiber surrounded by matrix to represent the real composite with repeating square array of fibers. Fully bonded fiber–matrix interface condition is considered and the displacement continuity and traction reciprocity are properly imposed to the interface. The cubic serendipity shape functions are used to convert the solution domain to a proper rectangular domain. A Least-squares based differential quadrature element method (DQEM) is used to obtain solutions for the governing partial differential equations of the problem. Results of the presented method for various stress and displacement components and thermal residual stresses show excellent agreement with finite element analysis. Furthermore, predicted overall properties also show good agreement with other available analytical and finite element results. Moreover, results also revealed that the presented model can provide highly accurate predictions with a few number of elements and grid points within each element.     

Elastic solutions of a cylindrical rod containing periodically distributed inclusions with axisymmetric eigenstrains

Summary. In this paper, we give the elastic solution for a special type of microstructure – a circular cylindrical rod containing periodically distributed inclusions along its axial direction. Each inclusion has the same uniform axisymmetric transformation strain (eigenstrain). Analytical elastic solutions are obtained for the displacements, stresses and elastic strain energy of the rod. The effects of microstructure and its evolution (growth of inclusions) on the elastic stress and strain fields as well as the strain energy of the rod are quantitatively demonstrated. As a result of such microstructure evolution nominal stress-strain relation with strain softening is derived for a rod under uniaxial tension.     

A comparison of homogenization and standard mechanics analyses for periodic porous composites

Composite material elastic behavior has been studied using many approaches, all of which are based on the concept of a Representative Volume Element (RVE). Most methods accurately estimate effective elastic properties when the ratio of the RVE size to the global structural dimensions, denoted here as , goes to zero. However, many composites are locally periodic with finite . The purpose of this paper was to compare homogenization and standard mechanics RVE based analyses for periodic porous composites with finite . Both methods were implemented using a displacement based finite element formulation. For one-dimensional analyses of composite bars the two methods were equivalent. Howver, for two- and three-dimensional analyses the methods were quite different due to the fact that the local RVE stress and strain state was not determined uniquely by the applied boundary conditions. For two-dimensional analyses of porous periodic composites the effective material properties predicted by standard mechanics approaches using multiple cell RVEs converged to the homogenization predictions using one cell. In addition, homogenization estimates of local strain energy density were within 30% of direct analyses while standard mechanics approaches generally differed from direct analyses by more than 70%. These results suggest that homogenization theory is preferable over standard mechanics of materials approaches for periodic composites even when the material is only locally periodic and is finite.     

Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate‐independent finite strain analysis of solids undergoing large elastic‐isochoric plastic deformations. The formulation relies on the introduction of a mixed‐variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed‐variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J₂ yield criterion and rate‐independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher‐order Padé approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi‐displacement finite element procedure. Finally, the numerical solution of finite strain elastic‐plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

Material properties of piezoelectric composites by BEM and homogenization method

Applications of boundary element method (BEM) to piezoelectric composites in conjunction with homogenization approach for determining their effective material properties are discussed in this paper. The composites considered here consist of inclusion and matrix phases. The homogenization model for composites with inhomogeneities is developed and introduced into a BE formulation to provide an effective means for estimating overall material constants of two-phase composites. In this model, a representative volume element (RVE) is used whose volume average stress and strain are calculated by the boundary tractions and displacements of the RVE. Thus BEM is suitable for performing calculations on average stress and strain fields of the composites. Numerical results for a piezoelectric plate with circular inclusions are presented to illustrate the application of the proposed micromechanics––BE formulation.     

Fourier transforms — An alternative to finite elements for elastic-plastic stress-strain analyses of heterogeneous materials

Summary The intent of this paper is to apply the technique of discrete Fourier transforms (DFT) to the computation of the stress and strain fields around holes in an externally loaded two-dimensional representative volume element (RVE). This is done to show that DFT is capable to handle geometries with rather sharp corners as well as steep gradients in material properties which is of importance for modeling changes in micro-morphology. To this end DFT is first briefly reviewed. In a second step it is applied to the appropriate equations which characterize a linear-elastic as well as a time-independent elastic-plastic, heterogeneous material subjected to external loads. The equivalent inclusion technique is used to derive a functional equation which, in principle, allows to compute numerically the stresses and strains within an RVE that contains heterogeneities of arbitrary shape and arbitrary stiffness (in comparison to the surrounding matrix). This functional equation is finally specialized to the case of circular and elliptical holes of various slenderness which degenerate into Griffith cracks in the limit of a vanishing minor axis. The numerically predicted stresses and strains are compared to analytical solutions for problems of the Kirsch type (a hole in an large plate subjected to tension at infinity) as well as to finite element studies (for the case of time-independent elastic/plastic material behavior).     

Embedded discontinuity finite element method for modeling of localized failure in heterogeneous materials with structured mesh: an alternative to extended finite element method

In this work we discuss the finite element model using the embedded discontinuity of the strain and displacement field, for dealing with a problem of localized failure in heterogeneous materials by using a structured finite element mesh. On the chosen 1D model problem we develop all the pertinent details of such a finite element approximation. We demonstrate the presented model capabilities for representing not only failure states typical of a slender structure, with crack-induced failure in an elastic structure, but also the failure state of a massive structure, with combined diffuse (process zone) and localized cracking. A robust operator split solution procedure is developed for the present model taking into account the subtle difference between the types of discontinuities, where the strain discontinuity iteration is handled within global loop for computing the nodal displacement, while the displacement discontinuity iteration is carried out within a local, element-wise computation, carried out in parallel with the Gauss-point computations of the plastic strains and hardening variables. The robust performance of the proposed solution procedure is illustrated by a couple of numerical examples. Concluding remarks are stated regarding the class of problems where embedded discontinuity finite element method (ED-FEM) can be used as a favorite choice with respect to extended FEM (X-FEM).     

Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size

The main objective of this paper is to present a generic meso-scale probability model for a large class of random anisotropic elastic microstructures in order to perform a parametric analysis of the Representative Volume Element (RVE) size. This new approach can be useful for a direct experimental identification of random anisotropic elastic microstructures when the standard method cannot easily be applied to anisotropic elastic microstructures. Such a RVE is used to construct the macroscopic properties in the context of stochastic homogenization. The probability analysis is not performed as usual for a given particular random microstructure defined in terms of its constituents. Instead, it is performed for a large class of random anisotropic elastic microstructures. For this class, the probability distribution of the random effective stiffness tensor is explicitly constructed. This allows a full probability analysis of the RVE size to be carried out and its convergence to be studied. The procedure of homogenization is based on a homogeneous Dirichlet condition on the boundary of the RVE. The probability model used for the stiffness tensor-valued random field of the random anisotropic elastic microstructure is an extension of the model recently introduced by the author for elliptic stochastic partial differential operators. The stochastic boundary value problem is numerically solved by using the stochastic finite element method. The probability analysis of the RVE size is performed by studying the probability distribution of the random operator norm of the random effective stiffness tensor with respect to the spatial correlation length of the random microstructure.     

Multiscale Nonlinear Constitutive Modeling of Carbon Nanostructures Based on Interatomic Potentials

Continuum-based modeling of nanostructures is an efficient and suitable method to study the behavior of these structures when the deformation can be considered homogeneous. This paper is concerned about multiscale nonlinear tensorial constitutive modeling of carbon nanostructures based on the interatomic potentials. The proposed constitutive model is a tensorial equation relating the second Piola-Kirchhoff stress tensor to Green-Lagrange strain tensor. For carbon nanotubes, some modifications are made on the planar representative volume element (RVE) to account for the curved atomic structure resulting a non-planar RVE. Using the proposed constitutive model, the elastic behavior of the graphene sheet and carbon nanotube are studied.     

Anti-plane Circular Nano-inclusion Problem with Electric Field Gradient and Strain Gradient Effects

As well known, gradient theories can describe size effects that are important in nano-scale problems. In this paper, we analyze the Eshelby-type anti-plane inclusion problem embedded in infinite dielectric body by considering both strain gradient and electric field gradient effects to account for the size effect and high-order electromechanical coupling effect. The size-dependent Eshelby and Eshelby-like tensor, strain, stress, electric field and electric displacement components are derived explicitly by means of Green's function method. Theoretical results indicate that strain and electric field are decoupled for anti-plane inclusion problem while stress field and electric displacement are coupled through strain gradient and electric field gradient. Based on the general form, the expressions for a special case of circular inclusion are obtained analytically. Numerical results reveal that when the inclusion radius becomes small, the gradient effects are significantly important and should not be ignored. The values approach asymptotically to classical solutions as increase of inclusion size. And the high-order electromechanical coupling effect in non-piezoelectric material (centrosymmetric dielectrics) can be equivalent to piezoelectricity of conditional piezoelectric materials when the inclusion size is small.     

A mixed finite element for the nonlinear analysis of in-plane loaded masonry walls

A mixed membrane eight-node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history-dependent 2D stress-strain constitutive law is used to model masonry material, the element derivation is based on a Hu-Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral-type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement-based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.     

Solution of the Eshelby-type anti-plane strain polygonal inclusion problem based on a simplified strain gradient elasticity theory

The Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion of arbitrary-shape polygonal cross-section is analytically solved using a simplified strain gradient elasticity theory that incorporates one material length scale parameter. The Eshelby tensor (with four nonzero components) is obtained in a general form in terms of two scalar-valued potential functions. These potential functions, as area integrals over the polygonal cross-section, are first converted to two line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former does not contain any elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle (inclusion) size effect. For homogenization applications, the area average of the new position-dependent Eshelby tensor over the polygonal cross-section is also provided in a general form. To illustrate the newly obtained Eshelby tensor, five types of regular polygonal inclusions (i.e., triangular, quadrate, hexagonal, octagonal, and tetrakaidecagonal) are quantitatively studied by directly using the general formulas derived. The components of the induced strain and the averaged Eshelby tensor inside the inclusion are evaluated. Numerical results reveal that the induced strain varies with both the position and the inclusion size. The values of the induced strain components in a polygonal inclusion approach from below those in a corresponding circular inclusion when the inclusion size or the number of sides of the polygonal inclusion increases. The results for the averaged Eshelby tensor components show that the size effect is significant when the inclusion size is small but may be neglected for large inclusions.     

Conference diary

A variational higher-order theory for bending and stretching of linearly elastic orthotropic beams including the deformations due to transverse shearing and stretching of the transverse normal fibre is presented. The theory assumes a linear distribution for the longitudinal displacement and a parabolic variation of the transverse displacement across the thickness. Additionally, independent expansions are introduced for the through-thickness displacement gradients with the requirement of a least-square compatibility for the transverse strains and the satisfaction of exact stress boundary conditions at the top/bottom beam surfaces. The theory is shown to be well suited for finite element development requiring simple C⁰- and C^?1- continuous displacement interpolation fields. To demonstrate the computational utility of the theory, a simple two-node stretching-bending finite element is formulated. The analytic and finite element results are obtained for a simple bending problem for which an exact elasticity solution is available. It is shown that the inclusion of the transverse normal deformation in the present theory enables improved displacement, strain and stress predictions, particularly, in the analysis of deep beams.     

考虑界面影响的混凝土弹性模量的数值预测

提出了一种考虑界面过渡层影响的混凝土弹性模量的数值预测方法。将球形骨料与包裹它的界面过渡层作为二相复合球结构的等效颗粒,由广义自洽方法计算不同粒径骨料与界面过渡层组成复合球的有效模量。然后由等效颗粒生成的随机骨料模型建立体积表征单元,施加均匀位移边界条件,通过数值方法计算该体积表征单元中的应力和应变场,由细观力学数值均匀化方法预测体积表征单元的有效弹性模量。计算结果表明:对于不同骨料含量的混凝土,有效弹性模量的预测值与试验值非常接近,界面过渡层的厚度对混凝土的整体弹性性质有较大影响。     

A finite deformation constitutive model for shape memory polymers based on Hencky strain

In many engineering applications, shape memory polymers (SMPs) usually undergo arbitrary thermomechanical loadings at finite deformation. Thus, development of 3D constitutive models for SMPs within the finite deformation regime has attracted a great deal of interest. In this paper, based on the classical framework of thermodynamics of irreversible processes, employing the logarithmic (or Hencky) strain as a more physical measure of strain, a 3D large-strain macromechanical model is presented. In the constitutive model development, we adopt a multiplicative decomposition of the deformation gradient into elastic and stored parts. In addition, employing the averaging scheme, the logarithmic elastic strain tensor is decomposed into the rubbery and glassy parts. The evolution equations for internal variables are introduced for both cooling and heating processes. The time-discrete form of the proposed model in the implicit form is also presented. Comparing the predicted results with experimental data reported in the literature, the model is validated. Finally, using the finite element method, two boundary value problems e.g., a 3D beam and a medical stent made of SMPs are numerically simulated.