Anisotropic Elastic Materials Research Articles

Crack propagation in anisotropic composite structures

A crack approaching a material interface between two elastic materials may stop or may advance by either penetrating the interface or deflecting into the interface (cf. N.Y. He and J.W. Hutchinson, Int. J. Solids Struct. 25 (1989), 1053-1067). Mathematical models for crack path prediction are based on the asymptotic behavior near the crack tip. In this work, an idea to compute the asymptotic expansion of the displacement field for structures composed of two dissimilar elastic anisotropic materials is shown, if the crack impinges the material interface. In contrast to the well-known case of isotropic materials, logarithmic terms can appear in the asymptotic decomposition of the displacement field in anisotropic composites. Based on this results, the energy release rate is calculated for different scenarios of crack propagation in composite structures.

Calculation of interfacial stresses in composites containing elliptical inclusions of various types

Abstract A volume integral equation method (VIEM) is used to study elastostatic problems in an unbounded elastic solid containing multiple elliptical inclusions of arbitrary orientation subject to uniform tensile stress at infinity. The inclusions are assumed to be long parallel elliptical cylinders composed of isotropic or anisotropic elastic materials and perfectly bonded to the isotropic matrix. The solid is assumed to be under plane strain on the plane normal to the cylinders. In contrast to previous studies cited in this paper where only one or a few specific types of inclusions were considered, a detailed analysis of the stress field at the matrix-inclusion interface for square and hexagonal packing arrays is carried out herein, taking into account different values for the number, aspect ratio, orientation angle and concentration of the elliptical inclusions. The accuracy and efficiency of the method are examined through comparison with results obtained from analytical and finite element methods.

Combining analysis of random elastic polycrystals with poroelasticity for granular composites having orthotropic porous grains and fluid-filled pores

Analysis of random polycrystals has typically been applied to solid grains of anisotropic elastic materials. Poroelastic analysis has similarly been applied to otherwise isotropic systems with pores having a variety of shapes and filled with fluids. Present effort is focused on combining these two types of geomechanical analyses by treating anisotropic (specifically orthotropic) poroelastic grains jumbled together to form an overall isotropic polycrystalline poroelastic material. The resulting problem is approximately twice as difficult to solve as the typical elastic polycrystal problem because the polycrystal analysis must be carried through twice: once for the drained (pore-fluid free to escape) poroelastic constants, and again for the undrained (pore-fluid trapped) poroelastic constants. As should be anticipated, poroelastic effects induced by trapped fluid are significantly stronger for the effective bulk modulus than for the effective shear modulus.

The topological gradient in anisotropic elasticity with an eye towards lightweight design

We derive a representation formula for the topological gradient with respect to arbitrary quadratic yield functionals and anisotropic elastic materials, thus laying the theoretical foundations for topological sensitivity analysis in lightweight design. For compliance, minimization involving general anisotropic materials and ellipsoidal perturbations, we give a closed formula for the topological gradient, enabling topology optimization of integrated designs involving several reinforced materials. If the materials are transversely isotropic and the perturbations are spheriodal, we even obtain an analytical formula. For general anisotropy, recent advances in the computation of Eshelby's tensor enable rapid numerical computation of the topological gradient.Restricting to isotropic materials and spheroidal inclusions, we obtain an analytical formula for minimizing isotropic yield functionals with applications to microscale‐scale sensitivity analysis of fiber reinforced composites or reinforcing analysis of brittle materials. Copyright © 2013 John Wiley & Sons, Ltd.

Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.

Multiple elliptical inclusions of arbitrary orientation in composites

Abstract A volume integral equation method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing multiple elliptical inclusions of arbitrary orientation subjected to uniform tensile stress at infinity. The inclusions are assumed to be long parallel elliptical cylinders composed of isotropic and anisotropic elastic material perfectly bonded to the isotropic matrix. The solid is assumed to be under plane strain on the plane normal to the cylinders. A detailed analysis of the stress field at the matrix – inclusion interface for square and hexagonal packing arrays is carried out, taking into account different values for the number, orientation angles and concentration of the elliptical inclusions. The accuracy and efficiency of the method are examined in comparison with results available in the literature.

A mesoscopic model for deep drawing of carbon fibre prepregs at large strains

The simulation of prepregs must regard highly anisotropic, viscoelastic and thermal-chemical properties. In this work we split the constitutive equations for the two component material into an anisotropic elastic material part and an isotropic, viscoelastic behaviour part, where the former represents the fibres and the latter represents the matrix. The second part also takes account of curing, which is dependent on time and temperature. During real deep-drawing processes large deformations up to 50% are occurring, which is accounted for in a large strain theory. For the elastic fibres we use a Neo-Hooke law enhanced by an anisotropic part. For the viscoelastic matrix we use Hencky-strains (or logarithmic strains) and the work-conjugate Hill-stresses. Moreover the identified parameters for the model are presented. These are validated by simulation of a simple block and a real deep drawing process forming a spherical part.

A contact model for orthotropic-viscoelastic materials

In many industrial applications, fibre-reinforced polymers are in contact with rigid surfaces. The contact behaviour of such polymers is both anisotropic and viscoelastic: the polymers exhibit viscoelastic behaviour and the addition of short fibres results in a directionality in the material behaviour of the composite. This work focuses on the contact problem of an orthotropic viscoelastic material in contact with a spherical rigid indenter, in which the radius of the contact area is much larger than the fibre size.A general form of anisotropic behaviour has been considered in order to describe materials with a high degree of anisotropy. By using separation of variables a solution for the quasi-static contact problem for viscoelastic and generally anisotropic materials is obtained by combining the linear theory of viscoelasticity with the Hertz solution for elastic anisotropic materials as derived by Willis. The developed contact model requires nine elastic material parameters in combination with one time dependent material parameter as input, in order to characterise the behaviour of the material. It is shown that the results of the developed model show good agreement with both isotropic and anisotropic materials described in literature. Furthermore, it is shown that ignoring the anisotropy of a viscoelastic material will result in an overestimate of the stiffness of the material.

Lattice instability of V2AlC at high pressure

We investigate the elastic and thermodynamic properties of nanolaminate V2AlC by using the ab initio pseudopotential total energy method. The axial compressibility shows that the c axis is always stiffer than a axis. The elastic constants revealed the structural instability at about 500 and 732 GPa. Furthermore, elastic constants C44 reached its maximum at about 550 GPa, differing with the other four C11, C12, C13 and C33 constants. The Poisson’s ratio investigations demonstrated that a higher ionic or weaker covalent contribution in intra-atomic bonding and the degree of ionicity increases with pressure. The G/B and B/C44 investigations revealed that V2AlC is brittle and the brittleness decreases with pressure. Also, we found that V2AlC is elastic anisotropic materials and the degree of anisotropy rapidly rises with pressure. Study on Debye temperature and Gruneisen parameter observed weak temperature and strong pressure responses, whereas the sensitive dependence in the thermal expansion coefficient and Helmholtz free energy are clearly seen.

Contact analysis for anisotropic elastic materials considering surface stress and surface elasticity

Contact analysis for an anisotropic elastic half-space is performed using the surface Green's function considering surface stress and surface elasticity. The discrete convolution and fast Fourier transform (DC–FFT) method and the conjugate gradient (CG) method was used to conduct the contact analysis. The influence coefficient in the contact analysis is calculated using the surface Green's function. The depth used employed in the surface Green's function considering surface stress and surface elasticity was decided such that the results of contact analysis of a rigid spherical indenter using Green's function for several values of depth agreed with the results of MD analysis. Several contact analyses were conducted in order to investigate the effects of material anisotropy and surface properties on the penetration depth for the ridge orientation in elliptic, Knoop, Vickers, and Berkovich indenters. When used to penetrate the (100) and (111) surfaces of Cu or Al, the penetration depth on indenter orientation clearly indicates material anisotropy in the elliptic and Knoop indenters, and indicates only a slight effect of material anisotropy in the Vickers and Berkovich indenters. The displacement field considering surface stress and surface elasticity is rounded in comparison with the shape of each crystal structure.

An attempt to separate elastic strain energy density of linear elastic anisotropic materials based on strains considerations

A promising new effort toward the decomposition of the elastic strain energy density of linear elastic anisotropic materials into a dilatational and a distortional part is presented. By assuming that volume changes must keep the material symmetries unchanged, a new physical perspective is presented and interesting definitions are drawn. This new perspective necessitates the introduction of a strain parameter m characteristic of the material’s anisotropy. This strain parameter besides easing the calculation of the dilatational and distortional energetic terms additionally accounts for the directional sensitivity of anisotropic materials.

Retraction Note: Crystal’s anisotropic properties and tensor representation: a discussion

For most applications, crystals need a careful selection to process an appropriate symmetry for a particular application. A crystal is innately symmetrical; hence it presents the same appearance from a number of different directions. Some of the physical properties of crystals which exhibit dependence on symmetry elements are presented. A method of using the inherent symmetry in order to simplify the formulation of the physical properties is needed. The use of tensors is one such tool. The authors review what tensors of different ranks are, and show how such tensors can be used to describe the directional variation of the physical properties within crystals. These arise when a tensor relates a vector to a second-rank tensor. Properties that involve third-rank tensors include the piezoelectric effect. However, to exhibit the reduction of the number of tensor components further, consider the fourth-rank tensors. Both stress and strain can be represented by second-rank tensors, the modulus can be represented by a fourth-rank tensor. Stress is the force acting in any direction divided by the area. The fourth-rank elasticity tensor of an anisotropic and linear elastic material of the material is considered.

Integral identities for a semi-infinite interfacial crack in anisotropic elastic bimaterials

The focus of the article is on the analysis of a semi-infinite crack at the interface between two dissimilar anisotropic elastic materials, loaded by a general asymmetrical system of forces acting on the crack faces. Recently derived symmetric and skew-symmetric weight function matrices are introduced for both plane strain and antiplane shear cracks, and used together with the fundamental reciprocal identity (Betti formula) in order to formulate the elastic fracture problem in terms of singular integral equations relating the applied loading and the resulting crack opening. The proposed compact formulation can be used to solve many problems in linear elastic fracture mechanics (for example various classic crack problems in homogeneous and heterogeneous anisotropic media, as piezoceramics or composite materials). This formulation is also fundamental in many multifield theories, where the elastic problem is coupled with other concurrent physical phenomena.

Transient dislocation emission from a crack tip in an anisotropic elastic material

The emission of a dislocation with a general Burgers vector from the tip of a stationary semi-infinite crack in an anisotropic elastic material is examined. The dislocation is assumed to leave the crack tip along the crack extension plane at constant speed. Explicit expressions for the transient shielding stress intensity factors at the crack tip and the drag forces on the dislocations are derived. Numerical results for a class of cubic materials and two hexagonal crystals, zinc and cobalt, are given. Dislocation emission under plane stress wave loading is discussed.

The Numerical Study on Errors of Stress in Anisotropic Linear Elastic Material when Simplified as Orthogonal one

Keywords: strain, stress, material constants, constitutive equation, anisotropy. Abstract. If the material is anisotropic, there are differences in stress distribution under the same boundary conditions when it was simplified as an orthotropic material. We established a simple finite element model for rectangular perforated planar material, in which one side was fixed, the opposite side was loaded with uniform force, and the other sides were set free. Based on this model we studied the difference of distribution of stress between anisotropic material and its simplified form, orthotropic material. The results showed differences in some cases quite large, the maximum relative error of extreme stress can reach 341%. In conclusions, this study does not support that the complex anisotropic materials are simplified to orthotropic materials. If researchers only concern the location of extreme stress, this study does not deny that the complex anisotropic materials can be simplified to orthotropic one.

A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact

We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new secular equation is obtained that covers all special cases in one equation. One special case is when a Rayleigh wave (called the R-wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S-wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.

Deficiencies in numerical models of anisotropic nonlinearly elastic materials

Incompressible nonlinearly hyperelastic materials are rarely simulated in finite element numerical experiments as being perfectly incompressible because of the numerical difficulties associated with globally satisfying this constraint. Most commercial finite element packages therefore assume that the material is slightly compressible. It is then further assumed that the corresponding strain-energy function can be decomposed additively into volumetric and deviatoric parts. We show that this decomposition is not physically realistic, especially for anisotropic materials, which are of particular interest for simulating the mechanical response of biological soft tissue. The most striking illustration of the shortcoming is that with this decomposition, an anisotropic cube under hydrostatic tension deforms into another cube instead of a hexahedron with non-parallel faces. Furthermore, commercial numerical codes require the specification of a 'compressibility parameter' (or 'penalty factor'), which arises naturally from the flawed additive decomposition of the strain-energy function. This parameter is often linked to a 'bulk modulus', although this notion makes no sense for anisotropic solids; we show that it is essentially an arbitrary parameter and that infinitesimal changes to it result in significant changes in the predicted stress response. This is illustrated with numerical simulations for biaxial tension experiments of arteries, where the magnitude of the stress response is found to change by several orders of magnitude when infinitesimal changes in 'Poisson's ratio' close to the perfect incompressibility limit of 1/2 are made.

Reciprocity principle for the detection of planar cracks in anisotropic elastic material

The functionality of elastic devices can be strongly reduced by cracks. Therefore, it is important to develop nondestructive techniques for testing whether cracks exist inside the device, and if they exist, what the position and size of these cracks are. In this paper, we focus on the inverse method of measurements of boundary data for anisotropic elastic fields under different loads in order to detect cracks. We draw on the paper of Andrieux et al (1999 Inverse Problems 15 59–65), where a reciprocity principle is used for the detection of plane cracks in isotropic linear elastic materials. Utilizing that the setup and significant parts of the analysis by Andrieux et al are formulated generally and remain valid in anisotropic materials, we generalize this method for two- and three-dimensional transversely anisotropic elastic devices and demonstrate it by numerical experiments. We start from a cracked domain and generate boundary data as artificial measurements by solving numerically a forward problem. The inverse computations show good agreement between the simulated crack and the original one.

On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials

In this paper, we show that the well known decomposition of a matrix into an orthogonal matrix (Q) and a upper triangular matrix (R), can be used to develop a remarkably simple representation for constitutive equations for anisotropic Green elastic materials. Using this decomposition (in which the upper triangular matrix is related to the Cholesky factorization of the Cauchy Green stretch tensor C), in lieu of the traditional polar decomposition, provides significant advantages: (1) the decomposition is very simple, fast and widely implemented in every major linear algebra package and does not require knowledge of eigenvalues. (2) The factorization allows us to decompose the total deformation into a sequence of simple, physically visualizable, meaningful elementary deformations. (3) It provides a remarkably simple representation of the stress response in terms of the derivatives of the strain energy (see Eqs. (26)–(28)) especially for anisotropic materials. (4) It provides a unified and simple way of considering triclinic, monoclinic, orthorhombic, and transversely isotropic materials and satisfies the “orthogonality” criterion described by Criscione (2005) in order to minimize experimental error propagation. The results, which were inspired by and are closely related to the developments by Criscione, Douglas, and Hunter (2001) and Criscione and Hunter (2003) and follows many of their ideas, are in a much simpler and computationally tractable form than that given by Criscione and coworkers while retaining every desirable feature of the decomposition presented by them.

Concentrated impact loading on one face of a semi-infinite crack in an anisotropic elastic material

The response of an unbounded anisotropic elastic body containing a semi-infinite crack subjected to a concentrated impact force on one of the crack faces is studied. An exact solution of the dynamic stress intensity factors is obtained from a linear superposition of the solution of Lamb’s problem and a solution of a dislocation emitting from the crack tip. The stress intensity factors exhibit square-root singularity upon the arrival of the Rayleigh wave at the crack tip. As the Rayleigh wave passes through the crack tip, the stress intensity factors either instantaneously assume the static values or gradually approach to zero. Several numerical examples are given for isotropic, cubic and orthotropic materials.