Determination of the Elastic Symmetry of a Monolithic Ceramic using Bulk Acoustic Waves

A nondestructive optimal determination of elastic properties from ultrasonic bulk wave velocity measurements on a monolithic ceramic plate immersed in water is presented. This procedure, that is applicable to flat plates with unknown material properties, is based on already established methods and includes discussions, using experimental data, on the reliability of the elastic property identification, such as the stiffness tensor and the material symmetry. By solving inverse propagation problems deduced from the Christoffel equation and depending on wave speed measurements, we show that the studied sample can be described by twenty-one dependent stiffness constants and that its intrinsic elastic material symmetry was hexagonal (or transversely isotropic).     

Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials

Summary Constitutive relations for incompressible (slightly compressible) anisotropic materials cannot (could hardly) be obtained through the inversion of the generalized Hooke's law since the corresponding compliance tensor becomes singular (ill-conditioned) in this case. This is due to the fact that the incompressibility (slight compressibility) condition imposes some additional constraints on the elastic constants. The problem requires a special procedure discussed in the present paper. The idea of this procedure is based on the spectral decomposition of the compliance tensor but leads to a closed formula for the elasticity tensor without explicit using the eigenvalue problem solution. The condition of nonnegative (positive) definiteness of the material tensors restricts the elastic constants to belong to an admissible value domain. For orthotropic and transversely isotropic incompressible as well as isotropically compressible materials the corresponding domains are illustrated graphically.     

An inverse procedure for determination of material constants of a periodic multilayer using Floquet wave homogenization

An inverse numerical method for periodic composite characterization is reported. The method utilizes the velocity data based on Floquet wave homogenization. The optimization procedure is performed on the basis of the gradient method. An efficient polynomial function is derived from Christoffel equation. The numerical procedure leads to an analytical form of the minimized function which is related to the whole Floquet data. The set of input data is collected from different azimuthal plane orientations inside the homogenization domain. The output results mainly include the effective elastic constants of the multidirectional composite and the reliability factor. The initialization of the elastic stiffness matrix is obtained by averaging the rigidity tensor corresponding to each layer orientation. This procedure is examined for [0/90] and [0/60/−60] composites; some of the obtained elastic constants are significantly dependent on the frequency. The agreement between the adopted Floquet velocities and the calculated ones is good; the reliability factor does not exceed 1%. Slight deviations are pointed out in the vicinity of the homogenization limits.     

An alternative model for anisotropic elasticity based on fabric tensors

Motivated by the mechanical analysis of multiphase or damaged materials, a general approach relating fabric tensors characterizing microstructure to the fourth rank elasticity tensor is proposed. Using a Fourier expansion in spherical harmonics, the orientation distribution function of a positive, radially symmetric microstructural property is approximated by a scalar and a symmetric, traceless second rank tensor. Following this approximation, a general expression of the elastic free energy potential is derived from representation theorems for anisotropic scalar functions. Based on a homogeneity assumption for the elastic constitutive law with respect to the microstructural property, a particular elasticity model is developed that involves three independent constants beside the fabric tensors. Strict positive definiteness of the corresponding elasticity tensor is ensured under explicit conditions on the independent constants for arbitrary fabric tensors.     

A symmetry invariant formulation of the relationship between the elasticity tensor and the fabric tensor

The fabric tensor is employed as a quantitative stereological measure of the structural anisotropy in the pore architecture of a porous medium. Earlier work showed that the fabric tensor can be used additionally to the porosity to describe the anisotropy in the elastic constants of the porous medium. This contribution presents a reformulation of the relationship between fabric tensor and anisotropic elastic constants that is approximation free and symmetry-invariant. From specific data on the elastic constants and the fabric, the parameters in the reformulated relationship can be evaluated individually and efficiently using a simplified method that works independent of the material symmetry. The well-behavedness of the parameters and the accuracy of the method was analyzed using the Mori–Tanaka model for aligned ellipsoidal inclusions and using Buckminster Fuller’s octet-truss lattice. Application of the method to a cancellous bone data set revealed that employing the fabric tensor allowed explaining 75–90% of the total variance. An implementation of the proposed methods was made publicly available.     

Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures

The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.     

A Relationship between Thermal Diffusivity and Finite Deformation in Polymers

The thermal diffusivity of elastomers (i.e., rubber-like materials) can change substantially with elastic finite deformation. Initially isotropic elastomers may be thermally anisotropic when deformed. Data from several experimental studies demonstrate significant changes in the thermal conductivity or diffusivity tensor with finite deformation. Formulating the thermal diffusivity tensor and deformation in terms of the reference configuration may aid in the development of constitutive relations by use of material symmetry. Illustrated here is a relationship between the diffusivity and deformation of representative materials during uniaxial and equibiaxial deformation. Each component of the diffusivity tensor appears to be related to the deformation in the direction of the component only. Paper presented at the Fifteenth Symposium on Thermophysical Properties, June 22–27, 2003, Boulder, Colorado, U.S.A.     

Conservation laws and the material momentum tensor for the elastic dielectric

It is shown that a Langrangian formulation of continuum mechanics can provide not only the equations of motion, but the conservation laws related to the material symmetries in a perfect continuum interacting with an external electric field. These conservation laws in the presence of defects lead to the path-independent integrals widely used in fracture mechanics. They are basically related to the “material force” on a defect in a continuum. The quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor. A material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupin's [1] formulations.     

Elastic stiffness model for the critical temperature Tc of Nb3Sn including strain dependence

A calculation is established for the critical temperature T_c of the superconductor Nb₃Sn that includes the dependence on applied mechanical strain. The calculation employs the formalism of strong coupling phonon superconductivity, as usually given in frequency space. The directional nature of strain is included by expressing the equations of strong coupling in wave vector space. The relation between wave number and frequency is provided by the dispersion relations incorporating the effective elastic constants for the symmetry directions of the cubic crystal. An analytical formalism is established in which the elastic constants are derived in a unified way from an assumed strain energy potential function. The form of the strain energy potential is governed by the cubic symmetry. The scalar invariants of the strain tensor under the cubic symmetry group are determined as a set of basis functions for the strain energy potential. In the harmonic approximation, the relation between the strain energy function and the elastic constants determines the harmonic amplitudes of the strain potential from the measured dispersion relations. The electron-phonon coupling characteristic is approximated in a simple analytic form determined by inspection of the experimentally determined tunneling and phonon density of states. The critical temperature is calculated, through the equations of strong coupling, as a sum over the crystal symmetry directions. The anharmonic terms of the strain energy potential are introduced as the source of the strain dependence of the critical temperature. The allowed form of the anharmonic terms is again governed by cubic symmetry. The amplitudes of the anharmonic terms are determined from the strain dependence characteristics of single crystals and composite superconductors. The calculations are found to represent the observed strain dependence well with a strain energy function that contains three scalar invariants of the strain tensor, including the spherical (hydrostatic) strain invariant, and the principle parts of the second and third invariants of the deviatoric strain tensor. The formalism is applied to the analysis of composite conductors. The characteristics of the strain dependence of wire and tape geometries under longitudinal and transverse loads are related to the symmetry of the conductor and direction of applied load. Implications of conductor symmetry and constraint on the measurement of the strain dependent properties are identified.     

The relationship between the elasticity tensor and the fabric tensor

An algebraic relationship between the fourth rank elasticity tensor of a porous, anisotropic, linear elastic material and the fabric tensor of the material is considered. The fabric tensor is a symmetric second rank tensor which characterizes the geometric arrangement of the porous material microstructure. In developing this result it is assumed that the matrix material of the porous elastic solid is isotropic and, thus, that the anisotropy of the porous elastic solid is determined by the fabric tensor. It is then shown that the material symmetries of orthotropy, transverse isotropy and isotropy correspond to the cases of three, two and one distinct eigenvalues of the fabric tensor, respectively.     

Evaluation of the in-plane effective elastic moduli of single-layered graphene sheet

In this paper, an equivalent continuum-structural mechanics approach is used to characterize the mechanical behaviour of nanostructured graphene. The in-plane elastic deformation of armchair graphene sheets is simulated by using finite element modelling. The model is based on the assumption that force interaction among carbon atoms can be modelled by load-carrying beams in a representative two-dimensional honeycomb lattice structure. The elastic properties of beam elements are determined by equating the energies of the molecular structure and the continuum beam model subjected to small strain deformation. Then an equivalent continuum technique is adopted to estimate effective elastic moduli from which elastic constants are extracted. A comparison of elastic constants obtained from current modelling concur with results reported in literature. With the multifunctional properties of graphene sheets as manifested in a broad range of industrial applications, determination of their elastic moduli will facilitate a better design of the corresponding materials at macroscopic level.     

Explicit relations between elastic and conductive properties of materials containing annular cracks

The impact of annular cracks on the effective elastic and conductive properties of a material is analysed. The compliance contribution tensor of an annular crack - the quantity that determines the increase in compliance of a solid due to introduction of such a crack - is derived analytically. The resistivity contribution tensor of an annular crack is calculated numerically. It is shown that an effective circular crack, i.e. a crack which yields the same change in elastic/conductive properties of a material as the given annular crack, can be chosen to match both of these tensors. Using this result, the explicit relation between elastic and conductive properties of a material containing annular cracks is obtained. The relation is derived using a non-interaction approximation. Applicability of the derived formulae to real materials (to plasma-sprayed coatings, in particular) is discussed.     

Computational homogenization of nonlinear elastic materials using neural networks

In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three‐dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two‐scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Characterization of all the elastic,dielectric, and piezoelectric constants of uniaxially oriented poled PVDF films

Polyvinylidene fluoride (PVDF), a piezoelectric material, has many useful applications, for example, as sensors, transducers, and surface acoustic wave (SAW) devices. Models of performance of these devices would be useful engineering tools. However, the benefit of the model is only as accurate as the material properties used in the model. The purpose of this investigation is to measure the elastic, dielectric and piezoelectric properties over a frequency range, including the imaginary part (loss) of these properties. Measurements are difficult because poled material is available as thin films, and not all quantities can be measured in that form. All components of the elastic stiffness, dielectric tensor, and electromechanical coupling tensor are needed in the models. The material studied here is uniaxially oriented poled PVDF that has orthorhombic mm2 symmetry. Presented are the frequency dependence of all nine complex elastic constants, three complex dielectric constants, and five complex piezoelectric constants. The PVDF was produced at Raytheon Research Division, Lexington, MA. Measurements were made on thin films and on stacked, cubical samples. The elastic constants c₄₄^D and c₅₅^D, the dielectric constants e₁₁^T and e₂₂^T , as well as the piezoelectric constants g₁₅ and g₂₄ reported here have not been published before. The values were determined by ultrasonic measurements using an impedance analyzer and a least square data-fitting technique     

Material characterization of laminated composite materials using a three-point-bending technique

A three-point-bending technique is presented for identifying the elastic constants of laminated composite materials. In the proposed technique, three strains in the axial, lateral, and 45° directions on the bottom surface at the mid-span of a symmetric angle-ply beam subjected to three-point-bending testing are measured for elastic constants identification. The narrow beam theory together with the trial elastic constants is used to predict the theoretical strains of the beam. The theoretical and experimental strains of the beams are then used in a stochastic optimization method to identify the elastic constants of the beam. The accuracy and applications of the proposed technique are demonstrated by means of a number of examples on the elastic constants identification of graphite/epoxy (Gr/ep) or glass/epoxy (Gl/ep) laminated composite materials. The effects of specimen aspect ratio and thickness on the accuracy of the proposed method are investigated.     

On stiffness and strength reduction of solids due to crack development

Modification of the effective elastic and plastic constants of initially homogeneous and isotropic material with regularly distributed cracks is considered in the paper. The stress-strain relation for linearly elastic range is formulated as a tensor function with two independent variables: the stress tensor and damage tensor describing the current state of the cracked solid. This equation made it possible to evaluate all the elastic constants and is a starting point in the analysis of the plastic behavior of the damaged material. The appropriate yield criterion is derived in the form of an isotropic scalar function with the same variables as in the elastic range. To choose the most important terms of the general representation of this function, the energy of the elastic strain was calculated for homogenized equivalent material. This was done employing the stress-strain relation of elasticity for damaged solid proposed in the paper. The theoretical considerations were verified experimentally. To this end the material constants determined theoretically in the elastic and plastic ranges were compared with those measured experimentally for the models simulating the damaged material.     

Mechanical properties of non-cohesive polygonal particle aggregates

We numerically investigate the effective material properties of aggregates consisting of soft convex polygonal particles, using the discrete element method. First, we construct two types of “sand piles” by two different procedures. Then we measure the averaged stress and strain, the latter via imposing a 10% reduction of gravity, as well as the fabric tensor. Furthermore, we compare the vertical normal strain tensor between sand piles qualitatively and show how the construction history of the piles affects their strain distribution as well as the stress distribution. In the next step, elastic constants are determined, assuming Hooke’s law to be locally valid throughout the sand piles. We determine the relationship between invariants of the stress and strain tensor, observing that the behaviour is nonlinear. While linear elastic behaviour near the centre of the pile is compatible with our data, nonlinearity signals the transition to plastic behaviour near its surface. A similar behaviour was assumed by Cantelaube et al. (Static multiplicity of stress states in granular heaps. Proc R Soc Lond A 456:2569–2588, 2000). We find that the macroscopic stress and fabric tensors are not collinear in the sand pile and that the elastic behaviour is anisotropic in an essential way.     

Apparent stiffness tensors for porous solids using symmetric Galerkin boundary elements

Representative volume elements (RVEs) from porous or cellular solids can often be too large for numerical or experimental determination of effective elastic constants. Volume elements which are smaller than the RVE can be useful in extracting apparent elastic stiffness tensors which provide bounds on the homogenized elastic stiffness tensor. Here, we make efficient use of boundary element analysis to compute the volume averages of stress and strain needed for such an analysis. For boundary conditions which satisfy the Hill criterion, we demonstrate the extraction of apparent elastic stiffness tensors using a symmetric Galerkin boundary element method. We apply the analysis method to two examples of a porous ceramic. Finally, we extract the eigenvalues of the fabric tensor for the example problem and provide predictions on the apparent elastic stiffnesses as a function of solid volume fraction.     

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.