Lame Moduli Research Articles

Propagation of elastic waves in fibrous composite with general anisotropy from two families of obliquely inclined fibers

AbstractAn isotropic elastic solid reinforced with two families of aligned fibres behaves anisotropic to any deformation. For arbitrary orientation of fibres in either family, one set of fibres may be inclined obliquely to the aligned fibres in other family. A plane that contains the two families of aligned fibres becomes the plane of symmetry for composite material to exhibit monoclinic anisotropy. For this fibrous anisotropy, a fourth‐order elastic tensor is derived to define the constitutive relations between stress tensor and strain tensor. Components of this elastic tensor involve two Lame's moduli, eight fibroelastic coefficients and the angular deviation between two obliquely inclined fibre‐families. The velocities of bulk waves are calculated for propagation along general direction in this anisotropic composite. Propagation of surface wave is considered along the stress‐free plane boundary of anisotropic elastic half‐space. In a coordinate system with coordinate planes deviating from the plane boundary of the medium and/or plane of anisotropic symmetry, the fibrous composite is treated as triclinic for mathematical convenience. Mathematical model is derived to calculate the phase velocity of the surface wave, which varies with direction on the plane boundary. Horizontal energy flux at the boundary is calculated to determine the group velocity and ray direction of the surface wave. Effects of fibre‐orientations are analysed, numerically, on the wave velocities and ray direction of surface wave.

Mechanical and Thermophysical Properties of ScM (M: Ru, Rh, Pd, Ag) Intermetallics

In this paper, we have investigated the elastic, mechanical, ultrasonic, and thermophysical properties of B2 structured scandium based intermetallic compounds ScM (M = Ru, Rh, Pd, Ag) at 300 K. Initially, the elastic constants were determined under potential model approach considering the interaction up to second nearest neighbors defined by Coulomb and Born–Mayer potential. The second order elastic constants (SOECs) were used to find the mechanical parameters such as shear modulus, bulk modulus, Poisson’s ratio, Young’s modulus, Pugh’s indicator, Zener ratio, Vicker’s hardness, Cauchy’s pressure, and Lame modulus in ScM intermetallic compounds. Further, SOECs were applied to compute the ultrasonic velocities and Debye velocity for wave transmission through ScM along ⟨100⟩ orientation. We have also evaluated thermal conductivity, thermal expansion coefficient, melting point, and thermal relaxation time of the chosen compounds. Finally, the ultrasonic attenuation was estimated using thermo-elastic relaxation and phonon–phonon interaction mechanisms at room temperature. The mechanical properties of ScM were discussed and analyzed on the basis of obtained elastic constants while their ultrasonic properties were discussed in connection with elastic and thermal properties.

Near optimal pentamodes as a tool for guiding stress while minimizing compliance in 3d-printed materials: A complete solution to the weak G-closure problem for 3d-printed materials

For a composite containing one isotropic elastic material, with positive Lame moduli, and void, with the elastic material occupying a prescribed volume fraction f, and with the composite being subject to an average stress, σ0, Gibiansky, Cherkaev, and Allaire provided a sharp lower bound Wf(σ0) on the minimum compliance energy σ0:ϵ0, in which ϵ0 is the average strain. Here we show these bounds also provide sharp bounds on the possible (σ0,ϵ0)-pairs that can coexist in such composites, and thus solve the weak G-closure problem for 3d-printed materials. The materials we use to achieve the extremal (σ0,ϵ0)-pairs are denoted as near optimal pentamodes. We also consider two-phase composites containing this isotropic elasticity material and a rigid phase with the elastic material occupying a prescribed volume fraction f, and with the composite being subject to an average strain, ϵ0. For such composites, Allaire and Kohn provided a sharp lower bound W˜f(ϵ0) on the minimum elastic energy σ0:ϵ0. We show that these bounds also provide sharp bounds on the possible (σ0,ϵ0)-pairs that can coexist in such composites of the elastic and rigid phases, and thus solve the weak G-closure problem in this case too. The materials we use to achieve these extremal (σ0,ϵ0)-pairs are denoted as near optimal unimodes.

СПЕКТРАЛЬНЫЕ РАЗЛОЖЕНИЯ В ДИНАМИЧЕСКИХ ЗАДАЧАХ ВЯЗКОУПРУГОСТИ

Theoretical relations obtained from solutions of dynamic problems of viscoelasticity represent an effective framework for experimental identification of dynamic rheological properties of materials. For the construction of such relations, closed solutions of boundary value problems (i.e. written in the form of convergent series or integrals) are preferred, because they(unlike solutions obtained by numerical methods) allow strict error estimates. However, the construction of analytical solutions is associated with the following difficulties. 1. As usual, the hypothesis of proportionality is accepted for relaxation operators corresponding to the first and second Lame moduli, which is equivalent to the hypothesis of a constant Poisson’s ratio. This significantly reduces the generality of consideration. 2. Representation of solutions for three-dimensional problems in the form of expansions in eigenfunctions makes it necessary to taking into account the large eigenvalues which in the vast majority of problems can be found only numerically, as the roots of transcendental equations, thus, it is likely to skip closely spaced and multiple roots. 3. Constructed series converge slowly. In this paper we suggest ways to overcome these difficulties. Solutions of initial boundary value problems are presented in the form of spectral expansions, but in contrast to the classical method of Fourier decomposition they are expanded over biorthogonal system of eigenfunctions of mutually conjugate pencils of differential operators. This pencils define generalized Sturm-Liouville problem with a polynomial spectral parameter. This eliminates the hypothesis of proportionality relaxation operators. Effective relations for the terms of spectral(in particular normalization factors) coordinate functions and asymptotic formulas for the initial approximations of eigenvalues excluding their omission in calculations are obtained. Power related ranking of elements of the spectral decomposition is proposed which allows achievingthe required accuracy of calculations on the partial sums of a low order.

Study of the dispersion properties of cylindrical waveguides with variable properties

A method is presented for analyzing the dispersion relations for a hollow inhomogeneous cylindrical waveguide with an arbitrary law of change in the Lame moduli (discontinuous or continuous). The boundary value problem is formulated, which contains two spectral parameters. An asymptotic analysis is performed for the long-wavelength approximation, the structure of the dispersion set is studied, and results of calculating the dispersion sets for various laws of change in the Lame moduli are presented.

A generalized constitutive elasticity law for GLPD micromorphic materials, with application to the problem of a spherical shell subjected to axisymmetric loading conditions

In this work we propose to replace the GLPD hypo-elasticity law by a more rigorous generalized Hooke's law based on classical material symmetry characterization assumptions. This law introduces in addition to the two well-known Lame's moduli, five constitutive constants. An analytical solution is derived for the problem of a spherical shell subjected to axisymmetric loading conditions to illustrate the potential of the proposed generalized Hooke's law.

Elastic and Seismic Properties of Dabie‐Sulu Ultrahigh Pressure Metamorphic Rocks

Abstract:Lamé modulus (λ) and shear modulus (μ) are among the most important, intrinsic, elastic constants of rocks. Using λ and μ could be much more advantageous than using P‐ and S‐wave velocities (Vp and Vs). Here we quantified these equivalent isotropic elastic moduli for 115 representative rocks from the ultrahigh pressure (UHP) metamorphic terrane of the Dabie‐Sulu erogenic belt (China) and their variations with pressure (P), temperature (T), density (p), Vp, Vs and minera logical composition. Both moduli increase nonlinearly and linearly with increasing pressure at low (<200–300 MPa) and high (>200–300 MPa) pressures, respectively. In the regime of high pressures, λ and μ decrease quasi‐linearly with increasing temperature with temperature derivatives dλ/dT and dμ/dT generally in the range of –10 × 10–3 to –1 × 10–3 GPa/°C. Dehydration of water‐bearing minerals such as serpentine in peridotites and chlorite in retrograde eclogites results in an abrupt drop in λ while μ remains almost unchanged. In λ‐ρ, μ‐ρ and λ‐μ plots, the main categories of UHP rocks can be characterized. Serpentinization leads to significant decreases in μ and λ as serpentine has extremely low values of λ, μ and ρ. Eclogites, common mafic rocks (mafic gneiss, metagabbro and amphibolite), and felsic rocks (orthogneiss and paragneiss) have high, moderate and low μ and λ values, respectively. For pyroxenes and olivines, λ increases but μ decreases with increasing Fe/Mg ratios. For plagioclase feldspars, both λ and μ exhibit a significant positive correlation with anorthite content. SiO2‐rich felsic rocks and quartzites are deviated remarkably from the general trend lines of the acid‐intermediate‐mafic rocks in Vs‐ρ, μ‐ρ, λ‐Vp, λ‐Vs and μ‐λ diagrams because quartz has extremely low λ (∼8.1 GPa) and p (2.65 g/cm3) but moderate μ (44.4 GPa) values. Increasing the contents of garnet, rutile, ilmenite and magnetite results in a significant increase in the λ and μ values of the UHP metamorphic rocks. However, either λ or μ is insensitive to the compositional variations for pyralspite (pyrope‐almandine‐spessartine) solution series. The results provide potentially improved constraints on characterization of crustal composition based on the elastic properties of rocks and in situ seismic data from deep continental roots.

Numerical analysis of elliptic inverse problems with interior data

A number of algorithms have been proposed and analyzed for estimating a coefficient in an elliptic boundary value problem when interior data is available. Most of the analysis has been done for the simple scalar BVP-Δ · a Δ u = f in Ω, a (∂ u / ∂ n) g on ∂ ΩHowever, some methods and the associated analysis have been extended to the problem of estimating the Lamé moduli in the system of linear, isotropic elasticity. Under certain idealized conditions, convergence of estimates to the exact Lame moduli has been proved for two techniques, the output least-squares method and a variational method similar to the equation error approach.

A General Framework for Localization of Classical Waves: II. Random Media

We study localization of classical waves in random media in the general framework introduced in Part I of this work. This framework allows for two random coefficients, encompasses acoustic waves with random position dependent compressibility and mass density, elastic waves with random position dependent Lame moduli and mass density, electromagnetic waves with random position dependent magnetic permeability and dielectric constant, and allows for anisotropy. We show exponential localization (Anderson localization) and strong Hilbert–Schmidt dynamical localization for random perturbations of periodic media with a spectral gap.

On Green's function for a three–dimensional exponentially graded elastic solid

The problem of a point force acting in an unbounded, three–dimensional, isotropic elastic solid is considered. Kelvin solved this problem for homogeneous materials. Here, the material is inhomogeneous; it is ‘functionally graded’. Specifically, the solid is ‘exponentially graded’, which means that the Lame moduli vary exponentially in a given fixed direction. The solution for the Green9s function is obtained by Fourier transforms, and consists of a singular part, given by the Kelvin solution, plus a non–singular remainder. This grading term is not obtained in simple closed form, but as the sum of single integrals over finite intervals of modified Bessel functions, and double integrals over finite regions of elementary functions. Knowledge of this new fundamental solution for graded materials permits the development of boundary–integral methods for these technologically important inhomogeneous solids.

A General Framework for Localization of Classical Waves: I. Inhomogeneous Media and Defect Eigenmodes

We introduce a general framework for studying the localization of classical waves in inhomogeneous media, which encompasses acoustic waves with position dependent compressibility and mass density, elastic waves with position dependent Lame moduli and mass density, and electromagnetic waves with position dependent magnetic permeability and dielectric constant. We also allow for anisotropy. We develop mathematical methods to study wave localization in inhomogeneous media. We show localization for local perturbations (defects) of media with a spectral gap, and study midgap eigenmodes.

Effective moduli of particulate solids: Lubrication approximation method

To efficiently calculate the effective properties of a composite, which consists of rigid spherical inclusions not necessarily of the same sizes in a homogeneous isotropic elastic matrix, a method based on the lubrication forces between neighbouring particles has been developed. The method is used to evaluate the effective Lame moduli and the Poisson's ratio of the composite, for the particles in random configurations and in cubic lattices. A good agreement with experimental results given by Smith (1975) for particles in random configurations is observed, and also the numerical results on the effective moduli agree well with the results given by Nunan & Keller (1984) for particles in cubic lattices.

Determination of complex moduli of isotropic viscoelastic materials

A procedure for determination of complex moduli of viscoelastic (polymeric) materials is described. The complex moduli of specimen's isotropic material are calculated from certain experimentally obtained data and with application of a 3-D finite element model of the experimental specimen. A system of two non-linear complex equations with respect to two complex Lame's moduli is formulated, and then solved by a successive approximation method. The convergence of the iterative procedure is observed for all cases. The complex moduli obtained for a series of polymeric materials are presented for 10–250 Hz frequency range.

Recovering Planar Lame Moduli from a Single-Traction Experiment

Under a simple nondegeneracy condition, the displacement and edge traction of a planar, isotropic, linearly elastic solid determine its Lame moduli. When these moduli are constant, they can be recovered exactly; this is demonstrated by a specific traction satisfying the nondegeneracy condition. Spatially varying moduli can be computed numerically by considering the equations of linear elasticity as a hyperbolic system for the unknown moduli. A stable finite difference scheme for solving this system is given; synthetic experiments demonstrate its efficacy.

On a transmission problem for two systems of elastostatics with polygonal interface

It is shown the invertibility of the boundary operator associated to a transmission problem for two systems of elastostatics with different Lame moduli, and where the interface is a piecewiseC1 curve in the plane.

An inverse boundary value problem for St. Venant-Kirchhoff materials

Finitely small deformations of a St. Venant-Kirchhoff material give rise to a well posed displacement boundary value problem for a 3*3 quasilinear elliptic system on a bounded domain Omega in R3. As a consequence using the solution of this boundary value problem, we can define the Dirichlet to Neumann map which maps each small displacement vector on the boundary to the corresponding traction vector on the boundary. In this paper we show that the Lame moduli of class Cinfinity ( Omega ) of the material can be determined from knowledge of the Dirichlet to Neumann map associated with this system. For a linear isotropic elastic material which is a St. Venant-Kirchhoff material under infinitesimal deformation, the corresponding result has already been proved in a forthcoming paper by Nakamura and Uhlmann entitled 'Global uniqueness for an inverse boundary problem arising in elasticity'. Our proof is based on establishing a relation between the Dirichlet to Neumann map for a St. Venant-Kirchhoff material and that for a linear isotropic material.

Extended Korn's inequalities and the associated best possible constants

We are concerned with the coerciveness of the strain energy E(u) (in linear elasticity) associated with a displacement vector u on the Sobolev space H1 (Ω) or its subspaces, a domain Ω in ℝn representing an isotropic elastic body—certain specific cases are called “Korn's inequalities”. Sufficient (and necessary) conditions on the “Lame moduli” for E(·) to be coercive (or uniformly positive) on such spaces are given, and the associated best possible constants are obtained for some cases.

Acoustic reflectivity of marine sandy sediment: Effect of the shear modulus gradient

Acoustic propagation loss in shallow water is sensitive to the reflection coefficient (R) of the sediment to incident plane sound waves. Unconsolidated marine sediment is usually a heterogeneous solid medium; the SV and P displacement potentials do not in general satisfy separate wave equations. Determination of R therefore requires a fundamental analysis of the equations of elastic motion, which is achieved using Richards' method of weakly coupled potentials. Marine sand is modeled as a half‐space in which density and the dilational Lame modulus (λ) are constant, whereas the shear modulus (μ) increases with depth in the manner prescribed by contact theory. Providing μ″ (0) = 0, quasi P and S potentials in the sediment satisfy (coupled) second‐order linear differential equations. By requiring the displacement and stress to satisfy boundary conditions at the interface, an expression is derived for R that includes, among other things, the gradient μ′ (0) and the ratio Z(0)/Z′(0), where Z(z) is the quasi P potential function. In the Born approximation, this ratio is unaffected by coupling to or from the S wave. The resulting equation for Z is transformed into a Helmholtz equation that is solved by an approximate method. Results for R are presented over the frequency band from 10 to 100 Hz.

Uniqueness in linear elastostatics for problems involving unbounded bodies

The problem of the uniqueness of elastostatic solutions to various boundary value problems involving unbounded two- and three-dimensional bodies is considered. The general boundary value problem is first considered for anisotropic bodies on which the elasticity tensor is uniformly positive definite. The displacement problem is then considered for the cases where the body is homogeneous and isotropic and the elasticity tensor is strongly-elliptic. Finally, the traction problem on homogeneous, isotropic bodies is considered with fairly little restriction placed on the Lame moduli. In the first two cases no restriction is placed on the geometry of the body other than the normal assumptions allowing for the use of the divergence theorem. For the traction problem, however, it is assumed that one component of the outward normal to the boudary of the body almost never vanishes. In each case it is shown that the desired uniqueness results hold with fairly mild restrictions placed on the displacement or stress fields (depending on the problem) in a neighborhood of infinity. Where possible, the results are shown to hold even though the solutions may possess the sort of discontinuities and singularities commonly found in problems involving cracks, corners, and bonded interfaces.

Determination of the Elastic Constants of Optical Glasses by an Ultrasonic Method

The values of the elastic constants of twelve samples of optical glass commercially produced in the United States and a sample of fused silica were calculated from the results of measurements made on these samples. The method used was a combination of the ultrasonic method developed by Schaefer-Bergmann with that of Hiedemann. The accuracies of the experimentally determined values of the longitudinal and shear wave velocities were 0.13% and 0.25%, respectively, while the accuracies in determining the elastic constants were 0.5% for the shear modulus, Lame's modulus, and the bulk modulus, 0.7% for Poisson's ratio, and 0.8% for Young's modulus. The values of the surface tensions of the samples were calculated using the results of the measurements and Auerbach's equation.