Anisotropic Solids Research Articles

Observation of phonon Poiseuille flow in isotopically purified graphite ribbons

In recent times, the unique collective transport physics of phonon hydrodynamics motivates theoreticians and experimentalists to explore it in micro- and nanoscale and at elevated temperatures. Graphitic materials have been predicted to facilitate hydrodynamic heat transport with their intrinsically strong normal scattering. However, owing to the experimental difficulties and vague theoretical understanding, the observation of phonon Poiseuille flow in graphitic systems remains challenging. In this study, based on a microscale experimental platform and the pertinent occurrence criterion in anisotropic solids, we demonstrate the existence of the phonon Poiseuille flow in a 5.5 μm-wide, suspended and isotopically purified graphite ribbon up to a temperature of 90 K. Our observation is well supported by our theoretical model based on a kinetic theory with fully first-principles inputs. Thus, this study paves the way for deeper insight into phonon hydrodynamics and cutting-edge heat manipulating applications.

Morphology Determines an Efficient Coherent Electron Transport for Push-Pull Organic Semiconductors Based on Triphenylamine and Dicyanovinyl Groups.

The morphology of the active layer in organic solar cells is fundamental for achieving high power conversion efficiency. However, the morphological characteristics for optimal performance are still being investigated. An atomistic computational approach is required to determine the relationship between active layer morphology and performance. Since the organic solar cell has multiple phases and interfaces, the computational modeling of charge generation and transport is challenging. We then used a set of push-pull semiconductors to illustrate how the electronic transmission spectrum, derived from the Landauer-Büttiker formalism, can be used to investigate the efficiency of coherent charge transport across anisotropic organic solids. The electronic transmission spectrum was calculated from the electronic band structure obtained using the density-functional-based tight-binding method. We found that coherent charge transport was more efficient along the direction parallel with the interface between the electron-acceptor and electron-donor moieties for a herringbone morphology.

Stable reconstruction of discontinuous solutions to the Cauchy problem in steady-state anisotropic heat conduction with non-smooth coefficients

We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L2–boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria.

Numerical evaluation of 3D time-harmonic elastodynamic Green's function and its derivatives for anisotropic solids

The paper presents a fast and accurate numerical technique for evaluation of the 3D time-harmonic elastodynamic Green's function and its derivatives for anisotropic solids. Following Wang & Achenbach the Green's function is presented in the form of the sum of singular (static) and regular parts, which are both reduced to integrals over a unit sphere. Singular part and its derivatives are then reduced to the integrals over a unit circle, which can be efficiently (fast and accurately) evaluated using the trapezoid rule, which is exponentially convergent for integrals over the period of a periodic integrand. The regular time-harmonic parts are presented through the double integrals. The outer integral is also efficiently evaluated using the trapezoid rule. A special quadrature is developed for evaluation of the inner integral, which accounts for the highly-oscillating behavior of the integrand. Numerical examples are presented, which shows the advantage of the proposed technique over others.

Measurement of Thermal Diffusivity in Principal Direction of Orthotropy Material by a Step Heating Method

Carbon fiber reinforced carbon composites (C/C composites) have been important structural materials at high temperatures. C/C composites are anisotropic solids and their thermal diffusivity is the second-order tensor of which components are called thermal diffusivity coefficients. Therefore, the purpose of study is to measure the thermal diffusivity in direction parallel to fiber and that in direction perpendicular to fiber by using the step heating method. This report demonstrates the correctness of the method to determine the thermal diffusivity in principal direction by comparing the time change of temperature observed by the experiment at room temperature for the C/C composite with the mathematical solution for the heat conduction under one dimension.

Multibody contact of two-dimensional anisotropic elastic/piezoelectric/magneto-electro-elastic solids

A unified expression for the generalized two-dimensional linear anisotropic elastic, piezoelectric, and magneto-electro-elastic (MEE) solids is established to simplify the connection between these three different kinds of materials. Here, the two-dimensional assumption includes the most general cases that in-plane and anti-plane deformations are allowed to couple each other. Based upon this unified expression, in this paper a recently developed boundary element method (BEM) for contact of two anisotropic elastic bodies is extended to solve the problems of multibody contact in which the bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids. When a special case of contact between multiple rigid punches and an anisotropic elastic, piezoelectric or MEE half-plane is considered, further reduction can be made by introducing three degrees of freedom (two rigid body translations and one rigid body rotation) for each punch. With this consideration, no meshes are required on the rigid punches. The usual contact modes of separation, stick, and slip all include the situation of conductor and insulator if any one of the multibody is piezoelectric or MEE. In other words, in this paper we successfully develop two BEMs to solve the generalized two-dimensional problems of multibody contact. One is for the general cases whose bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids, and the other is for the special cases of multiple rigid punches indenting on an anisotropic elastic, piezoelectric or MEE half-plane.

On the direct and reverse multiplicative decompositions of deformation gradient in nonlinear anisotropic anelasticity

In this paper we discuss nonlinear anisotropic anelasticity formulated based on the two multiplicative decompositions F=FeFa and F=FaFe. Using the Bilby–Kröner–Lee decomposition F=FeFa one can define a Riemannian material manifold (the natural configuration of an anelastic body) whose metric explicitly depends on the anelastic deformation Fa. We call this the global material intermediate configuration. Deformation is a map from this Riemannian manifold to the flat ambient space. Using the reverse decomposition F=FaFe, the reference configuration is a (flat) submanifold of the Euclidean ambient space, while the global intermediate configuration is a Riemannian manifold whose metric explicitly depends on the elastic deformation Fe. We call this the global spatial intermediate configuration. We show that the direct F=FeFa and reverse F=FaFe decompositions correspond to the same anelastic motion if and only if Fe and Fe are equal up to local isometries of the reference configuration. We discuss the constitutive equations of anisotropic anelastic solids in terms of both intermediate configurations. It is shown that the two descriptions of anelasticity are equivalent in the sense that the Cauchy stresses calculated using them are identical. We note that, unlike isotropic solids, for an anisotropic solid the material metric is not sufficient for describing the constitutive behavior of the solid; the energy function explicitly depends on Fa (or Fa) through the structural tensors.

Oscillatory sliding in the presence of asymmetric friction

Asymmetric friction corresponds to the case when different magnitudes of friction resist sliding in opposing directions. Asymmetric friction can also emerge in anisotropic solids (e.g. rocks) in sliding over fractures with an anisotropic face whose axis of anisotropy is inclined to the direction of sliding and constrained in the normal direction. It can therefore be important in geotechnical applications. Asymmetric friction can also be engineered using anisotropic blocks with an inclined anisotropy axis and used for designing new hybrid or metamaterials. Oscillations of a mass with asymmetric friction can be categorised under one of the following three cases: (1) steady state oscillations in the low friction direction, (2) oscillations in the low friction direction, with decreasing amplitude until the mass becomes stationary, and (3) stick–slip-type movement only in the low friction direction until the mass becomes stationary. In the case of high degree of asymmetric friction such that sliding in the hard direction is impossible, the spectra show peaks at both even and odd multiples of driving frequency.

A ray tracing approach to focusing ultrasonic beams in isotropic and anisotropic solids

Focal depths in solid materials are generally calculated using analytical expressions that rely on the paraxial approximation (i.e., small angles related to shallow focusing). However, with increasing use of highly focused probes for acoustic microscopy and backscattering studies, these assumptions are often challenged. Additional challenges are posed by the presence of material anisotropy within the solid. In this presentation, we present a ray tracing based approach to characterize ultrasonic beam focusing when using highly focused ultrasonic transducers. The results are compared to traditional methods to calculate focal regions to evaluate the range of applicability for isotropic and anisotropic materials. For isotropic solids, a non-paraxial focusing equation is derived and compared to the conventional focusing equation for both normal incidence and oblique incidence immersion setups. Both focusing equations are compared through ray diagrams where the proximity to the true geometric focus is explored. The proposed focusing equation results in a closer approximation to the geometric focus, a smaller beam cross-section, and a greater time convergence compared to the conventional focusing equation. Lastly, the results are expanded to anisotropic media where potential for multiple foci and complex beam behavior arise.

The Universal Program of Nonlinear Hyperelasticity

For a given class of materials, universal deformations are those that can be maintained in the absence of body forces by applying only boundary tractions. Universal deformations play a crucial role in nonlinear elasticity. To date, their classification has been accomplished for homogeneous isotropic solids following Ericksen’s seminal work, and homogeneous anisotropic solids and inhomogeneous isotropic solids in our recent works. In this paper we study universal deformations for inhomogeneous anisotropic solids defined as materials whose energy function depends on position. We consider both compressible and incompressible transversely isotropic, orthotropic, and monoclinic solids. We show that the universality constraints—the constraints that are dictated by the equilibrium equations and the arbitrariness of the energy function—for inhomogeneous anisotropic solids include those of inhomogeneous isotropic and homogeneous anisotropic solids. For compressible solids, universal deformations are homogeneous and the material preferred directions are uniform. For each of the three classes of anisotropic solids we find the corresponding universal inhomogeneities—those inhomogeneities that are consistent with the universality constraints. For incompressible anisotropic solids we find the universal inhomogeneities for each of the six known families of universal deformations. This work provides a systematic approach to study analytically functionally-graded fiber-reinforced elastic solids.

Stress intensity factor calculation of the cracks interacted by the oval-holes in anisotropic elastic solids under remote and non-uniform surface stresses

In underground rock engineering, natural defects (such as cracks, joints, cavities of arbitrary shape) and anisotropy of rock mass are main factors influencing the engineering strength. Calculating stress intensity factors (SIFs) of anisotropic elastic solids with multiple oval-holes and cracks is of great significance not only for designing the stop-holes to improve the stability of mining exploitation, but also for optimizing the hydraulic fracturing parameters to improve the productivity of geothermal/shale-gas utilization. Current studies are mainly focused on the loading case of remote stresses, regardless of the surface stresses applied onto the holes and cracks. In this paper, both remote uniform stresses and non-uniform surface stresses are taken into account to calculate the multi-crack SIFs interacted by the multiple oval-holes in anisotropic elastic solids. Two elementary solutions for traction components of single oval-hole and single crack subjected to the surface point-forces in the anisotropic solids are analytically deduced based on Cauchy integral approach in complex variable theory. A new integral equation approach is established to solve the interacting SIFs of multiple oval holes and cracks in the two-dimensional anisotropic elastic solids, without any limitation of the number, sizes, orientations and locations for oval-holes and cracks. Its accuracy and feasibility are validated by comparing our SIF solutions against the available exact solutions, approximated solutions and finite element solutions. The new method has high accuracy (with almost the same SIF solutions as the exact ones), simple formulation (without any singularity) and wide applicability (under both remote uniform stresses and non-uniform surface stresses). It can be further developed for three-dimensional anisotropic hole-crack problem under the same complex stress states.

Resonant ultrasound spectroscopy for crystalline samples containing initial strain

Resonant ultrasound spectroscopy (RUS) is a mature and robust technique for the nondestructive characterization of the elastic properties of solids capable of providing the elastic constants of anisotropic crystalline solids. The traditional method is based on assuming that the solid is linear elastic and, therefore, obeys a linearized Hookean constitutive relationship (Hooke’s law). In this article, an alternative constitutive law is provided that allows for an initial strain or prestrain to be present stemming from residual stress. Then, the constitutive relationship is integrated into the RUS framework. The model is demonstrated using a realistic prestrain field obtained by simulating shot-peening processing of a polycrystalline Cu sample. The sensitivity of the resonances to the developed prestrain is established and discussed. This work allows researchers to consider the influence of initial strain or residual stress in their samples and the potential influence on accurate elastic constant estimates. The model also supports the potential of RUS for the nondestructive characterization of prestrain in materials.

SOFTWARE FOR SIMULATION OF NON-STATIONARY HEAT TRANSFER IN ANISOTROPIC SOLIDS

The software «AnisotropicHeatTransfer3D» for the numerical solution of three-dimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4]. The dual reciprocity method with anisotropic radial basis functions is used to obtain particular solution, and the method of fundamental solutions is used to obtain a homogeneous solution of boundary-value problem.

On the Validity of Onsager Reciprocal Relations (ORR) for Heat Transfer in Anisotropic Solids

In this work, we investigate the validity of axioms such as Onsager Reciprocal Relations (ORR) for heat transfer in irreversible thermodynamics close to equilibrium. We show that the ORR for this case could be directly derived by introducing the widely accepted concept of heat transfer coefficients into the entropy production rate and by assuming that the thermal conductivity coefficients are uniquely defined. It is believed that this work can not only be used for pedagogical purposes but may also be generalized to other processes beyond heat transfer, thus leading to a generalized framework for transport phenomena and irreversible thermodynamics.

Omnidirectional wave polarization manipulation in isotropic polar solids

Mode conversion and anomalous wave polarization between shear and longitudinal waves have been mainly explored in anisotropic solids. However, the use of elastic anisotropy inevitably introduces fundamental limitation in operative directions, which is not desirable for applications. In this study, we propose to use two-dimensional (2D) isotropic polar media that exhibit polarity and chirality moduli to achieve the above behaviors in an omnidirectional way, thereby realizing omnidirectional wave polarization manipulation. First, we formulate analytically and numerically a nearly perfect wave mode conversion, enabled by the chirality modulus, through a polar slab. Then, we emphasize that the polarity modulus causes the omnidirectional anomalous wave polarization which features faster shear modes. Two supplementary examples of the anomalous wave polarization, including singly polarized transmission and anomalous wave refraction, are presented. The former corresponds to a “fluid-like” or a “shear-wave” solid, depending on the choice of the polarity modulus. While the latter shows a longitudinal incidence causes no shear refraction beyond a critical angle at the interface between a Cauchy and a polar media. All the cases investigated here are directionally independent, drastically differing from those in anisotropic systems. It is our hope that the isotropic polar media could represent a new platform for the exploration of versatile polarization manipulation devices.

深層学習を用いた異方性弾性体中の欠陥形状再構成

In this study, the shape reconstruction method of a cavity is developed for 2-D pure antiplane anisotropic elastodynamics. Elastic wave scattering by a cavity is implemented by using the convolution quadrature time-domain boundary element method. Scattered waves obtained at receiver points are prepared and utilized for the deep learning. The created deep learning model estimates position and size of a cavity in 2-D pure antiplane anisotropic solids. Numerical example shows that deep learning can be useful for inverse analysis of a defect in anisotropic elastic materials.

Surface lattice Green’s functions for high-entropy alloys

We study the surface elastic response of pure Ni, the random alloy FeNiCr and an average FeNiCr alloy in terms of the surface lattice Green’s function. We propose a scheme for computing per-site Green’s function and study their per-site variations. The average FeNiCr alloys accurately reproduces the mean Green’s function of the full random alloy. Variation around this mean is largest near the edge of the surface Brillouin-zone and decays as q −2 with wavevector q towards the Γ-point. We also present expressions for the continuum surface Green’s function of anisotropic solids of finite and infinite thickness and show that the atomistic Green’s function approaches continuum near the Γ-point. Our results are a first step towards efficient contact calculations and Peierls–Nabarro type models for dislocations in high-entropy alloys.

Family of the Atomic Radial Basis Functions of Three Independent Variables Generated by Helmholtz-Type Operator

The paper presents an algorithm for constructing the family of the atomic radial basis functions of three independent variables generated by Helmholtz-type operator, which may be used as basis functions for the implementation of meshless methods for solving boundary-value problems in anisotropic solids. Helmholtz-type equations play a significant role in mathematical physics because of the applications in which they arise. In particular, the heat equation in anisotropic solids in the process of numerical solution is reduced to the equation that contains the differential operator of the special form (Helmholtz-type operator), which includes components of the tensor of the second rank, which determines the anisotropy of the material. The family of functions is infinitely differentiable and finite (compactly supported) solutions of the functional-differential equation of the special form. The choice of compactly supported functions as basis functions makes it possible to consider boundary-value problems on domains with complex geometric shapes. Functions include the shape parameter , which allows varying the size of the support and may be adjusted in the process of solving the boundary-value problem. Explicit formulas for calculating the considered functions and their Fourier transform are obtained. Visualizations of the atomic functions and their first derivatives with respect to the variables and at the fixed value of the variable for isotropic and anisotropic cases are presented. The efficiency of using atomic functions as basis functions is demonstrated by the solution of the non-stationary heat conduction problem with the moving heat source. This work contains the results of the numerical solution of the considered boundary-value problem, as well as average relative error, average absolute error and maximum error are calculated using atomic radial basis functions and multiquadric radial basis functions.

A stable discontinuous Galerkin method for linear elastodynamics in 3D geometrically complex elastic solids using physics based numerical fluxes

Time-stable, high order accurate and explicit numerical methods are effective for hyperbolic wave propagation problems. As a result of the complexities of real geometries, internal interfaces and nonlinear boundary and interface conditions, discontinuities and sharp wave fronts may become fundamental features of the solution. Therefore, geometrically flexible and adaptive numerical algorithms are crucial for high fidelity and efficient simulations of wave phenomena in many applications. Adaptive curvilinear meshes hold promise to minimise the effort to represent complicated geometries or heterogeneous material data avoiding the bottleneck of feature-preserving meshing. To enable the design of stable DG methods on three space dimensional (3D) curvilinear elements we construct a structure preserving skew-symmetric coordinate transformation motivated by the underlying physics. Using a physics-based numerical penalty-flux, we develop a 3D provably energy-stable discontinuous Galerkin finite element approximation of the elastic wave equation in geometrically complex and heterogeneous media. By construction, our numerical flux is upwind and yields a discrete energy estimate analogous to the continuous energy estimate. The ability to treat conforming and non-conforming curvilinear elements allows for flexible adaptive mesh refinement strategies. The numerical scheme has been implemented in ExaHyPE, a simulation engine for parallel dynamically adaptive simulations of wave problems on adaptive Cartesian meshes. We present 3D numerical experiments of wave propagation in heterogeneous isotropic and anisotropic elastic solids demonstrating stability and high order accuracy. We demonstrate the potential of our approach for computational seismology in a regional wave propagation scenario in a geologically constrained 3D model including the geometrically complex free-surface topography of Mount Zugspitze, Germany.

Transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids

Consider the transient thermal stress analysis for a material that is anisotropic (direction-dependent), viscoelastic (time-dependent), and thermorheologically simple (temperature-dependent). To solve this problem, three fundamental theories are employed: (1) complex variable Stroh formalism for anisotropic elasticity, (2) elastic-viscoelastic correspondence principle (EVCP) or time-stepping method (TSM) for viscoelasticity, and (3) time-temperature superposition principle (TTSP) for thermorheologically simple materials. By using the Stroh formalism, all different kinds of anisotropic elastic materials can be analyzed by using the same formulae. With EVCP and TSM, the problem of viscoelasticity can be solved via its corresponding problem of elasticity. By TTSP, the temperature-dependent constitutive relation can be expressed by using the reference temperature and the reduced time defined through shift factor. Thus, through the combined use of these theories, the complicated problems of transient anisotropic thermo-viscoelasticity can be solved by using the available solution tool for anisotropic elasticity such as the boundary element method (BEM). Based upon this concept, two novel BEMs are developed for the transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids. One is BEM-EVCP, and the other is BEM-TSM. The final results show that both of them can provide accurate solutions, and BEM-TSM is much more efficient than BEM-EVCP.