Anisotropic Solids Research Articles

A novel hybrid boundary element for polygonal holes with rounded corners in two-dimensional anisotropic elastic solids

A novel hybrid boundary element is developed for polygonal holes in finite anisotropic elastic plates based on two different special fundamental solutions for holes. Since these special fundamental solutions satisfy traction-free condition along the hole's boundary, there is no mesh required on the boundary of polygonal holes. Various types of polygonal holes with rounded corners, such as triangles, rhombuses, ovals, pentagons, are considered by adding proper perturbation to an elliptical hole. The developed hybrid element is a mixture of two special boundary elements: one is based on the special fundamental solution derived through nonconformal mapping and the other is based on the solution derived through perturbation technique with conformal mapping. The special boundary element methods are combined through submodeling technique. First, the global model is solved using the perturbation solution. Then, using the displacements obtained from global model, an auxiliary submodel is set up and the results are evaluated with the nonconformal solution. The present method is compared and validated with conventional boundary element method and finite element method. The effect of hole curvature, material anisotropy, and loading condition on the stress distribution around the hole is presented.

High-harmonic spin-orbit angular momentum generation in crystalline solids preserving multiscale dynamical symmetry.

Symmetries essentially provide conservation rules in nonlinear light-matter interactions and facilitate control and understanding of photon conversion processes or electron dynamics. Since anisotropic solids have rich symmetries, they are strong candidates for controlling both optical micro- and macroscale structures, namely, spin angular momentum (circular polarization) and orbital angular momentum (spiral wavefront), respectively. Here, we show structured high-harmonic generation linked to the anisotropic symmetry of a solid. By strategically preserving a dynamical symmetry arising from the spin-orbit interaction of light, we generate multiple orbital angular momentum states in high-order harmonics. The experimental results exhibit the total angular momentum conservation rule of light even in the extreme nonlinear region, which is evidence that the mechanism originates from a dynamical symmetry. Our study provides a deeper understanding of multiscale nonlinear optical phenomena and a general guideline for using electronic structures to control structured light, such as through Floquet engineering.

Surface Green's functions for an anisotropic viscoelastic half-plane and their application to contact problems

In this paper, the elastic-like surface Green's functions for an anisotropic viscoelastic half-plane are derived using the time-stepping method. Using the elastic-like surface Green's functions as the core analytical solutions, we develop semi-analytical models (SAMs) and apply them to solve two different contact problems with anisotropic viscoelastic materials. As new modeling approaches, the SAMs developed here can provide fast and efficient approaches to solving contact problems. These methods enable us to consider contact problems with generally anisotropic viscoelastic solids, in which the contact surface is frictional and either smooth or rough, and the applied loads and boundaries can be time-variant. The correctness of the derived surface Green's functions is demonstrated by comparing the numerical results obtained by SAMs and those achieved from the analytical solutions or boundary element methods. Using the obtained numerical results, the impacts of time step size, anisotropy, frictional coefficient, roughness, and applied loads on the contact responses are further analyzed and discussed.

Development of governing partial differential equations of reinforcing thin films

In this paper, a novel interface model for elastic isotropic thin films perfectly bonded to the surrounding elastic media is introduced. Considering an average stress and displacement through the thin film thickness, the shear stress discontinuity across the interface is expressed in terms of spatial and time derivatives of displacement components. The derived interfacial equations are presented in a general form. As such, they are applicable to asymmetric elastodynamic problems of anisotropic solids with thin film interfaces. Employing the introduced interface model and Hankel transform technique, static Green's functions of an elastic isotropic half-space reinforced by a buried thin film are obtained. Results of some special cases, including axisymmetric loading, inextensible membrane, unreinforced half-space, and surface-coated half-space, are also presented. The robustness and effectiveness of the proposed interface model are shown by two solved numerical examples, and some elastic responses are plotted to show the impact of thin film stiffness. According to the numerical results, modeling elastic thin films as inextensible membranes can significantly alter the elastic responses.

An inverse method of fundamental solutions for the identification of 2D elastic properties of anisotropic solids

An inverse method of fundamental solutions for the identification of 2D elastic properties of anisotropic solids

Measuring the in-plane thermal diffusivity of anisotropic solids by lock-in infrared thermography: Laser-spot vs laser-line heating

A method is proposed to measure the in-plane thermal diffusivity of anisotropic solids along an arbitrary direction using laser-line lock-in thermography. In this method, the sample is heated by an intensity-modulated laser beam, which impinges on the sample surface with the shape of a narrow strip. The resulting temperature oscillations are recorded by a thermographic camera and a phase thermogram is obtained by lock-in processing the image sequence. The thermal diffusivity along the direction perpendicular to the laser line can be determined from the linear dependence of phase lag with the distance to the heating line using the slope method. Unlike laser-spot lock-in thermography, in which the slope method only provides the thermal diffusivity along the principal directions, in this configuration, the thermal diffusivity can be measured experimentally along any direction, just selecting the relative orientation of the sample with respect to the laser line. Moreover, this configuration allows averaging several parallel phase profiles leading to more reliable thermal diffusivity values. The experimental results are presented for a highly anisotropic material, namely, a unidirectional carbon fiber-reinforced polymer plate.

Custom lens design for ultrasonic inspection of immersed anisotropic parts

Inspection of anisotropic parts using immersion ultrasonic waves poses resolution challenges caused by directionally dependent wave properties. Currently, focused lenses with spherical or cylindrical geometries are the standard method for creating convergent, small cross-section, and high amplitude beams within solids. These lenses form a conical beam that uniformly refracts at the fluid-solid interface of the immersed part when the solid is isotropic, producing the desired focusing behavior. However, for anisotropic solids, refraction becomes directionally dependent, which distorts the beam and causes non-uniform focal profiles and, in some cases, multiple foci. In this work, the forward wave propagation for an immersed anisotropic solid is modeled and exploited to develop custom lens geometries. These custom lenses utilize inverted slowness profiles to produce conical beams within the anisotropic solid. By incorporating these lenses, the focused beam behavior is improved and becomes more circular, higher in amplitude, and smaller in the cross-sectional area. These lenses have the potential to improve inspection resolution in anisotropic solids, an effort beneficial to the additive, aerospace, and electronics industries.

Refined linearly anisotropic couple-stress elasticity

A recently developed, refined version of the conventional linear couple-stress theory of isotropic elasticity is extended to include the influence of anisotropic material effects. With this development, the implied refined theory (1) retains ability to determine the spherical part of the couple-stress and (2) is further furnished with constitutive ability to embrace modelling of linearly elastic solids that exhibit inherent polar material anisotropy of advanced levels that reach the class of locally monoclinic materials. This type of anisotropy embraces most of the structural material subclasses met in practice, such as those of general and special orthotropy, as well as the subclass of transverse isotropy. The thus obtained, enhanced version of the refined theory is furnished with ability to also handle structural analysis problems of polar fibrous composites reinforced by families of perfectly flexible fibres or, more generally, polar anisotropic solids possessing one or more material preference directions that do not possess bending resistance. A relevant example application considers and studies in detail the subclass of polar transverse isotropy caused by the presence of a single family of perfectly flexible fibres. By developing the relevant constitutive equation, and explicitly presenting it in a suitable matrix rather than indicial notation form, that application also exemplifies the way that the spherical part of the couple-stress is determined when the fibres are straight. It further enables this communication to initiate a discussion of further important issues stemming from (1) the positive definiteness of the full, polar form of the relevant strain energy function and (2) the lack of ellipticity of the final form attained by the governing differential equations.

Evaluation of mixed-mode stress intensity factors for anisotropic elastic solids with interface singularity

Abstract When a structure is composed of many different parts and each part is made of different materials, it is very possible that many interface corners appear in several local fields of the structure. Due to the mismatch of elastic properties, stress singularity may occur at the interface corners. The singular orders of stresses and their associated stress intensity factors (SIFs) are two important parameters for understanding failure initiation. Since the singular orders of general interface corners may be real or complex, distinct or repeated, several different definitions of SIFs have been proposed for different types of interfaces. In order to build a direct connection among all general interface corners (including cracks and interface cracks), a unified definition of SIFs was proposed in our previous studies. To calculate the uni-defined SIF accurately and efficiently, a special boundary element using the fundamental solution satisfying interface continuity conditions is established. The singular integrals involved in the associated boundary integral equations are solved by the setting of proper rigid body motions and the approximation through interpolation of nearby non-singular solutions. To avoid the unstable and inaccurate region near the corners, in this study we employ the path-independent H-integral, and derive the relations between H-integral and SIFs. Moreover, the auxiliary solutions of displacements and tractions required in the H-integral are provided for different integral paths. Several numerical examples, such as interface cracks, interface corners, single lap joints and delamination, are presented to demonstrate the methodology employed in this paper.

Constitutive relations for anisotropic porous solids undergoing small strains whose material moduli depend on the density and the pressure

Recently, Arumugam et al. (2023) developed a constitutive relation for the response of isotropic inhomogeneous compressible elastic solids in order to describe the response of the trabecular bone. Since porous solids such as bones, cement concrete, rocks, metallic alloys, etc., are anisotropic, in this short note we develop a constitutive relation for such bodies that exhibit transverse isotropy and also having two preferred directions of symmetry. Another characteristic of bones is that they exhibit different response characteristics in tension and compression, and hence any constitutive relation that is developed has to be capable of describing this. Also, the material moduli depend on both the density and the mean value of the stress (mechanical pressure), as is to be expected in a porous solid. In the constitutive relation that is developed in this paper, though the stress and the linearized strain appear linearly in the constitutive relation, the relationship is nonlinear. We also derive the response of such solids when undergoing uniaxial extension and compression, simple shear and torsion.

The Porothermoelastic Stress Evolution of Inhomogeneous Solid due to Near-Surface Reservoir

Anisotropic and heterogeneous solids, comprising a minimum of two or more elements with different properties, appear pervasively in rock materials including the pore structure and mineral composition of quartz and granite and usually have extensive applications in construction aggregates, dimension stone, geotechnical engineering, and petrology/geology. The elastic stress fields of inhomogeneous materials or composites inevitably change due to the presence of heterogeneous microstructure under applied external conditions, and hence the total mechanical and physical properties are affected by the temperature and pore pressure fluctuations beneath the surface. In this paper, the thermoporoelastic (TPE) approach on the basis of eigenstrain concept is introduced to predict the stress fields in fluid-saturated porous geologic materials like hydrocarbon reservoirs or aquifers by accounting for the coupling between thermal, poroelastic, and mechanical effects. It is an extension of the micromechanical theory that also incorporates thermal effects like pore pressure changes and temperature alternations. In addition, the TPE approach provides an important multiphysical modeling tool for understanding subsurface fluid-rock interactions and stress states in applications like unconventional oil/gas and geothermal energy.

Two-Dimensional Nye Figures for Some Micropolar Elastic Solids

The paper is devoted to problems related to two-dimensional Nye figures for micropolar continua. The representation technique known from studies on crystallography for 4th, 3rd and 2nd rank tensors is employed by two-dimensional matrices, supplied by relationships between their elements. Such representations are commonly used to simplify a script of the equations of athermic micropolar elastic solids. This method allows us to graphically represent micropolar constitutive tensors and pseudotensors in the form of specific two-dimensional blocks. The Nye figures for the micropolar elastic solids are obtained. The Nye figures enables us to discriminate anisotropic solids and figure out general anisotropic, hemitropic, isotropic, and ultraisotropic continua.

Spectral formulations in nonlinear solids: A brief summary

This summary focuses on spectral formulations developed by the authors for nonlinear anisotropic solids, where their advantages have been demonstrated in the literature. Some recent isotropic/anisotropic spectral formulations and results are highlighted here via a brief summary of the authors’ work.

Nonlinear mechanics of remodeling

In this paper, we present a large-deformation formulation of the mechanics of remodeling. Remodeling is anelasticity with an internal constraint—material evolutions that are mass and volume-preserving. In this special class of material evolutions, the explicit time dependence of the energy function is via one or more remodeling tensors that can be considered as internal variables of the theory. The governing equations of remodeling solids are derived using a two-potential approach and the Lagrange–d’Alembert principle. We consider both isotropic and anisotropic solids and derive their corresponding remodeling equations. We study a particular remodeling of fiber-reinforced solids in which the fiber orientation is time-dependent in the reference configuration—SO(3)-remodeling. We define an additional remodeling energy, which is motivated by the energy spent in collagen fiber-reinforced living systems to remodel to enhance stiffness or strength in the direction of loading. We consider the examples of a solid reinforced with either one or two families of reorienting fibers and derive their remodeling equations. This is a generalization of some of the proposed remodeling equations in the literature. We study three examples of material remodeling, namely finite extensions and torsion of solid circular cylinders, which are universal deformations for incompressible isotropic solids and certain anisotropic solids. We consider both displacement and force-control loadings. Detailed parametric studies are included for the effects of various material and loading parameters on fiber remodeling. It is observed that during remodeling, there is a competition between the action of the internal strain energy function and the remodeling energy. For a given material, a remodeling process dominated by strain energy works to align fibers in a direction that minimizes strain energy. On the other hand, a remodeling process dominated by the remodeling energy aligns fibers in the direction of the maximum principal strain according to a constitutive choice. We finally linearize the governing equations of the remodeling theory and derive those of linear remodeling mechanics.

Three-dimensional modeling and analysis of anisotropic materials with quasi-static deformation and dynamic fracture in non-ordinary state-based peridynamics

A general three-dimensional anisotropic model is proposed within the non-ordinary state-based peridynamic framework in this work. Three-dimensional peridynamic formulations for anisotropic solids are formulated, and two types of stress rate solutions, Green-Naghdi and Jaumann stress rate are implemented in this model, respectively. Moreover, the equivalent bond stress and corresponding equivalent bond strain are presented and calculated. The 3D-Hashin failure criterion is used among the numerical analysis of anisotropic materials through the equivalent bond strain. Various typical cases are performed by the proposed anisotropic peridynamic model and approach, including the numerical analyses of quasi-static deformation and dynamic fracture for anisotropic materials. The effectiveness and applicability of the proposed model are verified by precisely characterizing the quasi-static deformation and dynamic failure behavior.

Tensors with Constant Components in the Constitutive Equations of a Hemitropic Micropolar Solids

The present paper is devoted to elastic potentials and the constitutive equations of mechanics of anisotropic micropolar solids, the kinematics of which can be specified by two independent vector fields: a contravariant field of translational displacements and a contravariant pseudovector field of microrotations of weight +1. The quadratic stress potential is represented by three constitutive tensors of the fourth rank, two of which are pseudotensor in nature and can be assigned weights –2 and –1. Such a solid is completely specified by the 171st micropolar elastic modulus. The main attention is focused on the model of a hemitropic (half-isotropic, demitropic) micropolar elastic solid characterized by nine constitutive constants. The components of the constitutive pseudo-tensor of weight ‒1 turn out to be sensitive to mirror reflection transformations in three-dimensional space. A peculiar algebraic structure of the constitutive tensors of a hemitropic solid, more precisely, their absolute analogues obtained by multiplying by integer powers of a pseudoscalar unity, is studied. It is shown that these tensors can always be constructed from isomers (isomer) of a tensor with constant components (generally insensitive to any transformations of the coordinate system) and one additional fourth-rank tensor constructed, in turn, from the components of the metric tensor.

Universal displacements in inextensible fiber-reinforced linear elastic solids

For a given class of materials, universal displacements are those displacements that can be maintained for any member of the class by applying only boundary tractions. In this paper, we study universal displacements in compressible anisotropic linear elastic solids reinforced by a family of inextensible fibers. For each symmetry class and for a uniform distribution of straight fibers respecting the corresponding symmetry, we characterize the respective universal displacements. A goal of this paper is to investigate how an internal constraint affects the set of universal displacements. We have observed that other than the triclinic and cubic solids in the other five classes (a fiber-reinforced solid with straight fibers cannot be isotropic), the presence of inextensible fibers enlarges the set of universal displacements.

Dynamic crack propagation in anisotropic solids under non-classical thermal shock

Dynamic crack propagation in anisotropic solids under non-classical thermal shock

Anisotropic plates identification through analyses of dynamic behaviour

This manuscript presents an inverse method to identify the constitutive constants of anisotropic homogeneous elastic solids. The data on which the inverse method is essentially based are natural frequencies, whose ease of measurement and quality of the same is well recognized by the technical literature. The main objective is to evaluate the minimum number of engineering constants that can characterize the material. Identifying the engineering constants is justified by their physical significance in the static traction measurements, with which the present dynamic characterization directly compares. The method is theoretically introduced within the framework of classical plate theory and its performances are corroborated by numerical and experimental comparisons. However, the experimental number of tests herein represents the most corroborating aspect of the theoretical predictions. The dynamic characterization is also compared to the static measurements. The Riemannian distance from the closest isotropic modules has assisted in drawing relevant conclusions regarding the materials herein analyzed, which are usually supposed to be isotropic.

Optimal elastostatic cloaks

An elastic cloak hides a hole (or an inhomogeneity) from elastic fields. In this paper, a formulation of the optimal design of elastic cloaks based on the adjoint state method, in which the balance of linear momentum is enforced as a constraint, is presented. The design parameters are the elastic moduli of the cloak, and the objective function is a measure of the distance between the solutions in the physical and in the virtual bodies. Both the elastic medium and the cloak are assumed to be made of isotropic linear elastic materials. However, the proposed formulation can easily be extended to anisotropic solids. In order to guarantee smooth inhomogeneous elastic moduli within the cloak a penalization term is added to the objective function. Mixed finite elements are used for discretizing the weak formulation of the optimization problem. Several numerical examples of optimal elastic cloaks designed for both single and multiple loads are presented. We consider different geometries and loading types and observe that in some cases the optimal elastic cloaks for cloaking holes (cavities) are made of auxetic materials.