Classical Elasticity Research Articles

Hamiltonian System-Based Symplectic Framework for Analytical Vibration Analysis of Microplates

Abstract The classical elasticity is inadequate for the dynamic analysis of microplates due to the size effect. This study incorporates a higher-order strain gradient theory into the Hamiltonian system-based symplectic framework and derives new analytical solutions for the free vibration of microplates. The analytical solutions are obtained using rigorous mathematical techniques, including separation of variables, symplectic eigen expansion, and superposition, without relying on predetermined solution forms. Hence, they are not restricted to Lévy-type boundary conditions. Using these analytical solutions, we present comprehensive vibration results for microplates and perform detailed parametric studies to explore the impact of length scale parameters on the natural frequencies. Given the growing demand for microplates in advanced engineering applications, the obtained analytical solutions are expected to facilitate their design and performance optimization.

Anisotropic odd elasticity with Hamiltonian curl forces

Abstract A host of elastic systems consisting of active components exhibit path-dependent elastic behaviors not found in classical elasticity, which is known as odd elasticity. Odd elasticity is characterized by antisymmetric (odd) elastic modulus tensor. Here, from the perspective of geometry, we construct the Hamiltonian formalism to show the origin of the antisymmetry of the elastic modulus that is intrinsically anisotropic. Furthermore, both non-conservative stress and the associated nonlinear constitutive relation naturally arise. This work also opens the promising possibility of exploring the physics of odd elasticity in dynamical regime by Hamiltonian formalism.

Mathematical Models for Ultrasound Elastography: Recent Advances to Improve Accuracy and Clinical Utility.

Changes in biomechanical properties such as elasticity modulus, viscosity, and poroelastic features are linked to the health status of biological tissues. Ultrasound elastography is a non-invasive imaging tool that quantitatively maps these biomechanical characteristics for diagnostic and treatment monitoring purposes. Mathematical models are essential in ultrasound elastography as they convert the raw data obtained from tissue displacement caused by ultrasound waves into the images observed by clinicians. This article reviews the available mathematical frameworks of continuum mechanics for extracting the biomechanical characteristics of biological tissues in ultrasound elastography. Continuum-mechanics-based approaches such as classical viscoelasticity, elasticity, and poroelasticity models, as well as nonlocal continuum-based models, are described. The accuracy of ultrasound elastography can be increased with the recent advancements in continuum modelling techniques including hyperelasticity, biphasic theory, nonlocal viscoelasticity, inversion-based elasticity, and incorporating scale effects. However, the time taken to convert the data into clinical images increases with more complex models, and this is a major challenge for expanding the clinical utility of ultrasound elastography. As we strive to provide the most accurate imaging for patients, further research is needed to refine mathematical models for incorporation into the clinical workflow.

Buckling analysis of functionally graded nanobeams via surface stress-driven model

The manuscript investigates the buckling behaviour of Bernoulli-Euler nanobeams composed of Functionally-Graded (FG) materials with different cross-sectional shapes. This analysis is conducted using the surface stress-driven model of elasticity. The nonlocal governing equations for the elastostatic buckling problem are derived employing the principle of virtual work. The study also includes a parametric investigation, presenting and discussing the main results while varying the nonlocal parameter, material gradient index, the cross-sectional shapes and the constraints at the ends of the FG nanobeams. Critical loads are numerically calculated and compared with those obtained by other authors using the classical stress-driven model elasticity.

Thermoelastic damping in asymmetric vibrations of nonlocal circular plate resonators with Moore-Gibson-Thompson heat conduction

Precise estimation of the amount of thermoelastic damping (TED) in small-sized resonators is one of the crucial issues in their optimal design. According to several experimental, numerical and theoretical findings, the behavior of structures with such small dimensions in structural and thermal domains does not follow classical elasticity theory and the Fourier law of heat conduction. The objective of this work is to develop a nonclassical formulation for computing TED in asymmetric vibrations of circular nanoplates according to nonlocal theory (NT) and Moore-Gibson-Thompson (MGT) heat transfer model. To do so, first, the coupled equations of motion and heat are derived in the purview of NT and MGT model. Then, the real and imaginary parts of the system frequency are extracted from the frequency equation affected by thermoelastic coupling. Finally, by means of the existing definition for TED, a mathematical expression including nonlocal parameter of NT and nonclassical constants of MGT model is obtained to make an estimate of TED value. In the numerical examples section, to ensure the credibility of the provided formulation, the results obtained through this model are compared with those reported in the literature in the context of simpler models. Then, by presenting various graphical data, the type and extent of the role of different factors in TED changes are appraised. The obtained results indicate that employing NT and MGT model can substantially impact the amount of TED compared to the estimations of the classical formulation. Furthermore, the comparison of results reveals that the magnitude of TED in asymmetric vibration modes surpasses that in symmetric ones, and consequently incorporating asymmetric vibration modes is crucial for thorough analysis of TED in circular nanoplates. Thus, the formulation presented in this article can aid in achieving an optimal design for such plates, particularly in terms of minimizing energy loss.

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz theory.

Mixed finite element implementation of plane crack problems within strain gradient elasticity

AbstractAn approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method is used, where stresses, strains, and displacements are considered as independent variables. This significantly simplifies the pre‐requirement for approximating functions and allows one to use the simplest triangular finite elements with a linear approximation of displacements. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses in nodes are treated as local unknowns. This discrete problem is solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. The modeling plane crack problem (tension of a square plate with a central crack) is solved. The obtained solution agrees with the results available in the literature. Good convergence is achieved by refining the mesh for all scale parameters.

Atomistic and continuum Ascertainment of The crack tip stress fields in anisotropic elastic cubic media

The article provides a comparative analysis of stresses associated with the crack tip in an anisotropic elastic medium with cubic crystalline symmetry of elastic properties under mixed loading conditions obtained by two fundamentally different approaches: atomistic simulations and continuum fracture mechanics methods. The atomistic modelling approach is premised on the application of the molecular dynamics (MD) method implemented in the open source code program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The continuum mechanics approach is based on the classical elasticity theory of anisotropic media, in which the mechanical fields associated with the crack tip are represented by the crack tip stress and displacement series expansions generalizing the well-known classical asymptotical presentation of M. Williams to the case of anisotropic media. Using the MD method, a large series of computations of the loading of monocrystalline copper and aluminum plates with a face-centered cubic crystal lattice weakened by a central crack applying the embedded atom potential (EAM) was implemented. MD modeling is aimed at determining the atomic stresses and strains near the crack tip. The computed atomic stresses were compared with the stress field determined by the continuum elasticity theory of anisotropic media for crystal lattices with cubic symmetry of elastic properties. A comparative analysis was carried out for the angular dependencies of the stress and strain tensor components at various selected distances from the crack tip for the entire range of mixed deformation forms starting from pure tensile loading and ending with forms close to pure transverse shear. It is ascertained that the crack tip fields determined on the basis of two fundamentally different approaches, precisely, discrete and continuum, are completely consistent with each other. It is established that mathematical methods of continuum fracture mechanics can be applied to describe stress, strain and displacement fields at the atomic level.

Complete Solutions in the Dilatation Theory of Elasticity with a Representation for Axisymmetry

In this paper, we present certain complete solutions in the dilatation theory of elasticity. This model can be derived as a special case of Eringen’s linear theory of microstretched elastic solids when microrotations are absent. It is also a version of the theory of materials with voids. The dilatation theory can be considered the simplest theoretical model of microstructured materials and is suitable for investigating various phenomena that occur in engineering, geomechanics, and biomechanics. We establish three complete solutions to the displacement equations of equilibrium that are the counterpart of the Green–Lamé (GL), Boussinesq–Papkovich–Neuber (BPN), and Cauchy–Kowalevski–Somigliana (CKS) solutions of classical elasticity. The links between these BPN and CKS solutions are established. Then, we present a representation of the BPN solution in the case of axisymmetry. The results presented here are useful for solving axisymmetric problems in semi-infinite and infinite domains.

Computational homogenization of the elastic properties of polycrystalline fcc metals within Mindlin’s second strain-gradient theory

The lack of interest in Mindlin’s second strain-gradient theory, despite its success in characterizing various phenomena for which there is no explanation in the classical elasticity, is predominantly due to the numerous higher-order elastic constants involved therein. As an attempt to overcome this shortcoming, a computational homogenization method for determining such elastic constants of polycrystalline face-centered cubic (fcc) metals is proposed in the present study. In this homogenization method, using the Voigt-type averaging scheme, a polycrystalline material is modeled by an isotropic aggregate of randomly oriented single crystals without accounting for the grain size effect and the complexities due to grain boundaries and junctions. Subsequently, analytical expressions for the strain energy due to certain modes of loading are determined, and molecular simulations of a few fcc metals under such modes of loading are performed. Then, by fitting the corresponding analytical expressions to the results obtained from these simulations, the effective elastic constants and consequently the effective characteristic lengths of the fcc metals in their polycrystalline form are determined. Moreover, the free-surface-induced reconstruction in a thin layer of a solid is addressed both analytically and by molecular simulations, as a result of which the effective moduli of cohesion of the fcc metals in their polycrystalline form are calculated. In addition, the complete set of conditions for the positive-definiteness of the strain-energy-density function of isotropic materials within the adopted theory is derived, and subsequently a discussion of whether or not the numerical values obtained for the effective elastic moduli of the polycrystalline fcc metals satisfy such conditions is provided.

A Volterra edge dislocation problem in second-order elasticity theory

A displacement function technique has been developed to study plane strain problems in second-order elasticity theory for incompressible elastic materials. The formulation culminates in the development of a single inhomogeneous biharmonic equation of the form [Formula: see text] for the second-order displacement function [Formula: see text]. In the second-order, the isotropic stress associated with constitutive behaviour is governed by an inhomogeneous harmonic equation of the form [Formula: see text]. Both functions [Formula: see text] and [Formula: see text] depend only on the solution to the analogous problem in classical elasticity. While there are similarities to the Airy stress function, the displacement function approach enables the evaluation of the first- and second-order displacement fields purely through its derivatives, thus eliminating the introduction of arbitrary rigid body terms normally associated with formulations where the strains derived from a stress function approach need to be integrated. The displacement function approach is applied to develop a second-order elasticity solution to the problem of a Volterra edge dislocation.

A computational approach to integrate three-dimensional peridynamics and two-dimensional higher-order classical elasticity theory for fracture analysis

AbstractAs a nonlocal alternative of classical continuum theory, peridynamics (PD) is mathematically compatible to discontinuities, making it particularly attractive for failure prediction. The PD theory on the other side can be computationally demanding due to its nonlocal interactions. A coupling between PD and refined higher-order finite element method (FEM) integrates their salient features. The present study proposes a computational approach to couple three-dimensional peridynamics with two-dimensional higher-order finite elements based on classical elasticity. The bond-based PD modeling is considered in a region where damage might appear while refined finite element modeling is used for the remaining region. The refined finite elements employed in this study are based on the 2D Carrera Unified Formulation (CUF), which provides 3D-like accuracy with optimized computational efficiency. The coupling between PD and FEM is achieved through the Lagrange multiplier method which permits physical consistency and compatibility at the interface domain. An adaptive convergence check algorithm is also proposed to achieve predetermined accuracy in the solution with minimum computational effort. Simulations of quasi-static tension tests, wedge splitting tests and L-plate cracking tests are carried out for verification. In-depth analysis shows that the present approach can reproduce the linear deformation, material degradation and crack propagation in an effective way.

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

We study inextensional vibrations of the spherical shell using strain gradient elasticity theory under Kirchhoff-Love hypotheses. For admissibility of the inextensional deformations the shell is punctured by making a tiny hole. The shell is assumed to be made of linearly elastic, homogeneous, and isotropic material. The closed form expression for the frequencies of inextensional modes of vibration of spherical caps of various polar angles and that of nearly complete spherical shell are derived using Rayleigh's method. Towards this, the displacement and velocity fields are obtained by setting the membrane strains to zero. Further, to facilitate the existence of inextensional vibrations, boundaries are assumed to be traction free. The frequencies are computed for the positive and negative signs in the strain gradient term of the constitutive law. The computed frequencies, employing negative sign are found to be higher than those computed using classical elasticity theory. The opposite effect is seen when the frequencies are computed with the positive sign. Further, the frequencies are found to saturate when the positive sign is used in the constitutive law. Parametric studies viz. variation in frequencies with the circumferential wavenumbers for different values of length scale parameter, variation in frequencies with circumferential wavenumbers for different values of polar angles and a given length scale parameter, variation of frequency with increasing polar angle for a given circumferential wavenumber, variation of saturation wavenumber with increasing polar angle, and increasing length scale parameter are carried out. The frequencies of nearly a spherical shell with increasing circumferential wavenumber are also computed.

Elastic to plastic transition in amorphous solids.

The response of amorphous solids to mechanical loads is accompanied by plasticity that is generically associated with "non-affine" quadrupolar events seen in the resulting displacement field. To develop a continuum theory, one needs to assess when these quadrupolar events have a finite density, allowing the development of a field theory. Is there a transition, as a function of the material parameters and the nature of the loads, from isolated plastic events whose density is zero to a regime governed by a finite density? And if so, what is the nature of this transition? The aim of the paper is to explore this issue. The motivation for the present study stems from recent research in which it was shown that gradients of the quadrupolar fields act as dipole charges that can screen elasticity. Analytically soluble examples of mechanical loading that lead to screening and emergent length scales (that are absent in classical elasticity) have been analyzed and tested. However, "gradients of quadrupolar fields" make sense only when the density of quadrupoles is finite, and hence, the issue is central to this article. The article introduces a notion of polarizability under the strain of Eshelby quadrupoles and concludes that the onset of a density of such quadrupoles with random orientations can only appear when the polarizability is finite.

Finite deformation micropolar peridynamic theory: Variational consistency of wryness measure

This paper concerns developments in both classical and peridynamic (PD) finite deformation micropolar elasticity theory. The first part presents an alternative proof of a theorem that demonstrates variational consistency of the wryness measure in classical finite deformation micropolar elasticity which is one of the novel contributions of this work. The linearized version of the proposed wryness measure is compared with the wryness measure of small deformation micropolar elasticity. In the second part of this work, a finite deformation micropolar PD theory is presented. After proposing an additional integro-differential equation along with the standard PD equation of motion, the global balance of angular momentum is proved to be satisfied. The balance of virtual work is derived for the micropolar PD theory. Next, constitutive correspondence approach is used to relate PD force and moment states with their classical counterparts. The nonlocal versions of the strain and the proposed wryness measure are used in the constitutive correspondence. The constitutive equations of the classical micropolar theory are presented and how they can be incorporated in PD framework is discussed. We introduce a new bond breaking criterion for PD micropolar materials based on critical stretch and critical relative rotation. Quasi-static numerical simulations on deformations of plate with a hole and fracture of a double notched sheet are presented. Dynamic simulations concern wave propagation through a solid specimen and plate with a hole. Validation of the simulation results with the finite element solutions, boundary element solutions, and experimental observations demonstrate the potential of our approach.

Dynamic screening by plasticity in amorphous solids.

In recent work it was shown that elasticity theory can break down in amorphous solids subjected to nonuniform static loads. The elastic fields are screened by geometric dipoles; these stem from gradients of the quadrupole field associated with plastic responses. Here we study the dynamical responses induced by oscillatory loads. The required modification to classical elasticity is described. Exact solutions for the displacement field in circular geometry are presented, demonstrating that dipole screening results in essential departures from the expected predictions of classical elasticity theory. Numerical simulations are conducted to validate the theoretical predictions and to delineate their range of validity.

Continuum-molecular modeling of planar micropolar media: Anisotropy, chiral properties and length-scale effects

This paper presents a continuum-molecular formulation for bi-dimensional micropolar media within the mathematical formalism of a revised peridynamic theory with oriented material points. A variational procedure is considered to derive the fundamental governing equations of the model, which postulates that material points interact through pair potentials allowing generalized pairwise actions to be derived as energy conjugates to properly defined pairwise deformation measures. While continuity of mass is assumed, constitutive laws are produced for long-range micro-interactions to reproduce the effective behavior of the material at the macro-scale. The definition of proper micromoduli functions respecting material symmetries and invariance properties allows then to obtain a non-local micropolar continuum based on pseudo-discrete kinematics, while providing a mechanism-based description of material anisotropy and coupling behaviors uncovered by classical elasticity. An analytic micro-macro moduli correspondence procedure is also established, based on the formal analogy with the constitutive tensor of centrosymmetric planar micropolar continua. As an important result of this research, we show that distinctive chiral effects of micropolar elasticity can be reproduced by introducing a directional independent pseudo-scalar pair potential, which turns out to be analytically vanishing when the rotationally invariant part of the corresponding continuum elastic tensor is invariant to mirror reflections as well. Moreover, the proposed formulation demonstrates sensitivity to elastic bending size-effect as specific property of structured materials homogenized as micropolar continua, and related to the characteristic length of their underlying microstructure. The resulting constitutive non-locality, which plays an important role in fracture problems, is conceptually separated with respect to the intrinsic non-local character of the model related to the integral nature of its governing equations, and then does not vanish when the horizon reduces to zero. The theoretical findings and the effectiveness of the model are successfully verified through illustrative examples referred to representative cases of structured materials homogenized as micropolar continua, including length-scale dependent quasi-static crack propagation as well as the mechanism-based description of the coupling between bulk strain and micro-rotation in elastic bi-dimensional homogenized chiral lattices.

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. We avoid establishing exact micro–macro transition relations, as in classical homogenization theory, because defining a representative volume element is not feasible in the absence of scale separation, as such an element does not exist. The relaxed micromorphic model is a generalized continuum that utilizes the Curl of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature; one simplified formulation involving only one single characteristic length and a simplified isotropic curvature with three parameters. The relaxed micromorphic model demonstrates good agreement with the fully-resolved heterogeneous solution after optimizing only four parameters. The “simplified” full micromorphic model, which includes isotropic curvature and involves the optimization of seven parameters, does not achieve superior results, while the Cosserat model exhibits the poorest fitting.

Peridynamic correspondence model with strain gradient elasticity for microstructure dependent size effects

This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

The strain gradient viscoelasticity full field solutions for Mode-I and Mode-II crack problems

The impacts of size and viscosity become distinctly evident when considering micro- and nano-scale phenomena for advanced materials with micro- and nanostructures. In this study, the mechanical behavior of the advanced material is characterized by using the strain gradient viscoelasticity theory, and a novel solution is presented for the mode-I and mode-II cracks, which is formulated based on the strain gradient viscoelasticity theory, employing the Wiener-Hopf method. Besides, the gradient-dependent viscoelastic crack solutions are directly derived by applying the correspondence principle that aligns strain gradient viscoelasticity with strain gradient elasticity within the Maxwell's standard linear solid model. Relative to the influence of elastic strain gradient effects, the involvement of viscous gradient effects instigates a reinforcement to the stress field around the crack tip, thereby offering a more reasonable representation of advanced materials crack behavior. When the viscosity effect is omitted or as time tends to infinity, the solutions based on strain gradient viscoelasticity theory converges to those on the classical strain gradient elasticity theory.