Elastic Solids Research Articles

Rayleigh wave propagation in an initially stressed orthotropic elastic solid half-space lying under inviscid liquid and viscous liquid layers

PurposeThe purpose of this article is the propagation of Rayleigh waves in a homogeneous, isotropic, initially stressed orthotropic elastic solid half-space lying under a homogeneous, viscous, inviscid liquid layer of finite thickness.Design/methodology/approachIn the presence of both viscous and inviscid liquids, it derives the phase velocity, initial stress, and wave number-dependent frequency equation for an orthotropic elastic solid. With the help of the MATLAB program, the thickness effects of liquid layers, initial stress, and viscosity on the phase velocity and attenuation coefficient of the Rayleigh wave are explained for a particular model.FindingsThe phase velocity-dependent dispersion relation of Rayleigh waves at the interface of viscous liquid and solid half-space is a function of initial stress and wave number. Rayleigh waves along the free surface of an orthotropic elastic half-space are also derived as a particular case. The classical results of an inviscid liquid are achieved when the thickness of a viscous liquid approaches zero. Well-known classical results for initially stressed orthotropic elastic solids were also derived.Originality/valueSo far, many researchers have looked into the propagation of surface waves at the interfaces of solid–inviscid liquid, solid–solid, and multilayer interfaces. But in this article, the dispersion behavior of Rayleigh wave propagation in an initially stressed homogeneous orthotropic elastic solid half-space under a double layer of viscous liquid and inviscid liquid is studied.

Band gaps of elastic waves in 1-D dielectric phononic crystal with the flexoelectric and strain gradient effects consideration

The propagation of Bloch waves in one dimensional phononic crystal consisting of dielectric elastic solids is studied with consideration of the strain gradient, inertial gradient and flexoelectric effects. The transfer matrixes for single layer and one-cell of phononic structure are derived based on the constitutive equations and the governing equations of dielectric elastic solids. The dispersion equation for Bloch waves is obtained by the application of periodic conditions for the generalized displacements and tractions considered within strain gradient theory of electro-elasticity. Based on the numerical solution of the derived dispersion equation, the influences of the micro-stiffness length scale, micro-inertial length scale and flexoelectric coefficients on the dispersion and the bandgap are discussed.

On prescribing the number of singular points in a Cosserat-elastic solid

Abstract In a geometrically non-linear Cosserat model for micro-polar elastic solids, we prove that critical points of the Cosserat energy functional with an arbitrary large (finite) number of singularities do exist, whereas Cosserat energy minimizers are known to be locally Hölder continuous. To reach that goal, we first develop a technique to insert dipole pairs of singularities into smooth maps while controlling the amount of Cosserat energy needed to do so. We then use this method to force an arbitrary number of singular points into (weak) Cosserat-elastic solids by prescribing smooth boundary data. The boundary data themselves are given in such a way, that they contain no topological obstruction to regularity. Throughout this paper, we often exploit connections between harmonic maps and Cosserat-elastic solids, so that we are able to adapt and incorporate ideas of R. Hardt and F.-H. Lin for harmonic maps with singularities, as well as of F. Béthuel for dipole pairs of singularities.

A variational framework for Cahn–Hilliard-type diffusion coupled with Allen–Cahn-type multi-phase transformations in elastic and dissipative solids

This article presents a variational framework for coupled chemo-mechanical solids undergoing irreversible micro-structural changes at infinitesimal strains. The coupled problem is characterised by phenomena such as phase transitions, micro-structure coarsening and swelling. It is an extension of our previous work on variational inelasticity for a conserved chemo-mechanical setting to a unified conserved and non-conserved setting which include multi-phase transformations. The variational framework, again governed by continuous-time, discrete-time and discrete-space–time incremental variational principles, is outlined for coupled diffusion-phase transformation phenomena in elastic and dissipative solids. For the sake of simplicity, focus is restricted to isothermal conditions. It is shown that the governing macro- and micro-balance equations of the coupled problem appear as Euler equations of these minimisation and saddle point principles. In contrast to our previous work, extended variational principles (with the gradient of the chemical potential and phase fractions) are constructed that account for diffusion-phase transformation coupling. This is achieved by Legendre transformations. Note that the local–global solution strategy is still preserved and the resulting system of symmetric non-linear algebraic equations are solved by Newton–Raphson-type iterative methods. The applicability of the proposed framework is demonstrated by numerical simulations that qualitatively characterise lower bainitic micro-structure.

Application of J-integral to adhesive contact under general plane loading for rolling resistance

In the present work, a mechanical model for two-dimensional non-slipping adhesive contact between dissimilar elastic solids under general loading, namely, normal forces, tangential forces and moments is proposed. The general solutions are obtained analytically with the stresses at the contact edges exhibiting oscillatory singularity, similar to those at a bimaterial interface crack. The well-known J-integral under the current context is analyzed. Its application under the selected integration contour readily gives the relationship between the stress intensity factors and energy release rates at the contact edges. With the results rolling adhesion between two solids with parabolic profiles is considered further. The applied moment can be directly determined by the difference in energy release rates at the trailing and leading edges and hence the rolling resistance even for adhesive contact with cohesive zones. These results provide the foundation for understanding some tribological phenomena associated with adhesion.

Variational damage model: A novel consistent approach to fracture

The computational modeling of fractures in solids using damage mechanics faces challenge when dealing with complex crack topologies. One effective approach to address this challenge is by reformulating damage mechanics within a variational framework. In this paper, we present a novel variational damage model that incorporates a threshold value to prevent damage initiation at low energy levels. The proposed model defines fracture energy density (ϕ˜) and damage field (s) based on the energy density (ϕ), crack energy release rate (Gc), and crack length scale (ℓ). Specifically, if ϕ≤Gc2ℓ, then ϕ˜=ϕ and s=0; otherwise, ϕ˜=−Gc24ℓ21ϕ+Gcℓ and s=1−Gc2ℓ1ϕ. Furthermore, we extend the model with a threshold value to a higher-order version. Utilizing this functional, we derive the governing equation for fractures that evolve automatically with ease. The formulation can be seamlessly integrated into conventional finite element methods for elastic solids with minimal modifications. The proposed formulation offers sharper crack interfaces compared to phase field methods using the same mesh density. We demonstrate the capabilities of our approach through representative numerical examples in both 2D and 3D, including static fracture problems, cohesive fractures, and dynamic fractures. The open-source code is available on GitHub via the link https://github.com/hl-ren/vdm.

Investigating the Influence of the Substrate Temperature and the Organic Precursor on the Mechanical Properties of Low‐Pressure Plasma Polymer Films

ABSTRACTThis work aims to investigate the influence of the substrate temperature on low‐pressure plasma polymerization processes. Emphasis is placed on the mechanical properties and growth mechanism of the plasma polymer films (PPFs). For this purpose, two precursors are considered, differing only by their unsaturation degree: 2‐propen‐1‐ol (CH2═CH–CH2–OH) and propan‐1‐ol (CH3–CH2–CH2–OH). Although propan‐1‐ol‐based PPFs behave like hard elastic solids, 2‐propen‐1‐ol‐based coatings evolve from a liquid film to an elastic solid on increasing the substrate temperature. This behavior is understood considering the evolution of the glass transition temperature of PPFs. The latter is correlated with the cross‐linking degree of the polymeric network governed by the energy density of bombarding ions on the growing film.

Dynamic and modal analysis of nearly incompressible structures with stabilised displacement-volumetric strain formulations

This paper presents a dynamic formulation for the simulation of nearly incompressible structures using a mixed finite element method with equal-order interpolation pairs. Specifically, the nodal unknowns are the displacement and the volumetric strain component, something that makes possible the reconstruction of the complete stain at the integration point level and thus enables the use of strain-driven constitutive laws. Furthermore, we also discuss the resulting eigenvalue problem and how it can be applied for the modal analysis of linear elastic solids. The article puts special emphasis on the stabilisation technique used, which becomes crucial in the resolution of the generalised eigenvalue problem. In particular, we prove that using a variational multiscale method assuming the sub-grid scales to lie in the finite element space orthogonal to that of the approximation, namely the Orthogonal Sub-Grid Scales (OSGS), results in a convenient linear and symmetric generalised eigenvalue problem. The correctness, convergence and performance of the method are proven by solving a series of two- and three-dimensional examples.

Parallel isogeometric boundary element analysis with T-splines on CUDA

We present a framework for parallel isogeometric boundary element analysis (BEA) of elastic solids on CUDA. To deal with traction discontinuities, we propose a BEA model that supports multiple nodes and semi-discontinuous elements. The multiplicity of a node is defined by the number of regions containing any element influenced by the node. A region is a group of connected elements delimited by a closed crease curve. The default shape function of an element is determined by a linear operator applied to a set of basis functions. A BEA model is supposed to be generated from a watertight boundary representation of a solid. In this paper, we employ bicubic analysis-suitable T-splines. In this case, the shape of an element is defined by its Bézier extraction operator applied to the tensor product of Bernstein polynomials of degree 3. We describe the data structures of the BEA model and the main algorithms of the analysis pipeline on CUDA. In particular, we describe two strategies for parallel assembling of the linear system of equations. We also introduce a novel approach for inside integration based on the subdivision of the singular region in triangles with constant aspect ratio. In the T-splines context, we extend the Bézier extraction to handle unstructured T-meshes with crease edges. Moreover, we propose a scheme for embedding the influence of linked tangency handles on the shape of an element directly into the Bézier extraction operator. Such an embedding enables the removal of the corresponding nodes from the BEA model and the application of an alternative collocation method we discuss in the paper. We present several experiments to evaluate the accuracy and efficiency of the proposed framework. The results demonstrate that the GPU can be advantageously employed for parallelizing T-spline-based isogeometric analysis using boundary elements, achieving a speedup of up to 29x compared to the sequential code on a current laptop. We make the BEA code available in a prototype in MATLAB with a graphical interface that allows users to apply boundary conditions and visualize analysis results on the boundary.

Drag reduction in hypersonic flows with viscous compressible fluid–solid coupling: The role of elastic spikes and lateral jets

This article introduces a novel fluid–solid interaction (FSI) method designed for high-speed flow scenarios, which addresses the intricate interactions between viscous compressible fluids and elastic solids. The proposed method, grounded in the finite volume method, balances computational efficiency and stability while accurately capturing fluid dynamics and structural elasticity. Validation against experimental and numerical data from previous studies confirmed the algorithm's effectiveness. The validated FSI model is applied to study drag reduction in elastic spikes with lateral jets under hypersonic conditions, highlighting significant changes in flow characteristics due to structural deformation and lateral jets. The study extensively examined the effects of jet total pressure, jet orifice position, and spike material density on drag reduction, deformation, and flow field characteristics. Key findings include the influence of compressible FSI on temperature, pressure, and drag distribution, the benefits of increased jet pressure ratio for thermal protection, the impact of jet position on flow characteristics, and the relationship between spike deformation and material density. This study offers valuable perspectives and effective strategies for structure design and minimizing aerodynamic resistance in superspeed fluid situations. Nevertheless, there are still obstacles to overcome, such as non-linear deformation, thermal coupling, and computational precision, highlighting the necessity for further enhancement of FSI techniques.

An existence result for accretive growth in elastic solids

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality.

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.

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.

A Lumped Mass Meshfree Collocation Method with Frequency Error Estimates for Free Vibration Analysis of Elastic Solids

A Lumped Mass Meshfree Collocation Method with Frequency Error Estimates for Free Vibration Analysis of Elastic Solids

Physics-based discrete models for magneto-mechanical metamaterials

Magneto-mechanical metamaterials are emerging smart materials whose mechanical responses can be tailored through structure architecture and magnetic interactions. The latter provides additional freedom in the material design space and leads to novel behaviors due to its nonlocal nature. The enriched functionalities open new possibilities in various applications, such as actuators, energy absorbers, and soft robots. However, the nonlinear and nonlocal coupling between elastic and magnetic forces poses a great challenge in the modeling and simulation of these systems, further hindering theory-based rational design strategies. Here, we focus on a class of magneto-mechanical metamaterials comprising elastic solids embedded with rigid permanent magnets. The clear separation between elastic and magnetic forces simplifies the design and fabrication process, yet their nonlocal interplay still allows for complex behaviors. We present a simulation framework for such magneto-mechanical metamaterials by combining a lattice spring model for the elastic solid with the dipole model for the magnetic interactions and implementing it in the LAMMPS molecular dynamics software. We demonstrate the capabilities of our framework by simulating a few representative structures, including shape-locking lattice metamaterials, a soft cellular solid with controllable buckling, and a metamaterial chain with phase-transforming behavior. For the shape-locking lattice metamaterials, we successfully capture the magnetic-actuation-driven reconfiguration and the nonlinear mechanical response of the curved lattices. For the soft cellular solid, we identify its buckling patterns under external non-uniform magnetic fields and simulate a buckling evolution process consistent with experiments. For the metamaterial chain, we include the strong long-range interactions among the embedded magnets and reproduce the controllable phase transitions in the experiments. Our work provides a simple yet versatile simulation methodology to investigate the nonlinear mechanical behaviors in the presence of strong external and internal magnetic forces, which will facilitate the design and analysis of magneto-mechanical materials. It can also be applied to other magnetically-driven smart structures, such as soft robots.

Universal deformations and inhomogeneities in isotropic Cauchy elasticity

For a given class of materials, universal deformations are those deformations that can be maintained in the absence of body forces and by applying solely boundary tractions. For inhomogeneous bodies, in addition to the universality constraints that determine the universal deformations, there are extra constraints on the form of the material inhomogeneities—universal inhomogeneity constraints. Those inhomogeneities compatible with the universal inhomogeneity constraints are called universal inhomogeneities. In a Cauchy elastic solid, stress at a given point and at an instance of time is a function of strain at that point and that exact moment in time, without any dependence on prior history. A Cauchy elastic solid does not necessarily have an energy function, i.e. Cauchy elastic solids are, in general, non-hyperelastic (or non-Green elastic). In this paper, we characterize universal deformations in both compressible and incompressible inhomogeneous isotropic Cauchy elasticity. As Cauchy elasticity includes hyperelasticity, one expects the universal deformations of Cauchy elasticity to be a subset of those of hyperelasticity both in compressible and incompressible cases. It is also expected that the universal inhomogeneity constraints to be more stringent than those of hyperelasticity, and hence, the set of universal inhomogeneities to be smaller than that of hyperelasticity. We prove the somewhat unexpected result that the sets of universal deformations of isotropic Cauchy elasticity and isotropic hyperelasticity are identical, in both the compressible and incompressible cases. We also prove that their corresponding universal inhomogeneities are identical as well.

Micro-crack in solids evaluation based on zero-frequency component of the critically refracted longitudinal wave

Abstract Micro-crack in engineering materials may cause subtle changes in mechanical properties, which can lead to serious engineering accidents. It is of great significance to study the identification of micro-cracks. In this paper, it is discovered that when the critically refracted longitudinal (LCR) wave propagates in elastic solids containing micro-crack, the waveform of ultrasonic waves becomes distorted, resulting in the generation of a zero-frequency component. The generated zero-frequency component provides a new research idea for micro-crack detection. Firstly, the paper derives the wave equation for nonlinear LCR wave and provides a preliminary analytical solution with a zero-frequency component. Subsequently, through a combined approach involving numerical simulation and experimental observation, along with micro-crack modeling, the study delves deeper into the acoustic nonlinear characteristics of the zero-frequency component under the breathing behavior of micro-crack. Finally, both numerical simulation and experimental methods show that the zero-frequency component based on LCR wave is feasible to detect micro-crack, and it is also proved that the zero-frequency component has higher sensitivity to micro-crack damage than the second and third harmonics.

Spectral reordering for faster elasticity simulations

We present a novel method for faster physics simulations of elastic solids. Our key idea is to reorder the unknown variables according to the Fiedler vector (i.e., the second-smallest eigenvector) of the combinatorial Laplacian. It is well known in the geometry processing community that the Fiedler vector brings together vertices that are geometrically nearby, causing fewer cache misses when computing differential operators. However, to the best of our knowledge, this idea has not been exploited to accelerate simulations of elastic solids which require an expensive linear (or non-linear) system solve at every time step. The cost of computing the Fiedler vector is negligible, thanks to an algebraic Multigrid-preconditioned Conjugate Gradients (AMGPCG) solver. We observe that our AMGPCG solver requires approximately 1 s for computing the Fiedler vector for a mesh with approximately 50K vertices or 100K tetrahedra. Our method provides a speed-up between 10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10\\%$$\\end{document} – 30%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$30\\%$$\\end{document} at every time step, which can lead to considerable savings, noting that even modest simulations of elastic solids require at least 240 time steps. Our method is easy to implement and can be used as a plugin for speeding up existing physics simulators for elastic solids, as we demonstrate through our experiments using the Vega library and the ADMM solver, which use different algorithms for elasticity.

Elastic Solids with Strain-Gradient Elastic Boundary Surfaces

Elastic Solids with Strain-Gradient Elastic Boundary Surfaces

A general contact model for rough surfaces based on the incremental concept

It is now commonly accepted that real contact area rises linearly with increasing normal load for the contact of rough solids. However, the coefficient of proportionality is distinct in various models and simulations. This study advances a general method to determine the area-load proportionality based on the incremental contact concept. A pair of loose bounds on the coefficient of proportionality for general Gaussian random rough surfaces and a tight one working in the thermodynamic limit are derived. The resultant predictions are in accord with numerical results presented in literature. The current study might offer some deep insight into the role of surface morphologies in contact of rough elastic solids.