Compressible Elastic Solids Research Articles

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.

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.

A cell-based smoothed finite element method for finite elasticity

In this study, we present a displacement based polygonal finite element method for compressible and nearly-incompressible elastic solids undergoing large deformations in two dimensions. This is achieved by projecting the dilatation strain onto the linear approximation space, within the framework of volume averaged nodal projection method. To reduce the numerical integration burden over polytopes, a linear strain smoothing technique is employed to compute the terms in the bilinear/linear form. The salient features of the proposed framework are: (a) does not require derivatives of shape functions and complex numerical integration scheme to compute the bilinear and linear form and (b) volumetric locking is alleviated by adopting the volume averaged nodal projection technique. The efficacy, convergence properties and accuracy of the proposed framework is demonstrated through four standard benchmark problems.

On sub-surface stress caused by contact roughness in compressible elastic solids

Contact between elastic bodies with self-affine rough surfaces is mostly studied with a focus on determining surface fields, despite body fields are of great importance to establish, for instance, when and where elasticity breaks down. This work aims at analyzing the effect of contact roughness on the body fields of compressible frictionless solids modeled using Green's function molecular dynamics. Although area-load curves are insensitive to changes in the Hurst exponent as long as they are correctly normalized and are clearly not affected by compressibility, the Von-Mises stress is found to depend on both Hurst exponent and Poisson's ratio.

A stable added-mass partitioned (AMP) algorithm for elastic solids and incompressible flow

A stable added-mass partitioned (AMP) algorithm is developed for fluid-structure interaction (FSI) problems coupling viscous incompressible flow and compressible elastic solids. The hyperbolic system governing the solid is integrated using an explicit upwind (Godunov) scheme. The equations for the fluid are integrated using an implicit-explicit (IMEX) fractional-step scheme whereby the velocity is advanced in one step, treating the viscous terms implicitly, and the pressure is computed in a second step. The AMP interface conditions for the fluid arise from the outgoing characteristic variables in the solid, and they are partitioned into a Robin (mixed) interface condition for the pressure and interface conditions for the velocity. The latter conditions include an impedance-weighted average between fluid and solid velocities using a fluid impedance of a special form. A similar impedance-weighted average is used to define interface values for the solid. Deforming composite grids are used to effectively handle the evolving geometry and large deformations. The new algorithm is verified for accuracy and stability on a number of useful benchmark problems in two space dimensions, including a radial-piston problem for which exact solutions for radial and azimuthal motions are found and tested. Exact traveling wave solutions for a solid disk surrounded by an annular region of fluid are also derived and used to verify the AMP scheme. Fluid flow in a channel past a deformable solid annulus is computed, and the errors are estimated from a self-convergence grid refinement study. The AMP scheme is found to be stable and second-order accurate, without sub-time-step iterations, even for very difficult cases of very light solids when added-mass and added-damping effects are large.

A Stable Added-Mass Partitioned (AMP) Algorithm for Elastic Solids and Incompressible Flow: Model Problem Analysis

An analysis is made of a new partitioned scheme for solving fluid-structure interaction problems involving viscous incompressible flow and compressible elastic-solids. The new scheme is stable, wit...

An analysis of a new stable partitioned algorithm for FSI problems. Part I: Incompressible flow and elastic solids

Stable partitioned algorithms for fluid–structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. Importantly, these new added-mass partitioned (AMP) schemes are stable and retain full accuracy with no sub-iterations per time step, even in the presence of strong added-mass effects (e.g. for light solids). The numerical approach described here for bulk compressible solids extends the scheme of Banks et al. [1,2] for inviscid compressible flow, and uses Robin (mixed) boundary conditions with the fluid and solid solvers at the interface. The basic AMP Robin conditions, involving a linear combination of velocity and stress, are determined from the outgoing solid characteristic relation normal to the fluid–solid interface combined with the matching conditions on the velocity and traction. Two alternative forms of the AMP conditions are then derived depending on whether the fluid equations are advanced with a fractional-step method or not. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems; the full nonlinear case being left for future consideration. A normal mode analysis is performed to show that the new AMP algorithm is stable for any ratio of the solid and fluid densities, including the case of very light solids when added-mass effects are large. In contrast, it is shown that a traditional partitioned algorithm involving a Dirichlet–Neumann coupling for the same FSI problem is formally unconditionally unstable for any ratio of densities. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy solids.

An analysis of a new stable partitioned algorithm for FSI problems. Part II: Incompressible flow and structural shells

Stable partitioned algorithms for fluid–structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. The numerical approach described here for structural shells uses Robin (mixed) interface conditions for the pressure and velocity in the fluid which are derived directly from the governing equations. The resulting added-mass partitioned (AMP) algorithm is stable even for very light structures, requires no sub-iterations per time step, and is second-order accurate. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems. A normal mode analysis is performed to show that the new AMP algorithm is stable, even for the case of very light structures when added-mass effects are large. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy structures. A summary comparison of the AMP algorithm developed here and the one in Part I is provided.

An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids

The present paper is concerned with the propagation of Rayleigh waves in a pre-stressed elastic half-space coated with a thin pre-stressed elastic layer. The half-space and the layer are assumed to be compressible and in welded contact with each other. By using the effective boundary condition method, an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function. When the pre-strains are absent, the secular equation obtained coincides with the one for the isotropic case. Numerical investigation shows that the approximate secular equation obtained is a good approximation. Since explicit dispersion relations are employed as theoretical bases for extracting pre-stresses from experimental data, the secular equation obtained will be useful in practical applications.

On formulas for the Rayleigh wave velocity in pre-stressed compressible solids

In this paper, formulas for the velocity of Rayleigh waves in compressible isotropic solids subject to uniform initial deformations are derived using the theory of cubic equation. They are explicit, have simple algebraic forms, and hold for a general strain energy function. Unlike the previous investigations where the derived formulas for Rayleigh wave velocity are approximate and valid for only small enough values of pre-strains, this paper establishes exact formulas for Rayleigh wave velocity being valid for any range of pre-strains. When the prestresses are absent, the obtained formulas recover the Rayleigh wave velocity formula for compressible elastic solids. Since obtained formulas are explicit, exact and hold for any range of pre-strains, they are good tools for evaluating nondestructively prestresses of structures.

An original force–displacement relationship for spherical inclusions in multilayered viscoelastic finite media

This paper presents original solutions of the force–displacement relationships for a rigid spherical bead embedded in a composite medium made of n-isotropic linearly viscoelastic finite layers. Analytical solutions were provided for both compressible and incompressible elastic and viscoelastic solids, assuming no-slip conditions between the rigid spherical inclusion and its adjacent medium as well as between each layer of the composite medium. Thanks to these general formulas, we investigated the effect of finite size media on the force-bead displacement response and derived the exact relationship linking apparent and intrinsic elastic moduli of the medium. Such theoretical solutions can be interestingly applied to identify layer’s heterogeneities and to characterize accurately the mechanical properties of living material like cells when using translational microrheology assays. This point is especially illustrated by modeling animal cell cytoskeleton as a bilayer composite medium probed by magnetic tweezers. Interestingly, our results highlighted the influence of finite cell size effects, while allowing to distinguish viscoelastic properties of deep cell cytoskeleton from those of cellular cortex. Moreover, we established that translational microrheology experiments are well suited to characterize locally the viscoelasticity properties of the layer in contact with the probe as soon this layer thickness is larger than ten bead diameters.

An approach for obtaining approximate formulas for the Rayleigh wave velocity

In this paper, we introduce an approach for finding analytical approximate formulas for the Rayleigh wave velocity for isotropic elastic solids and anisotropic elastic media as well. The approach is based on the least-square principle. To demonstrate its application, we applied it in order to obtain an explanation for Bergmann’s approximation, the earliest known approximation of the Rayleigh wave velocity for isotropic elastic solids, and used it to establish a new approximation. By employing this approach, the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1] were found. By using the best approximate polynomial of the second order of the cubic power, we derived an approximate formula for the Rayleigh wave speed in isotropic elastic solids which is slightly better than the one given recently by Rahman and Michelitsch by employing Lanczos’s approximation. Also by using this second order polynomial, analytical approximate expressions for orthotropic, incompressible and compressible elastic solids were found. For incompressible case, it is shown that the approximation is comparable with Rahman and Michelitsch’s approximation, while for the compressible case, it is shown that our approximate formulas are more accurate than Mozhaev’s ones. Remarkably, by using the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1], we derived an approximate formula of the Rayleigh wave velocity in incompressible monoclinic materials, where the explicit exact formulas of the Rayleigh wave velocity so far are not available.

On Some Finite Deformations of Inhomogeneous Compressible Elastic Solids

We study several simple boundary value problems for a special class of inhomogeneous, isotropie, compressible elastic solids: inflation of a spherical shell; bending, stretching and shearing of a rectangular block; and straightening, stretching and shearing of a sector of a hollow cylinder. The purpose of the study is twofold: to establish new exact solutions to boundary value problems concerning inhomogeneous, isotropie, compressible elastic solids that have technological relevance (to the deformation of layered composites); and to reaffirm a thesis concerning an inherent difficulty with regard to the homogenization techniques that are in vogue that appeal to the stored energy in the homogenized body being the same as that for the inhomogeneous body. In addition to establishing new exact solutions for a class of inhomogeneous elastic solids, the study reinforces the thesis of Saravanan and Rajagopal that great care ought to be exercised in approximating even a very mildly inhomogeneous body by an equivalent homogeneous body when large deformations are involved.

On the equivalence of strong ellipticity in the material and spatial settings of finite elasticity

It is known from Shield’s theorem that the equilibrium equations for the inverse deformation in finite elasticity are equivalent to those for the primary deformation. We review this result for compressible elastic solids and then use it to provide a simple proof that strong ellipticity in the material setting is equivalent to that in the spatial setting. The result is then extended to incompressible materials.

Non-Isochoric Bending and Shearing

Two families of non-homogeneous finite deformations are treated. For each family, the governing system of second-order nonlinear partial differential equations reduces to a system of first-order non-linear ordinary differential equations for homogeneous compressible elastic solids with arbitrary isotropic or anisotropic response.

Tensile instabilities and ellipticity in fiber-reinforced compressible non-linearly elastic solids

In this paper we examine loss of ellipticity and associated failure for fiber-reinforced compressible non-linearly elastic solids under uniaxial plane deformations. We consider first fiber reinforcement that endows the material with additional stiffness only in the fiber direction. It is shown, in particular, that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and may require concavity of the Cauchy stress–stretch curve. Secondly we examine fiber reinforcement that introduces additional stiffness under shear deformations. In this case we find that loss of ellipticity again occurs at a turning point of the nominal stress, in contrast to the situation for incompressible materials.

Remarks on instabilities and ellipticity for a fiber-reinforced compressible nonlinearly elastic solid under plane deformation

In this paper we examine loss of ellipticity and associated failure for fiber-reinforced compressible nonlinearly elastic solids under plane deformation. The analysis concerns a material model that consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. We examine reinforcement that introduces additional stiffness under shear deformations. It is shown that loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. More particularly, under uniaxial loading in the fiber direction loss of ellipticity may be associated with different failure mechanisms. Under compression these include fiber kinking and fiber splitting, while under extension the relevant mechanism is matrix failure.

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.

Loading restrictions for the Blatz–Ko hyperelastic model—application to a finite element analysis

In this paper, we consider the Blatz–Ko constitutive model for compressible elastic solids. It is shown that the Cauchy stress tensor leads to a normal loading limitation. This limitation induces slow convergence of the Newton–Raphson algorithm near the maximum authorized normal loading value and divergence if this value is exceeded. In addition, convergence of the Newton–Raphson scheme also depends on ellipticity and strong ellipticity conditions. These various points are discussed in the case of a rectangular specimen subjected to a tensile load and modeled with finite elements.

Some universal deformations for a class of compressible elastic solids

Two finite deformations that carry a family of coaxial elliptic cylinders into another such family, or a family of concentric ellipsoids into another such family, are shown to be controllable or universal deformations for a class of homogeneous, isotropic, compressible elastic solids.