Cauchy Elasticity Research Articles

Tension/torsion of electroactive solid cylinders

In his pioneering work in nonlinear elasticity, Rivlin studied the tension-torsion of a solid isotropic homogeneous elastic cylinder and obtained expressions for the twisting moment as well as the normal force for all bodies belonging to that class in terms of the angle of twist and the stretch. The deformation considered by Rivlin was a universal controllable deformation in that it can be engendered by just the application of the appropriate surface tractions. In this short note we consider the tension-torsion of a solid electroelastic cylinder under the action of an electric field along the axis of the cylinder, and determine expressions for the twisting moment and normal force that depend on the angle of twist, stretch and the applied electric field. We obtain an expression for the torsional rigidity of the electroelastic solid cylinder that depends on the electric field, thereby implying its torsional rigidity can be “tuned” by varying the electrical field. While within the context of classical Cauchy elasticity one observes the compression of the isotropic solid cylinder, namely the POYNTING effect, we show that the cylinder can be made shorter or longer by controlling the electrical field.

Controllable deformations in compressible isotropic implicit elasticity

For a given material, controllable deformations are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, universal deformations are those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\sigma }$$\\end{document} and the left Cauchy-Green strain b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{b}$$\\end{document}, have the implicit form f(σ,b)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\ extsf{f}}(\\varvec{\\sigma },\ extbf{b})=\ extbf{0}$$\\end{document}. We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is constitutively admissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations.

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.

Tetramode Metamaterials as Phonon Polarizers.

In classical Cauchy elasticity, 3D materials exhibit six eigenmodes of deformation. Following the 1995 work of Milton and Cherkaev, extremal elastic materials can be classified by the number of eigenmodes, N, out of these six that are "easy". Using Greek number words, this leads to hexamode (N= 6), pentamode (N= 5), tetramode (N= 4), trimode (N= 3), dimode (N= 2), and monomode (N= 1) materials. While hexamode materials are unstable in all regards, the possibility of pentamode metamaterials ("meta-fluids") has attracted considerable attention throughout the last decade. Here, inspired by the 2021 theoretical work of Wei, Liu, and Hu, microstructured 3D polymer-based tetramode metamaterials are designed and characterized by numerical band-structure calculations, fabricated by laser printing, characterized by ultrasound experiments, and compared to the theoretical ideal. An application in terms of a compact and broadband polarizer for acoustical phonons at ultrasound frequencies is demonstrated.

Bridging scales between solid mechanics and surface chemistry

A continuum mechanics framework is used herein to model the strains induced in a micromechanical structure by surface phenomena such as adsorption. The resulting picture significantly differs from those of a liquid under surface tension. Considering a solid isotropic elastic material, it is shown that a sphere undergoes a non uniform deformation under surface adsorption. The direction of the surface’s displacement is additionally shown to depend on both the material and the sphere’s radius. It is also shown that modeling surface effects with an elastic membrane surrounding a Cauchy elastic material, the elastic energy is usually misestimated. The reported results also reveal that the overall response of a mechanical structure to surface adsorption strongly depends, at a given scaling, of the higher-grade elastic behavior of the material.

4D Pixel Mechanical Metamaterials with Programmable and Reconfigurable Properties

AbstractThe tension‐torsion coupling metamaterial (TTCM) exhibits torsional deformation while stretching, with the degree of freedom exceeding the Cauchy elasticity. However, the TTCM suffers from the limitations of the weak tension‐torsion coupling effect (TTCE), narrow deformation domain, and lack of adaptability. To address these limitations, mechanical pixel (MP) array design, helical microstructure, and 4D printing are introduced into the TTCM in this study, providing new strategies to exploit the deformation potential of the TTCM. The developed MP exhibits the tunability, programmability, and reconfigurability of the mechanical behaviors (stress–strain relationship, TTCE). The pixel mechanical metamaterial (PMM) with uncoupled deformation between MPs is obtained, which greatly enriches the design diversity of configurations and maintainability. The application prospects of the developed PMM for information encryptions, kinematics controllers, soft robots, and buffer devices are demonstrated. More interestingly, an egg falling on the PMM at the height of ≈1 m remains intact due to the excellent protection performance of the PMM.

Implicit nonlinear elastic bodies with density dependent material moduli and its linearization

We develop an implicit constitutive relation to describe the response of a compressible elastic solid, based on physical considerations, that captures all the characteristics exhibited by the popular Blatz–Ko model, but in addition presents some interesting novel features. The fact that the Cauchy stress appears linearly in the implicit constitutive relation between the stress and the left Cauchy–Green strain with the material moduli depending nonlinearly on the deformation gradient, allows us to capture several characteristic features of the response of rubber-like elastic solids. Interestingly, in the nonlinear implicit model that we develop, we find that it is possible to have the normal stress components of the stress influence the shearing motion at second order, when considering weakly nonlinear waves, that only occurs at third order within the case of the classical nonlinear Cauchy elasticity theory. Linearization of the constitutive relation under the assumption of small displacement gradient reduces the constitutive relation to one whose material moduli can depend on the trace of the linearized strain and hence the density in virtue of the balance of mass, such a feature is not possible within the context of the Blatz–Ko constitutive relation, or for that matter any Cauchy elastic body, as linearization leads to the classical linearized elastic constitutive relation that has constant material moduli.

Neuromorphic metamaterial structures

Computerized structural optimization methods are often used to design the shape of structures to achieve a desired function, such as a specific compliance. In the simplest case of a structure built from a material with small deformations obeying Cauchy elasticity, the compliance is constant and the relationship between the forces applied on the structure and its deformation is linear. This linearity severely limits the types of functions which can be achieved by such materials. Here we study mechanical metamaterials made of repeating unit cells, each with specific dimensional parameters and with one-sided contact non-linearities. We show that the force and displacement equilibrium configurations of these metamaterials are mathematically equivalent to the fixed points of a recurrent artificial neural network. By exploiting this equivalence, we demonstrate mechanical metamaterials that can be designed (trained) to implement complex non-linear functions, using a gradient descent algorithm as in artificial neural networks. One of our metamaterial structures has a higher compliance when it is pressed against a pattern of raised bumps corresponding to the vowels in the Braille alphabet, than when it is pressed against patterns for six consonants. As artificial neural networks are known to be efficient models for numerous problems in machine learning, our results reveal that beneficial features of neural networks can be transferred to physical objects (mechanical structures). These features include the design of systems with complex input–output relationships by using generic methods that only rely on the repetitive processing of pairs of inputs and desired outputs, as well as remarkable generalization capabilities. We anticipate our methodology to be a starting point for the transfer of some of the breakthroughs in the rapidly advancing field of machine learning to highly functional physical devices in applications that are constrained by energy, volume, data processing or response time.

On stretch-limited elastic strings

Motivated by the increased interest in modelling non-dissipative materials by constitutive relations more general than those from Cauchy elasticity, we initiate the study of a class of stretch-limited elastic strings : the string cannot be compressed smaller than a certain length less than its natural length nor elongated larger than a certain length greater than its natural length. In particular, we consider equilibrium states for a string suspended between two points under the force of gravity (catenaries). We study the locations of the supports resulting in tensile states containing both extensible and inextensible segments in two situations: the degenerate case when the string is vertical and the non-degenerate case when the supports are at the same height. We then study the existence and multiplicity of equilibrium states in general with multiplicity differing markedly from strings satisfying classical constitutive relations.

Parameterized level set method for structural topology optimization based on the Cosserat elasticity

When describing the mechanical behavior of some engineering materials, such as composites, grains, biological materials and cellular solids, the Cosserat continuum theory has more powerful capabilities compared with the classical Cauchy elasticity since an additional local rotation of point and its counterpart (couple stress) are considered in the Cosserat elasticity to represent the material microscale effects. In this paper, a parameterized level set topology optimization method is developed based on the Cosserat elasticity for the minimum compliance problem of the Cosserat solids. The influence of material characteristic length and Cosserat shear modulus on the optimized structure is investigated in detail. It can be found that the microstructural constants in the Cosserat elasticity have a significant impact on the optimized topology configurations. In addition, the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily. Furthermore, the optimized structure obtained by the developed Cosserat elasticity based parameterized level set method will degenerate to the result by using the classical Cauchy elasticity based parameterized level set method when the Cosserat shear modulus approaches zero. A parameterized level set topology optimization method is developed based on the Cosserat elasticity for the optimization of minimum compliance of the structures with micropolar materials. The influence of characteristic length and Cosserat shear modulus of the micropolar material on the optimized structure has been investigated in detail. It can be found that the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily.

Constitutively optimal governing equations for higher-grade elastic beams

A method is proposed herein to build beam equations for materials featuring higher-grade elasticity. As it is based on the minimization of the constitutive equation gap, static admissibility conditions are taken into account so that it naturally converges to the usual beam equations resulting from Cauchy elasticity when the beam dimensions are large enough. The method is exemplified, for Euler–Bernoulli beams, on first-strain gradient and second-strain gradient elasticity, which yield local and non-local beam equations, respectively. The solutions of these equations are computed for tensile and bending loads over a wide range of beam dimensions in order to assess the role of the different grades on the global behavior of simple mechanical structures. Under first-strain gradient elasticity, the proposed approach extends the validity of the equations obtained previously for tensile or bending loadings. Considering second-strain gradient elasticity, this additionally allows to distinguish two different regimes, depending on whether the elasticity is driven by the surface (it is proposed to denote this regime as the ecto-elastic regime) or by the bulk. It may also model the chemo-mechanical couplings. It finally suggests that second-strain gradient elasticity and first-strain gradient elasticity may affect the bending stiffness in the same dimensional range, so that both should always be considered simultaneously when analyzing experimental results.

Foundations of the Quaternion Quantum Mechanics.

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

Microtwist elasticity: A continuum approach to zero modes and topological polarization in Kagome lattices

The topologically polarized isostatic lattices discovered by Kane and Lubensky (2014, Nat. Phys. 10, 39–45) challenged the standard effective medium theories used in the modeling of many truss-based materials and metamaterials. As a matter of fact, these exhibit Parity (P) asymmetric distributions of zero modes that induce a P-asymmetric elastic behavior, both of which cannot be reproduced within Cauchy elasticity. Here, we propose a new effective medium theory baptized “microtwist elasticity” capable of rendering polarization effects on a macroscopic scale. The theory is valid for trusses on the brink of a polarized-unpolarized phase transition in which case they necessarily exhibit more periodic zero modes than they have dimensions. By mapping each periodic zero mode to a macroscopic degree of freedom, the microtwist theory ends up being a kinematically enriched theory. Microtwist elasticity is constructed thanks to leading order two-scale asymptotics and its constitutive and balance equations are derived for a fairly generic isostatic truss: the Kagome lattice. Various numerical and analytical calculations, of the shape and distribution of zero modes, of dispersion diagrams and of polarization effects, systematically show the quality of the proposed effective medium theory.

Mapping acoustical activity in 3D chiral mechanical metamaterials onto micropolar continuum elasticity

We compare the phonon band structures and chiral phonon eigenmodes of a recently experimentally realized three-dimensional (3D) cubic chiral metamaterial architecture to results from linear micropolar elasticity, an established generalization of classical linear Cauchy elasticity. We achieve very good qualitative agreement concerning the anisotropies of the eigenfrequencies, the anisotropies of the eigenmode properties of the acoustic branches, as well as with respect to the observed pronounced sample-size dependence of acoustical activity and of the static push-to-twist conversion effects. The size dependence of certain properties, that is, the loss of scale invariance, is a fingerprint of micropolar elasticity. We also discuss quantitative shortcomings and conceptual limitations of mapping the properties of finite-size 3D chiral mechanical metamaterials onto micropolar continuum elasticity.

Asymptotic Analysis of a Coupled System of Nonlocal Equations with Oscillatory Coefficients

In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous media that additionally accounts for short-range forces. We consider the vanishing nonlocality limit on the same length scale as the heterogeneity and show that the system's effective behavior is characterized by a coupled system of local equations that are elliptic in the sense of Legendre-Hadamard. This effective system is characterized by a fourth-order tensor that shares properties with Cauchy elasticity tensors that appear in the classical equilibrium equations for linearized elasticity.

Static chiral Willis continuum mechanics for three-dimensional chiral mechanical metamaterials

Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of Cauchy elasticity, the Milton-Briane-Willis equations. We show that in the homogeneous static cubic case only one additional parameter with respect to Cauchy elasticity results, which directly influences chiral effects. We show that the Milton-Briane-Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.

Normal and shear behaviours of the auxetic metamaterials: homogenisation and experimental approaches

The auxetic metamaterials exhibit attractive mechanical properties, including negative Poisson’s ratio and compressional resistance. Although auxetic metamaterials have been extensively investigated using experimental and computational approaches, the consistent estimation of shear properties has still not been clarified. According to Cauchy elasticity, the shear properties of an auxetic structure should be enhanced owing to the negative Poisson’s ratio. However, this study used homogenisation and experimental approaches to demonstrate that the shear elasticity is highly non-linear with respect to the characteristic geometrical parameters of a unit cell and that shear properties, particularly normal elasticity, are not always improved. Furthermore, the estimation of the shear elasticity based on the classical isotropic continuum elasticity relations can result in misleading values and should be avoided.

An experimental evidence of the failure of Cauchy elasticity for the overall modeling of a non-centro-symmetric lattice under static loading

Materials with coarse inner architecture being easily made with modern additive or folding processes, the question of their overall behavior rises. Do they behave like classical elastic continua, or do they exhibit additional higher-order effects ? Further, if present are those effects stable with respect to imperfections (geometry, constitutive material, ...) ? In this view, the current work is an experimental investigation for the need, in static, of a higher-order overall description. It comes from noticing that such behaviors are up to now nearly exclusively studied from a theoretical and numerical point of view. In the present study a non-centro-symmetric sample has been manufactured, based on an industrial honeycomb geometry used for aeronautic/aerospace composite materials. The geometrical anisotropy of the elementary cell and the scale separation ratio have been chosen in order to detect non-classical couplings. Samples are obtained by Fused Deposition Modeling (FDM), one of the most widespread 3D printing techniques. Simple experiments based on load controlled tests with full-field kinematic measurement have been performed. A distributed load control reveals that the overall behavior of the architectured material cannot be described within the realm of Cauchy elasticity.

A note on the linearization of the constitutive relations of non-linear elastic bodies

Within the context of the non-linear theory of Cauchy elastic bodies (hence Green elastic bodies which are a sub-set of Cauchy elastic bodies wherein the stress is derivable from a potential), linearization with regard to the gradient of displacement, in the sense that the squares of the norms of the gradient of displacement can be neglected in comparison tothe norm of the gradient of displacement, leads inexorably to the classical linearized elastic model. It is however common, especially in work related to inelastic bodies, to see expressions for the Cauchy stress as a nonlinear function of the linearized strain. Even though such models are outside the purview of purely elastic response, the nonlinear relationship between the stress and the linearized strain is also often assumed to hold in the elastic range also. While the linearized strain being a nonlinear function of the stress has no basis within the context of the classical Cauchy elasticity theory, we show that a proper justification can be provided for such models within the context of the new class of constitutive relations that have been developed to describe the response of elastic bodies by Rajagopal [19], and these models can be generalized to also describe the inelastic response in the small strain regime.

Identification of higher-order continua equivalent to a Cauchy elastic composite

A heterogeneous Cauchy elastic material may display micromechanical effects that can be modeled in a homogeneous equivalent material through the introduction of higher-order elastic continua. Asymptotic homogenization techniques provide an elegant and rigorous route to the evaluation of equivalent higher-order materials, but are often of difficult and awkward practical implementation. On the other hand, identification techniques, though relying on simplifying assumptions, are of straightforward use. A novel strategy for the identification of equivalent second-gradient Mindlin solids is proposed in an attempt to combine the accuracy of asymptotic techniques with the simplicity of identification approaches. Following the asymptotic homogenization scheme, the overall behaviour is defined via perturbation functions, which (differently from the asymptotic scheme) are evaluated on a finite domain obtained as the periodic repetition of cells and subject to quadratic displacement boundary conditions. As a consequence, the periodicity of the perturbation function is satisfied only in an approximate sense, nevertheless results from the proposed identification algorithm are shown to be reasonably accurate.