Nonlinear Elasticity Theory Research Articles

Nonlinear surface acoustic waves: Theory

A theoretical description is given for the propagation of surface acoustic wave pulses in anisotropic elastic media subject to the influence of nonlinearity. On the basis of nonlinear elasticity theory, an evolution equation is presented for the surface slope or the longitudinal surface velocity associated with an acoustic pulse. It contains a non-local nonlinearity, characterized by a kernel that strongly varies from one propagation geometry to another due to the anisotropy of the substrate. It governs pulse shape evolution in homogeneous halfspaces and the shapes of solitary surface pulses that exist in coated substrates. The theory describing nonlinear Rayleigh-type surface acoustic waves is extended in a straightforward way to surface waves that are localized at a one-dimensional acoustic waveguide like elastic wedges.

Symmetric hyperbolic equations in the nonlinear elasticity theory

The basic results are overviewed concerning the formulation of nonlinear elasticity equations in the form of symmetric hyperbolic systems in long-time studies performed under the direction of the first author. The underlying principles developed in those studies are stated, and some inaccuracies and errors are corrected. Procedures are described for computing the coefficient matrices of the equations from a given equation of state.

Geometrical nonlinear deformation model and its experimental study on bimorph giant magnetostrictive thin film

The geometrical nonlinearity of a giant magnetostrictive thin film (GMF) can be clearly detected under the magnetostriction effect. Thus, using geometrical linear elastic theory to describe the strain, stress, and constitutive relationship of GMF is inaccurate. According to nonlinear elastic theory, a nonlinear deformation model of the bimorph GMF is established based on assumptions that the magnetostriction effect is equivalent to the effect of body force loaded on theGMF.With Taylor seriesmethod, the numerical solution is deduced. Experiments on TbDyFe/Polyimide (PI)/SmFe and TbDyFe/Cu/SmFe are then conducted to verify the proposed model, respectively. Results indicate that the nonlinear deflection curve model is in good conformity with the experimental data.

Analysis of a stress singularity in a non-linear Flamant problem for certain models of a material

A generalized plane problem in the non-linear theory of elasticity is considered for a half-plane loaded on the boundary with a concentrated external force (the non-linear Flamant problem). The properties of the material of the half-plane are described by different (known) models, and each model of the non-linearly elastic material generates its own specific boundary-value problem. Analytical solutions of the problems are obtained for two models of an incompressible material: the neo-Hookean model and the Bartenev–Khazanovich model, and a model of a compressible semi-linear (harmonic) material. The dependence of the stress state as a whole on the adopted model of the material and the effect of the model of the material on the form of the stress singularity in the neighbourhood of a pole are investigated.

The equations of the geometrically non-linear theory of elasticity and momentless shells for arbitrary displacements

To validate earlier results for the case of arbitrary deformations and displacements in orthogonal curvilinear coordinates, kinematic and static relations of the non-linear theory of elasticity are set up which, in the limit of small deformations, lead, unlike the known relations, to correct and consistent relations. The same relations are also constructed for momentless shells of general form for the case of arbitrary displacements and deformations on the basis of which the problem of the static instability of a cylindrical shell with closed ends, made of a linearly elastic material and under conditions of an internal pressure (the problem of the inflation of a cylinder), is considered. It is shown that, in the case of momentless shells, the components of the true sheat stresses are symmetrical, unlike the three-dimensional case. All the above-mentioned relations are constructed for the loading of deformable bodies both by conservative external forces of constant directions and, also, by two types of “following” forces.

Mechanical Models of Artery Walls

The article presents the up to date review and discussion of approaches used to express mechanical behavior of artery walls. The physiology of artery walls and its relation to the models is discussed. Presented models include the simplest 0d and 1d ones but emphasis is put to the most sophisticated approaches which are based on the theory of 3d nonlinear elasticity. Also the alternative approach which consists in simple delinearization of the Koiter shell equations is presented.

Nonlinear elasticity, fluctuations and heterogeneity of nematic elastomers

Liquid crystal elastomers realize a fascinating new form of soft matter that is a composite of a conventional crosslinked polymer gel (rubber) and a liquid crystal. These solid liquid crystal amalgams, quite similarly to their (conventional, fluid) liquid crystal counterparts, can spontaneously partially break translational and/or orientational symmetries, accompanied by novel soft Goldstone modes. As a consequence, these materials can exhibit unconventional elasticity characterized by symmetry-enforced vanishing of some elastic moduli. Thus, a proper description of such solids requires an essential modification of the classical elasticity theory. In this work, we develop a rotationally invariant, nonlinear theory of elasticity for the nematic phase of ideal liquid crystal elastomers. We show that it is characterized by soft modes, corresponding to a combination of long wavelength shear deformations of the solid network and rotations of the nematic director field. We study thermal fluctuations of these soft modes in the presence of network heterogeneities and show that they lead to a large variety of anomalous elastic properties, such as singular length-scale dependent shear elastic moduli, a divergent elastic constant for splay distortion of the nematic director, long-scale incompressibility, universal Poisson ratios and a nonlinear stress–strain relation for arbitrary small strains. These long-scale elastic properties are universal, controlled by a nontrivial zero-temperature fixed point and constitute a qualitative breakdown of the classical elasticity theory in nematic elastomers. Thus, nematic elastomers realize a stable “critical phase”, characterized by universal power-law correlations, akin to a critical point of a continuous phase transition, but extending over an entire phase.

Model of bending of a hydrostatically compressed shell near its stability threshold

A circular cylindrical shell compressed from the outer side of the surface by a high-pressure fluid loses stability of its form at a certain value of the external pressure (“collapse” or nonlinear buckling occurs). As a result, bumps and dents extended along the shell generatrix appear on the shell surface; in the shell cross section, these structures alternate [1]. Deformations of the thin shell remain elastic and, as a whole, are consistent with its geometric bending [2]. Therefore, the initial stage of changes in the shell form can be described within the framework of the nonlinear theory of elasticity [3–5]. Equations of the nonlinear theory of elasticity take into account not only the geometric nonlinearity of the problem caused by nonlinearity of the strain tensor, but also the physical nonlinearity characterizing the material properties and described by the highest invariants of the strain tensor in the expansion of the expression for nonlinearly elastic energy of the medium. Correct allowance of the highest invariants is of principal importance, because the emergence of nonlinearity effects leads to localization of shell bendings. In the final analysis, formation of spatially localized patterns from dents on the shell surface at the initial stage of changes in its form is the result of interaction of nonlinearity and dispersion effects. One important feature of the problem is the absence of dispersion terms in the original equations of the nonlinear theory of elasticity. In simplified models of shells, dispersion terms appear owing to elimination of the “fast” variable characterizing the nonuniformity of strains along the normal to the shell surface. There are some novel methods [6] that allow obtaining simplified models, based on proven equations of the nonlinear theory of elasticity without using ap riorihypotheses and with controlled accuracy in terms of small parameters characterizing the shell size, magnitude of external stress, space and time scales of deformations, and geometric and physical nonlinearity of the problem. Such methods reveal latent dynamic symmetry of the problem; therefore, the simplified equations are universal and close to integrated models, which allows their solutions to be analyzed in detail by methods of the advanced theory of solitons. Nevertheless, the nonlinearly elastic dynamics of shells near their stability thresholds and the possibilities of its approximation by integrated models have not been studied yet. A variant of the reductive perturbation theory suitable for solving nonlinear boundary-value problems, where the final surface of the deformed shell is not known in advance and is found in the course of solving the problem, is proposed in the present paper. The expression for the initial nonlinearly elastic energy of the material is presented in the form of an expansion over all strain-tensor invariants admitted by medium symmetry. The method proposed allows automatic selection of invariants from equations of the nonlinear theory of elasticity and contributions of

Book Review: Larry A. Taber, Nonlinear Theory of Elasticity. Applications in Biomechanics

Book Review: <i>Larry A. Taber, Nonlinear Theory of Elasticity. Applications in Biomechanics</i>

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.

Nonlinear elasticity theory of dislocation formation and composition change in binary alloys in three dimensions

We present a Ginzburg–Landau theory of dislocation dynamics in binary alloys in which the elastic energy is a periodic function of the anisotropic strain components. The composition is twofold coupled to the elastic field via the lattice misfit and via the composition dependence of the elastic moduli (elastic inhomogeneity). We numerically solve the dynamic equations for the lattice displacement and the composition to describe various dislocation processes in three dimensions. On stretching in one-phase states, dislocations proliferate to form a tangle. They tend to be created near pre-existing dislocations. On stretching in two-phase states, dislocations appear in the interface region and glide into the soft region. They are detached from the interface to expand as closed loops, where hard precipitates are acting as dislocation mills. We also follow phase separation around screw dislocations.

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.

THREE-DIMENSIONAL MODELLING FOR REINFORCED CONCRETE BEAMS WITH OPENINGS BASED ON NONLINEAR ELASTIC-DAMAGE THEORY

The objective of this paper is presenting a new three-dimensional nonlinear elastic damage theory for simulating reinforced concrete elements. More attention was devoted for simulating of the nonlinear behaviour of the reinforced concrete beams with opening based on rigorous thermodynamic basis. The new developed theoretical approach takes into consideration the nonlinear elastic behaviour and the deteriorated state of the concrete materials. Finally, the introduced nonlinear concrete model is verified by various numerical simulation of experiments carried out in order to analyze the proposed behaviour of reinforced concrete beams under monotonic static loading. A comparison with experimental results available in literature is performed in order to validate the proposed model. A very good agreement was achieved.

Problems of geometric non-linearity and stability in the mechanics of thin shells and rectilinear columns

An analysis of the current state of the geometrically non-linear theory of elasticity and of thin shells is presented in the case of small deformations but large displacements and rotations, the ratios of which are known as the ratios of the non-linear theory in the quadratic approximation. It is shown that they required specific revision and correction by virtue of the fact that, when they are used in the solution of problems, spurious bifurcation points appear. In view of this, consistent geometrically non-linear equations of the theory of thin shells of the Timoshenko type are constructed in the quadratic approximation which enable one to investigate in a correct formulation both flexural as well as previously unknown non-classical forms of loss of stability (FLS) of thin plates and shells, many of which are encountered in practice, primarily in structures made of composite materials with a low shear stiffness. In the case of rectilinear elastic whereas, which are subjected to the action of conservative external forces and are made of an orthotropic material, the three-dimensional equations of the theory of elasticity are reduced to one-dimensional equations by using the Timoshenko model. Two versions of the latter equations are derived. The first of these corresponds to the use of the consistent version of the three-dimensional, geometrically non-linear relations in an incomplete quadratic approximation and the Timoshenko model without taking account of the transverse stretching deformations, and the second corresponds to the use of the three- dimensional relations in the complete quadratic approximation and the Timoshenko model taking account of the transverse stretching deformations. A series of new non-classical problems of the stability of columns is formulated and their analytical solutions are found using the equations which have been derived with the aim of analyzing their richness of content. Among these are problems concerning the torsional, flexural and shear FLS of a column in the case of a longitudinal axial, bilateral transverse and trilateral compression, a flexural-torsional FLS in the case of pure bending and axial compression together with pure bending and, also, a flexural FLS of a column in the case of torsion and a flexural-torsional FLS under conditions of pure shear. Five FLS of a cylindrical shell under torsion are investigated using the linearized neutral equilibrium equations which have been constructed: 1) a torsional FLS where the solution corresponding to it has a zero variability of the functions in the peripheral direction, 2) a purely beam bending FLS that is possible in the case of long shells and is accompanied by the formation of a single half-wave along the length of the shell while preserving the initial circular form of the cross-section, 3) a classical bending FLS, which is accompanied by the formation of a small number of half-waves in the axial direction and a large number of half-waves in a peripheral direction which is true in the case of long shells, 4) a classical bending FLS which holds in the case of short and medium length shells (the third and fourth types of FLS have already been thoroughly studied in the case of isotropic cylindrical shells), 5) a non-classical FLS characterized by the formation of a large number of shallow depressions in the axial as well as in the peripheral directions; the critical value of the torsional moment corresponding to this FLS is practically independent of the relative thickness of the shell. It is established that the well-known equations of the geometrically non-linear theory of shells, which were formulated for the case of the mean flexure of a shell, do not enable one to reveal the first, second and fifth non-classical FLS.

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves.

Threshold toughness of polymers in the ductile to brittle transition region by different approaches

The fracture behavior of polymers in the ductile-to-brittle region is neither completely brittle nor entirely ductile. Besides, scatter in toughness results impairs the situation. Consequently, conventional methods based exclusively either on linear elastic fracture mechanics theory (LEFM) or on non-linear elastic fracture mechanics theory (NLEFM) are not suitable. It was demonstrated previously, that Weibull statistical method could be successfully used to determine the toughness threshold of polymers displaying ductile-to-brittle behavior. The present study compares the threshold toughness value determined by the statistical approach with other critical values calculated following other different suitable approaches: Low temperature plane strain fracture toughness, Plastic zone corrected fracture toughness, Stable and unstable propagation combined model, J extrapolated at zero stable propagation value, and Quasi J–R curve. The analysis was carried out on data points taken from fracture tests performed on polypropylene homopolymer, PPH, and on a blend of PPH and an elastomeric polyolefin, PPH/POes. The results of this analysis indicate that statistical, stable and unstable propagation combined model, and the J extrapolated at zero stable propagation value methods yield to very similar toughness threshold values being practically equivalent. In this case, threshold value was slightly smaller than the minimum J displayed by the experimental replicas, suggesting that it is an actual representative material toughness. Among these methodologies, the Statistical Method is applicable even if stable crack growth is difficult to determine. On the other hand, the methodologies based on LEFM tended to underestimate the fracture toughness, being very conservative while Quasi J-R curve method based on NLEFM overestimated the PPH/POes toughness value.

Brain tissue deforms similarly to filled elastomers and follows consolidation theory

Slow, large deformations of human brain tissue—accompanying cranial vault deformation induced by positional plagiocephaly, occurring during hydrocephalus, and in the convolutional development—has surprisingly received scarce mechanical investigation. Since the effects of these deformations may be important, we performed a systematic series of in vitro experiments on human brain tissue, revealing the following features. (i) Under uniaxial (quasi-static), cyclic loading, brain tissue exhibits a peculiar nonlinear mechanical behaviour, exhibiting hysteresis, Mullins effect and residual strain, qualitatively similar to that observed in filled elastomers. As a consequence, the loading and unloading uniaxial curves have been found to follow the Ogden nonlinear elastic theory of rubber (and its variants to include Mullins effect and permanent strain). (ii) Loaded up to failure, the “shape” of the stress/strain curve qualitatively changes, evidencing softening related to local failure. (iii) Uniaxial (quasi-static) strain experiments under controlled drainage conditions provide the first direct evidence that the tissue obeys consolidation theory involving fluid migration, with properties similar to fine soils, but having much smaller volumetric compressibility. (iv) Our experimental findings also support the existence of a viscous component of the solid phase deformation. Brain tissue should, therefore, be modelled as a porous, fluid-saturated, nonlinear solid with very small volumetric (drained) compressibility.

Phragmen–Lindelof alternative for the displacement boundary value problem in a theory of non-linear micropolar elasticity

In this note we investigate the spatial behavior of the solutions for the displacement boundary value problem in a theory of non-linear micropolar elasticity. Under suitable conditions on the non-linear terms we obtain estimates for the solutions. The main tool is the energy method.

The non-linear Saint-Venant problem of the torsion, stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.

Solitary and compactlike shear waves in the bulk of solids

We show that a model proposed by Rubin, Rosenau, and Gottlieb [J. Appl. Phys. 77, 4054 (1995)] for dispersion caused by an inherent material characteristic length belongs to the class of simple materials. Therefore, it is possible to generalize the idea of Rubin, Rosenau, and Gottlieb to include a wide range of material models, from nonlinear elasticity to turbulence. Using this insight, we are able to fine-tune nonlinear and dispersive effects in the theory of nonlinear elasticity in order to generate pulse solitary waves and also bulk traveling waves with compact support.