Nonlinear Elasticity Theory Research Articles

Nonlinear large deformation of acoustomechanical soft materials

An acoustomechanical theory of soft materials is proposed to account for the nonlinear large deformation of soft materials triggered by both the ultrasound waves and mechanical forces. This theory is formulated by employing the nonlinear elasticity theory of the soft material and the theory of acoustic radiation force, which takes into consideration of the combination loading of mechanical forces and acoustic inputs. While the propagation of acoustic wave depends on material configuration, the radiation force generated by wave propagation deforms the material configuration. This complex interaction reaches a steady state when the mechanical stress and the acoustic radiation stress are able to balance with the elastic deformation stress. The acoustomechanical theory is employed to characterize the acoustomechanical behaviors of thin soft material layers under different boundary conditions (e.g., equal-biaxial forces, uniaxial force, and uniaxial constraint). Prestretches arising from these boundary conditions are shown to play significant roles in affecting the acoustomechanical repsonse of soft material: the same material actuated from different prestretches and boundary conditions exhibits different stretch-stress relations. This novel functionality enables innovative design of acoustic sensors and actuators based on soft materials.

Methodical fitting for mathematical models of rubber-like materials

A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.

Thermal Elastic Constitutive Equation of Orthotropic Materials

In the finite deformation range, the numbers of orthotropic 2n order elastic constants are studied on the basis of tensor function and of its representation theorem. On the basis of elastic constant research, the elastic orthotropic constitutive equation is derived by using the tensor method. Based on orthotropic elastic constitutive equations an in-depth study on the constitutive theory of orthotropic nonlinear thermal elasticity is carried out, and by considering the deformation produced by the coupling of temperature and load, nonlinear orthotropic thermoelastic constitutive equation is further derived with representation of the tensor invariant and scalar invariant. The constitutive equations could be used very convenient to the application in reality.

Application of the theory of reversible discontinuities to the investigation of equations describing waves in tubes with elastic walls

Abstract Equations which describe the propagation of waves in tubes with elastic walls are investigated, methods for calculating them are developed, and solutions containing reversible discontinuity structures are analysed in the case of a fluid-filled tube. The model of a tube with elastic walls constructed on the basis of the complete model of a membrane and the non-linear theory of elasticity is generalized. The viscosity and compressibility of the material, the possibility of filling the tube with a gas, and the flexural rigidity of the tube walls are taken into account. The problem of the decay of an arbitrary discontinuity is solved numerically in the case of a fluid-filled tube. The results obtained correspond to the previously developed theory of reversible discontinuities. Simplified hyperbolic equations of long waves, as well as equations for small-amplitude waves which do not take into account longitudinal elastic waves and are similar to the Boussinesq equations, are derived for cases when a tube is filled with a liquid and a gas. The possibility of overturning of the waves is analysed. A procedure for correcting the numerical schemes by adding terms with high-order derivatives to the equations is developed, and the order of approximation of the numerical scheme remains unchanged, enabling the performance of calculations with low schematic dissipation.

A Hyperelastic Two-Scale Optimization Model for Shape Matching

We suggest a novel shape matching algorithm for three-dimensional surface meshes of disk or sphere topology. The method is based on the physical theory of nonlinear elasticity and can hence handle large rotations and deformations. Deformation boundary conditions that supplement the underlying equations are usually unknown. Given an initial guess, these are optimized such that the mechanical boundary forces that are responsible for the deformation are of a simple nature. We show a heuristic way to approximate the nonlinear optimization problem by a sequence of convex problems using finite elements. The deformation cost, i.e., the forces, is measured on a coarse scale, while ICP-like matching is done on the fine scale. We demonstrate the plausibility of our algorithm through examples taken from different datasets.

A new hypothesis for foregut and heart tube formation based on differential growth and actomyosin contraction.

For decades, it was commonly thought that the bilateral heart fields in the early embryo fold directly towards the midline, where they meet and fuse to create the primitive heart tube. Recent studies have challenged this view, however, suggesting that the heart fields fold diagonally. As early foregut and heart tube morphogenesis are intimately related, this finding also raises questions concerning the traditional view of foregut formation. Here, we combine experiments on chick embryos with computational modeling to explore a new hypothesis for the physical mechanisms of heart tube and foregut formation. According to our hypothesis, differential anisotropic growth between mesoderm and endoderm drives diagonal folding. Then, active contraction along the anterior intestinal portal generates tension to elongate the foregut and heart tube. We test this hypothesis using biochemical perturbations of cell proliferation and contractility, as well as computational modeling based on nonlinear elasticity theory including growth and contraction. The present results generally support the view that differential growth and actomyosin contraction drive formation of the foregut and heart tube in the early chick embryo.

Effect of pressurization on helical guided wave energy velocity in fluid-filled pipes

The effect of pressurization stresses on helical guided waves in a thin-walled fluid-filled pipe is studied by modeling leaky Lamb waves in a stressed plate bordered by fluid. Fluid pressurization produces hoop and longitudinal stresses in a thin-walled pipe, which corresponds to biaxial in-plane stress in a plate waveguide model. The effect of stress on guided wave propagation is accounted for through nonlinear elasticity and finite deformation theory. Emphasis is placed on the stress dependence of the energy velocity of the guided wave modes. For this purpose, an expression for the energy velocity of leaky Lamb waves in a stressed plate is derived. Theoretical results are presented for the mode, frequency, and directional dependent variations in energy velocity with respect to stress. An experimental setup is designed for measuring variations in helical wave energy velocity in a thin-walled water-filled steel pipe at different levels of pressure. Good agreement is achieved between the experimental variations in energy velocity for the helical guided waves and the theoretical leaky Lamb wave solutions.

Nonlinear optical vibrations of single-walled carbon nanotubes

We demonstrate the new specific phenomenon of the long-time resonant energy exchange in the carbon nanotubes (CNTs) in the two optical branches - the Circumferential Flexure Mode (CFM) and Radial Btreathing Mode (RBM). It is shown that the modified nonlinear Schrödinger equation, obtained in the framework of nonlinear elastic thin shell theory, allows to describe the CNT nonlinear dynamics connected with considered frequency bands. Comparative analysis of the oscillations of the CFM and RBM branches shows the principal difference between nonlinearity effects. If the nonlinear resonant interaction of the low-frequency modes in the CFM branch leads to the energy capture in the some domain of the CNT, the same interaction in the RBM branch does not appear any tendency to the energy localization. The reason of such a distinction is the difference of the non-linear terms in the equations of motion. If the CFMs are specified by the soft power nonlinearity, the RBM dynamics is determined by the hard gradient nonlinearity. Moreover, in contrast to CFM the importance of nonlinearity in the case of RBM oscillations decreases with increasing of the length to radius ratio. The numerical integration of the thin shell theory equations confirms the results of the analytical study.

Bifurcation of pressurized functionally graded elastomeric hollow cylinders

Various bifurcation behaviors, including asymmetric axial buckling induced by end thrust, bifurcation under external pressure, and eversion without any loading, of a soft circular hollow cylinder (or cylindrical shell) composed of functionally graded incompressible elastomeric material are considered in a unified way. Based on the nonlinear elasticity theory of a deformable continuous body undergoing large deformation and the linearized incremental theory for a superimposed infinitesimal deformation, an efficient approach for buckling analysis is developed and applied to the cylinder subjected to a combination of axial end thrust and internal/external pressure. It is based on the state-space formalism, which naturally avoids the derivatives of instantaneous material constants, enabling a convenient and accurate numerical implementation. Along with the layerwise method, an analytical characteristic equation (i.e. the bifurcation criterion) governing the general asymmetric buckling of the cylinder is derived. Detailed parametric studies are then carried out for a pressurized soft functionally graded hollow cylinder using an incompressible Mooney-Rivlin material model. The effects of material gradient on bifurcation induced either by end thrust or external pressure are discussed through numerical examples. Not only does this study provide an efficient tool for buckling analysis of soft cylinders, some findings that are unique to a functionally graded material are also highlighted. All in all, it is found highly possible to tune the bifurcation behavior of soft elastomeric cylindrical shells by tailoring material composition and/or adjusting the pressures acting on the surfaces of the cylinder.

Effect of radial reaction force on the bending of circular plates resting on a ring support

In the classical analysis of bending of a circular plate resting on simple support when subjected to transverse loading, the effect of the radial component of the reaction force is often neglected. This paper analyzes the effect of the radial component of the reaction force on the bending of thin circular plates using the Kirchhoff plate theory. A nonclassical axisymmetric bending problem is studied. By solving the governing equation based on linearization of nonlinear theory of elasticity, two typical cases including uniformly distributed loading over a plate surface and a concentrated force at the plate center are considered. Expressions for the large- and small-scale deflection and rotation are obtained and a non-linear load-deflection relation is given. When neglecting the radial reaction force, a linearized model is reduced and its solution coincides with that of classical thin circular plates. The deflection and load-deflection response curve are graphically presented for centrally-loaded and uniformly-loaded circular plates, respectively. A comparison of the deflections with and without the radial reaction force is made. Obtained results are useful in safety design of linear and non-linear plates under complicated loading.

Higher-order elastic constants and megabar pressure effects of bcc tungsten:Ab initiocalculations

The general method for the calculation of $n\mathrm{th}$ ($n\ensuremath{\ge}2$) order elastic constants of the loaded crystal is given in the framework of the nonlinear elasticity theory. For the crystals of cubic symmetry under hydrostatic compression, the two schemes of calculation of the elastic constants of second, third, and fourth order from energy--finite strain relations and stress--finite strain relations are implemented. Both techniques are applied for the calculation of elastic constants of orders from second to fourth to the bcc phase of tungsten at a 0--600 GPa pressure range. The energy and stress at the various pressures and deformations are obtained ab initio in the framework of projector augmented wave+generalized gradient approximation (PAW+GGA) method, as implemented in Vienna Ab initio Simulation Package (VASP) code. Using the obtained results, we found the pressure dependence of Gr\uneisen parameters for long-wave acoustic modes in this interval. The Lam\'e constants of second and third order were estimated for polycrystalline tungsten. The proposed method is applicable for crystals with arbitrary symmetry.

Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy–Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.

Stochastic hyperelastic constitutive laws and identification procedure for soft biological tissues with intrinsic variability

In this work, we address the constitutive modeling, in a probabilistic framework, of the hyperelastic response of soft biological tissues. The aim is on the one hand to mimic the mean behavior and variability that are typically encountered in the experimental characterization of such materials, and on the other hand to derive mathematical models that are almost surely consistent with the theory of nonlinear elasticity. Towards this goal, we invoke information theory and discuss a stochastic model relying on a low-dimensional parametrization. We subsequently propose a two-step methodology allowing for the calibration of the model using standard data, such as mean and standard deviation values along a given loading path. The framework is finally applied and benchmarked on three experimental databases proposed elsewhere in the literature. It is shown that the stochastic model allows experiments to be accurately reproduced, regardless of the tissue under consideration.

Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics

We provide an easy approach to the geodesic distance on the general linear group $GL(n)$ for left-invariant Riemannian metrics which are also right-$O(n)$-invariant. The parameterization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated.

Effect of uniaxial stress on the propagation of higher-order Lamb wave modes

On the basis of the non-linear theory of elasticity and the invariant based formulation developed by Ogden, we analyse the effect of homogeneous stress on the propagation of Lamb waves. Using the theory of incremental deformations superimposed on large deformations, we derive the equations governing the propagation of small amplitude waves in a pre-stressed plate. By enforcing traction-free boundary conditions at the surfaces of the plate, we further obtain the characteristic equations for symmetric and anti-symmetric Lamb wave modes and investigate the effect of stress on the phase velocity, i.e. the acoustoelastic effect. A comparison with experimental data exhibits a better correlation than previously published results. The outcomes of this study can be utilised in the development of new techniques for the measurement of applied stresses based on the acoustoelastic effect. In particular, a strong sensitivity of the phase velocity to the applied stress near the cut-off frequencies of higher-order Lamb wave modes is a very promising option, which seems to have been overlooked in previous studies.

Beyond polyconvexity: an existence result for a class of quasiconvex hyperelastic materials

This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non‐interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split‐quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.

Shear strength features of unsaturated clayey sand by lab test

Abstract To study the shear feature of the clayey sand subgrade, the equations of soil–water characteristic curve with different net mean stresses were established based on laboratory test and non-linear elastic theory. The laboratory result shows that the net mean stress has significantly affected on the soil–water characteristic curve shape. The greater the net mean stress, the smaller moisture content of soil. The shear features of clayey sand were also figured out using triaxial equipment with mini air and water pressure sensors in non-drained and non-exhaust state. The results of research indicate that the net confining pressure and moisture content have significant effect on the strength, deformation, and yield feature of unsaturated clayey sand. The shear strength increases with the growth of net confining pressure in case of constant moisture content. The cohesion and friction angle of soil become larger with the increase of moisture content less than optimum content; while the water content is greater than the optimum water content, they show the opposite trend. The unsaturated clayey sand shows obvious dilatancy deformation in shearing. In the phase of shear contraction, the shear stress increases with the growth of suction and net confining pressure, which declines after it reach the peak point. Results of this research will be a benefit for the engineer.

Refined geometrically nonlinear equations of motion for elongated rod-type plate

We derive new refined geometrically nonlinear equations of motion for elongated rod-type plates. They are based on the proposed earlier relationships of geometrically nonlinear theory of elasticity in the case of small deformations and refined S. P. Timoshenko’s shear model. These equations allow to describe the high-frequency torsional oscillation of elongated rod-type plate formed in them when plate performs low-frequency flexural vibrations. By limit transition to the classical model of rod theory we carry out transformation of derived equations to simplified system of equations of lower degree.

Large deformation analysis and stability analysis of a cylindrical rubber tube under internal pressure

Rubber tubes under pressure can undergo large deformations and exhibit a particular nonlinear elastic behavior. In order to reveal mechanical properties of rubber tubes subjected to internal pressure, large deformation analysis and stability analysis have been proposed in this paper by utilizing a modified Gent’s strain energy function. Based on the nonlinear elastic theory, by establishing the theoretical model of a rubber tube under internal pressure, the relationship between the internal pressure and circumferential principal stretch has been deduced. Meanwhile stability analysis of the rubber tube has also been proposed and the relationship between the internal pressure and the internal volume ratio has been achieved. The effects on the deformation by different parameters and the failure reasons of the rubber tube have been discussed, which provided a reasonable reference for the design of rubber tubes.

Acoustomechanical constitutive theory for soft materials

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by combining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momentum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and analyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.