Nonlinear Elasticity Theory Research Articles

A constitutive description of the anisotropic response of the fascia lata

In this paper we propose a constitutive model to analyze in-plane extension of goat fascia lata. We first perform a histological analysis of the fascia that shows a well-organized bi-layered arrangement of undulated collagen fascicles oriented along two well defined directions. To develop a model consistent with the tissue structure we identify the absolute and relative thickness of each layer and the orientation of the preferred directions. New data are presented showing the mechanical response in uniaxial and planar biaxial extension. The paper proposes a constitutive relation to describe the mechanical response. We provide a summary of the main ingredients of the nonlinear theory of elasticity and introduce a suitable strain-energy function to describe the anisotropic response of the fascia. We validate the model by showing good fit of the numerical results and the experimental data. Comments are included about differences and analogies between goat fascia lata and the human iliotibial band.

Effective Properties of Composite Materials Strengthened by Orthotropic Fibers with Regard for the Physical Nonlinearity of the Components

UDC 539.3 A model of deformation of composite materials with physically nonlinear components is generalized to the case of materials in the form of a matrix reinforced by unidirectional infinite fibers. It is assumed that the matrix is isotropic and deforms nonlinearly and that the fibers are linearly elastic and have orthotropic symmetry. As initial equations, we use the stochastic differential equations of the physically nonlinear theory of elasticity. We transform these equations into integral equations and apply the method of conditional moments. As a result, the problem is reduced to a system of nonlinear algebraic equations solved by the method of simple iterations. On the basis of the proposed approach, we construct algorithms for the evaluation of the effective deformation properties of the composites with orthotropic components in which nonlinearity is caused by the nonlinear deformation of the matrix. We study the dependences of Poisson’s ratios on macrostrains, the diagrams of macrodefomation for different volume contents of the components, and the parameters characterizing the shape of the inclusions.

Equilibrium of a Thick-Walled Sphere of Inhomogeneous Nonlinear-Elastic Material

Paper considers the problem of nonlinear theory of elasticity for inhomogeneous thick-walled sphere. The problem is solved in centrally symmetric statement. In general, all the parameters of the nonlinear dependence between the intensities of the stresses and strains are functions of the radius. As an example, solve the problem of stress distribution in the soil mass with a spherical cavity.

On Gradient Elasticity and Discrete Peridynamics with Applications to Beams and Plates

In this paper two different nonlinear elasticity theories that account for (a) geometric nonlinearityand (b) microstructure-dependent size effects are revisited to establish the connection betweenthe two theories. The first theory is based on modified couple stress theory of Yang et al. [1]and the second one is based on Srinivasa–Reddy gradient elasticity theory [2]. The modified couplestress theory includes a material length scale parameter that can capture the size effect in a material.The gradient elasticity theory was developed for finitely deforming hyperelastic cosserat continuum,and it is a generalization of small deformation couple stress theories. The Srinivasa–Reddy theorycontains, as a special case, the first one. These two theories are used to derive the governing equationsof beams and plates. In addition, a discrete peridynamics idea as an alternative to the conventionalperidynamics is also presented.

Nonlinear elastic response of cubic crystals to biaxial strain

The biaxial Poissons ratio in the nonlinear elastic theory is an important material parameter. We report here an accurate and efficient means of calculating the nonlinear elastic response of thin, homogeneous films to biaxial strain in arbitrary planes using a continuum-elasticity theory model. The general analytic expressions were derived for the elastic energy and the Poissons ratio under biaxial strain for cubic crystals. The biaxial Poisson’s ratio did not remain constants, but showed a linear relationship with strain when second- and third-order elastic constants are considered. The expressions were verified with simulated biaxial Poisson’s ratio and elastic energy of copper by density-functional theory calculations for three high symmetry planes.

Nonlinear bending and buckling for strain gradient elastic beams

Nonlinear bending of strain gradient elastic thin beams is studied adopting Bernoulli–Euler principle. Simple nonlinear strain gradient elastic theory with surface energy is employed. In fact linear constitutive relations for strain gradient elastic theory with nonlinear strains are adopted. The governing beam equations with its boundary conditions are derived through a variational method. New terms are considered, already introduced for linear cases, indicating the importance of the cross-section area, in addition to moment of inertia in bending of thin beams. Those terms strongly increase the stiffness of the thin beam. The non-linear theory is applied to buckling problems of thin beams, especially in the study of the postbuckling behaviour.

Geometrically nonlinear higher-gradient elasticity with energetic boundaries

The objective of this contribution is to formulate a geometrically nonlinear theory of higher-gradient elasticity accounting for boundary (surface and curve) energies. Surfaces and curves can significantly influence the overall response of a solid body. Such influences are becoming increasingly important when modeling the response of structures at the nanoscale. The behavior of the boundaries is well described by continuum theories that endow the surface and curve with their own energetic structures. Such theories often allow the boundary energy density to depend only on the superficial boundary deformation gradient. From a physical point of view though, it seems necessary to define the boundary deformation gradient as the evaluation of the deformation gradient at the boundary rather than its projection. This controversial issue is carefully studied and several conclusions are extracted from the rigorous mathematical framework presented.In this manuscript the internal energy density of the bulk is a function of the deformation gradient and its first and second derivatives. The internal energy density of the surface is, consequently, a function of the deformation gradient at the surface and its first derivative. The internal energy density of a curve is, consequently, a function of the deformation gradient at the curve.It is shown that in order to have a surface energy depending on the total (surface) deformation gradient, the bulk energy needs to be a function of at least the first derivative of the deformation gradient. Furthermore, in order to have a curve energy depending on the total (curve) deformation gradient, the bulk energy needs to be a function of at least the second derivative of the deformation gradient. Clearly, the surface elasticity theory of Gurtin and Murdoch is intrinsically limited since it is associated with the classical (first-order) continuum theory of elasticity in the bulk. In this sense this contribution shall be also understood as a higher-gradient surface elasticity theory.

A comparative analysis of methods for solving equations in the nonlinear elasticity theory

An approach to solving a variational equation used in geometrically and physically nonlinear problems of deformable body mechanics is considered. This approach is based on the continuation of a solution with respect to the loading parameter. Large systems of nonlinear ordinary differential equations arise in such problems. Usually, these systems are solved by the Euler methods. It is proposed to use the Runge-Kutta and multistep methods and to estimate the total computational cost. A dependence of numerical errors on the number of integration steps is obtained. An optimal method for solving nonlinear problems is chosen on the basis of this dependence.

Elasticity of biopolymer filaments

Within the general one-dimensional theory of nonlinear elasticity we analyze the elasticity of biopolymer filaments. The approach adopted is purely mechanical but is reconciled with statistical physics approaches and allows for a proper formulation of boundary-value problems. By specializing the general framework we obtain force–extension relations for inextensible filaments and show how previous work on the biophysics of filaments fits within the framework. On the other hand, within the same framework, the theory of extensible filaments, which is appropriate for semi-flexible filaments such as F-actin, enables us to fit representative F-actin data. The specific formulas derived are relatively simple and the parameters involved have direct mechanical interpretations and are immediately related to the filament properties, including the initial end-to-end length, contour length and persistence length.

On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth

We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.

Experimental Study on Parameters of Duncan-Chang Model for Cement-Soil

The conventional triaxial tests for cement-soil show stronger soften characteristics of cement-soil under low confining pressure. The soften character is weaken as confining pressure decrease. For adopting Duncan-Chang hyperbola model to obtain parameters of cement-soil, the model cannot include the stress-strain soften characteristics of cement-soil, and must be discussed. The paper studies on parameters of Duncan-Chang model using generalized Hooke law and nonlinear elastic theory. The researches show the parameters of k and n are unrelated to soften characteristics of cement-soil; the relation of Rf and σ3 is given by the soften characteristics, Rf influence the soften process directly. The calculation method could be used in E-u model, and has good practical value.

Tone bursts in exponentially graded materials characterized by parametric plots

The propagation and reflection of the ultrasonic tone burst in the strongly inhomogeneous exponentially graded material are studied. Deformations of a specimen with two parallel boundaries are described by the five constant nonlinear theory of elasticity. The one-dimensional problem is considered. The influence of the variation in material properties on the profile of boundary oscillations is clarified by parametric plots. The obtained results may be useful in the ultrasonic nondestructive material characterization.

A Finite Element Method with Six-Node Isoparametric Element for Nonlinear Analysis of Cable Structures

This paper applied nonlinear elastic theory to the cable structures, presented a finite element method with six-node isoparametric element, used the polynomial of degree five as displacement functions and initial curve of the element. Assuming the cable to be ideal flexible, by means of Lagrangian method, the finite element equations has been derived, so can use this method to compute cable structures.

NUMERICAL RESEARCH OF THE RETAINING CONSTRUCTIONS DURING THE RECONSTRUCTION OF THE TRANSPORT STRUCTURES

The research analysis of interaction artificial retaining constructions with a non-uniform soil basis has been performed during the reconstruction of the transport structures. The mathematical model of soil semi-space research with the use of the nonlinear theory of elasticity is developed. The results of numerous researches of stress-strain state fencing constructions of the overpass in interaction with multilayer heterogeneous half-space and surfaces under the influence of band-pass loading are represented in the current paper.

The Anti-Plane Shear Problem in Nonlinear Elasticity Revisited

A classical problem in the framework of nonlinear elasticity theory is the characterization of the materials that may sustain a pure state of anti-plane shear in the absence of body forces. This problem has been solved by Knowles and by Hill in the framework of isotropic and incompressible elasticity in the seventies. Here we provide a simpler and shorter proof of these classical results. Moreover, we extend these results to nonlinear elastodynamics and we provide some new special solutions.

Dynamic Modeling for a Thin Plate with Large Deformation

In this paper, based on the nonlinear elastic theory, dynamic modeling for a thin plate with large deformation is proposed considering high-order deformation terms using hybrid-coordinate formulation. Through the calculation the potential energy and kinetic energy of the plate, rigid-flexible coupling dynamic equations is established. It is shown that the difference between the deformation results obtained by the present high-order dynamic model and those obtained by one-order approximate dynamic model is not significant in case of large deformation ,both them can calculation large deformation problem. However, the amplitude of vibration obtained by the present high-order dynamicmodel is smaller than that obtained by one-order approximate dynamic model. Furthermore, dynamic stiffening and softening effect of the rigid-flexible coupling system undergoing translational large overall motion is investigated. It is shown that with the reduce of the modulus of elasticity, the influence of the translational acceleration on the vibration frequency of the rigid-flexible coupling system is more significant.

Contact interaction of a circular punch with a preliminarily stressed isotropic layer

UDC 539.3 Using the linearized elasticity theory, we obtained the solution of the contact problem of pressure of a rigid circular punch of complex geometry on a preliminarily stressed isotropic layer. A numerical example of constructing the distribution function of contact stresses is considered. The effects of residual strains in the layer and the shape of a rigid punch on the distribution of contact stresses are analyzed. The enhancement of the reliability and durability of structures and mechanisms is one of the most urgent tasks of modern building and mechanical engineering. As is known [2], the residual deformations are almost always present in elements of structures and articles of machines. The nature of their appearance can be very diverse: irreversible deformations (plasticity, creep), structural transformations in materials, change of the aggregate state in separate places of structures, mechanical, chemical, and technological processes, etc. The stresses arising in these cases, like any other ones, can cause fracture and accelerate certain phase transitions and corrosion. Consideration of residual deformations in the calculations of critical elements of structures, machines, and buildings allows one to estimate more exactly the safety factor of a material and, hence, to decrease essentially its consumption, by preserving the necessary functional characteristics of elements as a whole. For this reason, studies of the contact interaction of elastic bodies with residual deformations are extremely crucial at the present time and will be such in the future. Studies of problems of contact interaction of preliminarily stressed bodies in our country and abroad appeared in significant numbers only at the end of the last century. This is related, in the first turn, to the fact that linear elasticity theory does not consider the presence of residual stresses in bodies. In the general case, the strict statement of such problems requires the application of the apparatus of nonlinear elasticity theory. However, if the initial stresses are sufficiently high, we may restrict ourselves to its linearized version. The present level of the linearized elasticity theory and mathematical methods together with the intense development of computers allow one to efficiently form various calculation models for a wide circle of problems. For example, the apparatus of the linearized elasticity theory was successfully used in works [5, 6] for the construction of a three-dimensional model of bounded elements and for studying the effects of the interaction of fibers during microdeformations in joints strengthened with isotropic and anisotropic fibers. A fairly complete description and the classification of works devoted to the theory of contact interaction of preliminarily stressed bodies with rigid punches can be found in [1]. However, the question on the interaction of ring punches of complicated configurations with the elastic half-space or a layer with residual deformations remains insufficiently studied. Let us consider the axisymmetric problem of the pressing of a rigid ring punch on a preliminarily stressed isotropic layer of thickness h that lies on a rigid absolutely smooth base. The problem will be solved within the framework of the linearized elasticity theory with the use of the terminology and notation of [3]. We assume that the elastic potentials are continuous twice differentiable functions of algebraic invariants of the Green’s tensor of deformations [3].

On the Infusion of a Therapeutic Agent Into a Solid Tumor Modeled as a Poroelastic Medium

The direct infusion of an agent into a solid tumor, modeled as a spherical poroelastic material with anisotropic dependence of the tumor hydraulic conductivity upon the tissue deformation, is treated both by solving the coupled fluid/elastic equations, and by expressing the solution as an asymptotic expansion in terms of a small parameter, ratio between the driving pressure force in the fluid system, and the elastic properties of the solid. Results at order one match almost perfectly the solutions of the full system over a large range of infusion pressures. Comparison with experimental results is acceptable after the hydraulic conductivity of the medium is properly calibrated. Given the uncertain estimates of some model constants, the order zero solution of the expansion, for which fluid and porous matrix are decoupled, yields acceptable values and trends for all the physical fields of interest, rendering the coupled analysis (in the limit of small displacements) of little use. When the deformation of the tissue becomes large nonlinear elasticity theory must be resorted to.

A nonlinear symmetry breaking effect in shear cracks

Shear cracks propagation is a basic dynamical process that mediates interfacial failure. We develop a general weakly nonlinear elastic theory of shear cracks and show that these experience tensile-mode crack tip deformation, including possibly opening displacements, in agreement with Stephenson's prediction. We quantify this nonlinear symmetry breaking effect, under two-dimensional deformation conditions, by an explicit inequality in terms of the first and second order elastic constants in the quasi-static regime and by semi-analytic calculations in the fully dynamic regime. Our general results are applied to various materials. Finally, we discuss related works in the literature and note the potential relevance of elastic nonlinearities for various problem, including frictional sliding.

Koiter’s Shell Theory from the Perspective of Three-dimensional Nonlinear Elasticity

Koiter’s shell model is derived systematically from nonlinear elasticity theory, and shown to furnish the leading-order model for small thickness when the bending and stretching energies are of the same order of magnitude. An extension of Koiter’s model to finite midsurface strain emerges when stretching effects are dominant.