Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

Third-order elastic,piezoelectric, and dielectric constants

The definitions of the third-order elastic, piezoelectric, and dielectric constants and the properties of the associated tensors are discussed. Based on the energy conservation and coordinate transformation, the relations among the third-order constants are obtained. Furthermore, the relations among the third-order elastic, piezoelectric, and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials. It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups, while all third-order dielectric constants are0 for 16 kinds of point groups as well as isotropic material. The reason is that some of the point groups are centrally symmetric, and the other point groups are high symmetry.These results provide the foundation to measure these constants, to choose material, and to research nonlinear problems. Moreover, these results are helpful not only for the study of nonlinear elastic and piezoelectric problems, but also for the research on flexoelectric effects and size effects.     

Evaluation of a simple nonlinear elastic constitutive form

A nonlinear, two constant stress-deformation form is deduced for elastic materials. At very large stretch ratios of greater than about 3 or 4, the model exhibits the strain stiffening behavior common to many elastomers. The constitutive form is very simple since the two material constants enter it as multiplying constants times certain nonlinear deformation terms. The model is evaluated with respect to data upon natural rubber under both uniaxial and bi-axial stress conditions. The model is also used to evaluate data obtained from a nonlinear membrane inflation experiment. The latter experimental capability and corresponding data are new.Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.     

Deformation Characteristics of Laminar Composites under Nonlinear Strains

It is shown that the type of governing relations in a composite can change, namely, a laminar composite formed by layers of physically linear materials under nonlinear strains should be described by nonlinear Hooke's law. Local stresses can be not proportional to elastic constants of the layers under nonlinear strains.     

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.     

A three-dimensional nonlinear model for dissipative response of soft tissue

A set of three-dimensional constitutive equations is proposed for modeling the nonlinear dissipative response of soft tissue. These constitutive equations are phenomenological in nature and they model a number of physical features that have been observed in soft tissue. The equations model the tissue as a composite of a purely elastic component and a dissipative component, both of which experience the same total dilatation and distortion. The stress response of the purely elastic component depends on dilatation, distortion and the stretch of material fibers, whereas the stress response of the dissipative component depends on distortional deformation only. The equations are hyperelastic in the sense that the stress is obtained by derivatives of a strain energy function, and they are properly invariant under superposed rigid body motions. In contrast with standard viscoelastic models of tissues, the proposed constitutive model includes the total deformation rate in evolution equations that can reproduce the observed physical feature that the hysteresis loops of most biological soft tissues are nearly independent of strain rate (Biomechanics, Mechanical Properties of Living Tissues, second ed. (1993)). Material constants are determined which produce good agreement with uniaxial stress experiments on superficial musculoaponeurotic system and facial skin.     

On the Nonlinearly Elastic Equilibrium of a Rectangular Multilayered Plate

A three-dimensional boundary-value problem of physically nonlinear elastic theory is solved for a multilayered plate. The nonlinear relationships between the stresses and small strains are assumed to be of the Kauderer form. The solution under given conditions is constructed as series in powers of a physical dimensionless small parameter. The physically nonlinear boundary-value problem is reduced to a recurrent sequence of linear boundary-value problems. The effect of the physical nonlinearity of the material on the stress–strain state of the plate is studied.     

On Saint-Venant's Principle in three-dimensional nonlinear elasticity

Consider a long elastic isotropic beam with a convex cross-section and a sufficiently smooth boundary. Suppose that a self-equilibrated load is applied at each end but the sides are stress-free and there are no internal body forces. It is proved in the context of three-dimensional, nonlinear elastostatics that if the first four derivatives of the displacement vector are a priori assumed to be everywhere sufficiently small with respect to the physical constants and the geometry of the cross-section, then the strains at any point decay exponentially with the distance of the point from the nearest end.This result is an extension of known results on Saint-Venant's Principle in linear and two-dimensional nonlinear elasticity.     

On non-singular GRADELA crack fields

A brief account is provided on crack-tip solutions that have recently been published in the literature by employing the so-called GRADELA model and its variants. The GRADELA model is a simple gradient elasticity theory involving one internal length in addition to the two Lame' constants, in an effort to eliminate elastic singularities and discontinuities and to interpret elastic size effects. The non-singular strains and non-singular (but sometimes singular or even hypersingular) stresses derived this way under different boundary conditions differ from each other and their physical meaning in not clear. This is discussed which focus on the form and physical meaning of non-singular solutions for crack-tip stresses and strains that are possible to obtain within the GRADELA model and its extensions.     

Approximate Analytical Solution of a Physically Nonlinear Spatial Problem on the Elastic Equilibrium of a Multilayer Rectangular Plate

A three-dimensional boundary-value problem for a multilayer rectangular plate is solved under the physically nonlinear theory of elasticity. Various types of interlayer contact conditions are considered. The nonlinear relation between stresses and small strains is assumed to be of the Kauderer form. The solution is represented by a series in powers of a dimensionless small parameter. The problem posed is reduced to a recurrent sequence of linear boundary-value problems. The stress distribution and the effect of physical nonlinearity on the elastic equilibrium of the plate are studied     

A fully coupled diffusional-mechanical formulation: numerical implementation,analytical validation,and effects of plasticity on equilibrium

A macroscopic coupled stress-diffusion theory which accounts for the effects of nonlinear material behaviour, based on the framework proposed by Cahn and Larché, is presented and implemented numerically into the finite element method. The numerical implementation is validated against analytical solutions for different boundary valued problems. Particular attention is payed to the open system elastic constants, i.e. those derived at constant diffusion potential, since they enable, under circumstances, the equilibrium composition field for any generic chemical-mechanical coupled problem to be obtained through the solution of an equivalent elastic problem. Finally, the effects of plasticity on the overall equilibrium state of the coupled problem solution are discussed.     

Plane nonaxisymmetric deformation of a viscoelastic cylinder with consideration of its physical and geometric nonlinearity

The nonaxisymmetric plane problem of the nonlinear theory of viscoelasticity is solved for a cylinder reinforced by an elastic circular shell. The cylinder has an internal cut resembling a Maltese cross in shape. The identification of the nonlinear endochronous theory of aging viscoelastic materials is conducted by a genetic algorithm method on the basis a nonmonotonic experimental stress-strain dependence. Some numerical results obtained for the stress-strain state of this cylinder under the action of internal pressure are discussed with consideration of the above physical nonlinearity and the finite logarithmic strains.     

Nonlinear primary resonance analysis of nanoshells including vibrational mode interactions based on the surface elasticity theory

The deviation from the classical elastic characteristics induced by the free surface energy can be considerable for nanostructures due to the high surface to volume ratio. Consequently, this type of size dependency should be accounted for in the mechanical behaviors of nanoscale structures. In the current investigation, the influence of free surface energy on the nonlinear primary resonance of silicon nanoshells under soft harmonic external excitation is studied. In order to obtain more accurate...     

The computation of dynamical moduli of solids under pressure

On the basis of the theory of finite strains, expressions are obtained in general form for the effective adiabatic second order elastic constants of crystals of any symmetry in terms of the isothermal elastic constants of second, third, and higher orders in the free energy decomposition. These expressions are used in the case of crystals of cubic symmetry under hydrostatic conditions to find the elastic wave velocities in mono- and polycrystals, and their pressure dependences. The polycrystal was considered as an isotropic body consisting of a large number of cubic monocrystals. The isotropic elastic constants were calculated from theoretical and experimental results for monocrystals in the Voigt-Reuss-Hill approximation. A method of applying this approximation to thermodynamic effective second order elastic constants is proposed. The results of a computation are compared with data of experiments to measure the sound velocity in polycrystalline NaCl and CsCl specimens under pressures to 100 kbar. The results of this comparison are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 162–170, July–August, 1972.     

Determination of hourglass coefficients in the theory of a Cosserat point for nonlinear elastic beams

The theory of a Cosserat point has recently been used [Int. J. Solids Struct. 38 (2001) 4395] to formulate the numerical solution of problems of nonlinear elastic beams. In that theory the constitutive equations for inhomogeneous elastic deformations included undetermined constants associated with hourglass modes which can occur due to nonuniform cross-sectional extension and nonuniform torsion. The objective of this paper is to determine these hourglass coefficients by matching exact solutions of pure bending and pure torsion applied in different directions on each of the surfaces of the element. It is shown that the resulting constitutive equations in the Cosserat theory do not exhibit unphysical stiffness increases due to thinness of the beam, mesh refinement or incompressibility that are present in the associated Bubnov–Galerkin formulation. Also, example problems of a bar hanging under its own weight and a bar attached to a spinning rigid hub are analyzed.     

Independent‐Stress Method for Analysis of Nonlinear Antiplane Strain

The stress field in a cylindrical body under antiplane strains is studied using the nonlinear theory of elasticity in actual variables under assumptions of the absence of body forces and weak nonlinearity of the elastic potential. The stresses are determined by solving the nonlinear boundaryvalue problem for two independent stresses in polar coordinates of the physical and stress planes. Analytical solutions of the nonlinear problems are obtained. The effect of potential nonlinearity is studied. It is shown that the nonlinear problem can be solved using the harmonicequation solution corresponding to the linear potential.     

航空有机玻璃高阶弹性常数超声测定

从有限变形弹性动力学理论出发,认为超声在经历初始静态形变的各向同性弹性体中的传播是叠加在有限变形上的微小扰动,假设应变是应力的三次函数,导出以材料高阶弹性柔度系数为参数的超声纵波传播速度与初始应力关系式,针对YB-3航空有机玻璃试件,通过一组应力-速度测量数据,利用非线性最小二乘法优化方法确定了该材料的前四阶弹性柔度系数,并对实验及模拟计算结果进行了讨论。     

Note on the expansion in powers of Poisson's ratio of solutions in elastostatics

In this paper we examine the power-series development with respect to Poisson's ratio of the solution to the second boundary-value problem (surface tractions prescribed) in three-dimensional classical elastostatics for the case of vanishing body forces. The individual terms of this expansion, by means of steady-state thermoelasticity theory, are found to admit an independent physical interpretation. This interpretation, in turn, permits certain conclusions concerning the dependence upon the elastic constants of solutions to space problems. In particular, it is shown that for a body of arbitrary connectivity all stresses are independent of the elastic constants if and only if the dilatation (and the mean normal stress) is a linear function of the cartesian coordinates. The feasibility of successive approximations of solutions appropriate to sufficiently small values of Poisson's ratio is also considered.The results communicated in this paper were obtained in the course of an investigation conducted under Contract Nonr 562(25) of Brown University with the Office of Naval Research, Washington, D.C.