Local Symmetry Group in the General Theory of Elastic Shells

We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Thermo-Chemo-Electro-Mechanical Formulation of Saturated Charged Porous Solids

A thermo-chemo-electro-mechanical formulation of quasi-static finite deformation of swelling incompressible porous media is derived from a mixture theory including the volume fraction concept. The model consists of an electrically charged porous solid saturated with an ionic solution. Incompressible deformation is assumed. The mixture as a whole is assumed locally electroneutral. Different constituents following different kinematic paths are defined: solid, fluid, anions, cations and neutral solutes. Balance laws are derived for each constituent and for the mixture as a whole. A Lagrangian form of the second law of thermodynamics for incompressible porous media is used to derive the constitutive restrictions of the medium. The material properties are shown to be contained in one strain energy function and a matrix of frictional tensors. A principle of reversibility results from the constitutive restrictions. Existing theories of swelling media should be evaluated with respect to this principle.     

A Micropolar Theory of Porous Media: Constitutive Modelling

The extension of the classical mixture theory by the concept of volume fractions leads to the theory of porous media. In this article, the theory of porous media is generalised to micropolar constituents. The kinematic relations and the balance equations for a porous medium are developed without restricting the number of constituents. Based on the entropy inequality, the general form of the constitutive equations are derived for a binary medium consisting of a porous elastic skeleton saturated by a viscous pore-fluid. Both constituents are assumed to be compressible. Handling the saturation constraint by a Lagrangian multiplier leads to a compatibility of the proposed model to so-called hybrid and incompressible models.     

Stability and finite strain of homogenized structures soft in shear: Sandwich or fiber composites,and layered bodies

The stability theories energetically associated with different finite strain measures are equivalent if the tangential moduli are transformed as a function of the stress. However, for homogenized soft-in-shear composites, they can differ greatly if the material is in small-strain and constant elastic moduli measured in small-strain tests are used. Only one theory can then be correct. The preceding variational energy analysis showed that, for sandwich columns and elastomeric bearings, respectively, the correct theories are Engesser’s and Haringx’s, associated with Green’s and Almansi’s Lagrangian strain tensors, respectively. This analysis is reviewed, along with supporting experimental and numerical results, and is then extended to arbitrary multiaxially loaded homogenized soft-in-shear orthotropic composites. It is found that, to allow the use of constant shear modulus when the material is in small strain, the correct stability theory is associated with a general Doyle–Ericksen finite strain tensor of exponent m depending on the principal stress ratio. Further it is shown that the standard updated Lagrangian algorithm for finite element analysis, which is associated with Green’s Lagrangian finite strain, can give grossly incorrect results for homogenized soft-in-shear structures and needs to be generalized for arbitrary finite strain measure to allow using constant shear modulus for critical loads at small strain.     

Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations

Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced, which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to tensor fields on another configuration.      

The General Mathematical Theory of Plasticity and the Il’yushin Postulates of Macroscopic Definability and Isotropy

The physical laws characterizing the relation between stresses and strains are considered and analyzed in the general modern theory of elastoplastic deformations and in its postulates of macroscopic definability and isotropy for initially isotropic continuous media. The fundamentals of this theory in continuum mechanics were developed by A.A. Il’yushin in the mid-twentieth century. His theory of small elastoplastic deformations under simple loading became a generalization of Hencky’s deformation theory of flow, whereas his theory of elastoplastic processes which are close to simple loading became a generalization of the Saint-Venant–Mises flow theory to the case of hardening media. In these theories, the concepts of simple arid complex loading processes arid the concept of directing form change tensors are introduced; the Bridgman law of volume elastic change and the universal Roche–Eichinger laws of a single hardening curve under simple loading are adopted; and the Odquist hardening for plastic deformations is generalized to the case of elastoplastic hardening media for the processes of almost simple loading without consideration of a specific history of deformations for the trajectories with small arid mean curvatures. In this paper we discuss the possibility of using the isotropy postulate to estimate the effect of forming parameters in the stress-strain state appeared due to the strain-induced anisotropy during the change of the internal structures of materials. We also discuss the possibility of representing the second-rank symmetric stress and strain tensors in the form of vectors in the linear coordinate six-dimensional Euclidean space. An identity principle is proposed for tensors and vectors.     

Nonlinear Eulerian thermoelasticity for anisotropic crystals

A complete continuum thermoelastic theory for large deformation of crystals of arbitrary symmetry is developed. The theory incorporates as a fundamental state variable in the thermodynamic potentials what is termed an Eulerian strain tensor (in material coordinates) constructed from the inverse of the deformation gradient. Thermodynamic identities and relationships among Eulerian and the usual Lagrangian material coefficients are derived, significantly extending previous literature that focused on materials with cubic or hexagonal symmetry and hydrostatic loading conditions. Analytical solutions for homogeneous deformations of ideal cubic crystals are studied over a prescribed range of elastic coefficients; stress states and intrinsic stability measures are compared. For realistic coefficients, Eulerian theory is shown to predict more physically realistic behavior than Lagrangian theory under large compression and shear. Analytical solutions for shock compression of anisotropic single crystals are derived for internal energy functions quartic in Lagrangian or Eulerian strain and linear in entropy; results are analyzed for quartz, sapphire, and diamond. When elastic constants of up to order four are included, both Lagrangian and Eulerian theories are capable of matching Hugoniot data. When only the second-order elastic constant is known, an alternative theory incorporating a mixed Eulerian–Lagrangian strain tensor provides a reasonable approximation of experimental data.     

Tangential continuity of elastic/plastic curvature and strain at interfaces

The continuity vs discontinuity of the elastic/plastic curvature & curvature rate, and strain & strain rate tensors is examined at non-moving surfaces of discontinuity, in the context of a field theory of crystal defects (dislocations and disclinations). Tangential continuity of these tensors derives from the conservation of the Burgers and Frank vectors over patches bridging the interface, in the limit where such patches contract onto the interface. However, normal discontinuity of these tensors remains allowed, and Kirchhoff-like compatibility conditions on their normal discontinuities across the concurring interfaces are derived at multiple junctions. In a simple plane case and in the absence of surface-disclinations, the compatibility of the normal discontinuities in the elastic curvatures assumes the form of a Young’s law between the grain-to-grain disorientations and the sines of the dihedral angles. Complete continuity of the plastic strain rate tensor at triple junctions also derives from the compatibility of the normal discontinuities in the plastic strain rates in such conditions.     

纳米圆轴在弹性介质中的扭转振动分析

基于非局部应变梯度理论,考虑周围弹性介质的影响,研究纳米圆轴的扭转自由振动。首先通过Hamilton原理推导纳米圆轴扭转振动的控制方程及边界条件,接着采用微分求积法得到控制方程及三类边界条件的离散形式,最后由数值计算结果分析扭转振动特性。讨论了两个小尺度参数和弹性介质刚度的变化对振动频率的影响,并通过小尺度参数比对振动频率的影响分析两个尺度参数的耦合作用。研究结果表明,扭转自由振动频率随非局部参数增加而减小,随应变梯度尺度参数、弹性介质刚度增加而增大;当非局部参数大于应变梯度尺度参数时,小尺度效应体现为非局部效应,相反则体现为应变梯度效应。     

Method of integro-differential relations in linear elasticity

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.     

Convex Fung-type potentials for biological tissues

The anisotropic, non-linear elastic behavior of biological soft tissue is typically accounted for by the hypothesis of hyperelasticity, i.e., the existence of an elastic potential. Fung-type potentials, based on the exponential of a quadratic form in the components of the Green-Lagrange strain, have been widely used in soft tissue modeling, and have inspired potentials in which the exponential was replaced by other monotonically increasing functions. It has been shown that simple fitting of the parameters of a Fung-type potential to experimental stress-strain curves may lead to non-convexity, with undesirable effects on the reliability of the algorithms used in Finite Element simulations. In this paper, we prove that the necessary and sufficient condition for the strict convexity of a Fung-type potential is that the quadratic form in the exponential is positive definite. This result provides a clear physical meaning for the parameters featuring in the quadratic form, and their relationship with the small-strain elastic moduli. This consistency relationship must be respected in order to guarantee that the Fung-type potential correctly reduces to the quadratic potential of classic linear elasticity in the small-strain approximation. Furthermore, we show that, when the conditions of convexity and consistency with the linear theory are respected, Fung-type potentials become a one-parameter family, and we discuss the consequences of this result for when fitting experimental data. An erratum to this article can be found at     

Basic Equations of Kelvin's Medium and Analogy with Ferromagnets

A nonlinear elastic Cosserat continuum of special kind whose particles have dynamical spins (Kelvin's medium) is considered. The basic equations for this medium are obtained on the basis of fundamental mechanical laws. Various complete sets of strain tensors are discussed. The analogies between Kelvin's medium and other types of media are discussed. First, we show the analogy between constitutive equations for Kelvin's medium and for elastic shells. The main difference is caused by the dimensionality of the media. Then, for a special case of fast proper rotation we establish the analogy between the dynamic equations of Kelvin's medium and elastic ferromagnetic saturated insulators. The most general way to take into account the coupling between magnetic and elastic subsystems is presented. All results can be interpreted in terms of a mechanical medium as well as in terms of ferromagnets. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the symmetries of 2D elastic and hyperelastic tensors

A thorough investigation is made of the independent point-group symmetries and canonical matrix forms that the 2D elastic and hyperelastic tensors can have. Particular attention is paid to the concepts relevant to the proper definition of the independence of a symmetry from another one. It is shown that the numbers of all independent symmetries for the 2D elastic and hyperelastic tensors are six and four, respectively. In passing, a symmetry result useful for the homogenization theory of 2D linear elastic heterogeneous media is derived.     

An elasto-plastic theory of dislocation and disclination fields

A linear theory of the elasto-plasticity of crystalline solids based on a continuous representation of crystal defects – dislocations and disclinations – is presented. The model accounts for the translational and rotational aspects of lattice incompatibility, respectively associated with the presence of dislocations and disclinations. The defects content relates to the incompatible plastic strain and curvature tensors. The stress state is described by using the conjugate variables to strain and curvature, i.e., the stress and couple-stress tensors. Defect motion is described by two transport equations. A dynamic interplay between dislocations and disclinations results from a disclination-induced source term in the transport of dislocations. Thermodynamic guidance provides the driving forces conjugate to dislocation and disclination velocity in a continuous context, as well as admissible constitutive relations for the latter. When dislocation and disclination velocity vanish, the model reduces to deWit’s elasto-static theory of crystal defects. It also reduces to Acharya’s linear elasto-plastic theory for dislocation fields when the disclination density is ignored. The theory is intended for use in instances where rotational defects matter, such as grain boundaries. To illustrate its applicability, a finite high-angle tilt boundary is modeled using a disclination dipole and its behavior under tensile loading normal to the boundary is shown.     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.     

Dynamics of deformation of an elastic medium with initial stresses

The constitutive equations of motion of an elastic medium with given initial stresses are formulated in the form of a hyperbolic system of first order differential equations. Equations describing the propagation of small perturbations in a prestressed isotropic medium with an arbitrary dependence of the elastic strain energy on the strain tensor are derived, and equations for the quadratic dependence of elastic strain energy on the strain tensor are given.     

Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory

Analytical solutions for bending, buckling, and vibration of micro-sized plates on elastic medium using the modified couple stress theory are presented. The governing equations for bending, buckling and vibration are obtained via Hamilton’s principles in conjunctions with the modified couple stress and Kirchhoff plate theories. The surrounding elastic medium is modeled as the Winkler elastic foundation. Navier’s method is being employed and analytical solutions for the bending, buckling and free vibration problems are obtained. Influences of the elastic medium and the length scale parameter on the bending, buckling, and vibration properties are discussed.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.