Plane strain bending of strain-stiffening rubber-like rectangular beams

This paper is concerned with investigation of the effects of strain-stiffening for the classical problem of plane strain bending by an end moment of a rectangular beam composed of an incompressible isotropic nonlinearly elastic material. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress–stretch response, the stresses and resultant moments are obtained explicitly. While such results are well known for classical constitutive models such as the Mooney-Rivlin and neo-Hookean models, our primary focus is on materials that undergo severe strain-stiffening in the stress–stretch response. In particular, we consider in detail two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of bending that beams composed of such materials can sustain is limited by the constraint. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Magnetostrictive composites in the dilute limit

We calculate the effective properties of a magnetostrictive composite in the dilute limit. The composite consists of well separated identical ellipsoidal particles of magnetostrictive material, surrounded by an elastic matrix. The free energy of the magnetostrictive particles is computed using the constrained theory of DeSimone and James [2002. A constrained theory of magnetoelasticity with applications to magnetic shape memory materials. J. Mech. Phys. Solids 50, 283-320], where application of an external field causes rearrangement of variants rather than rotation of the magnetization or elastic strain in a variant. The free energy of the composite has an elastic energy term associated with the deformation of the surrounding matrix and demagnetization terms. By using results from the constrained theory and from the Eshelby inclusion problem in linear elasticity, we show that the energy minimization problem for the composite can be cast as a quadratic programming problem. The solution of the quadratic programming problem yields the effective properties of Ni₂MnGa and Terfenol-D composite systems. Numerical results show that the average strain of the composite depends strongly on the particle shape, the applied stress, and the elastic modulus of the matrix.     

On the overall moduli of non-linear elastic composite materials

From the work of R. Hill on constitutive macro-variables it is known that for an inhomogeneous elastic solid under finite strain an overall elastic constitutive law may be defined. In particular, the volume average of the strain energy of the solid is a function only of the volume-averaged deformation gradient. In view of the importance of this result it is re-derived in this paper as a prelude to a discussion of composite materials. A composite material consisting of a dilute suspension of initially spherical inclusions embedded in a matrix of different material is considered. For second-order isotropic elasticity theory an expression for the overall bulk modulus of the composite material is obtained in terms of the moduli of the constituents. When the inclusions are vacuous a known result for the bulk modulus of porous materials is recovered. In certain situations the strengthening/ weakening effects of the inclusions are less pronounced in the second-order theory than in the linear theory.     

Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation

This paper is concerned with investigation of the effects of strain-stiffening on the classical limit point instability that is well-known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons) and circular cylindrical thin tubes composed of incompressible isotropic non-linearly elastic materials. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the non-monotonic character of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level. It is shown that for materials with sufficiently low extensibility no limit point instability occurs and so stable inflation is then predicted for such materials. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Saint-Venant's principle for linear elastic porous materials

Toupin's version of the Saint-Venant's principle in linear elasticity is generalized to the case of linear elastic porous materials. That is, it is shown that, for a straight prismatic bar made of a linear elastic material with voids and loaded by a self-equilibrated system of forces at one end only, the internal energy stored in the portion of the bar which is beyond a distance s from the loaded end decreases exponentially with the distance s.     

A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies

Strain-gradient elasticity is widely used as a suitable alternative to size-independent classical continuum elasticity to, at least partially, capture elastic size effects at the nanoscale. In this work, borrowing methods from statistical mechanics, we present mathematical derivations that relate the strain-gradient material constants to atomic displacement correlations in a molecular dynamics computational ensemble. Using the developed relations and numerical atomistic calculations, the strain-gradient constants are explicitly determined for some representative semiconductor, metallic, amorphous and polymeric materials. This method has the distinct advantage that amorphous materials can be tackled in a straightforward manner. For crystalline materials we also employ and compare results from both empirical and ab initio based lattice dynamics. Apart from carrying out a systematic tabulation of the relevant material parameters for various materials, we also discuss certain subtleties of strain-gradient elasticity, including: the paradox associated with the sign of the strain-gradient constants, physical reasons for low or high characteristic length scales associated with the strain-gradient constants, and finally the relevance (or the lack thereof) of strain-gradient elasticity for nanotechnologies.     

Material multipole mechanics of elastic dielectric composites

The overall mechanical and electrical behaviors of elastic dielectric composites are investigated with the aid of the concept of material multipoles. In particular, by introducing a statistical continuum material multipole theory, the effects of the electric-elastic interaction and the microstructure (size, shape, orientation,...) of inhomogeneous particles on the overall behaviors of the composites can be obtained. A basic solution for an ellipsoidal elastic inhomogeneity with electric polarization in an infinite elastic dielectric medium is first given, which shows that classical Eshelby ’s elastic solution is modified by the presence of electric-elastic interaction. The overall macroscopic constitutive relations and their overall macroscopic material parameters accounting for electroelastic interaction effect are then derived for the elastic dielectric composites. Some quantitative calculations on the problems with statistical anisotropy, the shape effect and the electric-elastic interaction are finally given for dilute composites.     

Constitutive model for third harmonic generation in elastic solids

In this article, we present a new constitutive model for studying ultrasonic third harmonic generation in elastic solids. The model is hyperelastic in nature with two parameters characterizing the linear elastic material response and two other parameters characterizing the nonlinear response. The limiting response of the model as the nonlinearity parameters tend to zero is shown to be the well-known St Venant–Kirchhoff model. Also, the symmetric response of the model in tension and compression and its role in third harmonic generation is shown. Numerical simulations are carried out to study third harmonic generation in materials characterized by the proposed constitutive model. Predicted third harmonic guided wave generation reveals an increasing third harmonic content with increasing nonlinearity. On the other hand, the second harmonics are independent of the nonlinearity parameters and are generated due to the geometric nonlinearity. The feasibility of determining the nonlinearity parameters from third harmonic measurements is qualitatively discussed.     

Second strain gradient elasticity of nano-objects

Mindlin's second strain gradient continuum theory for isotropic linear elastic materials is used to model two different kinds of size-dependent surface effects observed in the mechanical behaviour of nano-objects. First, the existence of an initial higher order stress represented by Mindlin's cohesion parameter, b₀, makes it possible to account for the relaxation behaviour of traction-free surfaces. Second, the higher order elastic moduli, c_i, coupling the strain tensor and its second gradient are shown to significantly affect the apparent elastic properties of nano-beams and nano-films under uni-axial loading. These two effects are independent from each other and allow for separated identification of the corresponding material parameters. Analytical results are provided for the size-dependent apparent shear modulus of a nano-thin strip under shear. Finite element simulations are then performed to derive the dependence of the apparent Young modulus and Poisson ratio of nano-films with respect to their thickness, and to illustrate hole free surface relaxation in a periodic nano-porous material.     

多层层合板圣维南问题的解析解

将哈密尔顿体系理论引入到多层层合板问题之中，建立了一套求解该问题的横向哈密尔顿算子矩阵的本征函数向量展开解法，并成功地求解出圣维南问题的解析解．进一步显示了弹性力学新求解体系的有效性及其应用潜力     

A kinetic interpretation of the section‐averaged Saint‐Venant system for natural river hydraulics

The classical Saint‐Venant system is well suited for the modeling of dam breaks, hydraulic jumps, reservoir emptying, flooding etc. For many applications, the extension of the Saint‐Venant system to the case of non‐rectangular channels is necessary and this section‐averaged Saint‐Venant system exhibits additional source terms. The main difficulty of these equations consists of the discretization of these source terms. In this paper we propose a kinetic interpretation for the section averaged Saint‐Venant system and derive an associated numerical scheme. The numerical scheme—2nd order in space and time—preserves the positivity of the water height, and is well‐balanced. Numerical results including comparisons with analytic and experimental test problems illustrate the accuracy and the robustness of the numerical algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion

The problem of an anisotropic ellipsoidal inclusion which undergoes a stress-free transformation strain (in the sense of J.D. Eshelby) is considered, and the following theorem is proved: If an ellipsoidal region in an infinite anisotropic linear elastic medium undergoes, in the absence of its surroundings, a stress-free transformation strain which is a polynomial of degree M in the position coordinates x_t, then the final stress and strain state in the transformed inclusion, when constrained by its surroundings, is also a polynomial of degree M in x_t.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

The electro-mechanical response of highly compliant substrates and thin stiff films with periodic cracks

We present results that describe the mechanical response of highly compliant substrates coated with ultra-thin stiff films, with thickness and elastic moduli differences spanning four orders of magnitude. Dimensional analysis based on shear-lag models of cracked films is used to identify key parameters that control the effective elastic properties of the cracked multi-layer, crack opening displacements, and the steady-state energy release rate for channeling crack formation. Analytical forms that describe multi-layer response in terms of film properties and crack spacing are presented and corroborated with numerical models for linear elastic materials. A key result is that the energy release rate scales with 1/(1 − α), where α is one of the Dundurs’ parameters describing elastic mismatch. The results can also be used to evaluate the performance of electrostrictive actuators comprised of cracked blanket electrodes and elastomer dielectrics. In this scenario, an interesting result is that ultra-thin cracked films can continue to distribute charge, since crack openings may be small enough to allow breakdown in air at typical operating voltages.     

The trousers test for tearing of soft biomaterials

The mechanical modeling of rubber-like materials within the framework of nonlinear elasticity theory is well established. The application of such modeling to soft biomaterials is currently the subject of intense investigation. For soft biomaterials it is well known that exponential strain energy density models are particularly useful as they reflect the typical J-shaped stress-stretch stiffening response that is observed experimentally. The most celebrated of these models for isotropic hyperelastic materials are those of Fung and Demiray which depends only on the first strain invariant and its generalization by Vito that depends on both strain invariants. In the limit as the strain-stiffening parameter tends to zero, one recovers the neo-Hookean and Mooney–Rivlin models that are linear functions of the invariants. Here we apply these models to the analysis of the fracture or tearing of soft biomaterials. Attention is focused on a particular fracture test namely the trousers test where two legs of a cut specimen are pulled horizontally apart out of the plane of the test piece. It is shown that, in general, the location of the cut in the specimen plays a key role in the fracture analysis, and that the effect of the cut position depends crucially on the constitutive model employed. This dependence is characterized explicitly for the strain-stiffening exponential constitutive models considered. In contrast to the situation for rubber, our findings show that the critical driving force and fracture toughness in tearing of some soft biomaterials in the trousers test are virtually independent of the cut position.     

Saint-Venant's principle in linear elasticity with microstructure

Toupin's version of the Saint-Venant principle in linear elasticity is generalized to the case of linear elasticity with microstructure. That is, it is shown that, for a straight prismatic bar made of an isotropic linear elastic material with microstructure and loaded by a self-equilibrated force system at one end only, the strain energy stored in the portion of the bar which is beyond a distance s from the loaded end decreases exponentially with the distance s.     

Constitutive modeling and the trousers test for fracture of rubber-like materials

The classical constitutive modeling of incompressible hyperelastic materials such as vulcanized rubber involves strain-energy densities that depend on the first two invariants of the strain tensor. The most well-known of these is the Mooney-Rivlin model and its specialization to the neo-Hookean form. While each of these models accurately predicts the mechanical behavior of rubber at moderate stretches, they fail to reflect the severe strain-stiffening and effects of limiting chain extensibility observed in experiments at large stretch. In recent years, several constitutive models that capture the effects of limiting chain extensibility have been proposed. Here we confine attention to two such phenomenological models. The first, proposed by Gent in 1996, depends only on the first invariant and involves just two material parameters. Its mathematical simplicity has facilitated the analytic solution of a wide variety of basic boundary-value problems. A modification of this model that reflects dependence on the second invariant has been proposed recently by Horgan and Saccomandi. Here we discuss the stress response of the Gent and HS models for some homogeneous deformations and apply the results to the fracture of rubber-like materials. Attention is focused on a particular fracture test, namely the trousers test where two legs of a cut specimen are pulled horizontally apart. It is shown that the cut position plays a key role in the fracture analysis, and that the effect of the cut position depends crucially on the constitutive model employed. For stiff rubber-like or biological materials, it is shown that the influence of the cut position is diminished. In fact, for linearly elastic materials, the critical driving force for fracture is independent of the cut position. It is also shown that the limiting chain extensibility models predict finite fracture toughness as the cut position approaches the edge of the specimen whereas classical hyperelastic models predict unbounded toughness in this limit. The results are relevant to the structural integrity of rubber components such as vibration isolators, vehicle tires, earthquake bearings, seals and flexible joints.     

The validity range of CPT and Mindlin plate theory in comparison with 3-D vibrational analysis of circular plates on the elastic foundation

The validity and the range of applicability of the classical plate theory (CPT) and the first-order shear deformation plate theory, also called Mindlin plate theory (MPT), in comparison with three-dimensional (3-D) p-Ritz solution are presented for freely vibrating circular plates on the elastic foundation with different boundary conditions. In order to achieve this purpose, a study of the 3-D elasticity solution is carried out to determine the free vibration frequencies of clamped, simply supported and free circular plates resting on an elastic foundation. The Pasternak model with adding a shear layer to the Winkler model is used for describing the elastic foundation. In addition to being employed the p-Ritz algorithm, the analysis is based on the linear, small strain and 3-D elasticity theory. In this analysis method, a set of orthogonal polynomial series in a cylindrical polar coordinate system is used to arrive eigenvalue equation yielding the natural frequencies for the circular plates. The accuracy of these results is verified by appropriate convergence studies and checked with the available literature and the MPT. Furthermore, the effect of the foundation stiffness parameters, thickness-radius ratio, and different boundary conditions on the ill-conditioning of the mass matrix as well as on the vibration behavior of the circular plates is investigated. Afterwards, the validity and the range of applicability of the results obtained on the basis of the CPT and MPT for a thin and moderately thick circular plate with different values of the foundation stiffness parameters are graphically presented through comparing them with those obtained by the present 3-D p-Ritz solution. Finally, the phenomenon of mode shape switching is investigated in graphical forms for a wide range of the Winkler foundation stiffness parameters.