Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

In this article we investigate several models contained in the literature in the case of near-incompressibility based on invariants in terms of polyconvexity and coerciveness inequality, which are sufficient to guarantee the existence of a solution. These models are due to Rivlin and Saunders, namely the generalized polynomial-type elasticity, and Arruda and Boyce. The extension to near-incompressibility is usually carried out by an additive decomposition of the strain energy into a volume-changing and a volume-preserving part, where the volume-changing part depends on the determinant of the deformation gradient and the volume-preserving part on the invariants of the unimodular right Cauchy–Green tensor. It will be shown that the Arruda–Boyce model satisfies the polyconvexity condition, whereas the polynomial-type elasticity does not. Therefore, we propose a new class of strain-energy functions depending on invariants. Moreover, we focus our attention on the structure of further isotropic strain-energy functions.     

On crack-tip strain energy release rate in non-principal directions of elasticity for simple layer plate of composite materials

In this paper,the fracture problem in non-principal directions of elasticity for a simple layer plate of linear-elastic orthotropic composite materials is studied.The formulae of transformation between characteristic roots,coefficients of elastic compliances in non-principal directions of elasticity and corresponding parameters in principal directions of elasticity are derived.Then,the computing formulae of strain energy release rate under skew-symmetric loading in terms of engineering parameters for principal directions of elasticity are obtained by substituting crack-tip stresses and displacements into the basic formula of the strain energy release rate.     

Curvature-dependent surface energy and implications for nanostructures

At small length scales, several size-effects in both physical phenomena and properties can be rationalized by invoking the concept of surface energy. Conventional theoretical frameworks of surface energy, in both the mechanics and physics communities, assume curvature independence. In this work we adopt a simplified and linearized version of a theory proposed by Steigmann–Ogden to capture curvature-dependence of surface energy. Connecting the theory to atomistic calculations and the solution to an illustrative paradigmatical problem of a bent cantilever beam, we catalog the influence of curvature-dependence of surface energy on the effective elastic modulus of nanostructures. The observation in atomistic calculations that the elastic modulus of bent nanostructures is dramatically different than under tension – sometimes softer, sometimes stiffer – has been a source of puzzlement to the scientific community. We show that the corrected surface mechanics framework provides a resolution to this issue. Finally, we propose an unambiguous definition of the thickness of a crystalline surface.     

Universal solutions and energy minimization in the saint-venant cylinder

In this paper we study the role played, from an energy view-point, by the explicit solution of Saint-Venant for a right circular cylinder. By taking into account a harmonic solution with constant divergence, we consider Maisonneuve's solution and investigate the assumptions under which the explicit solutions form a particular case.

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Sommario In questo lavoro si esamina, da un punto di vista energetico, il ruolo che le soluzioni esplicite di Saint-Venant hanno all'interno dell'intera classe di soluzioni equipollenti per un cilindro circolare retto. A tale scopo facendo riferimento alla soluzione ottenuta da Maisonneuve, che minimizza l'energia di deformazione nell'intera classe delle soluzioni equipollenti, si vuole studiare se esiste una intersezione non nulla tra la classe delle soluzioni di Maisonneuve e la classe delle soluzioni armoniche a divergenza costante.

     

Problems in determining the elastic strain energy function for rubber

Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress-strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress-strain relations by Treloar [7] [L.R.G. Treloar, Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59-70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285-1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251-288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress-strain relations is adduced or the applicability of a strain energy function is demonstrated.     

Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints

The purpose of this work is to provide a theoretical analysis of the mechanical behavior of the growth of soft materials under geometrical constraints. In particular, we focus on the swelling of a gel layer clamped to a substrate, which is still the subject of many experimental tests. Because the constrained swelling process induces compressive stresses, all these experiments exhibit surface instabilities, which ultimately lead to cusp formation. Our model is based on fixing a neo-Hookean constitutive energy together with the incompressibility requirement for a volumetric, homogeneous mass addition. Our approach is developed mostly, but not uniquely, in the plane strain configuration. We show how the standard equilibrium equations from continuum mechanics have a similarity with the two-dimensional Stokes flows, and we use a nonlinear stream function for the exact treatment of the incompressibility constraint. A free energy approach allows the extension both to arbitrary hyperelastic strain energies and to additional interactions, such as surface energies. We find that, at constant volumetric growth, the threshold for a wavy instability is completely governed by the amount of growth. Nevertheless, the determination of the wavelength at threshold, which scales with the initial thickness of the gel layer, requires the coupling with a surface effect. Our findings, which are valid in proximity of the threshold, are compared to experimental results. The proposed treatment can be extended to weakly nonlinearities within the aim of the theory of bifurcations.     

基于应变能法的单搭接螺栓剪切模型

为了简化复杂结构在冲击数值分析中的大量螺栓连接,可用等效的载荷位移模型代替复杂的螺栓连接关系,本文中针对单搭接螺栓连接在剪切载荷下建立了连接本构关系。首先通过对有预紧力的单搭接螺栓进行实验和精细有限元模拟,揭示了螺栓剪切载荷位移曲线的特征并针对不同特征阶段进行了相应的物理机理分析。在此基础上对于载荷位移曲线的界面黏结、部分滑移、整体滑移阶段提出了连接本构模型的基本形式和各阶段的参数估算方法。在部分滑移阶段考虑了4个方面的刚度贡献,其中部件对螺栓的支撑刚度是三维非轴对称变形问题,理论求解非常困难,本文中通过应力分布研究,采用应变能法解决了螺栓的支撑刚度的估算问题。提出的单搭接螺栓剪切模型物理含义明确,参数估算简单,准确度高。     

Stress multiaxiality factor for crack growth prediction using the strain energy density theory

The multiaxiality factor defined as the ratio of the von-Misses equivalent stress to the volumetric stress has been reported to be related to the initiation and progression of failure in structures. It is demonstrated in the present paper that the location around the crack tip where the multiaxiality factor obtains minimum value is an indicator of the direction of minimum material fracture resistance for crack propagation. It is also proposed that the location along the direction of crack propagation path where multiaxiality factor obtains minimum value is considered as the critical distance away from the crack tip, where the strain energy density should be evaluated and compared to its critical value. Theoretical predictions correlate well with the test results for the investigated cases.     

Generating functions for volume-preserving transformations

A general implicit solution for determining volume-preserving transformations in the n-dimensional Euclidean space is obtained in terms of a set of 2n generating functions in mixed coordinates. For n=2, the proposed representation corresponds to the classical definition of a potential stream function in a canonical transformation. For n=3, the given solution defines a more general class of isochoric transformations, when compared to existing methods based on multiple potentials. Illustrative examples are discussed both in rectangular and in cylindrical coordinates for applications in mechanical problems of incompressible continua. Solving exactly the incompressibility constraint, the proposed representation method is suitable for determining three-dimensional isochoric perturbations to be used in bifurcation theory. Applications in non-linear elasticity are envisaged for determining the occurrence of complex instability patterns for soft elastic materials.     

高应力岩体爆破卸荷过程中应变率及应变能特征

针对高应力岩体爆破开挖卸载问题,自制了一台轴向加、卸载实验测试平台,通过实验测试获得了爆破卸荷过程中岩杆的动态应变及应变率数据。实测数据表明：开挖面附近岩体的爆破加、卸载以及初始应力卸载应变率均在10⁻¹ s⁻¹量级以上,验证了高地应力区岩体爆破开挖卸荷是一动态过程。建立了初始应力卸载一维力学模型,揭示了卸载波的传播机制;通过分析爆破卸荷过程应变能密度的时空分布特征,建立了应变能密度与各阶段应变率变化规律的联系。结合实测数据,采用隐式-显式顺序求解方法,进一步分析了高应力区岩体爆破卸荷荷载各阶段应变率沿岩杆的变化规律。结果表明：爆破加载阶段的平均应变率沿杆件逐渐衰减,且衰减速度逐渐减小;爆破卸阶段平均应变率沿杆件也呈衰减趋势;而初始应力的应变能稳定释放,其平均应变率无衰减趋势。

     

Exact stationary probability density functions for non-linear systems under Poisson white noise excitation

The paper presents exact stationary probability density functions for systems under Poisson white noise excitation. Two different solution methods are outlined. In the first one, a class of non-linear systems is determined whose state vector is a memoryless transformation of the state vector of a linear system. The second method considers the generalized Fokker-Planck (Kolmogorov-forward) equation. Non-linear system functions are identified such that the stationary solution of the system admits a prescribed stationary probability density function. Both methods make use of the stochastic integro-differential equations approach. This approach seems to have some computational advantages for the determination of exact stationary probability density functions when compared to the stochastic differential equations approach.     

The continuum elastic and atomistic viewpoints on the formation volume and strain energy of a point defect

We discuss the roles of continuum linear elasticity and atomistic calculations in determining the formation volume and the strain energy of formation of a point defect in a crystal. Our considerations bear special relevance to defect formation under stress. The elasticity treatment is based on the Green's function solution for a center of contraction or expansion in an anisotropic solid. It makes possible the precise definition of a formation volume tensor and leads to an extension of Eshelby's [Proc. R. Soc. London Ser. A 241 (1226), 376] result for the work done by an external stress during the transformation of a continuum inclusion. Parameters necessary for a complete continuum calculation of elastic fields around a point defect are obtained by comparing with an atomistic solution in the far field. However, an elasticity result makes it possible to test the validity of the formation volume that is obtained via atomistic calculations under various boundary conditions. It also yields the correction term for formation volume calculated under these boundary conditions. Using two types of boundary conditions commonly employed in atomistic calculations, a comparison is also made of the strain energies of formation predicted by continuum elasticity and atomistic calculations. The limitations of the continuum linear elastic treatment are revealed by comparing with atomistic calculations of the formation volume and strain energies of small crystals enclosing point defects.     

Inspection of free energy functions in gradient crystal plasticity

The dislocation density tensor computed as the cud of plastic distortion is regarded as a new constitutive variable in crystal plasticity. The dependence of the free energy function on the dislocation density tensor is explored starting from a quadratic ansatz. Rank one and logarithmic dependencies are then envisaged based on considerations from the statistical theory of dislocations. The rele- vance of the presented free energy potentials is evaluated from the corresponding analytical solutions of the periodic two-phase laminate problem under shear where one layer is a single crystal material undergoing single slip and the second one remains purely elastic.     

Complementary energy release rates and dual conservation integrals in micropolar elasticity

The complementary energy momentum tensor, expressed in terms of the spatial gradients of stress and couple-stress, is used to construct the and conservation integrals of infinitesimal micropolar elasticity. The derived integrals are related to the release rates of the complementary potential energy associated with a defect translation or rotation. A nonconserved integral is also derived and related to the energy release rate that is associated with a self-similar cavity expansion. The results are compared to those obtained on the basis of the classical energy momentum tensor, expressed in terms of the spatial gradients of displacement and rotation, and the release rates of the potential energy. It is shown that the evaluation of the complementary conservation integrals is of similar complexity to that of the classical conservation integrals, so that either can be effectively used in the energetic analysis of the mechanics of defects. The two-dimensional versions of the dual conservation integrals are then derived and applied to an out-of-plane shearing of a long cracked slab.     

J^＊ INTEGRAL OF THE SPECIFIC DEVIATOR STRAIN ENERGY AND ITS APPLICATION

First the deviator strain energy is introduced, then the problem of plane-crack critical growth was discussed, a path independent line integral J^* was defined, furthermore its conservation was proved strictly. As application examples, Mode-Ⅰ stress intensity factors of cracked beams were obtained with present approach. The results are shown to agree well with those available in the open literature.     

Comparison of energy release rate and minimum strain energy density for potential failure of composite with bi-material interface

An analytical method is developed to describe the fields of stress and displacement in a bi-material strip specimen with an edge interfacial crack. All of the basic governing equations, boundary conditions on crack surfaces and conditions of continuity along the interface are satisfied by the eigenfunction expansion method. The other boundary conditions are satisfied by the generalized variational principle. The stress intensity factors are calculated for determining the energy release rate and minimum strain energy density factor S_min that is used the strain energy density criterion. Problems with oscillatory singularity and contact zone are discussed. Not only the effects of bi-material modulus ratio, thickness ratio, Poisson's ratio and crack length to S_min, but also the influences of bi-material modulus ratio, thickness ratio to phase angle are presented. Among these parameters, particular situations where S_min become jeopardously high and lead to failure are discussed.     

Research for the strain energy release rate of complex cracks by using point-by-point closed extrapolation approach

IntroductionTheresearchforcalculatingstrainenergyreleaserateisaveryimportanttaskforsolvingfracturemechanicsproblems.Inthepreviouscalculationforenergyreleaserate,externalforceworkwassubtractedbystrainenergy ,whichisthemethodoffreeenergy[1],thenodalforcesandnodaldisplacementsbetweentopandbottomsurfacesofcrackwasalsousedtocalculateenergyreleaserate[2 - 5 ].Theyarebothapproximatecalculations.Theformeriscomplex ,althoughtherearesomeadvisablefeaturesindefinition ,andawholecalculationforstrainenergy …