Necessary conditions at the boundary for minimizers in finite elasticity

Dedicated to Bernard Coleman on the occasion of his 60^th birthday     

Gradient elasticity and nonstandard boundary conditions

Gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality. The pertinent thermodynamic restrictions upon the gradient constitutive equations are derived, which are shown to include, besides the field (differential) stress–strain laws, a set of nonstandard boundary conditions. Consistently with the latter thermodynamic requirements, a surface layer with membrane stresses is envisioned in the strained body, which together with the above nonstandard boundary conditions make the body constitutively insulated (i.e. no long distance energy flows out of the boundary surface due to nonlocality). The total strain energy is shown to include a bulk and surface strain energy. A minimum total potential energy principle is provided for the related structural boundary-value problem. The Toupin–Mindlin polar-type strain gradient material model is also addressed and compared with the above one, their substantial differences are pointed out, particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations. A gradient one-dimensional bar sample in tension is considered for a few applications of the proposed theory.     

Some research directions in finite elasticity theory

Summary A number of topics in finite elasticity theory which appear to lend themselves to further development were briefly discussed. These include (i) the effect of kinematic constraints which are exactly, or approximately, satisfied; (ii) the mechanics of elastic membranes; (iii) the applicability of results in finite elasticity theory to problems involving stress relaxing materials; (iv) the development of necessary and sufficient conditions for material stability of isotropic elastic materials; (v) the conditions for bifurcation solutions to exist in deformed elastic bodies.

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Zusammenfassung Es werden einige Themen der nichtlinearen Elastizitätstheorie, die sich für weitere Entwicklungen anbieten, kurz besprochen. Darunter sind (i) die Wirkung kinematischer Zwangsbedingungen, die exakt oder näherungsweise erfüllt werden; (ii) die Mechanik elastischer Membranen; (iii) die Anwendbarkeit von Ergebnissen in der nichtlinearen Elastizitätstheorie auf Probleme, die Materialien mit Spannungsrelaxation einschließen; (iv) die Entwicklung notwendiger und hinreichender Bedingungen für materielle Stabilität isotroper elastischer Materialien; (v) die Bedingungen für die Existenz von Verzweigungslösungen in deformierten elastischen Körpern.

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Generalized stroh formalism for anisotropic elasticity for general boundary conditions

The Stroh formalism is most elegant when the boundary conditions are simple, namely, they are prescribed in terms of traction or displacement. For mixed boundary conditions such as there for a slippery boundary, the concise matrix expressions of the Stroh formalism are destroyed. We present a generalized Stroh formalism which is applicable to a class of general boundary conditions. The general boundary conditions include the simple and slippery boundary conditions as special cases. For Green's functions for the half space, the general solution is applicable to the case when the surface of the half-space is a fixed, a free, a slippery, or other more general boundary. For the Griffith crack in the infinite space, the crack can be a slit-like crack with free surfaces, a rigid line inclusion (which is sometimes called an anticrack), or a rigid line with slippery surface or with other general surface conditions. It is worth mention that the modifications required on the Stroh formalism are minor. The generalized formalism and the final solutions look very similar to those of unmodified version. Yet the results are applicable to a rather wide range of boundary conditions.     

Some new aspects of boundary conditions at cracks in piezoelectrics

Fracture mechanical investigations of piezoelectric materials as components of smart structures have become popular in the last 30?years. In the early years of research, boundary conditions at crack faces have been adopted from pure mechanical systems under the assumption that boundaries were traction free. From the electrostatic point of view, cracks have been assumed to be either free of charge or fully permeable. Later, limitedly permeable crack boundary conditions have become popular among the community, nevertheless still assuming traction-free crack faces. Recently, the theoretical framework has been extended to include electrostatically induced mechanical tractions in crack models yielding a significant crack closure effect. However, these models are still simple, neglecting, e.g., the piezoelectric field coupling. In this work, we present an extended model for crack surface tractions yielding some interesting effects. In particular, the orientation of the electrical field with respect to the poling axis becomes important. Furthermore, applying a collinear stress parallel to the crack faces influences the Mode-I stress intensity factor and a Mode-II shear loading couples to the Mode-I SIF.     

基于Hamilton体系的弹性力学问题的比例边界有限元方法

比例边界有限元方法是求解偏微分方程的一种半解析半数值解法。对于弹性力学问题,可采用基于力学相似性、基于比例坐标相似变换的加权余量法和虚功原理得到以位移为未知量的系统控制方程,属于Lagrange体系。但在求解时,又引入了表面力为未知量,控制方程属于Hamilton体系。因而,本文提出在比例边界有限元离散方法的基础上,利...     

Some problems in boundary element programming of axisymmetric elasticity

Axisymmetric problems in elasticity can be reduced to two dimensional ones, but they are a little more complicated than plane problems. Therefore, some special problems will be encountered in the boundary element programming of axisymmetric elasticity. In this paper, the methods to treat these problems and some remarks are given according to our experience in programming. Numerical examples are presented for the checking of these treatments.     

Closed form stress distribution in 2D elasticity for all boundary conditions

This paper applies a Hamiltonian method to study analytically the stress dis- tributions of orthotropic two-dimensional elasticity in(x,z)plane for arbitrary boundary conditions without beam assumptions.It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns.Since coordinates(x,z)can not be easily separated,an alternative symplectic expansion is used. Similar to the Hamiltonian formulation in classical dynamics,we treat the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian ma- trix differential operator.The exponential of the Hamiltonian matrix is symplectic.There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions.The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues(zero eigen-solutions) and that of the well-behaved nonzero eigenvalues(nonzero eigen-solutions).The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions associated with aver- aged global behaviors such as rigid-body translation,rigid-body rotation or bending.On the other hand,the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle.Completed numerical examples are newly given to compare with established results.     

Some refinements concerning the boundary conditions at the macroscopic level

The study of boundary effects initiated in a previous paper is continued. New assumptions regarding the geometrical structure of the boundary surface are introduced. Under these assumptions, it is shown that macroscopic Neumann conditions do not generally affect the determination of the macroscopic field in the case of the transport process considered — heat conduction. For this type of boundary condition, the boundary effect is generally confined within a thin layer near the boundary. When heat sources are taken into account within the porous domain, the result is different. In this case, making use of a Neumann boundary condition, expressed in terms of macroscopic variables, amounts to introducing an extra flux. Under normal circumstances, however, this additional flux is negligible.Roman Letters A cross-sectional area of a unit cell - A _e cross-sectional area of a unit cell at the boundary surface - A _sf interfacial area of the s-f interface contained within the averaging volume - surface area per unit volume (A _sf/ ) - A _sf interfacial area of the s-f interface contained within the macroscopic system - g closure vector - h closure vector - k heat transfer coefficient at the s-f interface - K_eff effective thermal conductivity tensor - _x unit cell length - n unit vector - n_e outwardly directed unit normal vector at the boundary - n_sf outwardly directed unit normal vector for thes-phase at f-s interface - q heat flux density - T ^* macroscopic temperature defined by the macroscopic problem - s closure variable - V volume of the macroscopic system - V boundary surface of the macroscopic domain - V ₁ macroscopic sub-surface of the boundary surface - x local coordinate Greek Letters _s,_f volume fraction - _s, gl_f microscopic thermal conductivities - true microscopic temperature - ^* microscopic temperature corresponding toT ^* - microscopic error temperature - vector defined by Equation (34) - < > spatial average     

Strong ellipticity conditions for the differential operator of the finite isotropic elasticity

Summary TheStrong-Ellipticity conditions of the operator expressing the problem of a state of finite elastic deformation superimposed on a distorted configuration are examined. Sufficient criteria ofLocal Strong-Ellipticity, depending only on the so called Elastic Moduli of the material and on the principal components of stretch in the initial distorted configuration are supplied.

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Sommario Vengono esaminate le condizioni diEllitticità Forte dell'operatore traducente il problema di uno stato di deformazione elastica sovrapposto a una configurazione deformata. In particolare sono dati criteri sufficienti diEllitticità Forte Locale, dipendenti solo dai cosiddetti moduli elastici del materiale e le componenti principali di dilatazione pura nella configurazione iniziale deformata.

     

Analytic methods in problems for finite bodies with improperly mixed boundary conditions

Elastic finite bodies whose surface consists of pieces of coordinate surfaces are considered. The boundary conditions posed on one of such pieces can be different from the boundary conditions on the other pieces. Such problems are said to be improperly mixed. An survey of analytic methods for solving such problems is given.     

Radiation boundary conditions for finite element solutions of generalized wave equations

On the basis of the dispersion relation of the generalized linear wave equation we derive a radiation boundary condition (RBC) that explicitly incorporates the physical parameters of the governing equation into the form of the boundary condition. Using finite element techniques we investigate the properties of the generalized RBC by examining forced and unforced solutions to the telegraph and Klein-Gordon equations in one dimension. The results show that within the limits of the physical parameters of the problem the generalized RBC is an improvement over the Sommerfeld RBC when the governing equation contains additional terms that influence the propagation. These gains are achieved without introducing any computational overhead. A two-dimensional example suggests that the 1D findings can generalize to higher dimensions.     

On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid

For incompressible Navier–Stokes equations in primitive variables, a method of setting absorbing outflow boundary conditions on an artificial boundary is considered. The advection equations used on the outflow boundary are convenient for finite difference (FD) methods, where a weak formulation of a problem is inapplicable. An unsteady viscous incompressible Navier–Stokes flow in a channel with a moving damper is modeled. An accurate comparison and analysis of numerical and mechanical situations are carried out for a variety of boundary conditions and Reynolds numbers. The proposed outflow conditions provide that the problem with Dirichlet boundary conditions should be solved on each time step.     

弹性力学问题的一阶有限元解法

求解弹性力学问题的应力时,如果采用常规的位移有限元法,需要先求得单元的节点位移,再经过求导运算得到。为了解决这种求解方式引起的应力精度下降的问题,提出了弹性力学问题的一阶多变量形式,使得应力与位移精度同阶,并推导了弱形式。采用有限元方法,对弹性力学问题给出了一阶解法的二维、三维数值算例,并且将一阶解法的结果与常规位移有限元法的解进行了比较。数值计算结果表明,一阶解法有效提高了应力的精度,并且应力的误差和节点位移的误差具有相同的收敛阶,验证了本文方法的有效性,为提高有限元法的应力精度提供了新的思路。     

局部有限元模型边界条件的直接传递方法

提出一种将整体分析得到的节点力或节点位移直接传递到精细化局部有限元模型的方法,即部分混合单元法。沿精细化局部有限元模型周边建立一组过渡单元,该组过渡单元采用与整体模型一致的单元类型和模拟方式,其外侧边界上的节点与整体模型节点的相对坐标对应,内侧边界与精细化局部有限元模型采用基于面约束的方式连接。在外侧边界上根据节点坐标对应施加整体分析获得的节点力或节点位移,过渡单元就可直接将边界条件传递到精细化局部有限元模型。通过贵州红水河特大桥钢-混结合段的精细化有限元分析,验证了本文方法的实用性和有效性。     

Energetic bounds in finite elasticity

Summary We study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.     

Inequalities sufficient for material stability in finite elasticity

Summary Some restrictions on the response function of isotropic elastic solids under finite strain are discussed. These restrictions follow when a strong localization of the classical Hadamard condition of infinitesimal stability is adopted as an a priori inequality under certain side conditions which are not fullfilled by all of the possible strained states.

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Sommario Vengono discusse alcune restrizioni sulla funzione di risposta dei solidi elastici isotropi in deformazioni finite. Queste restrizioni sono determinate da una versione locale della classica definizione di stabilità infinitesima di Hadamard, qui considerata come una diseguaglianzaa priori sotto certe condizioni che non sono verificate da tutte le possibili configurazioni deformate.

     

Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory

Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.