On Lagrange’s History of Mechanics

The birth of Lagranges Mechanics constitutes a milestone in the history of Mechanics. The genius of Lagrange should have been aware of this turning point inasmuch as he devoted a substantial part of his celebrated book to an account of the previous development of Mechanics. This historical part, located before the technical exposition in the Mécanique analytique, is the subject of the present paper. By directly consulting the original writings of the scientists that contributed to the development of Mechanics, Lagrange produced a first hand historical account. It largely influenced the historians of the XIX century and also those of the beginning of XX century. We interpret Lagranges motivations for writing his history and we suggest an interpretation of the foundations of his historical account.     

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

弹性体表面分层压缩失稳问题的数学弹性力学解

采用数学弹性力学的稳定平衡方程并结合富氏积分变换的方法研究了含表面平行裂纹的弹性体在压缩载荷下的表面分层失稳问题。导出了一级显式的精确齐次奇异积分方程组,然后.通过Gauss-Chebyshev积分公式,得到一组齐次代数方程组,从而求出临界压缩载荷。并将结果与经典的材料力学梁板稳定的研究方法所得结果进行了比较,指出经典方法误差太大而不适于求解此问题。最后,利用数学弹性力学解求出的等效弹性支承常数给出一个简单精确的临界压缩载荷计算公式。     

Stress gradient versus strain gradient constitutive models within elasticity

A stress gradient elasticity theory is developed which is based on the Eringen method to address nonlocal elasticity by means of differential equations. By suitable thermodynamics arguments (involving the free enthalpy instead of the free internal energy), the restrictions on the related constitutive equations are determined, which include the well-known Eringen stress gradient constitutive equations, as well as the associated (so far uncertain) boundary conditions. The proposed theory exhibits complementary characters with respect to the analogous strain gradient elasticity theory. The associated boundary-value problem is shown to admit a unique solution characterized by a Hellinger–Reissner type variational principle. The main differences between the Eringen stress gradient model and the concomitant Aifantis strain gradient model are pointed out. A rigorous formulation of the stress gradient Euler–Bernoulli beam is provided; the response of this beam model is discussed as for its sensitivity to the stress gradient effects and compared with the analogous strain gradient beam model.     

Pointwise errors in the classical and in Reissner's linear theory of plates,especially for concentrated loads

An infinite, horizontal, elastically isotropic plate is subjected to a distributed vertical, axisymmetric load, part of which is a body force and part of which is a surface traction. The resulting 3-dimensional stresses and displacements are found with the aid of Love's stress function and Hankel transforms. From these, the sum of the principal stress couples, the average rotation of radial fibers, and the average vertical deflection are computed and compared against the predictions of classical and Reissner's shear-deformation plate theory. Remarkably, the elasticity and plate theory predictions for the stress couples and the rotation agree if Poisson's ratio is zero. In general, for smoothly varying loads, the predictions of Reissner's theory are closer than those of classical theory to the predictions of elasticity theory. However, if a part of the load is (nearly) concentrated, then it is shown that the singularities in the sum of the principal stress couples and in the rotation predicted by Reissner's theory are too strong (because his theory accounts for normal stress effects based on smoothly varying loads). Moreover, if the concentrated part of the external load is a uniformly distributed line load through the thickness, then classical theory predicts the correct singularity in these variables, although with an erroneous strength. On the other hand, Reissner's theory correctly predicts the logarithmic singularity in the average vertical deflection (for any type of concentrated load), although with an erroneous strength.     

Energy theorems and the J-integral in dipolar gradient elasticity

Within the framework of Mindlin’s dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger–Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff–Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved.The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.     

Influence of couple-stresses on stress concentrations around the cavity

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Love solutions in the linear inhomogeneous transversely isotropic theory of elasticity

A general Love solution for the inhomogeneous transversely isotropic theory of elasticity with the elastic constants dependent on the coordinate z is proposed. This result may be considered as a generalization of the Love solutions we recently derived for the inhomogeneous isotropic theory of elasticity. The key steps of deriving the Love solution for the classical linear homogeneous transversely isotropic theory of elasticity are described for further use of the derivation procedure, which is then generalized to the inhomogeneous transversely isotropic case. Some particular cases of inhomogeneity traditionally used in the theory of elasticity are also examined. The significance of the derived solutions and their importance for the modeling of functionally graded materials are briefly discussed     

Foundation of Polar Linear Elasticity for Fibre-Reinforced Materials

This study is motivated by evidence suggesting that the equations of polar elasticity of fibre-reinforced materials are non-elliptic even within the regime of infinitesimal deformations. In its endeavour to resolve this issue, which in symmetric-stress elasticity emerges in the regime of finite deformations only, it lays the foundation for development of a second-gradient theory of linear elasticity. Complete formulation of this new theory is achieved for locally transverse isotropic materials; namely, materials having embedded a single unidirectional family of arbitrarily shaped fibres which are resistant in bending, stretching and twist. The associated analysis shows that, indeed, the obtained Navier-type displacement equations are not elliptic. They accordingly predict that there exist in the material weak discontinuity surfaces, which may indeed be activated within the infinitesimal deformation regime. Surfaces containing the fibres are certainly such surfaces of weak discontinuity; this result may be not irrelevant to numerous practical situations where straight metallic fibres in fibre-reinforced concrete structures emerge partially de-bonded and exposed from their concrete matrix. Nevertheless, the analysis reveals further that additional surfaces of weak discontinuity may well exist in the locally transverse isotropic material of interest. An extension framework is also outlined towards cases of fibrous composites containing two or more families of non-perfectly flexible fibres.     

Sh surface Waves in a Homogeneous Gradient-Elastic Half-Space with Surface Energy

The existence of SH surface waves in a half-space homogeneous material (i.e. anti-plane shear wave motions which decay exponentially with the distance from the free surface) is shown to be possible within the framework of the generalized linear continuum theory of gradient elasticity with surface energy. As is well-known such waves cannot be predicted by the classical theory of linear elasticity for a homogeneous half-space, although there is experimental evidence supporting their existence. Indeed, this is a drawback of the classical theory which is only circumvented by modelling the half-space as a layered structure (Love waves) or as having non-homogeneous material properties. On the contrary, the present study reveals that SH surface waves may exist in a homogeneous half-space if the problem is analyzed by a continuum theory with appropriate microstructure. This theory, which was recently introduced by Vardoulakis and co-workers, assumes a strain-energy density expression containing, besides the classical terms, volume strain-gradient and surface-energy gradient terms. This revised version was published online in August 2006 with corrections to the Cover Date.     

Generalized variational principles on nonlinear theory of elasticity with finite displacements

Two generalized variational principles on nonlinear theory of elasticity with finitedisplacements in which the σ_(ij),e_(i j)and u_i are all three kinds of independent functionsare suggested in this paper.It isproved that these two generalized variational principles areequivalent to each other if the stress-strain relation is satisfied as constraint.Some specialcases,i.e.generalized variational principles on nonlinear theory of elasticity with smalldeformation,on linear theory with finite deformation and on linear theory with smalldeformation together with the corresponding equivalent theorems are also obtained.All ofthem are related to the three kinds of independent variables.     

Critical Phenomena in Cracking of the Interface Between Two Prestressed Materials. 1. Problem Formulation and Basic Relations

Problems are formulated for critical phenomena accompanying the cracking of the interface between two different materials with initial stresses. The basic relations of the three-dimensional linearized dynamic theory of elasticity are used. Complex potentials are applied to a plane problem in the three-dimensional linearized dynamic theory of elasticity.     

弹性力学的一种正交关系

在弹性力学求解新体系中，将对偶向量进行重新排序后，提出了一种新的对偶微分矩阵，对于有一个方向正交的各向异性材料的三维弹性力学问题发现了一种新的正交关系．将材料的正交方向取为z轴，证明了这种正交关系的成立．对于z方向材料正交的各向异性弹性力学问题，新的正交关系包含弹性力学求解新体系提出的正交关系。     

Structure and properties of the solution space of general anisotropic laminates

The algebraic structure of the solution space of all types of anisotropic laminates is determined. The full space is shown to be the direct sum of a number of orthogonal eigenspaces, one for each simple or multiple eigenvalue, whose dimension equals the multiplicity. There are eight different types of eigenvalues, which combine to yield eleven distinct types of laminates with peculiar representations of the general solution. All such representations are explicitly obtained, along with the pseudo-metrics based on the binary product of the eigenvectors. This leads to the projection operators in the solution space, spectral sums and intrinsic tensors analogous to the Stroh–Barnett–Lothe tensors in 2-D elasticity. The present theoretical results are obtained by adopting a mixed formulation involving the deflection function and Airy’s stress function, and by using new laminate elasticity matrices different from the conventional stiffness matrices A, B and D. The new formulation also discloses an isomorphism relating each anisotropic laminate to an image laminate, such that every equilibrium solution of the former directly yields an image solution of the latter by interchanging the kinematical and kinetic variables and the in-plane and out-of-plane variables. This implies, in particular, that the classical bending theory of homogeneous plates and symmetric laminates is not a distinct subject, despite its historical development and pedagogical recognition, but is mathematically identical to the plane stress problem of anisotropic elasticity.     

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.     

Stress distribution instability in the plane problem of the theory of elasticity of heterogeneous bodies

A possibility of degeneration of relationships between stresses and their derivatives with respect to the coordinates in the plane problem of the elasticity theory is considered. For particular dependences of the parameters of elasticity on coordinates, curves are given, for which degeneration conditions are satisfied. It is shown that even a small inhomogeneity of the medium causes stress instability.     

Research on a systematic methodology for theory of elasticity

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薄板理论的正交关系及其变分原理

利用平面弹性与板弯曲的相似性理论，将弹性力学新正交关系中构造对偶向量的思路推广到 各向同性薄板弹性弯曲问题，由混合变量求解法直接得到对偶微分方程并推导了对应的变分 原理. 所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点. 发现了两个独立的、对称 的正交关系，利用薄板弹性弯曲理论的积分形式证明了这种正交关系的成立. 在恰当选择对 偶向量后，弹性力学的新正交关系可以推广到各向同性薄板弹性弯曲理论.     

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.     

Bounds for the torsional rigidity of elastic ring

A torsion problem of the elastic ring is formulated in the framework of the linear theory of elasticity. The meridian section of the ring-like body is bounded by coordinate lines of a plane orthogonal curvilinear coordinate system. The paper concentrates the torsional rigidity of the elastic ring which can be derived from Michell's theory. Upper and lower bound formulas for the torsional rigidity are presented, examples illustrate the application of bounding formulas obtained from two minimum theorems of elasticity. All expositions are based on the usual assumptions of the linear theory of elasticity.