Strain-gradient stress analysis in saint-venant bending

Light beam deflections caused by stress or strain gradients are investigated analytically and experimentally in homogeneous beam specimens which are subjected to a particular case of flexure with shear. This study is a generalization of the prior analytical-experimental examination of strain-gradient light deflections produced in stressed plates, which had concentrated on the simplest case where information of interest is collected along a line of symmetry of the stress field. Main purpose of the present investigation is to document the efficacy of the strain-gradient method in analysis of the general case of stress state. The most interesting stress state is that in a beam subjected to the Saint-Venant bending, where the transversal and the longitudinal axes of the beam are in pure shear. The obtained results are compared with the predictions of the developed analytical models and with the predictions of Filon's stress function. The procedures of evaluating the photoelastic and material coefficients using strain-gradient techniques were tested positively. Dedicated to Prof. Dr. Horst Lippmann, Muenchen, on the occasion of the 65th anniversary of his birthday The project was supported by the Natural Sciences and Engineering Research Council of Canada and the NATO Scientific Affairs Division     

Measurement of Orthogonal Stress Gradients Due to Impact Load on a Transparent Sheet using Digital Gradient Sensing Method

A full-field optical method called Digital Gradient Sensing (DGS) for measuring stress gradients due to an impact load on a planar transparent sheet is presented. The technique is based on the elasto-optic effect exhibited by transparent solids due to an imposed stress field causing angular deflections of light rays quantified using 2D digital image correlation method. The measured angular deflections are proportional to the in-plane gradients of stresses under plane stress conditions. The method is relatively simple to implement and is capable of measuring stress gradients in two orthogonal directions simultaneously. The feasibility of this method to study material failure/damage is demonstrated on transparent planar sheets of PMMA subjected to both quasi-static and dynamic line load acting on an edge. In the latter case, ultra high-speed digital photography is used to perform time-resolved measurements. The quasi-static measurements are successfully compared with those based on the Flamant solution for a line-load acting on a half-space in regions where plane stress conditions prevail. The dynamic measurements, prior to material failure, are also successfully compared with finite element computations. The measured stress gradients near the impact point after damage initiation are also presented and failure behavior is discussed.     

Basic theory and experimental techniques of the strain-gradient method

The theories of presently used experimental methods of stress and deformation analysis which employ radiant energy as a detector are based on the assumption that light propagates rectilinearly within both undeformed and deformed bodies which are initially homogeneous and isotropic when diffraction phenomena are negligible. This assumption is not correct: light propagation within deformed bodies is nonrectilinear in a general case. Although this has already been observed and applied practically by some researchers in photoelasticity, it has not so far been generally acknowledged and accepted in experimental mechanics. On the basis of empirical data produced by the authors in the period 1948–1983, we present theories and foundations of the techniques of a new experimental method which is based on the relations between stress/strain gradients and curvatures of light beams. This method is called the strain-gradient method or, less rigorously, gradient photoelasticity.     

Rectilinear and circular analysis of a plane slice optically isolated in a three-dimensional photoelastic model

Because of the coherence of scattered light, it is possible to produce a speckle image from a plane beam of light passing through a transparent model. When two plane parallel beams of light are transmitted through the model the slice between the beams is then optically isolated. The two speckle patterns corresponding to the two beams are superposed and provide optical data relative to the slice (principal stress directions, birefrengence), the data being collected on high contrast photographic plates or by optical filtering to obtain the square of the contrast. The isoclinic and isochromatic fringes are shown to exist. The concepts of rectilinear or circular analysis are extended to the observation of a plane slice in a three-dimensional model without freezing or cutting the model.     

Smooth static and dynamic indentation of a cantilever beam

The static and dynamic indentation of structural elements such as beams and plates continue to be intriguing problems, especially for scenarios where large area contacts are expected to occur. Standard methods of indentation analyses use a beam theory solution to obtain an overall load–displacement relationship and then a Hertzian contact solution to calculate local stresses under the indenter. However, these techniques are only applicable in a fairly limited class of problems: the stress distribution in the contact region will differ significantly from a Hertzian one when the contact length exceeds the thickness of the beam. The indentation models developed herein are improvements over existing GLOBAL/LOCAL models for static and dynamic indentation of cantilever beams. Maximum contact stresses, beam displacements, and contact force time histories are obtained and compared with the predictions of current static and dynamic indentation models. The validity of the solutions presented herein is further assessed by comparing the results obtained to the predictions of modified beam theory solutions.     

Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force

A realistic beam structure often exhibits material and geometrical non-linearity, in particular for those made of metals. The mechanical behaviors of a non-linear functionally graded-material (FGM) cantilever beam subjected to an end force are investigated by using large and small deformation theories. Young's modulus is assumed to be depth-dependent. For an FGM beam of power-law hardening, the location of the neutral axis is determined. The effects of depth-dependent Young's modulus and non-linearity parameter on the deflections and rotations of the FGM beams are analyzed. Our results show that different gradient indexes may change the bending stiffness of the beam so that an FGM beam may bear larger applied load than a homogeneous beam when choosing appropriate gradients. Moreover, the bending stress distribution in an FGM beam is completely different from that in a homogeneous beam. The bending stress arrives at the maximum tensile stress at an internal position rather than at the surface. Obtained results are useful in safety design of linear and non-linear beams.     

An Exact Theory for Circular, End-Loaded, Anisotropic Beams of Narrow Rectangular Cross Section

The exact theory of linearly elastic beams developed by Ladevèze and Ladevèze and Simmonds is limited to prismaticbeams. Herein, the theory is extended to circular beams of narrow rectangular cross section, using the equations of plane stress for an anisotropic elastic body. Explicit formulas are given for the cross-sectional material operators that appear in the overall beamlike stress–strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized strain). The extension to circular beams is non-trivial, especially for full anisotropy, because the analogues of the Saint-Venant solutions, that are key in the exact theory of straight beams, are more complicated.     

Two-dimensional complete rational analysis of functionally graded beams within symplectic framework

Exact solutions for generally supported functionally graded plane beams are given within the framework of symplectic elasticity. The Young’s modulus is assumed to exponentially vary along the longitudinal direction while the Poisson’s ratio remains constant. The state equation with a shift-Hamiltonian operator matrix has been established in the previous work, which is limited to the Saint-Venant solution. Here, a complete rational analysis of the displacement and stress distributions in the beam is presented by exploring the eigensolutions that are usually covered up by the Saint-Venant principle. These solutions play a significant role in the local behavior of materials that is usually ignored in the conventional elasticity methods but possibly crucial to the material/structure failures. The analysis makes full use of the symplectic orthogonality of the eigensolutions. Two illustrative examples are presented to compare the displacement and stress results with those for homogenous materials, demonstrating the effects of material inhomogeneity.     

Effective properties of a nonuniform beam under torsion

Effective characteristics are considered in the pure torsion problem for a nonuniform beam. The Saint-Venant semi-inverse method is used. A torsion stress function is introduced; this function can be found by solving a cross-sectional boundary value problem for a partial differential equation with variable coefficients. Two special boundary value problems are formulated for such an equation; after solving these problems, some effective characteristics are calculated in the case of torsion. It is shown that these effective characteristics satisfy the conditions of symmetry and positive definiteness. The case of an infinite in-plane layer of nonuniform thickness is discussed.     

A reciprocity theorem in linear gradient elasticity and the corresponding Saint-Venant principle

In many practical applications of nanotechnology and in microelectromechanical devices, typical structural components are in the form of beams, plates, shells and membranes. When the scale of such components is very small, the material microstructural lengths become important and strain gradient elasticity can provide useful material modelling. In addition, small scale beams and bars can be used as test specimens for measuring the lengths that enter the constitutive equations of gradient elasticity. It is then useful to be able to apply approximate solutions for the extension, shear and flexure of slender bodies. Such approach requires the existence of some form of the Saint-Venant principle. The present work presents a statement of the Saint-Venant principle in the context of linear strain gradient elasticity. A reciprocity theorem analogous to Betti’s theorem in classic elasticity is provided first, together with necessary restrictions on the constitutive equations and the body forces. It is shown that the order of magnitude of displacements are in accord with the Sternberg’s statement of the Saint-Venant principle. The cases of stretching, shearing and bending of a beam were examined in detail, using two-dimensional finite elements. The numerical examples confirmed the theoretical results.     

考虑剪应变的弹性薄壁杆件的振动理论

对于开口截面彈性薄壁杆件的振动問題,符拉索夫在1940年提出的理論,是基于下列两个基本假設:(1)截面具有剛硬不可变形的周綫;(2)中曲面无剪应变。其后,有些学者也曾提出过相同的理論(例如.R.海里格).由于忽略了中曲面的剪应变,我們可以估計得到,符拉索夫的理論只适用于研究低频率的固有振动和強迫振动,而不适用于研究高频瘁的振动。由于行动載荷和冲击载荷必然引起各个固有频率的振动,因此符拉索夫的理論也不适用于研究这些載荷作用下的振动問題。为了要更好地研究这些問題,必須要考慮剪应变的影响。     

Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force

The asymmetric buckling of a shallow initially curved stress-free micro beam subjected to distributed nonlinear deflection-dependent electrostatic force is studied. In order to highlight the symmetry breaking phenomenon and the approach to its analysis, the subsidiary simplified problem of a curved beam attached to a linearly elastic foundation, and subjected to uniformly distributed “mechanical” load, which is independent of deflections, is addressed first. The analysis is based on a two degrees of freedom reduced order (RO) model resulting from the Galerkin decomposition with linear undamped eigenmodes of a straight beam used as the base functions. Simple approximate expressions are derived defining the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the curve. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict non-symmetric buckling in electrostatically actuated bistable micro beams.     

A new application of differential interferometry for stress analysis

In the past, differential interferometry has found interesting applications in gas dynamics. The gradients of density could be measured in gas flows. Now, a first trial is made to extend this method to the experimental treatment of stress problems. A Wollaston prism with polarizing elements is used in the optical arrangement. This prism combines two beams of light which have penetrated the model at locally separated points. A field of interference fringes can be produced behind the Wollaston prism. The deflections of the different conjugated light beams, which are caused by the deformed elements of the model, lead to a shifting of the interference fringes. A Stress Differential-interferometer Law is derived theoretically in order to interpret the optical data According to this theory, the optical effect caused by the deflection in this arrangement is proportional to the gradient of the sum of principal stresses. A calibration test is performed by using a circular disk, this method is applied to a circular ring for measuring the stress gradients. Under special conditions, interference fringes could be produced which represent the loci of equal stress gradient. Plexiglas plane models are loaded diametrically by single loads. The experimental results verify the statements of the developed theory.     

Invariant solutions of the equations of a multicomponent charged-particle beam

The concept of the invariant-group solution (H-solution) was introduced and a general method for obtaining it was developed in [1–3]. The group properties of the equations of a monoenergetic charged-particle beam with the same value and sign of the specific charge, assuming univalency of the velocity vector V, were studied in [4–6], where all essentially different H-solutions were also constructed. Below, the results of [4–6] are extended to the case of a beam in the presence of a fixed background of density ₀ (§1), and also to the case of multivelocity (V is an s-valued function) and multicomponent beams (i.e., beams formed by particles of several kinds) (§2). A number of analytic solutions that describe some nonstationary processes in devices with plane, cylindrical, and spherical geometry —among them a continuous periodic solution for a plane diode with a period determined by the background density -are obtained in §1. A transformation that contains arbitrary functions of time and preserves Vlasov's equations is given (§2). The equations studied can be treated as the equations of a rarefied plasma in the magnetohydrodynamic approximation, when the pressure gradients are negligible as compared with forces of electromagnetic origin.     

Bending of a poroelastic beam with lateral diffusion

Bending an elastic beam leads to a complicated 3D stress distribution, but the shear and transverse stresses are so small in a slender beam that a good approximation is obtained by assuming purely uniaxial stress. In this paper, we demonstrate that the same is true for a saturated poroelastic beam. Previous studies of poroelastic beams have shown that, to satisfy the Beltrami–Michell compatibility conditions, it is necessary to introduce either a normal transverse stress or shear stresses in addition to the bending stress. The problem is further complicated if lateral diffusion is permitted. In this study, a fully coupled finite element analysis (FEA) incorporating the lateral diffusion effect is presented. Results predicted by the “exact” numerical solution, including load relaxation, pore pressure, stresses and strains, are compared to an approximate analytical solution that incorporates the assumptions of simple beam theory. The applicability of the approximate beam-bending solution is investigated by comparing it to FEA simulations of beams with various aspect ratios. For “beams” with large width-to-height ratios, the Poisson effect causes vertical deflections that cannot be neglected. It is suggested that a theory of plate bending is needed in the case of poroelastic media with large width-to-height ratios. Nevertheless, use of the approximate solution yields very small errors over the range of width-to-height ratios (viz., 1–4) explored with FEA.     

A Bending-Gradient model for thick plates,Part II: Closed-form solutions for cylindrical bending of laminates

In the first part (Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D (Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

Evaluating large plastic strains in shock-loaded beams

Conclusions The simple relations obtained make it possible to satisfactorily evaluate large plastic strains (deflections) of uniform and nonuniform beams with fixed ends in the case where the beam is made of a strain-rate-sensitive material and is subjected to static or purely impulsive transverse loads. With allowance for the method in [2], these relations can be used to determine combinations of the parameters P and I of different forms of shock load corresponding to a given level of large plastic strains in beams.Moscow. Translated from Prikladnaya Mekhanika, Vol. 22, No. 3, pp. 66–71, March, 1986.     

Timoshenko curved beam bending solutions in terms of Euler-Bernoulli solutions

Summary  This paper presents the exact relationships between the deflections and stress resultants of Timoshenko curved beams and that of the corresponding Euler-Bernoulli curved beams. The curved beams considered are of rectangular cross sections and constant radius of curvature. They may have any combinations of classical boundary conditions, and are subjected to any loading distribution that acts normal to the curved beam centreline. These relationships allow engineering designers to directly obtain the bending solutions of Timoshenko curved beams from the familiar Euler-Bernoulli solutions without having to perform the more complicated shear deformation analysis. Accepted for publication 26 July 1996     

Experimental study of oblique circular cylindrical apertures in plates

Ten steel plates weakened by the penetration of an oblique circular cylindrical aperture have been tested. The generator of the aperture makes 0-, 15-, 30- and 45-deg angles with the normal to the plate surface. In the case of the first seven models, the tractions were applied in the direction normal and parallel to the plane of symmetry. The strain distributions around the aperture are presented in nondimensional forms along three generators 45 deg apart. The last three models, with large aperture diameter, were tested to investigate the effect of thickness-to-diameter ratio for the three angles of skewness. The experimental results are compared with the theoretical predictions. It is noted that, for most of the plates, there was a fairly good agreement for the stress distribution throughout the thickness. The results of the experiments for uniaxial state of stress have been superimposed in order to obtain the response for various biaxial conditions.     

A meshless collocation method for the plane problems of functionally graded material beams and plates using the DRK interpolation

A meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the differential reproducing kernel (DRK) interpolation, in which the DRK interpolant is constructed by the randomly distributed nodes. A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach with excellent accuracy and has a fast convergence rate.