Saint-Venant End Effects Research Articles

Saint-Venant end effects in piezoceramic materials

The stress decay rate and characteristic decay length for anisotropic materials and fiber composites have been intensively investigated. However, those for piezoelectric materials have not been studied. In this paper, we examine the Saint-Venant end effects of piezoceramic materials by considering the problem of a semi-infinite piezoceramic strip polarized in the thickness direction and with applied voltages on the upper and lower surfaces. It is assumed that the gradient of electric potential in the axial direction is much smaller than that in the thickness direction. Thus, the governing equations in terms of the Airy stress function and electric potential function can be uncoupled. The governing equation in terms of the Airy stress function involves only two non-dimensional parameters after non-dimensionalization. Finally, the stress decay rates and characteristic decay lengths for a variety of piezoceramic materials are computed numerically, and their variation with the two non-dimensional parameters is presented.

End Effects for Anti-Plane Shear Deformations of Periodically Laminated STrips with Imperfect Bonding

The axial decay of Saint-Venant end effects is investigated for anti-plane shear deformations of semi-infinite generally laminated anisotropic strips. Imperfect bonding conditions are imposed at the interfaces. The analytical approach, using a displacement field which decays exponentially in the axial direction, gives rise to a transcendental equation for the real eigenvalues. The decay rate for the stresses is given in terms of the smallest positive eigenvalue. Laminated strips with periodic layout are then considered. In the presence of imperfect bonding, the effective shear elastic moduli, computed through a homogenization method, depend on the total number of slipping interfaces in the laminate. Numerical examples confirm that the decay lengths computed with effective shear moduli represent the asymptotic values (for an increasing number of layers) for those of periodically laminated strips.

End Effects in Anti-plane Shear for an Inhomogeneous Isotropic Linearly Elastic Semi-infinite Strip

The purpose of this research is to further investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This is carried out within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. In previous work [1], the elastic coefficients were assumed to be smooth functions of the transverse coordinate so that the material was inhomogeneous in the lateral direction only. Here we develop a new technique, based on a change of variable, to study generally inhomogeneous isotropic materials. The governing partial differential equation is transformed to a Helmholtz equation with a variable coefficient, which facilitates analysis of the influence of material inhomogeneity on the diffusion of end effects. For certain classes of inhomogeneous materials, an explicit optimal decay estimate is established. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials.

End effects in multilayered orthotropic strips with imperfect bonding

Abstract The axial decay of Saint-Venant end effects is investigated for plane strain deformations of semi-infinite generally laminated orthotropic strips, subject to self-equilibrated end loading. General imperfect bonding conditions are imposed at the interfaces. The analytical approach, using an Airy stress function which decays exponentially in the axial direction, gives rise to a transcendental equation for the (generally complex) eigenvalues. The decay rate for the stresses is given in terms of the eigenvalue of smallest positive real part. The decay rates are examined analytically and numerically in a number of special cases. These include a bimaterial orthotropic strip and the special subcases of a symmetric single lap joint made of the same material, symmetric orthotropic sandwich strips and the classical symmetric double lap joint. Asymptotic estimates for the decay rates, in the case of large slip, are obtained which predict extremely slow decay of end effects.

Saint-Venant decay rates for an isotropic inhomogeneous linearly elastic solid in anti-plane shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials.

Further analysis of end effects for plane deformations of sandwich strips

Saint-Venant end effects for plane deformations of sandwich strips are investigated. For two phase composite strips, with homogeneous isotropic layers that are subjected to self-equilibrated end loads, the exponential decay of end effects is characterized in terms of eigenvalues analogous to the Fadle-Papkovich eigenvalues for the homogeneous strip. An earlier study of this problem was carried out by Choi and Horgan (1978, Saint-Venant end effects for plane deformation of sandwich strips. Int. J. Solids Structures14, 187–195). Here we simplify and extend this analysis by using the results of Dundurs on reduced dependence on the elastic constants. The decay rate is shown to depend quadratically on the two Dundurs' constants α,β. Numerical results yield graphs for the decay rate in the α,β parallelogram, which are immediately applicable for design purposes. For any combination of materials in the two-phase sandwich strip, these plots allow one to determine the Saint-Venant decay length. For the case of a relatively compliant inner core, an asymptotic result shows that the characteristic decay length is much larger than that for a homogeneous isotropic strip.

End effects for anti-plane shear deformations of sandwich structures

The purpose of this research is to further investigate the effects of material inhomogeneity and the combined effects of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects. Saint-Venant decay rates for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The problem is governed by a second-order, linear, elliptic, partial differential equation with discontinuous coefficients. The most general anisotropy consistent with a state of anti-plane shear is considered, as well as a variety of boundary conditions. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement which depends only on the in-plane coordinates. They can be thought of as complementary deformations to those of plane elasticity. While these deformations have received little attention compared with the plane problems of linear elasticity, they have recently been investigated for anisotropic and inhomogeneous linear elasticity. In the context of linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that quickly decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Self-equilibrated loads are a class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity significantly affects the practical application of Saint-Venant's principle to sandwich structures.

Decay rates in a piezoelectric strip

Saint-Venant end effects can be characterized mathematically by the so-called decay analysis. With this decay analysis, Saint-Venant's principle can be extended to modern materials and structures beyond the traditional homogeneous and isotropic elastic solids. In the present paper, the decay rate in a piezoelectric strip is considered. In order to overcome the difficulties raised by general anisotropy, the Stroh formalism is applied here for the two-dimensional decay analysis in piezoelectricity.

Decay rates in a bimaterial circular cylinder

Decay rates in a bimaterial circular cylinder under axisymmetric torsion loading are considered via an eigen-expansion near the end of the cylinder. The decay rates depend on the shear modulus ratio of the materials and the radius ratio of inner and outer cylinders. Following the derivation of the traditional Saint-Venant end effect of an isotropic bimaterial cylinder, cases of anisotropic material (transversely isotropic material) and non-traditional Saint-Venant end effect (displacement prescribed on the side surface) are considered. This study sheds some light on the decay studies for other geometric configurations and the deformation modes of composite structures.

Antiplane Shear Deformations for Homogeneous and Inhomogeneous Anisotropic Linearly Elastic Solids

Antiplane shear deformations of a cylindrical body, with a single displacement field parallel to the generators of the cylinder and independent of the axial coordinate, are one of the simplest classes of deformations that solids can undergo. They may be viewed as complementary to the more familiar plane deformations. Antiplane (or longitudinal) shear deformations have been the subject of the considerable recent interest in nonlinear elasticity theory for homogeneous isotropic solids. In contrast, for the linear theory of isotropic elasticity, such deformations are usually not extensively discussed. The purpose of the present paper is to demonstrate that for inhomogeneous anisotropic linearly elastic solids the antiplane shear problem does provide a particularly tractable and illuminating setting within which effects of anisotropy and inhomogeneity may be examined. We consider infinitesimal antiplane shear deformations of an inhomogeneous anisotropic linearly elastic cylinder subject to prescribed surface tractions on its lateral boundary whose only nonzero component is axial and which does not vary in the axial direction. In the absence of body forces, not all arbitrary anisotropic cylinders will sustain an antiplane shear deformation under such tractions. Necessary and sufficient conditions on the elastic moduli are obtained which do allow an antiplane shear. The resulting boundary value problems governing the axial displacement are formulated. The most general elastic symmetry consistent with an antiplane shear is described. There are at most 15 independent elastic coefficients associated with such a material. In general, there is a normal axial stress present, which can be written as a linear combination of the two dominant shear stresses. For a material with the cylindrical cross-section a plane of elastic symmetry (monoclinic, with 13 moduli), the normal stress is no longer present. For homogeneous materials, it is shown how the governing boundary value problem can be transformed to an equivalent isotropic problem for a transformed cross-sectional domain. Applications to the issue of assessing the influence of anisotropy and inhomogeneity on the decay of Saint-Venant end effects are described.

Saint-Venant end effects in composite structures

Thin-walled structures, such as rifle barrels, rocket casings, helicopter blades and containment vessels, are often constructed of layers of anisotropic, filament or fiber-reinforced materials which must be designed to remain elastic. A proper assessment of end or edge effects in such structures is of fundamental technological importance. The extent to which local stresses, such as those produced by fasteners and at joints, can penetrate girders, beams, plates and shells must be understood by the designer. Thus a distinction must be made between global structural elements (where Strength of Materials or other approximate theories may be used) and local elements which require more detailed (and more costly) analyses based on exact elasticity. Moreover, it must be recognized that it is impossible, in general, to refine global approximate theories (such as various so-called higher-order plate and shell theories) without a simultaneous consideration of local effects. The neglect of end effects is usually justified by appeals to some form of Saint-Venant's principle, and years of experience with homogeneous isotropic elastic structures have served to establish this standard procedure. Saint-Venant's principle also is the fundamental basis for static mechanical tests of material properties. Thus property measurements are made in a suitable gage section where uniform stress and strain states are induced and local effects due to clamping of the specimen are neglected by an appeal to Saint-Venant's principle. Such traditional applications of Saint-Venant's principle require major modifications when strongly anisotropic and composite materials are of concern. For such materials, local stress effects persist over distances far greater than are typical for isotropic materials. In this paper, we describe some problems of static and dynamic elasticity where anisotropy induces such extended Saint-Venant end zones. The paper is a review and a comprehensive list of references is given to original work where details of the analysis may be found. The consideration of such extended end zones due to anisotropy is essential in the proper analysis and design of structures using advanced composite materials.

The Eigenvalues for a Self-Equilibrated, Semi-Infinite, Anisotropic Elastic Strip

The linear theory of elasticity is used to study an homogeneous anisotropic seminfinite strip, free of tractions on its long sides and subject to edge loads or displacements that produce stresses that decay in the axial direction. If one seeks solutions for the (dimensionless) Airy stress function of the form φ = e−γxF(y), γ constant, then one is led to a fourth-order eigenvalue problem for F(y) with complex eigenvalues γ. This problem, considered previously by Choi and Horgan (1977), is the anisotropic analog of the eigenvalue problem for the Fadle-Papkovich eigenfunctions arising in the isotropic case. The decay rate for Saint-Venant end effects is given by the eigenvalue with smallest positive real part. For an isotropic strip, where the material is described by two elastic constants (Young’s modulus and Poisson’s ratio), the associated eigencondition is independent of these constants. For transversely isotropic (or specially orthotropic) materials, described by four elastic constants, the eigencondition depends only on one elastic parameter. Here, we treat the fully anisotropic strip described by six elastic constants and show that the eigencondition depends on only two elastic parameters. Tables and graphs for a scaled complex-valued eigenvalue are presented. These data allow one to determine the Saint-Venant decay length for the fully anisotropic strip, as we illustrate by a numerical example for an end-loaded off-axis graphite-epoxy strip.

The effect of constitutive law perturbations on finite antiplane shear deformations of a semi-infinite strip

This paper is concerned with assessing the effects of small perturbations in the constitutive laws on antiplane shear deformation fields arising in the theory of nonlinear elasticity. The mathematical problem is governed by a second-order quasilinear partial differential equation in divergence form. Dirichlet (or Neumann) boundary-value problems on a semi-infinite strip, with nonzero data on one end only, are considered. Such problems arise in investigation of Saint-Venant end effects in elasticity theory. The main result provides a comparison between two solutions, one of which is a solution to a simpler equation, for example Laplace’s equation. Three examples involving perturbations of power-law material models are used to illustrate the results.

Saint-Venant End Effects for Materials With Negative Poisson’s Ratios

In this paper we analyze Saint-Venant end effects for materials with negative Poisson ’s ratios. We present an example of slow decay of stress arising from selfequilibrated stress at the end of a circular cylinder of elastic material with a negative Poisson’s ratio. By contrast, a sandwich panel containing rigid face sheets and a compliant core exhibits no anomalous effects for negative Poisson’s ratio, but exhibits slow stress decay for core Poisson’s ratio approaching 0.5. In sandwich panels with stiff but not perfectly rigid face sheets, slow decay of stress is known to occur; a negative Poisson’s ratio results in end stress decay, which is faster than it would be otherwise.

Saint-Venant End Effects for Incremental Plane Deformations of Incompressible Nonlinearly Elastic Materials

This paper is concerned with assessing the extent of Saint-Venant end effects within the theory of small deformations superposed on a large deformation for plane strain of homogeneous, isotropic, incompressible materials. The problem considered is that of plane deformation of a body which in its undeformed configuration, occupies a semi-infinite strip. The long sides of the strip are free of traction while the short side is subjected to prescribed normal and shear tractions. A purely normal tensile traction is applied uniformly at the remote end. For the case of slightly nonuniform end tractions at the near end, it is shown that the resulting stress distribution differs from that of homogeneous uniaxial tension by an exponentially decaying function of the distance from the end of the strip. The decay rate is characterized explicitly in terms of the strip width, the remotely applied tensile load, and constitutive parameters. Numerical results are provided for the Mooney-Rivlin material and power-law materials which either harden or soften in tension.

Local stress analyses of composite materials using finite element methods. 1st report Saint-Venant's end effects of laminate media.

Saint-Venant's end effects to composite materials are frequently different from those to isotropic and homogeneous materials, therefore, the estimation of its effects is an important subject in relation to the evaluation of strength of composite materials. This paper is intended to establish the general method to estimate this Saint-Venant's end effects to composite materials, and the contents are as follows: (1) The finite element formulation to estimate Saint-Venant's end effects is established using the principle of virtual work and some example problems of laminate media are analysed by deyeloped method. It is shown, through these analyses, that the newly-defined characteristic length le is useful to evaluate the rate of attenuation of non-uniform local stress distribution. (2) The conventional elastic-plastic finite element analyses are carried out to the same examples, and the availability of developed method is confirmed. The end effects in case that plastic deformation occurs, are also discussed.

Saint-Venant end effects for plane deformation of sandwich strips

The purpose of this paper is to draw attention to the fact that the routine application of Saint-Venant's principle in the solution of elasticity problems for sandwich type structures is not justified in general. This is illustrated in the context of the plane problem of elasticity for a sandwich strip composed of two dissimilar isotropic materials. The exponential decay of end effects is characterized in terms of a complex eigenvalue. For the case of a sandwich with relatively soft middle core, the characteristic decay length is shown to be much greater than that for an homogeneous isotropic strip. The results are analogous to those obtained previously by the authors for highly anisotropic and composite materials.