Saint-Venant Principle Research Articles

A Saint-Venant principle for shear band localization

A type of Saint-Venant principle is derived for a two-dimensional model of shear band formation in thermoviscoplastic solids. To establish that the thermal energy generated during the formation process remains highly localized, a spatially decaying upper bound on the temperature is derived. It is found that the temperature bound decays exponentially along the direction perpendicular to the band, with a rate that decreases in time. The result is established by using maximum principles for second-order nonlinear parabolic partial differential equations.

Mechanics of Solids and Shells: Theories and Approximations

Mechanics of Solids and Shells: Theories and Approximations

Development of the Spectral Sturm–Liouville Method for the Solution of a Boundary-Value Problem for the Biharmonic Equation

We generalize the spectral Sturm–Liouville method for the solution of the biharmonic equation. The characteristic equation for the determination of eigenvalues is investigated and eigenfunctions are constructed. We determine the stress-strain state for a rectangular plate loaded by arbitrary forces on its sides. For an arbitrary external load, we obtain a relation for the stress-strain state in the form of a series in eigenfunctions. A method of integral moments for the determination of the coefficients of the series is proposed. The Saint-Venant principle is verified.

The Saint-Venant principle and an impact load acting on an elastic half-space

The Saint-Venant principle and an impact load acting on an elastic half-space

Source signature and elastic waves in a half‐space under a momentary shear line impulse

AbstractThe transient deformation of an elastic half‐space under a line‐concentrated impulsive vector shear load applied momentarily is disclosed in this paper. While in an earlier work, the author gave an analytical–numerical method for the solution to this transient boundary‐value problem, here, the resultant response of the half‐space is presented and interpreted. In particular, a probe is set up for the kinematics of the source signature and wave fronts, both explicitly revealed in the strained half‐space by the solution method. The source signature is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Fourteen wave fronts exist behind the precursor shear wave S: four concentric cylindrical, eight plane, and two relativistic cylindrical initiated at propagating centres that are located on the stationary boundaries of the solution domain. A snapshot of the stressed half‐space reveals that none of the 14 wave fronts fully extend laterally. Instead, each is enclosed within point bounds. These wave arresting points and the two propagating centres of the relativistic waves constitute the source signature. The obtained 14 wave fronts are further combined into 11 disparate wave fronts that are grouped into four categories: an axis of symmetry wave—so named here by reason of being a wave front that is contiguous to the axis of symmetry, three body waves, five surface waves and two inhibitor waves—so named here by reason that beyond them the material motion dies out. Of the three body waves, the first is an unloading shear wave, the second is a diffracted wave and the third is a reflected longitudinal two‐branch wave. Of the two inhibitor waves, the first is a two‐joint relativistic wave, while the second is a two‐branch wave. The wave system, however, is not the same for all the dependent variables; a wave front that appears in the behaviour of one dependent variable may not exist in the behaviour of another. It is evident from this work that Saint–Venant's principle for wave propagation problems cannot be formulated. Therefore, the above results are valid for the particular proposed model for the momentary line‐concentrated shear load. The formulation of the source signature, the wave system, and their role in the half‐space transient deformation are presented here. Copyright © 2003 John Wiley & Sons, Ltd.

208 Saint Venant の原理と開閉断面を持つ卵型構造に関する基礎的研究

208 Saint Venant の原理と開閉断面を持つ卵型構造に関する基礎的研究

Torsional load transfer from a rigid shaft to an elastic plane with a slit

A plane elastostatic problem for an elastic wedge loaded by a concentrated moment at its apex provides an example of violation of the Saint-Venant principle for apex angles 2α larger than π. Considering the problem for a truncated wedge, Neuber demonstrated the method of construction of an applicable solution for any apex angles in the range π≤2α≤2π, despite the failure of the Saint-Venant principle. In the present paper the particularly important case of the truncated-wedge problem is examined. The truncated wedge degenerates into a slitted elastic plane, while a rigid circular shaft, acted upon by a torsional moment, is inserted into the plane. The analytical solution of the mixed boundary-value problem is obtained. Numerical results turn out to be in complete agreement with Neuber's results for the slitted elastic plane.

The biharmonic problem and progress in the development of analytical methods for the solution of boundary-value problems

A comparative analysis of harmonic and biharmonic boundary-value problems for 2D problems on a rectangle is given. Some common features of two types of problems are emphasized and special attention is given to the basic distinction between them. This distinction was thoroughly studied for the first time by L. N. G. Filon with respect to some plane problems in the theory of elasticity. The analysis permits to introduce an important aspect of the general solution of boundary-value problems. The procedure for solving the biharmonic problem involves both the method of homogenous solutions and the method of superposition. For some cases involving self-equilibrated loadings on one pair of sides of the rectangle, the complete solution, including calculation of the quantitative characteristics of the displacements and stresses, is given. The efficiency of the numerical implementation of the general solutions is shown. The analysis of the quantitative data allows to elucidate some main points of the Saint-Venant principle.

SAINT-VENANT'S PRINCIPLE FOR A MICROPERIODIC COMPOSITE THERMOELASTIC SEMISPACE: THE DYNAMICAL REFINED AVERAGE THEORY

A Saint-Venant's principle associated with a one-dimensional dynamic coupled thermoelastic effective modulus theory (EMT) for a microperiodic layered semispace was presented in J. Thermal Stresses, vol. 23, pp. 1-14, 2000. It was shown there that a thermoelastic energy associated with a solution to an initial boundary value problem of the theory decays exponentially as a distance x from the thermomechanical load region goes to infinity and that its decay length depends on the time t, an effective velocity d c 1 * ¢ , an effective time unit d T* ¢ , and an effective thermoelastic coupling parameter d k * ¢ . In the present article the Saint-Venant's principle is extended to include a refined averaged theory (RAT) for a microperiodic layered thermoelastic semispace in which a microstructural length is taken into account (see [IFTR Report #25, pp. 1-158, 1995]). It is shown that for such an extended theory, a similar exponential decay estimate for a thermoelastic energy holds true. In the refined estimate the thermoelastic energy depends on a number of microstructural parameters while its decay length is independent of these parameters; and the decay length for small (large) times is comparable to that of a pure thermal (elastic) energy for a rigid (elastic) semispace for every time t > 0 .

Saint-Venant's Principle in the Asymptotic Analysis of Elastic Rods with One End Fixed

The authors take advantage of an already stated Saint-Venant's principle for a linear elastic free rod to obtain a generalization in the case of the rod fixed on one end. The result is established for star-shaped cross sections and assumes some regularity on the axial forces. Also, an exposition of the technique and its conjecture for the case of the rod fixed on both of its ends is given.

Construction of elementary solutions to a plane elastic problem for a rectangular domain

Necessary existence conditions are established for the solution to a plane elastic problem for a rectangular domain given with external boundary and bulk forces. Elementary solutions satisfying Saint-Venant's and superposition principles are constructed.

Saint-Venant's Principle for Antiplane Shear Deformations of Linear Piezoelectric Materials

This paper is concerned with investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. We examine the decay of Saint-Venant end effects in the context of antiplane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are shown to be a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential $\varphi$. The traction boundary-value problem with prescribed surface charge is formulated as an oblique derivative boundary-value problem...

ASYMPTOTIC JUSTIFICATION OF THE TREATMENT OF A METALLOSTATIC PRESSURE TYPE BOUNDARY CONDITION IN AN ALUMINIUM CASTING

We present an asymptotic analysis to justify the treatment of a metallostatic pressure type boundary condition on a thermal free boundary arising from the thermomechanical modelling of aluminium alloy castings. The problem is solved by means of a fictitious domain method. Using the Saint-Venant's principle we shall justify that for a suitable choice of the Lamé coefficients in the fictitious domain, the solution obtained with this methodology approaches the initial problem solution "far" from the free boundary.

Analysis of laminated circular cylinders of materials with the most general form of cylindrical anisotropy.: II. Flexural deformations

This second paper of a two-part series is devoted to flexural stresses and deformations in a laminated anisotropic cylinder of perfectly bonded materials possessing the most general form of cylindrical anisotropy. The loading conditions include pure bending, flexure by a transverse force, and applied surface tractions that lead to resultant transverse loads and bending moments of uniform or linear variation along the axis. These loading conditions cause strain and stress fields that are uniform, linear and quadratically varying along the axis. The solutions herein are based on the relaxed formulation of the Saint-Venant's and Almansi–Michell problems where conditions on the ends of the cylinder are satisfied on an integral basis rather than on a point-wise basis. Differences in the stress distributions between these integral conditions with any point-wise specification are self-equilibrated states that decay with distance into the interior, i.e., Saint-Venant's principle. Means to account for such effects are discussed.

Analysis of laminated circular cylinders of materials with the most general form of cylindrical anisotropy.: I. Axially symmetric deformations

A procedure is presented for the analysis of stresses and deformations in a laminated circular cylinder of perfectly bonded materials with the most general form of cylindrical anisotropy. Attention is focused in this paper on axisymmetric loads, and a companion paper will be devoted to loads producing flexural deformations. Axisymmetric loading conditions include an axial force and torque on the ends as well as a general distribution of surface tractions of pressure, circumferential shear and longitudinal shear. These surface tractions are represented by a power series in the axial coordinate. Only the leading two terms are considered herein; they relate to uniform and linear variations of the normal pressures and uniform circumferential and longitudinal surface shears. From the methodology for these two terms, extension of the analysis procedure for higher terms will be readily apparent. The solutions herein fit descriptions of (1) the relaxed formulation of Saint-Venant's problem where prescribed tractions on the ends on a point-wise basis are replaced by integral conditions relating to an axial force and torque and (2) the Almansi–Michell problem. Differences in the stress distributions between these integral conditions with any point-wise specification are self-equilibrated states that evanesce with distance from the ends, i.e., Saint-Venant's principle, and some remarks are offered for treatment of such effects.

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ϕ. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal \({\bar 4}\) crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d0,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d0. In the limit as d0→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal \({\bar 4}\) class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.

Spatial decay of transient end effects in functionally graded heat conducting materials

The purpose of this research is to investigate the influence of material inhomogeneity on the spatial decay of end effects in transient heat conduction for isotropic inhomogeneous heat conducting solids. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The spatial decay of solutions to an initial-boundary value problem with variable coefficients on a semi-infinite strip is investigated. It is shown that the spatial decay of end effects in the transient problem is faster that that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. Explicit decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation with constant coefficients. The results may be interpreted in terms of a Saint-Venant principle for transient heat conduction in inhomogeneous solids.

Computational analysis of thermally loaded duplex stainless steels: the role of the free surfaces and the microstructure

Hot-forged ferritic–austenitic duplex stainless steels possess complicated structural and thermomechanical features which necessitate numerical analysis when modeling their deformational behavior. The microstructure of these steels consists of two phases with different temperature-dependent thermomechanical properties. As a consequence, purely thermal cycling in the temperature interval between 900°C and 20°C can generate residual strains/stresses even in the absence of an external mechanical load. Stress gradients, which are to be expected near traction-free surfaces of specimens due to Saint–Venant's principle are responsible for considerable deviations from the uniformity conditions and thus also influence the character of spatial distributions of strains and stresses. If one aims at the modeling of entire specimen, this excludes a utilization of traditional concepts (e.g., homogenization or representative volume elements (RVE)) based on the assumption of the absence of macroscopic stress gradients. Three model representations for a cylindrical two-phase specimen with a low aspect ratio are introduced and analyzed by means of numerical (finite element) simulation in order to study the effect of free surfaces on the macroscopic response of ferritic–austenitic duplex steels to purely thermal loading. The numerical analysis is carried out for two different types of matrix–inclusion morphology covering a simplified and a random distribution of the phase domains. The solution of a complementary problem provides a general information on the effect of the geometrical features of both the whole specimen (e.g., its aspect ratio) and of the microstructure element on the length of the free-end zone. Such zones in vicinity of the specimen's end faces influence not only the stress distribution but also the level of the irreversible axial strain increment per thermal cycle. It is shown that this increment is very sensitive to the type of the matrix–inclusion topology. The perimeter surface of cylindrical specimen, in contrast, does not affect the axial and the circumferential stress components. The width of the zone measured in radial direction of varying radial stress is extends only the layer of microscopic elements closest to this surface.

Consider Saint-Venant's principle by means of chain model

A precise background theory of computational mechanics is formed. Saint- Venant's principle is discussed in chain model by means of this precise theory. The classical continued fraction is developed into operator continued fraction to be the constrictive formulation of the chain model. The decay of effect of a self-equilibrated system of forces in chain model is decided by the convergence of operator continued fraction, so the reasonable part of Saint- Venant's principle is described as the convergence of operator continued fraction. In case of divergence the effect of a self-equilibrated system of forces may be non-zero at even infinite distant sections, so Saint- Venant' s principle is not a common principle.

Saint-venant principle for paraboloidal elastic bodies

We study the asymptotic behavior of solutions to the linear problem of elasticity in a domain Ω with paraboloidal exit at infinity. Properties of solutions and the condition of the existence of solutions depend on a parameter γ∈[0,1] characterizing the velocity of extending the paraboloid (a cylinder and a cone correspond to the cases γ=0¤γ=1 respectively). Asymptotic formulas are deduced for displacement fields generating forces and moments “applied at infinity”. The Saint-Venant principle is verified for “oblong” bodies such as paraboloids, cylinders, and narrow cones. The following question turns out to be a key one: What rigid displacements belong to the energy space obtained by completion of\(C_0^\infty (\bar \Omega )^3 \) by the energy norm? The dimension dγ of the lineal Rγ of rigid energy displacements is computed (in this case, d0=6, d1=0, and the function γ↦dγ has jumps at the points γ=1/4, 1/2, 3/4). We also clarify the reasons why it is necessary to distinguish the notions “energy solution” and “solution with finite energy”. We also discuss the phenomenon of a boundary layer that appears near the endpoints of spindle-like rods and is described by energy solutions in paraboloids. As is shown, in order to have the well-posed formulation of the boundary conditions in one-dimensional models of such rods, it is necessary to use the weakened Saint-Venant principle, i.e., replace R0 with Rγ: for γ>1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem. Bibliography: 71 titles.