Saint Venant's principle in orthotropic planar elasticity: rates-of-diffusion for stress
For plane deformations generated by an arbitrary distribution of tractions applied in a small region on the boundary of an elastic half-plane, the rates-of-decay for displacements, stresses and strain energy density are obtained as functions of complexity of the load distribution. The rates-of-decay increase in proportion to the complexity of the load distribution; i.e., they increase with the order of the smallest nonvanishing moment of the traction distribution. In orthotropic materials the elastic moduli differ in two perpendicular directions of principal stiffness; in this case as the modulus ratio Ei/E\ increases, the angular distributions of the displacement and energy density fields become channeled towards the direction of the larger elastic modulus. Introduction. Self-equilibrated tractions acting on a small region of an elastic solid result in stresses that rapidly decay with increasing distance from the region where the tractions are applied. This behaviour is characteristic of systems with elliptic equations of equilibrium. In practice, general knowledge about the variation of stresses with radial distance from a small region of load application is widely used to claim that at large distances from the loaded region (i.e., large in comparison with the size of the loaded region), the distribution of tractions is unimportant—the only significant effect on the stresses comes from any resultant force and couple that are equivalent to the distributed tractions. If distances are not very large, however, even self-equilibrated tractions generate stresses that may depend on details of the traction distribution. To explain how the stress field depends on the distribution of tractions, we examine the effect of any local distribution of tractions on the rate-of-decay of stresses in an orthotropic elastic solid. The anisotropy of the solid affects the circumferential distribution of stress around the loaded area but not the rate-of-decay. It will be shown that the rate-of-decay (or rate-of-diffusion) of stress depends on the complexity of the traction distribution but is independent of whether the applied tractions act normal or tangential to the loaded surface; i.e., for tractions applied in a small area, the rate-of-decay does not depend on the Received December 16, 1997. 1991 Mathematics Subject Classification. Primary 35,76. ©1999 Brown University 741 742 W. J. STRONGE AND M. KASHTALYAN moment of the applied tractions, but rather on the moment of the spatial distribution of the applied tractions. The statement that self-equilibrated systems of tractions result in stresses that decay rapidly with distance from the region where tractions are applied is known as St. Venant's Principle; this principle expresses the property of diffusivity in systems represented by elliptic equations of equilibrium and compatibility. Barre St. Venant (1855) proposed this principle on the basis of his analyses of end effects in slender elastic beams and shafts. The principle was generalized to include tractions acting on a body of infinite extent by Boussinesq who stated that, "An equilibrated system of external forces applied to an elastic body, all the points of application lying within a given sphere, produces deformations of negligible magnitude at distances from the sphere which are sufficiently large compared to its radius." While Boussinesq's expression of St. Venant's principle is quite general, it gives no sense of how the rate-of-diffusion depends on details of the traction distribution and/or the configuration of the body. After examining the stress distribution results from low-order sets of tangential forces acting in a small region on the surface of a half-space, von Mises (1945) noted that the rate-of-decay of stresses did not directly correspond with whether or not the set of forces was self-equilibrated. He proposed that the rate-of-decay depends also on whether the self-equilibrated system of forces is also in astatic equilibrium, (von Mises introduced the term astatic equilibrium to characterise a self-equilibrated set of forces that act at specific points on a surface if this set remains in equilibrium when the forces are rotated through an arbitrary angle tjj.) A general proof of Saint-Venant's principle as stated by von Mises was given by Sternberg (1954), who examined an elastic body loaded by surface tractions applied in a small region of radius e. Sternberg proved that with increasing radius from the region of load application, the decay of stresses is more rapid if the applied forces are not merely self-equilibrated but are also in astatic equilibrium. Proofs of these statements were reviewed by Gurtin 1973. Later investigations established an exponential rate-of-decay with axial distance for 2-D stresses generated by self-equilibrated tractions acting over the end of a semi-infinite slender strip or cylinder composed of anisotropic elastic material (Everstine and Pipkin 1971, Horgan 1972, Horgan and Knowles 1983, Horgan and Simmonds 1994, Arimitsu et al 1995, Durban and Stronge 1995, Horgan 1996). For an orthotropic half-plane, Stronge and Kashtalyan 1997 found the distribution of stresses and the rates-of-decay arising from some distributions of traction in a small area on the edge of a half-plane. 1. Stress field from a discrete force. Previous investigations of St. Venant's principle were based on consideration of particular sets of traction distributions involving a small number of discrete forces. Here we consider an orthotropic elastic half-space where 2-D stress and displacement fields result from an arbitrary distribution of tractions acting in a small region e on the surface y = 0. Let the force F = Fe act on the surface of an orthotropic elastic half-plane y < 0 at an angle from the normal, e = sinf2ex + cosf2ey as shown in Fig. 1. The half-plane has Young's moduli Ex and Ey, in directions tangential ex and normal ey to the surface. For a polar coordinate system with origin at the point of application, the force results in ORTHOTROPIC PLANAR ELASTICITY 743 Fig. 1. Force F on surface of orthotropic half-plane radial stress only and the stress field can be expressed as arr(r, 6) = -kF • ip(9), ar6 = age = 0, (la) r (ui + U2) K, — , 7T ip(0) sin# L ex + R\'2 cos 9 (ib) (lc) A(0) = sin4 6 + 2R\ sin2 6 cos2 6 + R2 cos4 9, (Id) where u\ and U2 are roots of a characteristic equation given by Lekhnitskii (1981), u4 2R\u2 + i?2 = 0. (2) Most orthotropic materials have moduli giving roots U\,U2 that are real and distinct but the roots can be complex or, for isotropic materials, real and repeated (Matemilola et. al, 1995). In Eqs. (1) and (2) the parameters R\ and R2 are ratios of material properties that depend on Young's moduli Ex,Ey, the shear modulus G and Poisson's ratio vxy. For plane stress these material constants are defined as E y En „ E r V = R2 = (3) ■L-Jy J-Jl] In an isotropic material these modulus ratios are identical, R\ = R2 = 1. 744 W. J. STRONGE and M. KASHTALYAN If the point of force application is offset a distance e along the surface from the origin of the coordinate system, the stress field generated by F will be arr = — kF • <p{9e) cos2(9 — 9.) re ^-kF-^,6), (4a) r are = -kF • <pre(£,Q), (4b) r aee = -kF • <pee(£,6), (4c) where re = (r2 + e2 — 2resmd)1'2,r£ cos9e = rcos9, and £ = e/r. Here the angular distribution of stress and the offset £ of the origin from the point of application of the force are represented solely by {1^ } * + ^ { "'" (5a) m [£[-£ +sin <9] [l-£ sin 0] cos 0] 1/2 \ -£(1 £ sin 0) cos2 = \ WT) /G" + R2 1 (5b) //\\ f £2 [£ — sin 01 cos2 0 1 1/2 f£2cos30"| *»«■«>—h+e^ (W)e" (5c) A(£, 9) = (—£ + sin 9)4 + 2R\ (—£ + sin 9)2 cos2 6 + R-2 cos4 6. (5d) It is useful to recognize that both components of the vector function iptJ have the same form; i.e., the normal and tangential components can each be expressed as a rational function of the offset £ = e/r. 2. Stress field from set of parallel forces. Rather than a single force, consider a set of n parallel forces F= F^e, k = 1,n, each force acting on the surface at a spatial coordinate eu relative to the origin of the coordinate system and acting in a direction e. Superposing the effects of these surface tractions gives components of stress 1 = ~K22Fk ' VijUkJ))(6) k=1 If = £k/r 6 (—£,<£) and ( < 1 the distribution function ^pvj in the expression above can be expanded in a Maclaurin series in terms of powers of i.e., m=0 where the coefficients ^■ = (7a)