Self-equilibrating Tractions Research Articles

Saint-Venant decay rates for an inhomogeneous isotropic linear thermoelastic strip

In this paper we consider the state of plane strain in an isotropic and inhomogeneous thermoelastic material occupying a rectangular strip. Such a strip is maintained in equilibrium under self-equilibrated traction applied on one of the heated edges, while the other three edges are thermally insulated and traction-free. Our aim is to derive some explicit spatial estimates describing how certain appropriate measures of the Airy stress function and temperature evolve with respect to the distance from the loaded and heated edge, provided specific assumptions are made upon the derivatives of the thermoelastic coefficients. The results of the present paper prove how the spatial decay rate varies with the inhomogeneous constitutive profile.

On Saint-Venant's principle in a poroelastic arch-like region

In this paper we consider the state of plane strain in an elastic material with voids occupying a curvilinear strip as an arch-like region described by R: a<r<b, 0<θ<ω, where r and θ are polar coordinates and a, b, and ω (<2π) are prescribed positive constants. Such a curvilinear strip is maintained in equilibrium under self-equilibrated traction and equilibrated force applied on one of the edges, whereas the other three edges are traction free and subjected to zero volumetric fraction or zero equilibrated force. In fact, we study the case when one right or curved edge is loaded. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The results of the present paper prove how the spatial decay rate varies with the constitutive profile. For the problem corresponding to a loaded right edge, we are able to establish an exponential decay estimate with respect to the angle θ. Whereas for the problem corresponding to a loaded curved edge, we establish an algebraical spatial decay with respect to the polar distance r, provided the angle ω is lower than the critical value . The intended applications of these results concern various branches of medicine as for example the bone implants. Copyright © 2010 John Wiley & Sons, Ltd.

On Saint-Venant's principle for a linear poroelastic material in plane strain

In this paper we consider the state of plane strain in an elastic material with voids occupying a rectangular strip. Such a strip is maintained in equilibrium under self-equilibrated traction and equilibrated force applied on one of the edges, while the other three edges are traction-free and subjected to zero volumetric fraction or zero equilibrated force. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The both cases of homogeneous and inhomogeneous poroelastic materials are considered. The results of the present paper prove how the spatial-decay rate varies with the constitutive profile.

End effects for a generalized biharmonic equation with applications to functionally graded materials

In this paper we consider a generalized biharmonic equation modeling two-dimensional inhomogeneous elastic state in the curvilinear rectangle a < r < b , 0 < θ < α , where ( r , θ ) denote plane polar coordinates. Such an arch-like region is maintained in equilibrium under self-equilibrated traction applied on one of the edges, while the other three edges are traction free. Our aim is to derive some explicit spatial estimates describing how some appropriate measures concerning the specific Airy stress function evolve with respect to the distance to the loaded edge. Two types of smoothly varying inhomogeneity are considered: (i) the elastic moduli vary smoothly with the polar distance, (ii) they vary smoothly with the polar angle. Such types of smoothly varying inhomogeneous elastic materials provide a model for technological important functionally graded materials. The results of the present paper prove how the spatial decay rate varies with the constitutive profile.

Edge Effects in Laminated Composite Plates

The attenuation of self-equilibrated edge stress states into the interior of a laminated plate composed of an arbitrary number of bonded, elastic, anisotropic layers is investigated in the context of Saint-Venant’s principle using the exponential decay results of Toupin, Knowles, and Horgan. To model the plate’s behavior, a semianalytical method is used with finite element interpolations over the thickness and exponential decay into the plate’s interior. The formulation leads to a second-order algebraic eigensystem whose eigenvalues are the characteristic inverse decay lengths, and corresponding right eigenvectors depict the displacement distributions of self-equilibrated traction states. Orthogonality relations between these right and left eigenvectors of the adjoint system are established. An eigenvector expansion for representing arbitrary self-equilibrated edge tractions is then presented. This approach is useful in revealing the interlaminar effects and their decay rates in a laminated composite plate under plane strain. Two examples are provided where the interlaminar phenomena due to eigenstates of self-equilibrated edge stress are illustrated.

Some aspects of Saint Venant's principle

When self-equilibrating tractions act on an edge of a rectangle, the stresses decay exponentially with distance from the loaded edge. This decay is illustrated for a number of important cases. Results are also given for the half plane loaded by self-equilibrating edge tractions. In this case the stresses decay as integral powers of the distance. The paper concludes with an illustration of the effect on the maximum stress of the rapidity of shear traction variation.

Thermal Stresses in a Partially Clamped Elastic Half-Plane

Numerical methods, such as those of Southwell, have made amenable to approximate analysis a large class of problems in the theory of elasticity. However, certain loadings must occasionally be considered which may give rise to theoretical stress singularities. In such cases an analytical study, at least in the immediate neighborhood of the singularities, would appear to be mandatory. One such problem is that of determining the thermal stresses in a partially clamped elastic half-plane. The solution given in this paper assumes self-equilibrating tractions over the clamped portion of boundary so that the results apply approximately also to large finite areas.