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

Abstract

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<h<x, where r and θ are polar coordinates and a, b, and ɷ (<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 ɷ is lower than the critical value π√2. The intended applications of these results concern various branches of medicine as for example the bone implants

Reflection of Plane Waves from Surface of a Generalized Thermo-Viscoelastic Porous Solid Half-Space with Impedance Boundary Conditions

“…and t rs are the components of the stress tensor, H r are the components of the equilibrated stress vector, g is the intrinsic equilibrated body force, η is the entropy per unit mass, Q r are the components of the heat flux vector, e rs are the components of the strain tensor, ϱ is the mass density of the medium, K * is the equilibrated inertia, u r are the components of the displacement vector, ϕ is the void volume fraction, θ is the change in temperature from the constant reference temperature T 0 and δ rs are the components of the Kronecker delta,λ and µ are well known Lame's constant parameters, b, α, ξ and ξ * are the constant parameters corresponding to voids present in the medium, β, τ * , m, κ, ζ and a are the constant thermal parameters and λ * , µ * , b * , α * and γ * are the constant viscoelastic parameters, τ 0 is thermal relaxation time. Equations (10) to (12) are specialized in x-z plane as:…”

“…Scalia [5], Chirita and Scalia [6], Chirita et al [7], Iesan and Nappa [8], Chirita and D'Apice [9,10] and Ciarletta et al [11] have studied various outstanding dynamical problems in theory of thermoelasticity with voids. Various problems on plane wave propagation in elasticity and thermoelasticity with voids were also studied.…”

Reflection of Plane Waves from Surface of a Generalized Thermo-viscoelastic Porous Solid Half-space with Impedance Boundary Conditions

“…In what follows x 1 and x 3 are denoted as x and z, respectively. Equations (10) to (12) are specialized in x − z plane as the equations (13) to (16) result in the following equations…”

“…Various dynamical problems and plane strain problems in the theory of elasticity and thermoelasticity with voids have appeared in literature. For example, Iesan [4], Ciarletta and Scalia [5], Chirita and Scalia [6], Chirita et al [7], Iesan and Nappa [8], Chirita and D'Apice [9,10] and Ciarletta et al [11] have studied various outstanding dynamical problems in the theory of thermoelasticity with voids. Various problems on plane wave propagation in elasticity and thermoelasticity with voids were also studied by, for example, Puri and Cowin [12], Chandrasekharaiah [13,14], Singh [15], Ciarletta and Straughan [16], Ciarletta, et al [17] and Bucur et al [18].…”