TY - JOUR

T1 - Saint-venant principle for paraboloidal elastic bodies

AU - Nazarov, S. A.

AU - Slutskii, A. S.

PY - 2000/1/1

Y1 - 2000/1/1

N2 - 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 ∞ (Ω̄) 3 by the energy norm? The dimension d γ of the lineal R γ of rigid energy displacements is computed (in this case, d 0 = 6, d 1 = 0, and the function γ → d γ 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 R 0 with R γ : for γ > 1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem.

AB - 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 ∞ (Ω̄) 3 by the energy norm? The dimension d γ of the lineal R γ of rigid energy displacements is computed (in this case, d 0 = 6, d 1 = 0, and the function γ → d γ 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 R 0 with R γ : for γ > 1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem.

UR - http://www.scopus.com/inward/record.url?scp=54649084399&partnerID=8YFLogxK

U2 - 10.1007/BF02355387

DO - 10.1007/BF02355387

M3 - Article

AN - SCOPUS:54649084399

VL - 98

SP - 717

EP - 752

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -