Ciarlet P.G. — Mathematical elasticity. Volume II: Theory of plates
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Название: Mathematical elasticity. Volume II: Theory of plates
Автор: Ciarlet P.G.
Аннотация:
The objective of Volume II is to show how asymptotic methods, with the thickness as the small parameter, indeed provide a powerful means of justifying two-dimensional plate theories. More specifically, without any recourse to any a priori assumptions of a geometrical or mechanical nature, it is shown that in the linear case, the three-dimensional displacements, once properly scaled, converge in H 1 towards a limit that satisfies the well-known two-dimensional equations of the linear Kirchhoff-Love theory; the convergence of stress is also established.
In the nonlinear case, again after ad hoc scalings have been performed, it is shown that the leading term of a formal asymptotic expansion of the three-dimensional solution satisfies well-known two-dimensional equations, such as those of the nonlinear Kirchhoff-Love theory, or the von K?rm?n equations. Special attention is also given to the first convergence result obtained in this case, which leads to two-dimensional large deformation, frame-indifferent, nonlinear membrane theories. It is also demonstrated that asymptotic methods can likewise be used for justifying other lower-dimensional equations of elastic shallow shells, and the coupled pluri-dimensional equations of elastic multi-structures, i.e., structures with junctions. In each case, the existence, uniqueness or multiplicity, and regularity of solutions to the limit equations obtained in this fashion are also studied.
Flexural equations (of the linear Kirchhoff — Love theory)6775 Flexural equations (of the linear Kirchhoff — Love theory), existence of solutions4867 Flexural equations (of the linear Kirchhoff — Love theory), regularity of solutions4867 Flexural equations (of the linear Kirchhoff — Love theory), uniqueness of solutions4867 Flexural rigidity of a plate67319386404 Flexural theory (inextensional, large deformation, nonlinear flexural theory)343345 Folded plates175180199 Formal asymptotic method82 Frame-indifference19323328337364365407 Frame-indifferent flexural theory343 Frame-indifferent membrane theory340353355365 Frame-indifferent plate theory335340 Functionalsee "Energy""Minimization General constitutive equation325 General elastic material325407 General nonlinearly elastic material325 Generalized Korn's inequality126144224231236239 Generic character of the nonlinear Kirchhoff — Love plate theory327 Generic character of the von Karman equations407 Green — St Venant strain tensor260267326 Hierarchic plate theory87 Hilbert uniqueness method (HUM)118203 Holes (plates with holes)85438 Homogeneous elastic material17260325 Homogenization theory85437 Hooke's law197688 Hybrid finite element method24 Hyper-elastic material262 Hyperelastic material262 Implicit function theorem263323364450 In-plane displacement64313404 In-plane displacement of the middle surface64 Inextensional deformation343 Inextensional flexural theory343 Infimizing sequence of an energy287 Infinite energy176 Integration across the thickness81 Integration across the thickness of a plate81 Inverted constitutive equation294375 Isotropic elastic material17260325 Junction between a plate and a rod175 Junction between a plate and a shallow shell245 Junction between a plate and a three-dimensional structure163168 Junction between a shallow shell and a plate245 Junction between a three-dimensional structure and a plate163168 Junction between a three-dimensional structure and a rod176 Junction between a three-dimensional structure and a shallow shell175178 Junction between nonlinearly elastic substructures175 Junction between plates175 Junction between plates and rods175 Junction between rods175199 Junction between shells180 Junction conditions for a coupled, multi-dimensional problem148170172175197 Junction conditions for a three-dimensional problem139 Justification of assumptions on the data94329 Justification of the displacement-stress approach313 Justification of the linear Kirchhoff — Love plate theory72 Justification of the nonlinear Kirchhoff — Love plate theory321334 Justification of the scalings of the displacements93329 Justification of the scalings of the stresses58120 Kirchhoff — Love displacement field6567120173188243314322406 Kirchhoff — Love hypothesis7681120 Korn's inequality13 Korn's inequality with boundary conditions10223654119135144214224 Korn's inequality without boundary conditions1054119 Kuhn — Tucker relations125486 Lagrange multiplier77125172 Lame constants1719261 Laminated plate8588 Large deformation nonlinear flexural theory343 Large deformation nonlinear membrane theory340353 Lateral face of a plate15 Lateral face of a shallow shell439 Lax — Milgram lemma2123395055 Leading term in a formal asymptotic expansion269 Limit displacement inside the plate6476313404 Limit displacement of the middle surface64313403404 Limit displacements inside the plate6476313404 Limit displacements of the middle surface64313386 Limit energy69171 Limit energy of an elastic multi-structure (linearly elastic multi-structure)171 Limit energy of an elastic multistructure171 Limit equations of an elastic multi-structure (linearly elastic multi-structure)163168 Limit scaled energy44 Limit scaled stress5862302387444 Limit scaled stresses5862302387444 Limit scaled three-dimensional equations34 Limit scaled three-dimensional equations for a linearly elastic clamped plate34 Limit scaled three-dimensional equations for a nonlinearly elastic clamped plate277 Limit scaled three-dimensional equations for a von Karman plate382 Limit stress6976174319322328361404 Linear elastodynamics104112189199 Linear Kirchhoff — Love plate theory677275798694342 Linear Kirchhoff — Love plate theory as a small displacement theory73335 Linear Kirchhoff — Love plate theory, eigenvalue problem111 Linear Kirchhoff — Love plate theory, existence of solutions67 Linear Kirchhoff — Love plate theory, regularity of solutions67 Linear Kirchhoff — Love plate theory, uniqueness of solutions67 Linear Kirchhoff — Love theory of a plate677275798694342 Linear membrane equation341432446 Linear small displacement theory73335 Linearized bending moment70 Linearized boundary condition of traction18 Linearized displacement-traction problem17 Linearized elasticity (three-dimensional linearized elasticity)1324 Linearized elasticity (three-dimensional linearized elasticity), eigenvalue problem105 Linearized elasticity (three-dimensional linearized elasticity), time-dependent problem104112 Linearized elastodynamics104112189199 Linearized strain19 Linearized strain tensor19 Linearized strains19 Linearized stress couple7076 Linearized stress resultant7076 Linearized stress tensor18 Linearly elastic clamped plate problem20 Linearly elastic clamped plate, eigenvalue problem105 Linearly elastic clamped plate, scaled three-dimensional equations28 Linearly elastic clamped plate, scaled two-dimensional energy43 Linearly elastic clamped plate, scaled two-dimensional equations4257 Linearly elastic clamped plate, three-dimensional boundary conditions18 Linearly elastic clamped plate, three-dimensional energy23 Linearly elastic clamped plate, three-dimensional equations20 Linearly elastic clamped plate, three-dimensional Hellinger — Reissner variational principle2382120 Linearly elastic clamped plate, three-dimensional principle of virtual work23 Linearly elastic clamped plate, time-dependent problem112 Linearly elastic clamped plate, two-dimensional boundary conditions66181 Linearly elastic clamped plate, two-dimensional energy69 Linearly elastic clamped plate, two-dimensional equations67 Linearly elastic clamped plate, two-dimensional flexural equations67111 Linearly elastic clamped plate, two-dimensional Hellinger — Reissner variational principle7176 Linearly elastic clamped plate, two-dimensional membrane equations67169 Linearly elastic clamped plate, two-dimensional principle of virtual work71 Linearly elastic material19 Linearly elastic multi-structuressee "Elastic multi-structure" Linearly elastic rod8385112 Linearly elastic shallow shell242 Linearly elastic shallow shell in Cartesian coordinates242 Linearly elastic shallow shell in curvilinear coordinates219 Linearly elastic shell213 Linearly elastic von Karman plate122 Lions (lemma of J.L. Lions)911226 Live load327408424448 Locking phenomenon87 Lower face of a plate15 Lower face of a shallow shell214 Marguerre — von Karman equations446 447 Marguerre — von Karman equations, bifurcation of solutions446 Marguerre — von Karman equations, degeneracy into the linear membrane equation446 Marguerre — von Karman equations, existence of solutions446 Marguerre — von Karman equations, regularity of solutions446
-dependent constitutive equation 
-convergence 
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