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.
Marguerre — von Karman shallow shell439 Marguerre — von Karman shallow shell, scaled two-dimensional equations443 Marguerre — von Karman shallow shell, three-dimensional equations440 Marguerre — von Karman shallow shell, two-dimensional equations446 mass density104 Membrane (tension in a membrane)342 Membrane energy69317 Membrane equation (linear membrane equation)341432 Membrane equations (of the Kirchhoff — Love theory)6776112402 Membrane equations (of the Kirchhoff — Love theory), existence of solutions5367 Membrane equations (of the Kirchhoff — Love theory), regularity of solutions5467 Membrane equations (of the Kirchhoff — Love theory), uniqueness of solutions5367 Membrane theory (large deformation, nonlinear membrane theory)340353355365 Membrane theory (large deformation, nonlinear membrane theory), existence of solutions340 Method of formal asymptotic expansions8290268356358381442 Metric (change of metric tensor)319 Micro-vibrations199 Middle surface of a plate16 Middle surface of a shell213 Min-max principle107109 Minimization of a functional2122424369135171243262263284336349352416 Minimizer of an energysee "Minimization of a functional" Minimum principle107109 Mixed finite element method24 Modal synthesis by substructuring method198 Model multi-structure204 Moderately slanted plate245247 Moderately slanted plate theory245247 Moderately thin plate86 Moderately thin plate theory86 Monge — Ampere equation416 Monge — Ampere form391 Multi-dimensional problem163170175197 Multi-structuresee "Elastic multi-structure" Multiplicity of solutionssee "Nonuniqueness of solutions" Multiply connected domain402420448 Naghdi plate theory87 Naghdiplate theory87 Nash — Moser implicit function theorem324 Natural state1719260 Nikodym set388395 Non-frame-indifferent plate theory19323364407 Nonconforming finite element method24 Nonhomogeneous elastic material85 Nonlinear elasticity (three-dimensional nonlinear elasticity)263 Nonlinear elastodynamics327 Nonlinear flexural theory (large deformation nonlinear flexural theory)343 Nonlinear flexural theory (large deformation nonlinear flexural theory), existence of solutions345 Nonlinear Kirchhoff — Love plate theory317321326344356364 Nonlinear Kirchhoff — Love plate theory as a small displacement theory324335 Nonlinear Kirchhoff — Love plate theory, existence of solutions316363 Nonlinear Kirchhoff — Love plate theory, regularity of solutions316 Nonlinear Kirchhoff — Love theory of a plate317324326327334344364 Nonlinear membrane theory (large deformation nonlinear membrane theory)327340353355365 Nonlinear Reissner — Mindlin plate theory328 Nonlinear small displacement theory324335362 Nonlinearly elastic clamped plate problem262 Nonlinearly elastic clamped plate, limit scaled three-dimensional equations277 Nonlinearly elastic clamped plate, scaled three-dimensional equations265295 Nonlinearly elastic clamped plate, scaled two-dimensional energy283 Nonlinearly elastic clamped plate, scaled two-dimensional equations279 Nonlinearly elastic clamped plate, three-dimensional boundary conditons262 Nonlinearly elastic clamped plate, three-dimensional constitutive equation260 Nonlinearly elastic clamped plate, three-dimensional energy263 Nonlinearly elastic clamped plate, three-dimensional equations262294 Nonlinearly elastic clamped plate, three-dimensional equations of equilibrium259 Nonlinearly elastic clamped plate, three-dimensional principle of virtual work259 Nonlinearly elastic clamped plate, time-dependent problem327 Nonlinearly elastic clamped plate, two-dimensional boundary conditions315317 Nonlinearly elastic clamped plate, two-dimensional constitutive equations317 Nonlinearly elastic clamped plate, two-dimensional energy283316 Nonlinearly elastic clamped plate, two-dimensional equations279315 Nonlinearly elastic clamped plate, two-dimensional equations of equilibrium317 Nonlinearly elastic clamped plate, two-dimensional principle of virtual work317 Nonlinearly elastic clamped shallow shell356 Nonlinearly elastic clamped shallow shell, scaled two-dimensional equations358 Nonlinearly elastic clamped shallow shell, three-dimensional equations357 Nonlinearly elastic clamped shallow shell, two-dimensional equations360 Nonlinearly elastic material260 Nonlinearly elastic multi-structure175 Nonlinearly elastic rod325328335 Nonlinearly elastic shallow shell356 Nonlinearly elastic shallow shell in Cartesian coordinates361 Nonlinearly elastic von Karman platesee "von Karman plate" Nonlinearly elastic von Karman plate problem374 Nonuniqueness of solutions288313426 Numerical approximation2487179180198438 Obstacle (plate-lying on an obstacle)5385124438 One-dimensional rod theory8385112 Operator of linearized elasticity (three-dimensional linearized elasticity)18105 Orientation-preserving condition217336 Partially clamped plate17 Periodic boundary conditions84323 Perturbed bifurcation of solutions of the von Karman equations435436 Piezoelectric plate86118 Place (boundary condition of place)16258 Plate lying on an obstacle5385124438 Plate theory derived by a constraint method88 Plate theory derived by integration across the thickness81328 Plate theory with rapidly varying thickness85327 Plate theory with varying thickness85123 Plate with holes85438 Plate with rapidly varying thickness85327 Plate with stiffeners175 Plate with varying thickness85123 Plate with varying thickness of a plate85123 Poisson ratio19 Pressure load424 Properly invariant plate theory335 Quasi-convex envelope351352354 Quasi-convex function351 Quasi-static problem116 Quasilinear equations322340 Rapidly varying thickness of a plate85327 Rayleigh quotient107110 Reduced von Karman equation411426429434435 Reference configuration16 Regularity of solutions of partial differential equations4854290302316404420446 Reissner — Mindlin plate theory8486 Rigidity of an elastic material17 Rotational inertia term118 Rotor blade177246 Satellite177 Scaled Airy stress function390444 Scaled deformation337350 Scaled displacement258389125351441 Scaled eigenfunction108193 Scaled eigensolution108193 Scaled eigenvalue108193 Scaled energy304397283351 Scaled flexural equations (of the linear Kirchhoff — Love theory)3548 Scaled inverted constitutive equation295300380 Scaled Kirchhoff — Love displacement field47110115142195276278333344384 Scaled linearized strain32 Scaled linearized strains32 Scaled Marguerre — von Karman equations445 Scaled membrane equations (of the Kirchhoff — Love theory)3553162 Scaled principle of virtual work295380 Scaled strains32 Scaled stress575883295 Scaled three-dimensional equations218 Scaled three-dimensional equations for a linearly elastic clamped plate28 Scaled three-dimensional equations for a linearly elastic multi-structure140 Scaled three-dimensional equations for a linearly elastic shallow shell237 Scaled three-dimensional equations for a nonlinearly elastic clamped plate265295 Scaled three-dimensional equations for a von Karman plate380 Scaled two-dimensional equations229 Scaled two-dimensional equations for a linearly elastic clamped plate4257 Scaled two-dimensional equations for a linearly elastic shallow shell229 Scaled two-dimensional equations for a Marguerre — von Karman shallow shell443
Scaled two-dimensional equations for a nonlinearly elastic clamped plate279289 Scaled two-dimensional equations for a nonlinearly elastic shallow shell358 Scaled two-dimensional equations for a von Karman plate384 Scaled vector field27295 Scaled von Karman equations390 Scaling of stresses5783295378 Scalings of the deformations337350 Scalings of the displacements258389125137183215264295329357378441 Scalings of the stresses5783295378 Second Piola — Kirchhoff stress tensor18259260266448 Semilinear equations322407 Shallow rod85246 Shallow shell245 Shell212 Simply connected domain389 Simply connected domain in 389 Simply supported plate102103122 Singular perturbations46 Singularities of solutions of partial differential equations245356174 Sliding edges323 Small displacement linear plate theory73 Small displacement nonlinear plate theory324 Small displacement theory73324 Sobolev space858261291 Solar panel of a satellite177 Spectral theory106427 St Venant — Kirchhoff elastic material260 St Venant — Kirchhoff material260262263354357374 Stabilization of elastic multi-structures203 Stabilization of elastic multi-structures (linearly elastic multi-structures)203 Stationary point of a functional262283361 Stationary solution105189 Stiff eigenvalue problem198 Stiff elliptic problem171 Stiff time-dependent problem202 Stiffener (plate with stiffeners)175 Stored energy function336349 Stored energy function of a St Venant — Kirchhoff material336 strain260 Strain tensor319 Strain tensor of the middle surface of a plate68319 Stress18259 Stress couple321 Stress function (Airy stress function)see "Airy stress function" Stress resultant321406 String328348 Substructure133 Substructure of an elastic multi-structure (linearly elastic multi-structure)133 Substructuring method (modal synthesis by substructuring method)198 Surface force (applied surface force)16 Tension in a membrane342 Term of order p in a269 Term of order p in a formal asymptotic expansion269 Thermo-elastic material85420 Thermoelastic plate85 Thickness of a plate16 Thickness of a shell213 Thin-walled rod175 Three-dimensional constitutive equation260 Three-dimensional eigenvalue problem for a plate105 Three-dimensional elasticity tensor2024261318 Three-dimensional energy23336 Three-dimensional equations213 Three-dimensional equations for a linearly elastic clamped plate20 Three-dimensional equations for a linearly elastic multi-structure135 Three-dimensional equations for a linearly elastic shallow shell213 Three-dimensional equations for a Marguerre — von Karman shallow shell440 Three-dimensional equations for a nonlinearly elastic clamped plate262 Three-dimensional equations for a nonlinearly elastic shallow shell357 Three-dimensional equations for a von Karman plate376 Three-dimensional equations of an elastic multi-structure (linearly elastic multi-structure)135 Three-dimensional equations of equilibrium259 Three-dimensional Hellinger — Reissner variational principle2382120 Three-dimensional principle of virtual work23259294375 Three-dimensional response function261 Three-dimensional set with a two-dimentional slit163 Time-dependent coupled multi-dimensional problem199 Time-dependent flexural equations (of the linear Kirchhoff — Love theory)115117 Time-dependent linear Kirchhoff — Love plate theory115 Time-dependent membrane equations (of the Kirchhoff — Love theory)116 Time-dependent nonlinear Kirchhoff — Love plate theory327 Time-dependent problem for a von Karman plate420 Time-dependent problem for an elastic multi-structure199 Time-dependent problem for an elastic multi-structure (linearly elastic multi-structure)199 Time-dependent problem for folded plates203 Time-dependent problem in three-dimensional linearized elasticity112 Time-dependent problems for folded plates203 Time-dependent von Karman equations420 Traction (boundary condition of traction)259 Transmission conditions136 Transmission problem136 Transverse displacement64313403 Transverse displacement of the middle surface64 Tranverse displacement64313403 Two-dimensional boundary conditions of a linearly elastic clamped plate66 Two-dimensional boundary conditions of a nonlinearly elastic clamped plate315317 Two-dimensional boundary conditions of a von Karman plate403407 Two-dimensional boundary conditions of clamping317 Two-dimensional constitutive equation317322 Two-dimensional constitutive equations of a plate317322 Two-dimensional eigenvalue problem for a plate111 Two-dimensional elasticity tensor68319 Two-dimensional energy69283316361 Two-dimensional equations242 Two-dimensional equations for a linearly elastic clamped plate67 Two-dimensional equations for a linearly elastic shallow shell242 Two-dimensional equations for a Marguerre — von Karman shallow shell446 Two-dimensional equations for a nonlinearly elastic clamped plate315 Two-dimensional equations for a nonlinearly elastic shallow shell360 Two-dimensional equations for a von Karman plate386 Two-dimensional equations of equilibrium317322364 Two-dimensional flexural equations6775172 Two-dimensional Hellinger — Reissner variational principle7176 Two-dimensional membrane equations6776112 Two-dimensional plate theory7981 Two-dimensional principle of virtual work71317 Two-dimensional response function318 Two-dimensional slit in a three-dimensional open set163 Two-dimensional strain tensor318322 Two-dimensional time-dependent problem for a linearly elastic plate115117 Two-dimensional time-dependent problem for a nonlinearly elastic plate327 Two-scale convergence85 Uniqueness of solutions214853164238247288328426450 Unperturbed bifurcation of solutions of the von Karman equations433434 Upper face of a plate15 Upper face of a shallow shell214 Vekua plate theory87 Viscoplastic plate85 von Karman equation (reduced von Karman equation)411426429434435 Von Karman equations403404 von Karman equations, bifurcation of solutions433434435436437 von Karman equations, degeneracy into the linear membrane equation432 von Karman equations, equivalence with a two-dimensional displacement problem389 von Karman equations, existence of solutions404416420448 von Karman equations, nonuniqueness of solutions426 von Karman equations, regularity of solutions404420 von Karman equations, uniqueness of solutions426450 von Karman plate375376425447 von Karman plate lying on an obstacle438 von Karman plate with holes438 von Karman plate, scaled three-dimensional equations380 von Karman plate, scaled two-dimensional equations384 von Karman plate, thermoelastic scaled two-dimensional equations420 von Karman plate, three-dimensional boundary conditions374376 von Karman plate, three-dimensional constitutive equations374 von Karman plate, three-dimensional equations374 von Karman plate, two-dimensional boundary conditions386 von Karman plate, two-dimensional equations386403 Weak lower semi-continuity284347418449 Weak lower semi-continuity of a functional284347418449
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