{"product_id":"homogenization-of-heterogeneous-thin-and-thick-plates-hardback-9781848216525","title":"Homogenization of Heterogeneous Thin and Thick Plates (Hardback) 9781848216525","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eHomogenization of Heterogeneous Thin and Thick Plates\u003c\/font\u003e\u003cbr\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003c\/p\u003e\n\u003cp\u003e\u003cfont size=\"4\"\u003eKaram Sab (Author), Arthur Lebée (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9781848216525, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 6 October 2015\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e294 pages\u003cbr\u003e24.1 x 16.5 x 2.4 cm, 0.635 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003e\u003cp\u003eThis book gives new insight on plate models in the linear elasticity framework tacking into account heterogeneities and thickness effects. It is targeted to graduate students how want to discover plate models but deals also with latest developments on higher order models. Plates models are both an ancient matter and a still active field of research. First attempts date back to the beginning of the 19th century with Sophie Germain. Very efficient models have been suggested for homogeneous and isotropic plates by Love (1888) for thin plates and Reissner (1945) for thick plates. However, the extension of such models to more general situations --such as laminated plates with highly anisotropic layers-- and periodic plates --such as honeycomb sandwich panels-- raised a number of difficulties. An extremely wide literature is accessible on these questions, from very simplistic approaches, which are very limited, to extremely elaborated mathematical theories, which might refrain the beginner. Starting from continuum mechanics concepts, this book introduces plate models of progressive complexity and tackles rigorously the influence of the thickness of the plate and of the heterogeneity. It provides also latest research results. The major part of the book deals with a new theory which is the extension to general situations of the well established Reissner-Mindlin theory. These results are completely new and give a new insight to some aspects of plate theories which were controversial till recently.\u003c\/p\u003e\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e\u003cp\u003eIntroduction xi\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 1. Linear Elasticity 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1. Notations 1\u003c\/p\u003e \u003cp\u003e1.2. Stress 3\u003c\/p\u003e \u003cp\u003e1.3. Linearized strains  6\u003c\/p\u003e \u003cp\u003e1.4. Small perturbations 8\u003c\/p\u003e \u003cp\u003e1.5. Linear elasticity 8\u003c\/p\u003e \u003cp\u003e1.6. Boundary value problem in linear elasticity 10\u003c\/p\u003e \u003cp\u003e1.7. Variational formulations. 11\u003c\/p\u003e \u003cp\u003e1.7.1. Compatible strains and stresses 11\u003c\/p\u003e \u003cp\u003e1.7.2. Principle of minimum of potential energy 13\u003c\/p\u003e \u003cp\u003e1.7.3. Principle of minimum of complementary energy 14\u003c\/p\u003e \u003cp\u003e1.7.4. Two-energy principle 15\u003c\/p\u003e \u003cp\u003e1.8. Anisotropy 15\u003c\/p\u003e \u003cp\u003e1.8.1. Voigt notations  15\u003c\/p\u003e \u003cp\u003e1.8.2. Material symmetries 17\u003c\/p\u003e \u003cp\u003e1.8.3. Orthotropy 20\u003c\/p\u003e \u003cp\u003e1.8.4. Transverse isotropy 22\u003c\/p\u003e \u003cp\u003e1.8.5. Isotropy 23\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart 1. Thin Laminated Plates 27\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 2. A Static Approach for Deriving the Kirchhoff–Love Model for Thin Homogeneous Plates 29\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1. The 3D problem 29\u003c\/p\u003e \u003cp\u003e2.2. Thin plate subjected to in-plane loading  32\u003c\/p\u003e \u003cp\u003e2.2.1. The plane-stress 2D elasticity problem 33\u003c\/p\u003e \u003cp\u003e2.2.2. Application of the two-energy principle 34\u003c\/p\u003e \u003cp\u003e2.2.3. In-plane surfacic forces on ∂Ω ± 336\u003c\/p\u003e \u003cp\u003e2.2.4. Dirichlet conditions on the lateral boundary of the plate 38\u003c\/p\u003e \u003cp\u003e2.3. Thin plate subjected to out-of-plane loading 40\u003c\/p\u003e \u003cp\u003e2.3.1. The Kirchhoff–Love plate model 41\u003c\/p\u003e \u003cp\u003e2.3.2. Application of the two-energy principle 47\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 3. The Kirchhoff–Love Model for Thin Laminated Plates 53\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1. The 3D problem 53\u003c\/p\u003e \u003cp\u003e3.2. Deriving the Kirchhoff–Love plate model 55\u003c\/p\u003e \u003cp\u003e3.2.1. The generalized plate stresses 55\u003c\/p\u003e \u003cp\u003e3.2.2. Static variational formulation of the Kirchhoff–Love plate model 56\u003c\/p\u003e \u003cp\u003e3.2.3. Direct formulation of the Kirchhoff–Love plate model 58\u003c\/p\u003e \u003cp\u003e3.3. Application of the two-energy principle  59\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart 2. Thick Laminated Plates 65\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 4. Thick Homogeneous Plate Subjected to Out-of-Plane Loading  67\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1. The 3D problem 67\u003c\/p\u003e \u003cp\u003e4.2. The Reissner–Mindlin plate model. 69\u003c\/p\u003e \u003cp\u003e4.2.1. The 3D stress distribution in the Kirchhoff–Love plate model 69\u003c\/p\u003e \u003cp\u003e4.2.2. Formulation of the Reissner–Mindlin plate model 71\u003c\/p\u003e \u003cp\u003e4.2.3. Characterization of the Reissner–Mindlin stress solution 72\u003c\/p\u003e \u003cp\u003e4.2.4. The Reissner–Mindlin kinematics 73\u003c\/p\u003e \u003cp\u003e4.2.5. Derivation of the direct formulation of the Reissner–Mindlin plate model 74\u003c\/p\u003e \u003cp\u003e4.2.6. The relations between generalized plate displacements and 3D displacements 76\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 5. Thick Symmetric Laminated Plate Subjected to Out-of-Plane Loading 81\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1. Notations 81\u003c\/p\u003e \u003cp\u003e5.2. The 3D problem 82\u003c\/p\u003e \u003cp\u003e5.3. The generalized Reissner plate model 85\u003c\/p\u003e \u003cp\u003e5.3.1. The 3D stress distribution in the Kirchhoff–Love plate model 85\u003c\/p\u003e \u003cp\u003e5.3.2. Formulation of the generalized Reissner plate model 90\u003c\/p\u003e \u003cp\u003e5.3.3. The subspaces of generalized stresses 91\u003c\/p\u003e \u003cp\u003e5.3.4. The generalized Reissner equilibrium equations 95\u003c\/p\u003e \u003cp\u003e5.3.5. Characterization of the generalized Reissner stress solution 97\u003c\/p\u003e \u003cp\u003e5.3.6. The generalized Reissner kinematics 98\u003c\/p\u003e \u003cp\u003e5.3.7. Derivation of the direct formulation of the generalized Reissner plate model 100\u003c\/p\u003e \u003cp\u003e5.3.8. The relationships between generalized plate displacements and 3D displacements 102\u003c\/p\u003e \u003cp\u003e5.4. Derivation of the Bending-Gradient plate model 106\u003c\/p\u003e \u003cp\u003e5.5. The case of isotropic homogeneous plates 109\u003c\/p\u003e \u003cp\u003e5.6. Bending-Gradient or Reissner–Mindlin plate model? 111\u003c\/p\u003e \u003cp\u003e5.6.1. When does the Bending-Gradient model degenerate into the Reissner–Mindlin’s model? 112\u003c\/p\u003e \u003cp\u003e5.6.2. The shear compliance projection of the Bending-Gradient model onto the Reissner–Mindlin model 113\u003c\/p\u003e \u003cp\u003e5.6.3. The shear stiffness projection of the Bending-Gradient model onto the Reissner–Mindlin model 115\u003c\/p\u003e \u003cp\u003e5.6.4. The cylindrical bending projection of the Bending-Gradient model onto the Reissner–Mindlin model 116\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 6. The Bending-Gradient Theory 117\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1. The 3D problem 117\u003c\/p\u003e \u003cp\u003e6.2. The Bending-Gradient problem 119\u003c\/p\u003e \u003cp\u003e6.2.1. Generalized stresses 119\u003c\/p\u003e \u003cp\u003e6.2.2. Equilibrium equations 121\u003c\/p\u003e \u003cp\u003e6.2.3. Generalized displacements 122\u003c\/p\u003e \u003cp\u003e6.2.4. Constitutive equations 122\u003c\/p\u003e \u003cp\u003e6.2.5. Summary of the Bending-Gradient plate model 123\u003c\/p\u003e \u003cp\u003e6.2.6. Field localization 123\u003c\/p\u003e \u003cp\u003e6.3. Variational formulations 125\u003c\/p\u003e \u003cp\u003e6.3.1. Minimum of the potential energy 126\u003c\/p\u003e \u003cp\u003e6.3.2. Minimum of the complementary energy 127\u003c\/p\u003e \u003cp\u003e6.4. Boundary conditions 128\u003c\/p\u003e \u003cp\u003e6.4.1. Free boundary condition  129\u003c\/p\u003e \u003cp\u003e6.4.2. Simple support boundary condition  130\u003c\/p\u003e \u003cp\u003e6.4.3. Clamped boundary condition 131\u003c\/p\u003e \u003cp\u003e6.5. Voigt notations 131\u003c\/p\u003e \u003cp\u003e6.5.1. In-plane variables and constitutive equations 131\u003c\/p\u003e \u003cp\u003e6.5.2. Generalized shear variables and constitutive equations 132\u003c\/p\u003e \u003cp\u003e6.5.3. Field localization 135\u003c\/p\u003e \u003cp\u003e6.6. Symmetries 136\u003c\/p\u003e \u003cp\u003e6.6.1. Transformation formulas  136\u003c\/p\u003e \u003cp\u003e6.6.2. Orthotropy 139\u003c\/p\u003e \u003cp\u003e6.6.3. π\/2 invariance  140\u003c\/p\u003e \u003cp\u003e6.6.4. Square symmetry 140\u003c\/p\u003e \u003cp\u003e6.6.5. Isotropy 140\u003c\/p\u003e \u003cp\u003e6.6.6. The remarkable case of functionally graded materials 142\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 7. Application to Laminates 145\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1. Laminated plate configuration 145\u003c\/p\u003e \u003cp\u003e7.2. Localization fields  146\u003c\/p\u003e \u003cp\u003e7.2.1. In-plane stress unit distributions (bending stress) 147\u003c\/p\u003e \u003cp\u003e7.2.2. Transverse shear unit distributions (generalized shear stress) 148\u003c\/p\u003e \u003cp\u003e7.3. Distance between the Reissner–Mindlin and the Bending-Gradient model  149\u003c\/p\u003e \u003cp\u003e7.4. Cylindrical bending 150\u003c\/p\u003e \u003cp\u003e7.4.1. Closed-form solution for the Bending-Gradient model 152\u003c\/p\u003e \u003cp\u003e7.4.2. Comparison of field distributions 155\u003c\/p\u003e \u003cp\u003e7.4.3. Empirical error estimates and convergence rate 160\u003c\/p\u003e \u003cp\u003e7.4.4. Influence of the bending direction 161\u003c\/p\u003e \u003cp\u003e7.5. Conclusion 163\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart 3 Periodic Plates 167\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 8. Thin Periodic Plates 169\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1. The 3D problem 169\u003c\/p\u003e \u003cp\u003e8.2. The homogenized plate problem 173\u003c\/p\u003e \u003cp\u003e8.3. Determination of the homogenized plate elastic stiffness tensors 174\u003c\/p\u003e \u003cp\u003e8.4. A first justification: the asymptotic effective elastic properties of periodic plates 181\u003c\/p\u003e \u003cp\u003e8.5. Effect of symmetries 184\u003c\/p\u003e \u003cp\u003e8.5.1. Symmetric periodic plate  185\u003c\/p\u003e \u003cp\u003e8.5.2. Material symmetry of the homogenized plate 186\u003c\/p\u003e \u003cp\u003e8.5.3. Important special cases 187\u003c\/p\u003e \u003cp\u003e8.5.4. Rectangular parallelepipedic unit cell 189\u003c\/p\u003e \u003cp\u003e8.6. Second justification: the asymptotic expansion method 194\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 9. Thick Periodic Plates 205\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1. The 3D problem 206\u003c\/p\u003e \u003cp\u003e9.2. The asymptotic solution 208\u003c\/p\u003e \u003cp\u003e9.3. The Bending-Gradient homogenization scheme 209\u003c\/p\u003e \u003cp\u003e9.3.1. Motivation and descrition of the approach 210\u003c\/p\u003e \u003cp\u003e9.3.2. Introduction of corrective terms to the asymptotic solution 210\u003c\/p\u003e \u003cp\u003e9.3.3. Identification of the localization tensors 212\u003c\/p\u003e \u003cp\u003e9.3.4. Identification of the Bending-Gradient compliance tensor 214\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 10. Application to Cellular Sandwich Panels 219\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1. Introduction 219\u003c\/p\u003e \u003cp\u003e10.2. Questions raised by sandwich panel shear force stiffness 220\u003c\/p\u003e \u003cp\u003e10.2.1. The case of homogeneous cores 221\u003c\/p\u003e \u003cp\u003e10.2.2. The case of cellular cores 223\u003c\/p\u003e \u003cp\u003e10.3. The membrane and bending behavior of sandwich panels 225\u003c\/p\u003e \u003cp\u003e10.3.1. The case of homogeneous cores 225\u003c\/p\u003e \u003cp\u003e10.3.2. The case of cellular cores 226\u003c\/p\u003e \u003cp\u003e10.4. The transverse shear behavior of sandwich panels 229\u003c\/p\u003e \u003cp\u003e10.4.1. The case of homogeneous cores 229\u003c\/p\u003e \u003cp\u003e10.4.2. A direct homogenization scheme for cellular sandwich panel shear force stiffness 230\u003c\/p\u003e \u003cp\u003e10.4.3. Discussion 232\u003c\/p\u003e \u003cp\u003e10.5. Application to a sandwich panel including Miura-ori 235\u003c\/p\u003e \u003cp\u003e10.5.1. Folded cores  236\u003c\/p\u003e \u003cp\u003e10.5.2. Description of the sandwich panel including the folded core 237\u003c\/p\u003e \u003cp\u003e10.5.3. Symmetries of Miura-ori 238\u003c\/p\u003e \u003cp\u003e10.5.4. Implementation 239\u003c\/p\u003e \u003cp\u003e10.5.5. Results 241\u003c\/p\u003e \u003cp\u003e10.5.6. Discussion on shear force stiffness  250\u003c\/p\u003e \u003cp\u003e10.5.7. Consequence of skins distortion 255\u003c\/p\u003e \u003cp\u003e10.6. Conclusion 257\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 11. Application to Space Frames 259\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1. Introduction 259\u003c\/p\u003e \u003cp\u003e11.2. Homogenization of a periodic space frame as a thick plate 261\u003c\/p\u003e \u003cp\u003e11.2.1. Homogenization scheme 261\u003c\/p\u003e \u003cp\u003e11.3. Homogenization of a square lattice as a Bending-Gradient plate  268\u003c\/p\u003e \u003cp\u003e11.3.1. The unit-cell  268\u003c\/p\u003e \u003cp\u003e11.3.2. Kirchhoff–Love auxiliary problem  269\u003c\/p\u003e \u003cp\u003e11.3.3. Bending-Gradient and Reissner–Mindlin auxiliary problems 270\u003c\/p\u003e \u003cp\u003e11.3.4. Difference between Reissner–Mindlin and Bending-Gradient constitutive equation 273\u003c\/p\u003e \u003cp\u003e11.4. Cylindrical bending of a square beam lattice 274\u003c\/p\u003e \u003cp\u003e11.4.1. Lattice at 0° 274\u003c\/p\u003e \u003cp\u003e11.4.2. Lattice at 45°  276\u003c\/p\u003e \u003cp\u003e11.5. Discussion 282\u003c\/p\u003e \u003cp\u003e11.6. Conclusion 283\u003c\/p\u003e \u003cp\u003eBibliography 285\u003c\/p\u003e \u003cp\u003eIndex  293\u003c\/p\u003e\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mechanical engineering \u0026amp; materials [\u003ca title=\"See our other books on Mechanical engineering \u0026amp; materials\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mechanical%20engineering%20\u0026amp;%20materials%20%5BTG%5D%22\"\u003eTG\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Wiley-ISTE","offers":[{"title":"Brand New","offer_id":52449386889496,"sku":"9781848216525","price":100.57,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/files\/9781848216525.jpg?v=1785197647","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/homogenization-of-heterogeneous-thin-and-thick-plates-hardback-9781848216525","provider":"Freshly Printed Books","version":"1.0","type":"link"}