Contents Chapter 1 Introduction 1.1 An overview of higher-order continuum theory 1.2 Basic equations of the modified gradient elasticity (MGE) 1.2.1 Modified constitutive equations of gradient elasticity 1.2.2 Principle of virtual work: equilibrium equation and boundary conditions 1.3 Outline of this book Chapter 2 Micro-scale Bernoulli-Euler beam model based on MGE 2.1 Purpose of developing a micro-scale Bernoulli-Euler beam model 2.2 The governing equation and the boundary conditions for the bending problem 2.3 Numerical example of cantilever beams 2.3.1 Boundary conditions statement 2.3.2 Case 1: bending moment loading 2.3.3 Case 2: concentrated force loading 2.4 Comparison and discussion on the size effect 2.4.1 The comparison of micro-beam models 2.4.2 The influence of the internal length scales compared in different direction 2.4.3 Features Chapter 3 Thermal buckling of micro-scale Bernoulli-Euler beams based on MGE 3.1 Purpose of developing the thermal buckling model 3.2 The governing equation and boundary conditions 3.3 Numerical examples of different supported beams 3.3.1 Case 1: hinged-hinged micro-beams 3.3.2 Case 2: clamped-hinged micro-beams 3.3.3 Case 3: clamped-clamped micro-beams 3.4 A comparison of the thermal buckling model with other models 3.5 Chapter summary Chapter 4 Buckling of micro-scale thin-walled Bernoulli-Euler beams based on MGE 4.1 Purpose of developing the buckling model 4.2 Formulations and solution methodology 4.2.1 Governing equations and boundary conditions 4.2.2 Understanding of the governing equations 4.2.3 Solution methodology 4.3 Size effect of the critical buckling load and the buckling modes 4.4 A comparison of the buckling model with other models 4.5 Chapter summary Chapter 5 Thermal post-buckling of micro-scale Bernoulli-Euler beams based on MGE 5.1 Purpose of developing the thermal post-buckling model 5.2 The governing equations and boundary conditions 5.3 General solution of the thermal post-buckling 5.3.1 General solution for micro-beams with immovable axial boundary condition 5.3.2 Analytical solution for hinged-hinged micro-beams 5.3.3 Analytical solution for clamped-clamped micro-beams 5.4 Thermal post-buckling behavior of different supported micro-beams 5.5 Size effect and geometrically nonlinear effect on the thermal post-buckling 5.6 Chapter summary Chapter 6 Thermoelastic damping of micro-scale Bernoulli-Euler beams based on MGE 6.1 Purpose of developing the thermoelastic damping model 6.2 The governing equations and boundary conditions 6.3 Heat conduction equation considering strain gradients
6.4 Exact expression of the thermoelastic damping 6.5 Effects of thermal performance parameters and the size effect on thermoelastic damping 6.6 Chapter summary Chapter 7 Micro-scale Timoshenko beam model based on the MGE 7.1 Purpose of developing the model 7.2 The governing equations and boundary conditions for the bending problem 7.3 Solution methodology: A corresponding finite difference method 7.4 Results and discussion for types of beams 7.4.1 Case 1: a doubly clamped homogeneous micro-beam 7.4.2 Case 2: a clamped-simply supported homogeneous micro-beam 7.4.3 Case 3: a simply supported AFG micro-beam Chapter 8 Thermal buckling of micro-scale Timoshenko beams based on MGE 8.1 Purpose of developing the model 8.2 Modeling 8.3 Solution methodology 8.3.1 General solution for the thermal buckling of micro-beams 8.3.2 Analytical form for the thermal buckling of doubly clamped micro-beams 8.3.3 Analytical form for the thermal buckling of doubly hinged micro-beams 8.4 Results and discussion 8.4.1 The shearing effect on the thermal buckling of micro-beams 8.4.2 Coupling of the shearing and size effects on the thermal buckling of micro-beams 8.5 Chapter summary Chapter 9 Effects of local defects on the buckling of micro-scale Timoshenko beam based on MGE 9.1 Purpose of studying the buckling of micro-beam with local thickness defects 9.2 The governing equations and boundary conditions 9.3 Solution methodology 9.4 Effects of the local defects on the buckling behavior 9.4.1 The effect of defect shape on k_α and buckling modes 9.4.2 The effect of defect position on k_α Chapter 10 Extension of MGE 10.1 A stress analytical solution for Mode III crack within MGE 10.1.1 Problem statement and basic equations 10.1.2 Stress field 10.1.3 Numerical results and discussion 10.2 Discrete and finite element formulation of MGE 10.3 Shear boundary layer analysis based on MGE 10.3.1 Numerical tests 10.3.2 The double 1D-Hermite shape functions for rectangle element Appendix A Appendix B Afterword(後記)
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