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Chapter 1 Fundamentals of Elastodynamics 1.1 Basic Hypothesis of Elastodynamics 1.1.1 Continuity Hypothesis The continuity hypothesis holds that the research object of elastic mechanics is the elastic deformable body which is a continuous medium filled with material points without any voids inside. In fact�� all matter is composed of atoms and molecules�� and matter is not continuous at the microscopic level. Even at the macroscopic level�� the existence of internal cavities and cracks cannot be avoided. Continuity is only an idealized model. When studying the macroscopic phenomena and motion laws of objects�� the continuity assumption makes it easier to deal with the problem. For example�� the physical quantities of stress�� strain and displacement are all continuous functions of coordinates�� so that mathematical tools such as calculus can be used to establish and solve mathematical models of dynamic problems. 1.1.2 Elasticity Hypothesis Under the action of external load�� the object will generate stress field and strain field inside. When the amplitude of the stress field does not exceed the elastic limit of the material�� after the external load is removed�� the stress field and the strain field will disappear accordingly. This property is called the elastic property of the material. When the external load is large enough�� for example�� the amplitude of the stress field generated inside the object exceeds the yield limit of the material�� there will be residual deformation (i.e. plastic deformation) existing inside the object and cannot be recovered when the external load is removed. The properties are called the plastic properties of the material. The elasticity hypothesis assumes that the internal stress field of the object is always in the elastic range under the action of external load. 1.1.3 Small Deformation Hypothesis Small deformation means that the deformation at various points within the object due to external load is small relative to the size of the object. In other words�� the strain components (including line strain and shear strain) are all quantities much smaller than 1 �� Their second powers and products are small quantities of higher order relative to the first-order quantity and can be ignored without large precision loss. Due to the small deformation assumption�� the stress field and strain field can be thought to satisfy the generalized Hooke��s law. Moreover�� when establishing the equilibrium equation�� the geometry before deformation (initial configuration) can be used instead of the geometry after deformation (current configuration). 1.1.4 Homogeneous Hypothesis Homogeneous hypothesis means that all points inside the object have the same elastic properties�� namely the material elastic parameters do not change with the spatial coordinates. For non-homogeneous materials�� the elastic parameters of the material are functions of coordinates�� such as functional gradient materials�� where the material parameters are continuous functions of coordinates. Another example is the fiber or particle reinforced composites�� where the material parameters are discontinuous functions of coordinates or piecewise continuous functions. The assumption of uniformity makes the mechanical properties of the material not depend on the location�� but does not guarantee that the material properties do not depend on the direction. The direction-dependent character of the material properties is described by isotropy or anisotropy. Therefore�� the homogeneous assumption is not the same thing as the isotropic assumption. L1.5 Isotropic Hypothesis The isotropic means that each point inside object has the same elastic properties along different directions�� namely the material elastic parameters do not change with the change of direction. For isotropic materials�� only two independent elastic parameters are needed to describe the elastic properties of the material. The commonly used elastic parameters are: modulus of elasticity E�� shear modulus G��Lame constants �� and /x�� Poisson��s ratio v. However�� they are not independent of each other�� and there are only 2 independent parameters. For completely anisotropic materials�� where the material has different properties along different directions�� 21 independent parameters are needed to describe the elastic properties of the material. Usually�� materials have certain symmetry�� are not completely anisotropic�� and their independent material parameters are between 2 and 21. For example�� transverse isotropic materials have five independent elastic parameters; Orthotropic anisotropic materials have nine independent elastic parameters; Cubic crystalline systems have three independent elastic parameters; triangular crystalline systems have seven independent elastic parameters�� etc. 1.1.6 Zero Initial Stress Hypothesis The zero initial stress assumes that the object is in its natural state before the external load is applied and that there