Study On Elastic Wave Propagation Based On The Theory Of Dipolar Strain Gradient Elasticity

In the classical elastic theory, the stress at a material point is assumed to be dependent upon the strain at same material point, and no characteristic length is included in the constitutive relations. Therefore, the classical elastic theory cannot reflect the size effect of the material. According to the classical elastic theory, elastic wave propagation in homogeneous material is nondispersive. In fact, the microstructure effects appear gradually as the incident wavelength is close to the characteristic length of microstructure, which makes the elastic wave propagation exhibiting a dispersion characteristic. In order to make up for some defects that the classical elastic theory does not suffice for an accurate and detailed description of corresponding mechanical behavior in the range of micro and nano-scales, the generalized continuum theories were proposed successively. The dipolar gradient elastic theory is derived according to Mindlin’s linear elastic theory with microstructure which the relative deformation between macro and micro material is neglected in this dissertation. The problems of reflection and transmission at the interface between two different dipolar gradient elastic solids, and through a microstructured slab sandwiched by two half-spaces, and the dispersion characteristics of Bloch waves and band gap properties of elastic waves in the periodic laminated structure formed by periodically repeating of two different gradient elastic solids, and the propagation of thermo-elastic waves in a dipolar gradient thermo-elastic solid are studied based on the dipolar gradient elastic theory proposed. The specific contents of the dissertation include:(1) The dipolar gradient elastic theory is derived in detail based on Mindlin’s linear elastic theory with microstructure. Two additional parameters and are introduced apart from the classic elastic lame constants in this theory. The parameter reflects the elastic properties of the microstructure. The parameter reflects the inertia properties of the microstructure. Because of the influence of and, there are six kinds of wave modes in the dipolar gradient elastic solid, namely, P wave, SV wave, SH wave, SP wave, SS wave and SHH wave.(2)When the incident wave impinges the interface between two different dipolar gradient elastic solids, the dispersion bulk waves and the dispersion surface waves which propagate along the interface and attenuate in normal direction of interface are created. The problem of reflection and transmission is studied based on the continuous interface conditions, namely, the displacements, the normal derivative of displacements, the mono polar tractions and dipolar tractions are continuous across interface. The influences of incident wavelength and microstructure parameters on the reflection and transmission are discussed. The reflection and transmission coefficients in the energy flux ratio are calculated and the numerical results are validated by the energy conservation law.(3) The boundary conditions of dipolar gradient elastic solid include the displacements, the normal derivative of displacements, the monopolar tractions and the dipolar tractions. There are five kinds of possible interface conditions based on all possible situations of the displacement and the normal derivative of displacements. The conception of double energy transfer channels is first proposed in order to explain the influences of the five kinds of possible interface conditions on the refection and the transmission coefficients. The first channel relates monopolar tractions while the second channel relates dipolar tractions.(4) The problem of reflection and transmission through a microstructured slab sandwiched by two half-spaces are studied. The reflection and transmission coefficients determined by the interface conditions between two different dipolar gradient elastic solids and between dipolar gradient and classical elastic solids are calculated numerically in term of the energy flux ratio. The influences of the incident wavelength, the microstructure parameters in the slab and the thickness of slab on reflection and transmission are discussed.(5) The dispersion characteristics and the band gap properties of Bloch wave propagation in a periodic laminated structure formed by the periodical and infinite repeat of two different gradient elastic solids are studied. The in-plane Bloch waves and the anti-plane Bloch waves are both considered in the present work. The oblique propagation situation and the normal propagation situation are also considered, respectively. The influences of microstructure parameters and the ratio of microstructure parameters on the dispersion characteristics of Bloch wave and the band gap properties of elastic waves are discussed.(6) By appropriately postulating the functions of strain energy density, the surface effects of gradient elastic solid are incorporated directly into the constitutive relations. The governing equations and boundary conditions are derived in detail in the case of considering the surface effects. The reflection and transmission coefficients in the energy flux ratio are calculated numerically and the influences of microstructure parameters and the surface parameters on the reflection and transmission coefficients at the interface of two different gradient solids are discussed.(7) The propagation of thermo-elastic waves in the dipolar gradient solid is studied when the thermo-mechanical coupling effects is taken into consideration. The energy conservation equation is derived for dipolar gradient thermo elastic solid. The constitutive relations of dipolar gradient thermo-elastic solid and the heat conduction equation are derived by postulating the functions of free energy density. The four kinds of thermo-mechanical coupling interfaces, namely, the ideal interface, the isothermal interface, the adiabatic interface and the heat resistance interface, are given according to four possible situations of temperature and heat flow density. The reflection and transmission coefficients are calculated and the influences of the all kinds of thermo-mechanical coupling interfaces are discussed.The influence of the microstructure effect, surface effect and thermal-mechanical coupling effect on high frequency elastic wave propagation in dipolar gradient solid material is revealed through the above researches. The research can be used to guide the design of the acoustic microscope, the surface acoustic wave devices, the resonator of bulk acoustic wave and the acoustic metamaterial.