Some Dynamical Problems in Elastic Continua with Microstructures

Abstract

Chapter 1 contains the introduction of basic laws of classical theory of elasticity, short comings of classical elasticity and evolution of microcontinuum theories including consistent couple stress theory. Basic governing equation of motion and constitutive relations of couple stress theory are given in this chapter. It also contains discussion on various types of waves in elastic medium like body and surface waves, including their literature. Basics of viscoelastic materials with their applications are also discussed in this chapter. Chapter 2 deals with the propagation of Lamb waves in a stress/couple stress free elastic plate. Profiles of Lamb waves are studied using consistent couple stress theory to capture the effects of a microstructural parameter called characteristic length involved in the theory. The governing equation of motion of couple stress theory and constitutive relations involving stresses and couple stresses are solved using various boundary conditions to find dispersion relation of Lamb waves propagating in a plate. Effects of varying characteristic length as compared to cell size have been observed. Phase velocity profiles for different Lamb wave modes are modified under the effect of couple stress theory. Results obtained under couple stress theory are compared graphically with the results of classical theory of elasticity. Since bones have an internal microstructure, so the study has been carried out for a cortical bone type material and this work will be quite useful for evaluation of characteristics of bones and the material exhibiting microstructures. This study may also find possible applications in the fields of non destructive testing techniques. Chapter 3 is an extension of chapter 2 and it contains the study of propagation of Lamb waves in a plate with internal microstructure and loaded with an inviscid liquid on both sides using consistent couple stress theory. Lamb waves are guided waves, propagating in a traction free plate surface, however if the surface of the plate is in contact with the fluid or liquid, a part of energy will leak into the liquid. The dynamical characteristics of a structure get affected when it is surrounded by a fluid or liquid. Dispersion equation of Lamb waves propagating in an elastic plate loaded with a layer of an inviscid liquid of finite thickness is derived. The impact of liquid loadings is studied on the propagation of Lamb waves. Effect of characteristic length 𝑙 is also studied on the phase velocity of Lamb waves in plate for various modes in the presence of liquid loadings. Three different cases for the thickness of liquid loaded on both side of the plate are considered. It is observed that with the increase in the thickness of the loadings the phase velocity tends to decrease. Here, three different values of characteristic length are considered, which are comparable with the internal cell size of the material and their effect is studied on the phase velocity of the Lamb waves. It is observed that with the increase in the value of this parameter the phase velocity of Lamb wave also increases. The considered solid plate has material properties similar to cortical bone, so it also enhances the applicability of this model in the characterisation of the properties of bones loaded with different type of fluids. As the physical model of the problem consists of thin plate loaded with inviscid liquid on both sides, it is of practical use in ultrasonic immersion testing of plates. Chapter 4 contains investigation of propagation of shear horizontal wave (SH) in a layered structure. The propagation of SH waves is studied in viscoelastic layer overlying a couple stress elastic half space. As theory of seismic wave propagation was developed within the frame work of linear elasticity, but later developments showed that earth should be more correctly regarded as a dissipative medium. To encounter dissipation of energy considerations and to overcome the shortcomings of linear elasticity the near sub surface of earth is modelled as linearly viscoelastic material. The geological evidence of heterogeneity within the earth are provided by the wide variations of rocks erupted from volcanoes. The scattering of high frequency seismic waves also support the existence of small scale heterogeneity in the earth lithosphere. Hence, for characterising the internal microstructure of solid earth, heterogeneity and viscoelasticity of the material composition of the earth subsurface has to be taken into account. In this problem, we have investigated shear horizontal wave propagation in a layered structure, consisting of granular macromorphic rock (Dionysos Marble) substrate underlying a viscoelastic layer of finite thickness. Dionysos Marble is a white fine-grained metamorphic marble with a saccharoidal microstructure. SH wave characteristics are affected by the material properties of both underlying substrate and the coating. Dispersion equation for propagation of SH waves using consistent couple stress theory is derived. The effects of microstructural parameter characteristic length of the substrate, along with heterogeneity, internal friction and thickness of viscoelastic layer are studied on the dispersion curves. Real and damping phase velocities of SH waves are studied against dimensionless wave number, for different combinations of various parameters involved in the problem and it is observed that these parameters have significant effects on the propagation of SH waves. The numerical results are also presented graphically for various combinations of parameters involved in the problem. The theoretical consideration of study concerning microstructural effects of the substrate and effects of other parameters of viscoelastic layer on propagation of SH waves, may find possible applications in seismology, exploration geophysics, non destructive testing techniques and in designing chemical and biochemical sensors coated with surface bound receptive layers possessing viscoelastic properties which are used to detect compounds in liquid or gases. Chapter 5 contains study of propagation of SH waves in viscoelastic layer over a couple stress substrate with imperfect bonding at the interface. This chapter is an extension of chapter 4, where we studied SH waves in a layered structure for a perfectly bonded interface between two media, but this condition is difficult to achieve. Due to many reasons like thermal mismatch or some faults in manufacturing process, cracks or defects may appear at the interface which leads to an imperfect interface between two media. Components of displacement field are not continuous at the common boundary of two media in case of imperfect interface. The difference in displacement fields is assumed to depend linearly upon traction vector. These imperfections at the common boundary may affect the propagation of SH waves. Dispersion equation of SH waves in a viscoelastic layer overlying a couple stress substrate with imperfect interface between them has been obtained. Dispersion equations for propagation of SH waves with perfectly bonded interface and slippage interface between two media are also obtained as particular cases. Effects of degree of imperfectness of the interface are studied on the phase velocity of SH waves. Dispersion curves are plotted and effects of material properties of both couple stress substrate and viscoelastic layer are studied. Effects of internal microstructures of couple stress substrate in terms of characteristic length of the material are presented. Effects of heterogeneity, friction parameter and thickness of viscoelastic layer are also studied on the propagation of SH waves. The numerical results are presented graphically. It is found that imperfectness factor at the interface between two media has a significant effect on the phase velocity of SH waves. It is observed that with the decreasing value of imperfectness at the interface, phase velocity of SH waves increases and phase velocity is highest when interface becomes perfectly bonded. In comparison to perfectly bonded interface model for SH waves, this model is more realistic and further the consideration of microstructural effects on the propagation of SH waves, may provide possible applications in the fields of non-destructive testing, semiconductor industry, seismology or geomechanics engineering. Chapter 6 contains study of leaky Rayleigh waves in a homogeneous isotropic elastic solid half space loaded with homogeneous inviscid liquid layer of finite thickness (H) or a liquid half-space. It is interesting to study the changes in the profiles of Rayleigh type waves when homogeneous elastic half space is loaded with liquid. This study becomes more application oriented when the material of half space exhibits internal microstructures. Equations of couple stress theory are solved and dispersion equations for the propagation of leaky Rayleigh waves in a homogeneous solid elastic half space loaded with liquid layer of finite thickness or liquid half space are obtained. As a special case, dispersion equation for propagation of Rayleigh waves in a stress free homogeneous solid elastic half space using consistent couple stress theory is also derived. Rayleigh type waves at the solid-liquid interface are found to be dispersive in this considered model. Phase velocity of leaky Rayleigh waves is studied for three different values of characteristic length (𝑙) which are of the order of internal cell size of the material. It is found that with the increase in the characteristic length of the material, phase velocity of Rayleigh waves also increases. Effects of thickness of liquid layer are also studied. It is observed that phase velocity of Rayleigh waves decreases with the increase in thickness of liquid layer. The properties of leaky Rayleigh waves like velocity, dispersion are affected by both, loaded liquid and underlying solid half space. Material properties of underlying solid can be indirectly obtained by studying profiles of these leaky Rayleigh waves. The applications of these waves range from geophysics to non destructive testing of structures. Generally, most of the non destructive methods for the detection of defects are based on classical model, so this proposed model may provide modifications to these existing methods by considering the microstructural effects of the underlying half space. Chapter 7 Seeing the importance of Rayleigh waves to the fields of geophysics and seismology, in this chapter we have extended our work to study the effects of gravity together with microstructures of substrate and liquid loadings on the propagation of leaky Rayleigh waves. We have solved the problem of propagation of leaky Rayleigh waves in a model consisting of couple stress half space loaded with inviscid liquid layer of finite thickness or a liquid half space under the effects of gravity. Dispersion relations for leaky Rayleigh waves in couple stress half space loaded with inviscid liquid layer of finite thickness or a liquid half space under the effects of gravity are derived using consistent couple stress theory. The effects of gravity, thickness of liquid layer and characteristic length are studied on propagation of leaky Rayleigh waves.