## Wave propagation in elastic solids

Contents

## Lec 6: propagation of elastic waves in continuum

The scattering from a fluid loaded I-beam and the resulting behavior due to fluid-structure interaction are both simulated in this thesis. The first chapter offers a summary of the problem and explains the solid and fluid properties, as well as periodicity, boundary conditions, and the coupling of the two media.

The non-local radiation boundary condition, as well as the formulae for these equations, are derived. Difference formulas for standard solid and corner node boundary points are also derived. Appendix C contains all of the finite difference formulae that were used. The numerical results are found in Chapter IV. In Chapter V, conclusions are drawn, and areas of the issue that need further analysis are addressed.

## Mod-03 lec-15 l15-3 dimensional wave propagation, waves

Elastic Solids Wave Propagation

### How to simulate a 2d wave propagation in a steel plate with

Linearized theory and perfectly elastic media are the subject of Wave Propagation in Elastic Solids.

### Elastic stress wave propagation

The one-dimensional motion of an elastic continuum, the linearized theory of elasticity, elastodynamic theory, and elastic waves in an unbounded medium are all discussed in this book. There is also discussion of plane harmonic waves in elastic half-spaces, harmonic waves in waveguides, and forced motions of a half-space. This book also covers transient waves in layers and rods, slit diffraction of waves, thermal and viscoelastic effects, and anisotropy and nonlinearity effects. A description of equations in rectangular coordinates, time-harmonic plane waves, approximate theories for rods, and transient in-plane motion of a sheet are among the other topics covered. This book is an excellent resource for students and researchers interested in wave propagation in elastic solids.

### Module 4.1 elastic waves in solids

are the coefficients that make up a constitutive system. The integers are represented by Latin subscripts (1,2,3). Participation with respect to the corresponding Cartesian coordinates is indicated by subscripts followed by a comma. Partially differentiated with respect to time t is denoted by a superposed dot. Method assumes that the constitutive coefficients and the microinertia tensor obey the following symmetry relations. A homogeneous transversely isotropic microstretch solid half space is considered. The coordinate system’s origin is on the free surface, and the negative z axis is pointing normally into half-space, which is denoted by z0. The medium is said to be transversely isotropic, with the planes of isotropy perpendicular to the z-axis. If we limit our research to plane strain parallel to the xz plane with displacement and a microrotation vector of the form u=(u 2, u 2, u 2, u 2, u 2, u 2, u 2, u 2, u 2, u 2, u 2, u 2, u 2, u 2, (A *=D 1+D 2+D 3+D 4+D 5+D 6+D 7+D 8+D 9+D 10+D *_3+D* 3+D* 3+D* 3+D* 3+D*_ 4* (B) *=D 1D 2+D 1D 1D 1D 1D 1D 1D 1D 1D 1D 1D 1D 1D 1D 1 *_3+D 1D* 3+D 1D* 3+D 1D* 3+D 1D* 3+D *_4+D 2D* 4+D 2D* 4+D 2D* 4+D 2D* 4+D *_3+D 2D* 3+D 2D* 3+D 2D* 3+D 2D* 3+D _4+D* 4+D* 4+D* 4+D* 4+D*_ 3D 3D 3D 3D 3 _4-K 1K _4-K 1K _4-K 1K _4-K 1K _4-K *_1 _cos _cos _cos _cos _cos _cos _cos -K 2K -K 2K -K 2K -K 2K -K 2K -K 2K -K 2 *_2sin2theta -D 11D* 2sin2theta -D 11D* 2sin2theta -D 11D* 2sin2theta -D 11D* 2sin *_11sin 2theta -D 33D* 11sin 2theta -D 33D* 11sin 2theta -D 33D* 11sin 2theta -D (C), (*_33cos 2theta -L2), *=D 1D 2D=D 1D 2D=D 1D 2D=D 1D 2D=D 1D_ *_3+D 1D 2D* 3+D 1D 2D* 3+D 1D 2D* 3+D 1D 2D* 3 *_4+D 1D* 4+D 1D* 4+D 1D* 4+D 1D* 4+D 3D 3D 3D 3D 3 *_4+D 2D* 4+D 2D* 4+D 2D* 4+D 2D* 4+D 3D 3D 3D 3D 3 *_4-(K 2K)* 4-(K 2K)* 4-(K 2K)* 4-(K 2K * 2D 1+D 11D 2D 1+D 11D 2D 1+D 11D 2D 1+D 11D 2 *_11D 2+D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D _11D _11D _11D _11D _11D _11D 3 +K 2K 3 3 3 3 3 3 3 3 3 2D 2D 2D 2D 2 *_4*)sin 2 theta -(D 33D) *_33D 1+K 1K* 33D 1+K 1K* 33D 1+K 1K* 33D 1+K 1K* 33 * 1D 2+D 33D 1D 2+D 33D 1D 2+D 33D 1D 2+D 33D 1 33D 33D 33D 33D 33 * 3+K 1K* 3+K 1K* 3+K 1K* 3+K 1K* 3+ _1D _1D _1D _1D _1D _1D *_4*)cos 2 theta +L(D) *_11D 33+D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D 11D * 33+K* 33+K* 33+K* 33+K* 33+K* *_1K 2+K 1K 1K 1K 1K 1K 1K 1K 1K 1K 1K 1K 1K 1K *_2(D)sin theta -L2(D)sin theta -L2(D)sin theta -L2(D)sin theta -L2(D)sin theta -L2(D)sin theta -L2 *_3+D* 3+D* 3+D* 3+D* 3+D*_ (D) (*_4), (D) (D) (D) (D) (D) (D) (D) (D) (D) (D *=D 1D 2D=D 1D 2D=D 1D 2D=D 1D 2D=D 1D_ 3D 3D 3D 3D 3 *_4+L(K)* 4+L(K)* 4+L(K)* 4+L(K _1K 2D _1K 2D _1K 2D _1K 2D _1K 2D *_4+K 1K* 4+K 1K* 4+K 1K* 4+K 1K* 4+K 2D 2D 2D 2D 2 _4+D* 4+D* 4+D* 4+D* 4+D*_ _11D 33D _11D 33D _11D 33D _11D 33D _11D 33D *_3+D 11D* 3+D 11D* 3+D 11D* 3+D 11D* 3+D 33D 33D 33D 33D 33 *_3)sin theta cos theta -(D 1D)sin theta -(D 1D)sin theta -(D 1D)sin theta -(D 1D)sin theta -(D 1D)sin the _3D 33D _3D 33D _3D 33D _3D 33D _3D 33D *_33+K 1K* 33+K 1K* 33+K 1K* 33+K 1K* 33+K _1D 2D _1D 2D _1D 2D _1D 2D _1D 2D *_4*)cos 2 theta -(D 2D) 3D 11D 3D 11D 3D 11D 3D 11D _11+K 2K _11+K 2K _11+K 2K _11+K 2K _11+K _2D 1D _2D 1D _2D 1D _2D 1D _2D 1D *_4*sin 2 theta -(K 1K*sin 2 theta -(K 1K*sin 2 theta -(K 1K*sin 2 theta -(K 1K*sin 2 theta 2D 2D 2D 2D 2 _11D 33+K _11D 33+K _11D 33+K _11D 33+K _11D_ _1K 2D 11D _1K 2D 11D _1K 2D 11D _1K 2D 11D _1K_ *_33) cos 2 theta sin 2 theta +K 1K* 1D 33D* 33) cos 4 theta +K 2K* 1D 33D* 33) cos 4 theta +K 2K _2D 11D _2D 11D _2D 11D _2D 11D _2D 11D *_11sin 4 theta – L2 D* 3 D* 4 *_11sin 4 theta – L2 D* 3 D* 4 *_11sin 4 theta – L2 D* 3 D* 4 *_11sin 4 theta – L2 D* 3 D* 4 *_11sin 4 theta – L2 D*

### 05 elastic waves & density of states

Seismic waves cause elastic deformation in the subsurface along their propagation course. Hooke’s law and Newton’s second law of motion are used to derive the equation for wave propagation in elastic solids. To begin, we’ll look at the stress-strain relationship for elastic solids.

Solid bodies, such as rocks, have the ability to propagate forces acting on them. As shown in Figure L-1, imagine an infinitesimally small volume surrounding a point within a solid body with dimensions (dx, dy, dz). The concept of stress is force per unit area. In general, the stress acting on one of the surfaces, say dy dz, can be directed in any direction. Pxx, which is normal to the surface, and Pxy and Pxz, which are tangential to the surface, can be decomposed into three components. The first subscript denotes the direction of the surface’s natural, while the second denotes the direction of the stress component. The normal stress components are the stress components that are normal to the surfaces associated with the infinitesimal volume shown in Figure L-1, whereas the shear stress components are the stress components that are tangential to the surfaces. If a normal stress component is positive, it is tensional, and if it is negative, it is compressional. The nine stress components that make up the stress tensor are needed to keep the solid volume depicted in Figure L-1 in equilibrium.