Simulation of Tsunami Effect by Seismic Wave Propagation in Hypoplastic Medium at Vicinity of Free Boundary

A theoretical study of seismic waves propagation in a soil layer with a free surface has a great importance for a prediction in engineering decisions. Wave packets are radiated from an earthquake source and transfer energy. A transformation and a selection of wave packets occur in a process of wave propagating that why waves which arrive in a layer have a length considerably greater than a variation scale of heterogeneity in a medium in a layer near free surface. In the case, when the properties of different layers affect a relatively small degree on a behavior of the waves, an approximation of effective medium gives a fairly good solution. A model of a hypoplastic medium is used for a describing of some effects, which are observed in the time of seismic wave propagation. The model of hypoplastic medium allows describing many effects which are observed in granular soils. We consider a successive application of effective medium and ray methods in order to receive of approximate analytical solutions wishing to describe shear wave propagation in stratified layer, which lies on a half-space.


Introduction
The model of elastic stratified medium widely is applied in seismology geotechnique [1][2][3][4]. In the case, when a medium constitutes from discrete layers, then it is necessary to solve the boundary problem for each layer. It is enough a laborious investigation [5]. The dynamic equations for inhomogeneous media are the differential equations with variable coefficients. As it is known for solving of these equations there are no general analytical methods. The most famous methods which are applied for solving of differential dynamical equations for inhomogeneous media are a ray method [6] and method of effective medium [7]. Some time ago a hypoplastic model was applied for an investigation of seismic wave propagation in a soil [8,9]. The model of hypoplastic medium allows describing many effects which are observed in granular soils. We consider a successive application of effective medium and ray methods in order to receive of approximate analytical solutions for description of shear wave propagation in stratified layer, which lies on a half -space.
As it is known a real soil is inhomogeneous, is usually stratified in a depth. We take the model of effective medium which has the same macroscopic properties as real inhomogeneous medium. The effective model can be received on a basic of experimental results in the form of phenomenological theory or on a basic of theoretical accounts. We take as effective model a hypoplastic medium [8,9], which describes a medium with initial stresses increased in depth linearly. It is correct if a thickness of each layer is comparatively less than a thickness of a great layer.
For a solving of constitutive phenomenological equations, we apply a ray method. An application of this method is correct if a wave length (or a width of a wave packet) is less than a variation scale of effective properties. It is mean that macroscopic properties change monotonically in a depth.

Formulation of Problem
The layer of stratified granule medium lies on the surface 10 1 x x ≥ x < A propagation of wave packet in continuum is described of the first law of thermodynamic [10] where is a kinetic energy, is a power of generalized mass forces k F and surface forces t , Q is heat power.
For constant density and isothermical processes where ij t is stress tensor.
Let V be a volume of current pipe, S is a surface of current pipe (Fig. 2) then we obtain Figure 2. The ray pipe, which is formed by rays Used a theorem about an average to the integrals in Eq. 1, we obtain [11] ( ) We have used the Eq. 3 in pure mechanical formulation (without a registration of a mechanic energy transformation to thermal energy and either kinds of energy). Universal Journal of Mechanical Engineering 6(1): 9-20, 2018 11 The Umov's -Poyinting's vector P (а density of flow energy) describes a direction of energy transmission in a medium.
Lines of energy flow are found from the equation [12] , P P ds is radius vector of a point on an energy flow line, s is a distance along a flow line, τ is unit tangential of the vector to a flow line, ij t is stress tensor, i ϑ is a velocity vector.
A variation of the vector τ along a flow line is where n is normal vector, k is curvature of a flow line, R is curvature radius of a flow line.
A flow line curvature depends on medium parameters. We must set boundary conditions for each layer.

Shear Wave Propagation in Stratified Granular Layer with Effective Hypoplastic Properties
Let a layer of a thickness L be on a half-space 10 1 x x ≤ . The layer L constitutive from some layers of thicknesses ) ,..., 2 , For example, it may be an interchange of layers of sand and clay (Fig. 1).
Let a shear wave (SH) be to incidences on a plane boundary 10 1 x x = and is propagated from the half-space 10 1 x x < . If we want to solve the problem exactly we must set boundary conditions (or connected conditions) and describe wave propagation for each layer. There are very unwieldy expressions especially for multiple scattered waves.
A method of effective medium allows obtaining a solution of this problem [7]. Applied an averaging method (method of homogenization or energy continuation) we obtain a phenomenological model of inhomogeneous medium [7].
Suppose that macroscopic (effective) conditions of layer medium are described of the equations of a hypoplastic medium [8,9].
In general case the equations of wave propagation in hypoplastic medium have the form [6,7] , Div The kinetic equations we write in the form of constitutive equations of a hypo plasticity [8,9] ), , , where  T is Jaunman's derivative time. , in (8) T  is material derivative time, D and ω are a tensor of rate and a spin tensor respectively. , 2 (10) and pore quantity e satisfies the equation We represent the each field value  (13) Here and in the future the sign ~ we do not write. Let Eq. The expression (14) is written in nonindex form (directly designation). In the index form the members of equation (14) have trT ,  where a e is the minimal possible void ratio, c e is the critical void ratio, i e is void ratio in the least state, L, β, n, h are material parameters, c ϕ is the friction angle in critical state. For example, in Table 1 the constitutive hypoplastic parameters of Hochstetten sand are given [8,9]. Table 1. Hypoplastic parameters of Hochstetten sand [8,9] ] [°ϕ The equation for disturbances follows from Eq. 9 for case when Therefore the Eqs. 5, 6, 8, 10 describe a disturbance propagation in hypoplastic medium.

Shear Wave Propagation in Effective Hypoplastic Medium
Set initial and boundary conditions on plane The coefficients i K we write analogous [6,7] Combined Eq. 24, Eq. 25, Eq. 21 we obtain the equations for wave disturbance propagation in the layer with effective properties

Solving of Equations of Shear Wave Propagation in Layer
The differential equations Eq. 26, Eq. 27 have variable coefficients. There are no general analytical methods for a solving of similar equations. The ray method is the most effective among different asymptotic methods for solving of differential equations with variable coefficients [6].
For nonstationary waves an application of this method is correct, if a wave length λ is much less than a variable of a scale of effective layer parameters. It has place if an inhomogeneity of a layer changes in a depth monotonic.
It is known that usually a stiffness of a layer changes local nonmonotonic in a depth, but effective stiffness is monotonic function of spatial coordinate in a depth.
Let the coefficients   (33) (34) Substituted Eq. 32, Eq. 33 into Eq. 26, Eq. 27 we obtain where ( ) In the formula Eq. 40 we take the sign "+" for waves which propagate in the direction + 1 x and the sign "-" for waves in direction -1 x .
Substituted Eq. 40 into Eq. 37, Eq. 38 we obtain the equations of transfer for In formulas Eq. 41, Eq. 43 the members at 0 = n are principal because they describe main part of wave energy. Supposed 0 , 1 − = n in the equations Eq. 37, Eq. 38 we obtain the equations Eq. 39, Eq. 40 and the equations of transfer or transformed we obtain the law of conversation of energy along ray pipes where S is closed surface of ray pipe, n is normal vector to S [17] From Eq. 51 follow the conditions where dS is the area of a base of ray pipe at Let initial impulse be an impulse function where the function The boundary conditions by The wave which comes in the point 1 x is described with a registration Eq. 55 in the formulas The conditions Eq. 54 allow to obtain a variation of wave profile. With a registration Eq. 58, Eq. 59 we have (61) 4 1 1 3 ( then a wave profile takes place a compression. Therefore, the rate displacement wave has the compression and the shear stress wave has the decompression (Fig. 4).

Oblique Wave Incidence on Boundary of Layer
Consider the oblique wave incidence on the boundary 10 1 x x = from the half space 10 1 x x < . In this case the rays will be curve lines which satisfy to the equations [17].
where τ is a tangential vector to ray trajectory, n is a refraction coefficient, s is a length along a ray.
For the systems of differential equations Eq. 61 we set boundary conditions on initial surface (64) If ray trajectory is found so surface (eiconal) ψ is calculated along a ray accordingly do formula (67) where ( ) ) , ( / , The behavior of rays for different angles between an axis 1 x and rays is depicted on Fig. 6. How it follows from Eqs. We set the boundary conditions for the direct wave ( ) The refracted wave may be written with the help of the formulas [7,19] in the form where ( ) 1 x R is a function which satisfies to Riccatti's equations [18,19] ( ),  x R allow taking into account an effect of a decreasing of an energy of direct wave and increasing of reflective wave energy. As it is known [18,19] , 1 2 Therefore, the Eqs. 77, 78, 80, 82, 83 gives us the solution of the problem about the propagation of reflection wave in the hypoplastic layer.
On Fig. 7 is depicted the dependency of  Fig. 8 is depicted the dependency of

Conclusions
Shear waves arises in sources of earthquakes and propagate in direction of the earth surface. In the paper is described the process of shear wave which is arisen and propagates in stratified layer in free surface vicinity. A change of wave amplitude depends on wave resistance of effective medium which decreases in direction of free surface. In this case amplitudes of a displacement and a rate displacement increase but an amplitude of shear stress decreases. From energy law conversation, in the case a disturbance in the form of impulse function we obtain, that an impulse high increases for displacements and displacement rates but a width decreases. It is on contrary for a stress disturbance. An increasing of displacement and displacement rate is analogous to tsunami effect in case when a wave goes to a bank, however, a bend of a wave comb is absent. A wave, reflected from a free surface, propagates in the direction -1 x and is summed from a direct and a reflective wave. From the energy conservation law, it follows that an amplitude of a direct wave decreases and an amplitude of a reflective wave accumulates and increases in the direction of a free surface. A combined application of effective medium method and of ray method allows solving the problems of wave propagation in a stratified medium. A model of a hypoplastic medium is applied in the capacity of an effective medium. This approach is correct if thickness i l of each layer is more less then L ( )