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SIMULATION OF SHOCK COMPRESSION AND THERMAL EXPANSION OF SIMPLE SUBSTANCE AND COMPLEX COMPOUNDS

Mathematical modeling of physical processes using modern computer technology is applied for studying the behavior of continuous media with dynamic loads. The system of equations for the mechanics of a continuous medium is the laws of conservation of mass, momentum, and energy. This system of equations is closed by the equation of state constructed for a elementary substance or a complex chemical compound. A new method of the formulation of state equations in the area of not very big pressures is offered at this paper. Basic parameters of the equation of state are selected for complex chemical compounds using the original version of the Simplex-method. The method uses linear dependences of the shock wave velocity against the substance velocity and thermal expansion data under constant pressure. For the target function a sum of quadratic differences between calculated and experimental values has been selected. coincidence of theoretical and experimental values is obtained. Satisfactory coincidence of theoretical and experimental values is obtained. The paper was supported by RFBR, Grant 13-01-00072.


Introduction
Now in literature [1][2][3] a large number of the equations of a state of substances (EOS) from very simple to very difficult is described. When studying the behavior of continuous media that are exposed to dynamic loads, mathematical modeling is used, which is performed on computers.The number of components in difficult models of multicomponent media with chemical reactions and phase transitions can reach several tens, and even hundreds. When calculating a mixture in such models, computer time can be stretched for a considerable time even with the use of modern computers. The proposed low-parametric EOS will significantly reduce the calculation time on the computer of complex physical processes. The paper was supported by RFBR. Grant 13-01-00072.

Equations at the Surface of a Strong Discontinuity
The laws of conservation for mass, pulse, and energy at the surface of a strong discontinuity (shock wave) in the case of an ideal medium (the deviator of the stress tensor is zero and there is no thermal conductivity) have the following form Values without an index in equations (1)-(3) characterize the state behind the discontinuity, D -the velocity of the shock wave. Values with index 0, such as ρ 0density, U 0 -mass velocity, P 0 -pressure, E 0 -specific internal energy describe the state of the substance before discontinuity. Let us consider a shock wave in a substance at rest. According to [1], the equation of state was considered under the assumption that the quantities P 0 and E 0 are negligible in comparison with P and E. Equations (1)-(3) take the form of P 0 = 0, E 0 = 0 and U 0 = 0.
The system consists of three equations (4), (5) and contains five quantities P , ρ, E, U, D. If any two of them are obtained form experiments, the remaining quantities can be found from expressions (4) and (5). The point on the Hugoniot adiabat is completely determined. The dependence between experimentally measured and known for more than fifty years and is satisfactorily described by a linear ratio On the basis of a large number of processed experimental data, the reference book [2] contains information on the D(U) ratio for elements and compounds.
Pressure is defined by the caloric equation of a state P = P (ρ, E) form in case ρ and E are independent thermodynamic variables. Complex and labor-consuming modern EOS are considered in [3,4]. However, for the express calculations it is possible to use rather simple EOS.
In [1] the EOS of a form is considered where C 0 -sound speed before shock wave for P = P 0 , ρ = ρ 0 , S -enthropy, n = const, γ = const, x = ρ 0 /ρ . The dependencies P X (x), E X (x) and P T (E T , x) follow from (8) and (9) EOS is obtained by substituting (10) into (7), which takes the following form where According to [1], equation (11) is called an "equation with matched γ and n" if γ = n. Thus, equation (11) takes the form Comparing expression (13) to expression (6), it is possible to see that these dependencies differ. The smallest differences are in area As the assumption of constancy n in EOS (12) leads to a nonlinear dependence in expression (13) which, in turn, does not agree with the linear ratio D (U) (6), we do the opposite. Linear ratio D (U) (6) is taken as a basis. Values P and E are expressed through C 0 and b from equations (4)- (6). After that, we will substitute the received expressions in (11) and we will find the equation in which n along a Hugoniot adiabat can't be a constant quantity

Equation of State
The behavior of the substance in place of EOS (12) will be described by an equation in which the assumption that n depends on x In the expression (15) ρ 0K , C 0K is density and sound speed at the point of P = 0, T = 0, In [5][6][7], EOS of the type (15) was used, where the value n depends on x. In equation (14), the quantity n is linearly dependent x on the range D and U in which relation D (U) (6) is valid. From the expressions (4), (5) and (15) follows the dependence n(x) (14), hereinafter named as experimental.
The function n(x) must have a maximum in the neighborhood x ≈ 1 according to [2] and [3]. Thus, if x = 0 and n = n 0 , then it is true that n m > n 0 . We will use a simple function of the form.
n (x) = n 0 + (n m − n 0 ) where Knowing that for x = 0, and x = ∞ n (x) = n 0 also for x = x m n (x) = n m , from (16), we find the derivative n (x) in the form The derivative n(x) is equal to zero for x = 0, x = ∞ and x = x m . Value n 0 , as well as values n m , x m we select from the condition of the best description of the experimental data. It is made because the area of applicability of low-parametrical EOS is limited to final compression. We take the function φ (x) from (15) in the form closest to expression (12) φ We divide the pressure and energy into cold and thermal components to determine the temperature and heat capacity In accordance with [7], we take the dependence E T (x, T ) in the form where A -individual characteristics of the substance, for simple substances close to 3R µ ; R -universal gas constant; µ -molecular mass. Differentiating expression (20), we obtain the heat capacity at constant x In accordance with [7], the expression for the thermal pressure will have the form Knowing that the equation is fair we obtain the characteristic function θ(x), which is related to equation n(x) The dependence θ(x) follows from expression (24) and (16) where B = 2 + (n m − n 0 ) 2 − (n m − n 0 ) .
In addition, we make the condition that the isobar P = 10 −4 GPa pass through a point characterizing the normal state (P = 10 −4 GPa, T = 293 K, ρ = ρ 0 , C P = C P 0 ), and also through the melting point at P = 10 −4 GPa (T = T m , C P = C P m , ρ = ρ m ). In describing the dependence C P (T ) for P = const must be expressed C P in terms of T and x. To this end, we use the equation from [8] Knowing that ∂P X ∂T X = 0, we differentiate expression (23). The derivative ∂P ∂T X is written in the form Since expression (19) is valid, the derivative ∂P ∂x T will have the form The equation of state of a substance is determined by the set of equations (15)-(17), (25) and contains 7 parameters: ρ 0K , C 0K , A, n 0 , n m , x m and θ 0 . The numerical values of these parameters are determined in such a way as to best describe the behavior of the Hugoniot adiabat state of matter at a point characterizing the normal state of P = 10 −4 GPa, T = 293 K, ρ 0 = ρ (P 0 , T 0 ), C P 0 = C P (ρ 0 , T 0 ), C P = C P 0 and at the melting point P = 10 −4 GPa, T 1 = T m , ρ 1 = ρ m (T m , P 0 ), C P 1 = C P (T m , P 0 ).