The one-dimensional transient perturbation of the bi-component elastic diffusion layer

Аuthors

Goykhburg D. M.^1*, Zemskov A. V.^2**, Tarlakovsky D. V.^3***

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. ,
3. Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia

*e-mail: dmgburg@gmail.com
**e-mail: azemskov1975@mail.ru
***e-mail: tdvhome@mail.ru

Abstract

The stress-strain state of a bicomponent layer is considered taking into account the structural variations caused by diffusion flows. The effect of diffusion to stress-strain state of materials is modeled by use of the locally equilibrium state model of elastic diffusion including the dynamic equations for elastic body and equations of mass transfer. The geometrically linear model of elasticity with diffusion is used.
The bi-component homogeneous elastic layer (an alloy) with the surfaces parallel to the plane x₁Ox₂ of Cartesian coordinate system is considered. It is assumed that at the surfaces x₃= 0,L the displacements, the heat transfer to the environment and diffusion flows are known. The mathematical model can be formulated with respect to Cartesian coordinates as follows:

where t is time, x_i are Cartesian coordinates, u_i are displacements vectors components, L is the thickness of the layer, n^(q) =n^(q) ̶ n₀^(q) is the concentrations variation, n^(q) is the concentration, n₀^(q) is the initial concentration of the q-th component, q in the composition of an two-component solid solution (medium), C_ijkl are components of elastic constants tensor, α_ij^(q) are the components of tensor defined by crystal structure type, D_ij^(q) are the components of self-diffusion tensor; g^(q) is Darken thermodynamic constant, R is the universal gas constant, T₀ is temperature. The coefficients Ʌ₃₃₃₃^(q) are defined by the following formulae:

It is assumed that at the surfaces of the layer the displacement and diffusion flows are set:

where diffusion flows.
The initial terms are assumed zero:

The solution is based on the integral representation of the unknown functions. The right-hand sides of the equations are assumed as delta functions and the problem can be solved using Fourier expansions and Laplace integral transform in time domain. The received Laplace representations are rational functions depending on the transform parameter. Expanding of these functions into common fractions the inverse transforms are found and the fundamental solution of the problem is constructed. The final solution can be obtained as an integral convolution. The constant and exponentially decreasing diffusion flows at the surfaces are investigated as the examples. The numerical calculations have been made in the Maple 13 software.

Keywords:

elastic diffusion, transient problems, Fourier series, the Laplace transform

References

1. Van Byuren Kh.G. Defekty v kristallakh (Defects in crystals), Moscow, lnostrannaya Literatura, 1962, 584 p.
2. Dankov P.D., Churaev P.V. Doklady AN SSSR, 1950, vol. 73, no. 6, pp. 1221-1125.
3. Kubashevskii O., Gopkins B. Okislenie metallov i splavov (Oxidation of metals and alloys), Moscow, Metallurgiya, 1965, 428 p.
4. Prussin S. J. Appl. Physics, 1961, vol. 32, 1876 p.
   
5. Eremeev V.S. Diffuziya i napryazheniya (Diffusion and tension), Moscow, Energoatomizdat, 1984, 182 p.
6. Knyazeva A.G. Vvedenie v lokalno-ravnovesnuyu termodinamiku fiziko-khimicheskikh prevrashchenii v deformiruemykh sredakh (Introduction in local and equilibrium thermodynamics of physical and chemical transformations in deformable environments), Tomsk, 1996, 146 p.
7. Podstrigach Ya.S., Pavlina V.S. Fiziko-khimicheskaya mekhanika materialov, 1965, no. 4, pp. 383-389.
8. Gachkevich A.R., Zemskov A.V., Tarlakovskii D.V. Materialy XVII Mezhdunarodnogo simpoziuma «Dinamicheskie i tekhnologicheskie problemy mekhaniki konstruktsii i sploshnykh sred» im. A.G.Gorshkova, Moscow, TR-print, 2011, vol.3, pp. 96-106.
9. Afanaseva O.A., Zemskov A.V., Tarlakovskii D.V. Nestatsionarnye protsessy deformirovaniya elementov konstruktsii, obuslovlennye vozdeistviem polei razlichnoi prirody, Lvov, Institut prikladnykh problem mekhaniki i matematiki im. Ya.S.Podstrigacha NAN Ukrainy, 2012, pp. 12-16.