ON A CLASS OF BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY FOR AN ORTHOTROPIC BODY IN THE RECTANQULAR CARTESIAN COORDINATE SYSTEM

I. Khomasuridze

I.Vekua Institute of Applied Mathematics I.Javakhishvili Tbilisi State University

The basic equations system describing the thermoelastic equilibrium of a plane-stressed, orthotropic homogeneous body has the form [1]

\begin{matrix}Δ_{21}φ_{1} = \frac{g}{γ_{4}} \frac{∂^{2}T̃}{∂z^{2}}, \\ Δ_{22}Φ_{1} = −\frac{γ_{2}}{γ_{4}} \frac{∂^{2}φ_{1}}{∂z^{2}} + \frac{γ_{5}γ_{2}}{γ_{4}} \frac{∂^{2}T̃}{∂z^{2}} \end{matrix}  (1)

Further we use the following notation

a = \sqrt{γ_{1}γ_{4} − γ_{2}^{2}}i, if γ_{2}^{2}−γ_{1}γ_{4} < 0, i = \sqrt{−1}; a = \sqrt{γ_{2}^{2} − γ_{1}γ_{4}}, if γ_{2}^{2}−γ_{1}γ_{4} ≥ 0,

a_{1} = a − \frac{1}{2μ}, a_{2} = a + \frac{1}{2μ}, g = γ_{2}γ_{5} − aγ_{5} − γ_{3}γ_{4},

Δ_{21} = \frac{γ_{2} − a}{γ_{4}} \frac{∂^{2}}{∂z^{2}} + \frac{∂^{2}}{∂x^{2}} , Δ_{22} = \frac{γ_{2} + a}{γ_{4}} \frac{∂^{2}}{∂z^{2}} + \frac{∂^{2}}{∂x^{2}} , Δ_{23} = \frac{λ_{3}}{λ_{1}} \frac{∂^{2}}{∂z^{2}} + \frac{∂^{2}}{∂x^{2}} ,

γ_{1} = \frac{E_{1} − E_{2}ν_{12}^{2}}{E_{1}^{2}} , γ_{2} = \frac{E_{1} − 2μ (ν_{13} + ν_{23}ν_{12})}{2μE_{1}}, γ_{3} = − (\frac{E_{2}ν_{12}k_{2}}{E_{1}} + k_{1}) ,

γ_{4} = \frac{E_{2} − E_{3}ν_{23}^{2}}{E_{3}E_{2}} , γ_{5} = − (ν_{23}k_{2} + k_{3}) ,

where E_{1}, E_{2}, E_{3} are Young’s modulus along Ox, Oy, Oz; ν_{12} is Poisson’s ratio in the plane Oxy; ν_{13} is Poisson’s ratio in the plane Oxz; ν_{23} is Poisson’s ratio in the plane Oyz; k_{1}, k_{2}, k_{3} are temperature linear expansion coefficients along Ox, Oy, Oz respectively; λ_{1}, λ_{2}, λ_{3} are heat conductivity coefficients along

Ox, Oy, Oz respectively and \frac{∂^{2}T̃}{∂z^{2}} = T is a temperature in the elastic body

(Δ_{23}T = 0, Δ_{23}T̃ = 0).

For the overwhelming majority of orthotropic bodies

μ = \frac{E_{1}E_{2}}{E_{1} (1 + 2ν_{12}) + E_{2}} ,

where μ is a shift modulus in the plane Ozx; The following dependences

E_{1}ν_{21} = E_{2}ν_{12}, E_{2}ν_{32} = E_{3}ν_{23}, E_{3}ν_{13} = E_{1}ν_{31}.

If a ≠ 0, γ_{3} = \frac{λ_{3}}{λ_{1}} γ_{5}, then solution of the system (1) will take the form

φ_{1} = φ̃ + γ_{5}T̃, Φ_{1} = Φ̃ + G_{2}φ̃  (2)

where G_{2} = −\frac{γ_{2}}{2a}, Δ_{21}φ̃ = 0, Δ_{22}Φ̃ = 0.

Taking into consideration (1) , (2), components of the displacement vector U⃗ and components of the stress tensor could be presented by means of the functions Φ̃, φ̃, T̃ in the following form

\begin{matrix}u = (1 + \frac{a_{1}G_{2}}{γ_{2}} ) \frac{∂φ̃}{∂x} + \frac{a_{1}}{γ_{2}} \frac{∂Φ̃}{∂x} + γ_{5} \frac{∂T̃}{∂x}, \\ w = − (1 + \frac{a_{2}G_{2}}{γ_{2}} ) \frac{∂φ̃}{∂z} − \frac{a_{2}}{γ_{2}} \frac{∂Φ̃}{∂z} − γ_{5} \frac{∂T̃}{∂z}, \end{matrix}  (3)

Y_{y} = k_{r}E_{r} \frac{∂^{2}T̃}{∂z^{2}},

Z_{z} = \frac{1}{γ_{2}} \frac{∂^{2}Φ̃}{∂x^{2}} + \frac{G_{2}}{γ_{2}} \frac{∂^{2}φ̃}{∂x^{2}},

X_{x} = \frac{1}{γ_{2}} \frac{∂^{2}Φ̃}{∂z^{2}} + \frac{G_{2}}{γ_{2}} \frac{∂^{2}φ̃}{∂z^{2}},

Z_{x} = −\frac{1}{γ_{2}} \frac{∂^{2}Φ̃}{∂x∂z} − \frac{G_{2}}{γ_{2}} \frac{∂^{2}φ̃}{∂x∂z} .

We consider thermoelastic equilibrium of a plane stressed body which occupies the domain Ω_{p} = {0 < x < x_{1}, 0 < z < z_{1}}.

Let us consider the following boundary value conditions

If x = x_{0} = 0 then X_{x} = 0, Z_{x} = 0, T = f_{1} (z) ,
If x = x_{1} then X_{x} = 0, Z_{x} = 0, T = f_{2} (z) ,
If z = z_{0} = 0 then Z_{z} = 0, Z_{x} = 0, T = f_{3} (x) ,
If z = z_{1} then Z_{z} = 0, Z_{x} = 0, T = f_{4} (x) .

(4)

The solution of problem (1), (4) (solution for temperature problem in the domain Ω_{p} is trivial and can be easily constructed using the method of separation of variables) could be presented assuming in (3) φ̃ = 0, Φ̃ = 0 . But in this case everywhere in the domain the components of the stress tensor

\begin{matrix}Z_{z} = 0, \\ X_{x} = 0, \\ Z_{x} = 0, \end{matrix}

regardless of temperature field, but Y_{y}, u, w depend on the function T, and this is the generalization for orthotropic bodies of temperature effect obtained by N.Muskhelishvili [2].

34 Bulletin of TICMI

R e f e r e n c e s

[1] Khomasuridze N., Khomasuridze I., Two-dimensional Problems of Thermoelasticity in Cylindricall and Rectangular Coordinates. Proceedings of I.Vekua Institute of Applied Mathematics, 48, 1998.

[2] Muskhelishvili N., Some Basic Problems of the Mathematicals Theory of Elasticity. Nauka, Moscow, 1966, (Russian).