L. Giorgashvili and G. Karseladze

Georgian Technical University Tbilisi, Georgia

A GENERAL REPRESENTATION FORMULA FOR A SOLUTION OF THE SYSTEM OF STATICS OF ELASTIC

MIXTURES

The system of differential equations of statics of the linear theory of elastic mixtures (in the case of two-dimensions) reads as follows

a_{1}Δu′ + b_{1} grad div u′ + cΔu′′ + d grad div u′′ = 0,

cΔu′ + d grad div u′ + a_{2}Δu′′ + b_{2} grad div u′′ = 0,

(1)

where u′ = (u′_{1}, u′_{2}) and u′′ = (u′′_{1}, u′′_{2}) are the partial displacement vectors,

a_{1} = μ_{1} − λ_{5}, a_{2} = μ_{2} − λ_{5}, c = μ_{3} + λ_{5}, b_{1} = μ_{1} + λ_{5} + λ_{1} − \frac{ρ_{2}}{ρ} α′_{2},

b_{2} = μ_{2} + λ_{2} + λ_{5} + \frac{ρ_{1}}{ρ} α′_{2}, d = μ_{3} + λ_{3} − λ_{5} − \frac{ρ_{1}}{ρ} α′_{2}, α′_{2} = λ_{3} − λ_{4},

ρ = ρ_{1} + ρ_{2}, ρ_{1} and ρ_{2} are the partial densities, λ_{1}, λ_{2}, …, λ_{5}, μ_{1}, μ_{2}, μ_{3} are elastic constants.

The following result is proved: a solution to the system (1) can be represented by means of four harmonic functions

u′(x) = grad Φ_{1}(x) +

+ r^{2} grad \{[(α_{1} + \frac{1}{2} )(r \frac{∂}{∂r} + 2) − 1]Φ_{2}(x) + β_{1}(r \frac{∂}{∂r} + 2)Φ_{3}(x)\} −

− xr \frac{∂}{∂r} [(2α_{1} − 1)Φ_{2}(x) + 2β_{1}Φ_{3}(x)] + χ′(x),

u′′(x) = grad Φ_{4}(x) +

+ r^{2} grad \{α_{2}(r \frac{∂}{∂r} + 2)Φ_{2}(x) + [(β_{2} + \frac{1}{2} )(r \frac{∂}{∂r} + 2) − 1]Φ_{3}(x)\} −

− xr \frac{∂}{∂r} [2α_{2}Φ_{2}(x) + (2β_{2} − 1)Φ_{3}(x)] + χ′′(x),

where ΔΦ_{j}(x) = 0, j = 1, 2, 3, 4, r \frac{∂}{∂r} = (x · grad),

χ′(x) = \begin{cases}A_{0}x + B_{0}x̃, & x ∈ Ω^{+}, \\ 1/r^{2}(A_{0}x + B_{0}x̃), & x ∈ Ω^{−}, \end{cases} χ′′(x) = \begin{cases}C_{0}x + D_{0}x̃, & x ∈ Ω^{+}, \\ 1/r^{2}(C_{0}x + D_{0}x̃), & x ∈ Ω^{−}, \end{cases}

x = (x_{1}, x_{2}), x̃ = (−x_{2}, x_{1}), A_{0}, B_{0}, C_{0}, and D_{0} are arbitrary constants, Ω^{+} is a bounded domain, Ω^{−} = ℝ^{2} \ \overline{Ω^{+}},

α_{1} = \frac{1}{2Δ_{1}} (cd − b_{1}a_{2} − Δ_{1}), β_{1} = \frac{1}{2Δ_{1}} (cb_{2} − a_{2}d),

α_{2} = \frac{1}{2Δ_{1}} (cb_{1} − a_{1}d), β_{2} = \frac{1}{2Δ_{1}} (cd − a_{1}b_{2} − Δ_{1}), Δ_{1} = a_{1}a_{2} − c^{2}.