system of pde (solid mechanics)

system of pde (solid mechanics)



Please I am currently solving the following system of Partial Differential

Equations as follows;

-Mx/Dxy = (∂w^2)/〖∂x〗^2 + v

(∂w^2)/〖∂y〗^2 ……………………………………………………………………………….1

-My/Dxy = v (∂w^2)/〖∂x〗^2 +

(∂w^2)/〖∂y〗^2

……………………………………………………………………………………..2

-〖Mxy〗_ /Dxy = (1-v)

(∂w^2)/〖∂x∂y〗^

………………………………………………………………………………3

equation 1 + equation 3 and 2+3 gives; -(Mx + Mxy)/Dxy =

(∂w^2)/〖∂x^2] +

(1-v)(∂w^2)/〖∂x∂y〗 + v

(∂w^2)/{∂y^2}…………4

-(My+〖Mxy〗 )/D_xy = v (∂w^2)/〖∂x^2] +

(1-v) (∂w^2)/〖∂x∂_y〗+

(∂w^2)/〖∂y^2] ………………………………..5

eqn 4 - eqn 5 gives;

(∂w^2)/〖∂x^2] - (∂w^2)/〖∂y^2] = (My-

Mx〗)/((1-v)Dxy ) ………………………………………………………………………………6

complementary solution ; D^2 –Di^2 = 0 CE ; m^2 -1 = 0 ; m = ± 1 CF = w =

f1(y+x) + f2(y-x)

Let R = (My-Mx)/((1-v)Dxy ) P.I = R/(D^2-Di^2 ) a = b = 0 (fail case) = (

R/〖 2D〗) ∬dxdx a = 0

PI = ( x^2 R)/( 2) And so w = f1(y+x) + f2(y-x) + (x^2R)/(2)

the boundary condition is assumed as; w(x,0) = 0 w(0,y) = 0 at x =a and y

= an w(0,an) = 0 w(a,an) = 0 also w(0,0) = 0

substituting these initial boundary values does not lead to specific value

for f1 (x,y) or f2(x,y) so as to give the expression for w.

Please guide me through. Thanks.