Question

3 ) Consider a small Ax by Ay material element in a thin plate. The state of stress in the plate is specified by four components ơxx, Ơyy and Ơxy-o,r, and the state of strain is specified by Exx. Eyy and Exy Eyx. The other components are zero. Let the x-component of the body force acting on the body be Fx, show that the equation of equilibrium for the element in the x-direction is (ii) Given the constitutive relation between the stress tensor and the strain tensor and that the displacement of the point (x, y) in the plate is given by u(x, y) in the x-direction and v(x, y) in the y-direction, show that the equation of equilibrium derived above can be written as

Answer 1

Let ABCD is the thin plate of gide Delta x & Delta y suppose , sigma_yy, sigma_xy, sigma_yx are components of stresses and otger component are zero i.e y z - direction similarly Let sigma_xx sigma_yy sigma_xy = sigma_yx are strain component y body force in x - direction therefore force acting on si de BC in the x -deirction is here partial differential/partial differential z ma theta matical 0 sigma_xx + partial differential sigma_xx/partial differential x Delta x similarly other force given by sigma_xy + partial differential sigma^_xy/partial differential y Delta x, partial differential_yyx +partial differential sigma_8x/partial differential y Delta y and by same volume but -ve sigin in -ve direction of a and y. Now applying equilibrium at any point in x -direction we have sum of all force a long x -direction = 0 rightarrow (sigma_xx + partial differential sigma_xx/partial differential x Delta x) Delta y - sigma_xx Delta y + (sigma_yx + partial differential sigma_yx/ \r\n