Consider a square plate of side length X (see Figure 1) made of a microstructured material...

 

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Consider a square plate of side length X (see Figure 1) made of a microstructured material so that homo- geneous deformations may result in heterogeneous stress states. The plate has thickness t (not shown in the figure) with t <A so that a state of plane stress may be assumed, i.e., z = 0yz = 0zz = 0, and stresses are constant across the thickness (i.e., there is no dependence on z). The state of stress in the thin plate is unknown except for the stress component Tyy(x, y) = σ₁ x y² in the Cartesian orthonormal basis B = (î, ĵ, k). The plate is in static equilibrium with no body forces. The only known surface traction is shown in Figure 1, i.e., on the red surface, the traction vector only has a tangential component go in the direction shown. The shown traction is assumed to be constant on the (red) surface. (a) (10 points) Use the equations of equilibrium to obtain general expressions for the remaining stress components Oxx(x, y) and ory (X, Y). (b) (10 points) Use the provided traction boundary condition to determine any unknowns from the answers of part (a), thus obtaining specific expressions for Orx(x, y) and oxy (x, y). Be sure to check your result by confirming that the stress tensor a satisfies static equilibrium. (c) (15 points) Determine the traction vectors t(n) on the remaining three surfaces of the plate. x 00 Y Y₁ P Consider a square plate of side length X (see Figure 1) made of a microstructured material so that homo- geneous deformations may result in heterogeneous stress states. The plate has thickness t (not shown in the figure) with t <A so that a state of plane stress may be assumed, i.e., z = 0yz = 0zz = 0, and stresses are constant across the thickness (i.e., there is no dependence on z). The state of stress in the thin plate is unknown except for the stress component Tyy(x, y) = σ₁ x y² in the Cartesian orthonormal basis B = (î, ĵ, k). The plate is in static equilibrium with no body forces. The only known surface traction is shown in Figure 1, i.e., on the red surface, the traction vector only has a tangential component go in the direction shown. The shown traction is assumed to be constant on the (red) surface. (a) (10 points) Use the equations of equilibrium to obtain general expressions for the remaining stress components Oxx(x, y) and ory (X, Y). (b) (10 points) Use the provided traction boundary condition to determine any unknowns from the answers of part (a), thus obtaining specific expressions for Orx(x, y) and oxy (x, y). Be sure to check your result by confirming that the stress tensor a satisfies static equilibrium. (c) (15 points) Determine the traction vectors t(n) on the remaining three surfaces of the plate. x 00 Y Y₁ P