University of Massachusetts, Amherst CEE/MIE 630: Advanced Solid Mechanics

HW 01: Due Sept. 22 at the beginning of class

Graded: Problem 1: For the state of stress σx = 2, τxy = −3, σy = −1 Compute the eigenvector associated with the principal stress σp = −2.85 and draw a figure indicating this direction and the associated principal stress on a properly oriented element of material.

Ungraded: Problem 2: Use an eigenvalue approach to solve for the principal stresses and directions for the state of stress shown in Fig. P1.26a. Show the principal stresses on a properly oriented element of material.

Problem 3: Derive the differential equation of equilibrium for stresses the x-direction in three dimensions.

Problem 4: Consider the strain field given by

εx = y2 + y4

εy = x2 + x4

γxy =4xy(x2 +y2 +1)

(a) Compute the corresponding displacement field by integrating appropriately

(b) How would you determine constants/functions that appear in the answer to (a)?

Problem 5: Consider the triangular piece of material shown below. Write down the traction boundary condition on the inclined face in the x-y coordinate system. The applied stress σ0 is uniform and is applied normal to the face.

Problem 6: Consider the displacement field

u(x, y) = c1y + c2x

v(x, y) = c3x + c4y

with c1 =1e−3, c2 =−1e−3, c3 =−2e−3, c4 =1e−3.

(a) Compute the strains

(b) Compute the stresses

(c) Check compatibility and equilibrium