Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 62

e 2

- σ

e 1

σ

Fig. 3.6 An illustration for problem 3.3.1

Fig. 3.7 An illustration for problem 3.3.2

equal to as one would expect, and when the normal to the plane n coincides with

the e 2 direction, the stress component t 2 is zero. The shear stress t

s

m ¼ s cos y sin y

has maxima of s /2 at y ¼ p /4 and 3 p /4.

Problems

3.3.1. The flat plate shown in Fig. 3.6 has only two stresses acting on it. There is a
uniform tensile stress s in the e 2 direction and a uniform compressive

stress— in the e 1 direction. Find the stress vector and the components of

the stress tensor acting on the planes whose normals are

s

1
p ð Þ

1
p ð

1
p ð

1
p ð

ð Þ e 1 þ e 2 Þ; ð e 1 þ e 2 Þ; ð Þ e 1 e 2 Þ; ð Þ e 1 þ e 2 Þ

a b c d

3.3.2. The flat plate shown in Fig. 3.7 has shearing stresses acting on each of its

four faces. Find the stress vector and the stress components acting on the
planes whose normals are

t

1
p ð

1
p ð ð a Þ n ¼ e 1 þ e 2 Þ; ð b Þ n ¼ e 1 e 2 Þ