Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 151

Fig. 6.4 See Problem 6.3.11

the stresses in terms of the components of the displacement vector into the
equations of motion ( 6.18 ).
6.3.8. Use ( 6.23 ) and (5.34H) to determine the restrictions on the coefficients l

0 for all nonzero strains E .
6.3.9. Use the result of the previous problem and Table 6.2 to obtain restrictions
on the values of the isotropic Poisson's ratio

so that E C E

and m >

.
6.3.10. Show that, in an isotropic elastic material the principal axes of stress and
strain always coincide. {Hint: Recall that the principal axes of stress (strain)
are characterized by the vanishing of the shearing stresses (shearing
strains)}.
6.3.11. Prove that the principal axes of stress and the principal axes of strain cannot
coincide in a triclinic material.
6.3.12. Under what conditions do the principal axes of stress and the principal axes
of strain not coincide for an orthotropic elastic material?
6.3.13. Verify that the conditions of compatibility in terms of strain (2.53) or (2.54)
are identities by substituting the components of (2.49) or (2.52) into (2.54).
6.3.14. The bar shown in Fig. 6.4 is made of an orthotropic material. It is fixed to a
rigid surface at x 3 ¼

n

0; the origin of the coordinate system is positioned at
the center of the bar where the bar is attached to the rigid surface. It is
subjected to constant stress acting in the x 3 direction along its lower

surface. The orthotropic elastic constants appear in the strain-stress
equations ( 6.27 ) and ( 6.28 ).
Assume a stress state in the bar and then calculate the strain state in the
bar. Next calculate the displacement field. If this calculation takes more
time than you have, write out the steps you would take to find the solution
and guess what the solution is for the displacement field.
6.3.15. As an extension of Example 6.3.1, show that if the beam in Fig. 6.3 is bent
about both the x 1 and the x 2 axes, and if is also subjected to an axial tensile
load of magnitude P , then T 33 ¼

s

M 2 x 1 / I 22 .
6.3.16. This problem is a plane stress problem, which means that the stressed
domain is a thin plate of thickness h under the action of forces applied at
the boundary and organized so that their directions all lie in the plane of the
plate. The domain is a rectangular region of length L and width d .

P / A þ M 1 x 2 / I 11