Questions 5,1-5,10:
strain and the theory of
elasticity
- I What is the meaning of the first stress invariant and the first strain
invariant?
45.2 The differential equations of force equilibrium were the subject of
Q3.9. The equivalent equations for displacement and strain are the com-
patibility equations; these equations ensure that the normal and shear
strains are compatible, so that no holes, tears or other discontinuities
appear during straining. Show that the following compatibility equation
is valid:
45.3 Draw a Mohr circle for strain, indicating what quantities are on
the two axes, how to plot a 2-D strain state, and the location of the
principal strains, EI and 82.
45.4 Show why the shear modulus, Young's modulus and Poisson's
ratio are related as G = E/2(1 + u) for an isotropic material. This
equation holds for an isotropic material but not for an anisotropic
material - why? Hence explain why five elastic constants are required
for a transversely isotropic material rather than six.
Q5.5 (a) How can the strain in a particular direction be found from the
strain matrix components and hence how can a strain gauge rosette be
used to estimate the state of strain at a point, and hence the state of stress
at a point?
(b) Assume that strains measured by a strain gauge rosette are
~p = 43.0 x and that the
gauges make the following angles to the x-direction: Op = 20", OQ = 80"
and OR = 140". Determine the principal strains and their orientations and
then, using values for the elastic constants of E = 150 GPa and u = 0.30,
determine the principal stresses and their orientations.
EQ = 7.8 x and ER = 17.0 x