Engineering Rock Mechanics

Questions and answers: strain and the theory of elasticity 61

45.2 The differential equations of force equilibrium were the subjed

of 43.9. The equivalent equations for displacement and strain are

the compatibility 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.

a2e= a2e, a2Yv

-+-=-

ay2 ax2 axay’

A5.2 The strain is the rate of change of the displacement, u, so

au, au, au, au,

ax aY ay ax

E,, = -, E,, = -, and vxy = - + -.

Double partial differentiation of E,, = au,/ax and E,,, = au,/ay with

respect to y and x gives

a2&,, a3u, a2Eyy a3uy

ay2 axap ax2 ayax2.

-- - and - - -

Adding these,

The term in parentheses is the definition of yxy, and hence the right hand

side can be written as a2yx,/axay. Thus, we arrive at the compatibility

equation

a2G, -+- a2eyy - a2yxy

ay2 ax2 axay‘

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, el and e2.

A5.3 - y12 the principal strains

occur where the shear

strains are zero

EY?

E.u -

E, normal strain

+

the two ends of a Mohr’s

circle diameter represent

V (Eu’+Exy) a 2-D strain state

+ y/2, shear strain

45.4 Show why the shear modulus, Young’s modulus and Poisson’s

ratio are related as 0 = €/2(1 + w) for an isotropic material. This

equation holds for an isotropic material but not for an anisotropic