Page 220 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 193

gg. The yield function of hardening materials only depends the stress state:
a. True
b. False
hh. Write the general mathematical formulation of the yield function of materials
characterised by hardening and an elastic limit that is considered to be sensitive
to temperature variations.
ii. Write the condition of consistency for materials characterised by hardening and
an elastic limit that is considered to be sensitive to temperature variations.
jj. Describe the concept of critical state.
kk. The critical state theory can be employed to describe the mechanical behaviour
of both coarse- and fine-grained materials.
ll. Write the mathematical expressions of the CSL in relevant planes.
mm. Describe the main features of multisurface and bounding surface plasticity.

Solutions

a. The variable that characterises most problems involved in mechanics is
the displacement vector.
b. The infinitesimal strain tensor is a symmetric tensor characterised by
nine components that in three-dimensional rectangular Cartesian coordi-
nates reads:

2 3
ε xx ε xy ε xz
ε ij 5 4 ε yx ε yy ε yz 5
ε zx ε zy ε zz
where the diagonal components ε kk [-] are called normal strains and repre-
sent stretching of an element and the off-diagonal components ε kl [-] are
called infinitesimal shear strains and measure angular distortion. The infini-
1
tesimal shear strains are one half of the engineering shear strains, ε kl 5 γ .
2 kl
c. The normal strains are the diagonal components ε kk of the infinitesimal
strain tensor. They represent the stretching of an element. In a two-
dimensional rectangular Cartesian coordinate system ðx;yÞ, the normal
strains read:

@u
ε xx 52
@x

@v
ε yy 52
@y
where u and v [m] are the horizontal and vertical displacements,
respectively.