Page 219 - Analysis and Design of Energy Geostructures

192 Analysis and Design of Energy Geostructures

v. Consider a three-dimensional body subjected to a uniform temperature variation ΔTðtÞ

[ C], while zero displacements are maintained at all point of the bounding surface.
i. Write the solution of the corresponding thermoelastic problem in terms of
displacements u; v; w [m], stresses σ ij [Pa] and strains ε ij [ ], in extended form.
ii. Write the surface stress components, X; Y ; Z [Pa] that are needed to maintain
the considered state of stress.
w. Calculate the deformation of a steel bar characterised by a linear thermal expan-

sion coefficient of α 5 12 με= C and subjected to a temperature variation of

ΔT 5 20 C assuming that free deformation is ensured and referring to one-
dimensional conditions. Assuming that the bar is 20 m in length, calculate the
associated thermally induced displacement. Which would be the magnitude of
this displacement in a completely restrained case?
x. Calculate the stress that is generated in a completely restrained steel bar charac-

terised by a linear thermal expansion coefficient of α 5 12 με= C and Young’s
modulus of E 5 210 GPa, subjected to a temperature variation of ΔT 5 20 C.

Refer to one-dimensional conditions.
y. How is the degree of freedom, DOF, of an energy pile mathematically defined?
Specify the meaning of the terms involved in its definition.
z. Which is the magnitude of the displacement of a bar subjected to a temperature
variation in a completely restrained case?
aa. Define the property of plasticity.
bb. In reality, a plastic behaviour directly follows an elastic behaviour, that is the
transition is abrupt:
a. True
b. False
cc. When plasticity is addressed in the context of nonisothermal conditions, two
main approaches can be considered to model the behaviour of materials.
Describe these approaches.
dd. Define the physical meaning of a yield criterion. Express it mathematically.
ee. Write the generalised formulation for a flow rule in situations in which the unit
normal to the plastic potential approaches a finite number of linearly indepen-
dent limiting values as the stress point approaches the singular point in
question.
ff. Materials behaving according to the theory of hardening plasticity are charac-
terised by a size, location or shape of the yield surface that may change under
the development of plastic deformations:
a. True
b. False