Page 218 - Analysis and Design of Energy Geostructures
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Deformation in the context of energy geostructures 191
n. Consider a body subjected to a given force field. Assume that at a certain time an
equal but opposite force field is applied to the same body. Which is the resulting
displacement, strain and stress state characterising the considered body?
o. Can a material free to deform due to the application of a temperature variation be
subjected to stress?
p. Write the stress strain relations for a thermoelastic material in compact and
extended forms.
q. In a three-dimensional case, to which parameter is proportional the thermally
induced stress? Justify this answer.
r. Calculate the magnitude of the thermally induced stress for a reinforced concrete
cubic sample subjected to a temperature variation of ΔT 5 10 C, characterised
by a bulk modulus of K 5 20 GPa and a linear thermal expansion coefficient of
α 5 10 με= C, which is completely restrained to deform. Repeat the calculation
for a soil cubic sample characterised by a bulk modulus of K 5 30 MPa.
s. The stress tensor at a point in a thermoelastic material reads
2 3
10 0 0
σ ij 5 4 0 5 0 ½kPa
5
0 0 2
i. Find the principal stresses and the orientation of the principal planes.
ii. Find the three principal invariants.
t. The stress tensor at a point in a thermoelastic material reads
2 3
1 1 0
σ ij 5 1 21 0 ½kPa
4
5
0 0 1
Consider the surface passing through this point whose normal vector is parallel
to ½1; 2; 3.
i. Find the components of the stress vector that acts on this surface.
ii. Find the magnitudes of normal and shear stress that act on this surface.
iii. Find the principal stresses of the stress tensor on this surface.
u. Prove that the temperature distribution in a three-dimensional body is a linear
function of the rectangular Cartesian coordinates. That is prove that if
Tx; y; z; tÞ 5 atðÞ 1 btðÞx 1 ctðÞy 1 dtðÞz [ C], then all the stress components are
ð
identically zero throughout the body, provided that all external restraints, body
forces and displacement discontinuities are absent. Under the same conditions,
prove that the considered temperature distribution is the unique one for which all
stress components are identically zero.
