Page 217 - Analysis and Design of Energy Geostructures

190 Analysis and Design of Energy Geostructures

Questions and problems
Statements
a. Which is the fundamental kinematic variable that characterises most problems
involved in mechanics?
b. Write the infinitesimal strain tensor. Define the relationship between the infinites-
imal shear strain and the engineering shear strain.
c. Consider a square body subjected to a uniform temperature change in a two-
dimensional rectangular Cartesian coordinate system ðx; yÞ. Write the normal
strains in terms of displacement that are generated by the application of the tem-
perature variation to the material.
d. Which are the two fundamental terms governing the thermally induced strain of
materials subjected to thermal loads?
e. The unit measure of the linear thermal expansion coefficient is:
i. C

ii. m/ C

iii. C/m

iv. 1/ C

f. Write the infinitesimal strain tensor for an element that is free to deform and
solely subjected to a thermal load. Specify if the produced strain is deviatoric or
spherical.
g. Which is the physical meaning of the compatibility equations?
h. Define the features of the stress tensor and write it in matrix form.
i. The linear mapping t 5 σ ji n i specifies that when the components σ ij acting on
n
i
any three mutually perpendicular planes through a point O are known, the stress
vector on any plane through O can be determined. Prove that this is the case
with reference to the equilibrium of a small portion of a continuum material in
the shape of a tetrahedron. This demonstration will constitute the so-called
Cauchy’s theorem.
j. Write in compact form the stress tensor in terms of its deviatoric and spherical
components.
k. Which are the portions that constitute the total strain in a thermoelastic problem?
l. Which can be the origin of stress in a thermoelastic problem?
m. Consider a body subjected to a mechanical force field, A (e.g. directed downward
along the positive z coordinate of a Cartesian system), and to a temperature varia-
tion, B. Assume that the mechanical force field induces a displacement, strain and
stress field termed u i;A ; ε ij;A and σ ij;A respectively. Suppose that the temperature
variation induces a displacement, strain and stress field termed u i;B ; ε ij;B and σ ij;B
respectively. Which is the resulting displacement, strain and stress field governing
the considered body? To which fundamental principle of mechanics must be
addressed this fact?