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202 Analysis and Design of Energy Geostructures
where ν [ ] is the Poisson’s ratio, α ½με= C is the linear thermal expan-
sion coefficient, E [Pa] is the Young’s modulus, T [ C] is the tempera-
ture, x;y and z [m] are the Cartesian coordinates, σ kk [Pa] are the normal
stresses and σ kl [Pa] are the shear stresses. For the considered tempera-
ture distribution Tx;y;z;tÞ 5 at ðÞ 1 bt ðÞx 1 ct ðÞy 1 dt ðÞz, all equations are
ð
homogeneous. Therefore the solution is
σ xx 5 σ yy 5 σ zz 5 σ xy 5 σ yz 5 σ zx 5 0
If the solution of the previous problem is assumed to apply, the sub-
stitution in the compatibility equations then shows that the temperature
distribution must satisfy the relations:
2 2 2 2 2 2
@ T @ T @ T @ T @ T @ T
5 5 5 5 5 5 0
@x 2 @y 2 @z 2 @x@y @y@z @z@x
and the temperature distribution must then be linear.
v. i. The solution of the thermoelastic problem is:
u 5 v 5 w 5 0
ε xx 5 ε yy 5 ε zz 5 ε xy 5 ε yz 5 ε zx 5 σ xy 5 σ yz 5 σ zx 5 0
Eα
σ xx 5 σ yy 5 σ zz 52 3KαΔT 52 ΔT
1 2 2υ
where x;y and z are the Cartesian coordinates, u;v and w [m] are the
displacements in the associated directions, ε kk [ ] are the normal
strains, ε kl [ ] are the shear strains, σ kk [Pa] are the normal stresses,
σ kl [Pa] are the shear stresses, K [Pa] is the bulk modulus, α ½με= C
is the linear thermal expansion coefficient, T [ C] is the temperature,
E [Pa] is the Young’s modulus and ν [ ] is the Poisson’s ratio.
ii. The surface stress components X;Y and Z that are needed to maintain
the considered state of stress are:
X Y Z Eα
5 5 52 ΔT
n x n y n z 1 2 2υ
where n x , n y and n z are the coordinate vectors in the directions x, y
and z.
