Page 198 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 171

The term ε is generally not known a priori but needs to be estimated based on
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the results of similar case studies or preferably to be calculated. During preliminary
analysis and design stages where the term ε would not be known yet, the aforemen-
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tioned formulations are still powerful tools to get acquainted with the orders of mag-
nitude of the thermally induced strain and stress that may characterise the energy pile
under free or completely restrained conditions, respectively. Orders of magnitude are
given, for example, by Bourne-Webb et al. (2016b).
The previously proposed framework implicitly refers to energy piles embedded in
typical soil deposits for which the soil-pile thermal expansion coefficient ratio
X 5 α soil =α EP # 1, where α soil is the linear thermal expansion coefficient of the soil. In
rare cases where X 5 α soil =α EP . 1, typically at successive stages of geothermal opera-
tions, the temperature variation applied to an energy pile and its thermal expansion
coefficient do not satisfy inequality (4.79), that is
ε . ε th ð4:83Þ
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o f
The above occurs because when the linear thermal expansion coefficient of the soil
is greater than that of the energy pile, the thermally induced deformation of energy
piles is dominated by that of the soil rather than by the deformation of the piles. This
phenomenon becomes more pronounced as wider soil regions are affected by temper-
ature variations (Bourne-Webb et al., 2016a; Rotta Loria and Laloui, 2017).
For example inequality (4.83) indicates that heating thermal loads applied to
energy piles can induce tensile stress. This phenomenon has been confirmed by
full-scale experimental evidence and numerical analyses (Rotta Loria and Laloui,
2017, 2018).

4.10 Plasticity and thermoplasticity
4.10.1 Yield criterion

The condition that defines the limit of elasticity and the onset of plasticity through the
concept of yield limit is known as the yield condition or yield criterion. The yield crite-
rion defines all of the possible stress states that are associated with a reversible mechani-
cal behaviour of the material as well as those that are associated with an irreversible
mechanical behaviour of the material (depending on the loading situation).
The graphical representation of a yield criterion assumes different forms depending
on the dimensions considered for any given problem (Yu, 2006). In one dimension,
the yield criterion is represented by a point. In two dimensions, the yield criterion is
represented by a curve. In three dimensions, the yield criterion is represented by a sur-
face. The most comprehensive representation of a yield criterion is through a surface
represented in stress space, which is generally termed as yield surface.