Page 199 - Analysis and Design of Energy Geostructures

172 Analysis and Design of Energy Geostructures

The limit of the elastic domain characterising the mechanical behaviour of a mate-
rial can be expressed mathematically through the following general yield surface (or
yield function)

f 5 0 ð4:84Þ

4.10.2 Elastic and plastic strains

The existence of a yield criterion indicates that increments of stress that lie within the
yield surface produce elastic deformation, while those that lie on the yield surface pro-
duce plastic deformation for further loading. In this context, the total strain of any
material can be decomposed in an elastic ε and a plastic ε portion:
p
e
kl
kl
dε kl 5 dε 1 dε p ð4:85Þ
e
kl kl
The elastic part of the strain produced by a given load is recoverable when the
load is removed, while the plastic part of the strain is irrecoverable when the load is
removed. In this sense, the work done by the plastic strain must be positive and this
corresponds to the condition of irreversibility postulated by Prager (1949).
Decomposing the total strain with an elastic and a plastic component represents the
first essential step for deriving a complete stress strain relation for materials charac-
terised by plasticity, irrespective of whether isothermal or nonisothermal conditions
are considered.

4.10.3 Flow rule
A key aspect when addressing plasticity is to determine plastic strains, which are
induced by the phenomenon of plastic flow (Yu, 2006). The increments of plastic
strain can be expressed as (von Mises, 1928; Melan, 1938)

dε 5 dλ p @g ð4:86Þ
p
ij @σ ij
Eq. (4.86) is often referred to as flow rule. In this equation, dλ is a positive scalar
p
called plastic multiplier, which represents the magnitude of plastic strain, and g is termed
plastic potential function, which defines the direction of plastic strain vectors. Eq. (4.86)
defines ratios of the components of plastic strain rate. In particular, @g=@σ ij gives the
components of normal vector to g in the stress space. In other words, the potential
function defines the direction of plastic strains (and to which incremental plastic strain
vectors are orthogonal).
Eq. (4.86) also indicates that only one plastic mechanism is considered. However,
several plastic mechanisms may be considered, each one referring to a relevant plastic