Page 251 - The Combined Finite-Discrete Element Method

234 TRANSITION FROM CONTINUA TO DISCONTINUA

L−s s s L−s

e s
e e
∆ L L ∆

Figure 7.3 Strain softening bar in tension – assumed strain field.

strain-softening segment while the rest of the bar has strains smaller than the strain
corresponding to the peak stress, for the equilibrium of stress field it is required that

σ(ε s ) − σ(ε e ) = 0 (7.4)

and for this equilibrium to be stable it is required that

∂ ∂σ(ε s ) ∂σ(ε e ) ∂ε e
(σ(ε s ) − σ(ε e )) > 0; i.e. − > 0; (7.5)
∂ε s ∂ε s ∂ε e ∂ε s
which is equivalent to
∂σ(ε s ) ∂σ(ε e ) s
+ > 0
∂ε s ∂ε e L − s
It follows that the only stable segment is the segment of length larger than
∂σ(ε s )

∂ε s
s> − L (7.6)
∂σ(ε e ) ∂σ(ε s )
−
∂ε e ∂ε s
Since
∂σ(ε s ) ∂σ(ε e )
< 0and > 0 (7.7)
∂ε s ∂ε e

which means that all segments smaller than one given by (7.6) are unstable, which leads
to the conclusion that the actual length of the strain-softening segment is zero.
Localisation is the intense straining of a material within thin bands. It is associated with
material instability, resulting in bifurcation connected to the loss of ellipticity (in static
problems) or to the loss of hyperbolicity (in dynamic problems), and in both dynamic
and static problems the strain-softening constitutive law in a classical (local) continuum
(i.e. defined in terms of strains), although not mathematically meaningless, dissipates no
energy. This is not the case for known strain-softening materials such as rock, which is
characterised by finite strain-softening regions. The strains for real engineering softening
materials are localised over a relatively small (far smaller than the size of the actual
physical or engineering problem) yet finite lengths (characteristic lengths) that reflect the
microstructure of the rock, and energy dissipation is therefore well defined.
In finite element based applications, these instabilities are manifested in mesh size and
mesh orientation dependency of the solution; the localisation zone width corresponds to the