Geoscience Reference
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such that the constitutive models are subtle and the numerical analyses can be lengthy.
Macroelement modelling can provide a shortcut for parametric study of system behav-
iour.Theunderlyingprinciplesoftheconstitutivemodelsguidethechoiceofappropriate
features forthe macroelement models.
2. Constitutive modelling framework
Desirable features of a constitutive model appropriate for dynamic problems are shown
intermsofthedependenceofincrementalsoilstiffnessonhistory(Figure6.1).Afterany
corner in the strain path, the incremental stiffness immediately increases by an amount
dependent on the sharpness of the corner and then decays as deformation continues
(Figure6.1a,b).Seismicloadingisquintessentiallyaboutstrainpathswithmultipleirreg-
ularcorners:inanyseismicloadingeventwithvariableamplitudeofcyclesofloadingthe
stiffness will vary constantly. Contrast this with the ability of an elastic-perfectly plastic
model such as the widely used Mohr-Coulomb model (Figure 6.1c, d) which only has
two values of incremental stiffness: thefull elasticstiffness and zero.
Ingredients necessary for a successful model include: hardening plasticity to model the
steady change in stiffness with strain; and kinematic yielding to imprint on the model
some memory of recent history. Figure 6.2 illustrates the way in which kinematic yield-
ing can function in an extended Mohr-Coulomb model in two views of effective stress
space. Figure 6.2a shows mean effective stress
p
and distortional stress
q
; Figure 6.2b
shows adeviatoric view of principal stressspace. Ineach diagram there is a smallregion
shear
stress
shear
stress
a.
c.
shear strain
reloading
initial
loadi
ng
tangent
shear
stiffness
tangent
shear
stiffness
unloading
reloading
unloading
b.
d.
perfect
plasticity
initial
loading
shear strain
Fig. 6.1. (a) Typical irreversible stress-strainresponse and (b)typical variation
of incremental stiffnessforsoil;(c) stress-strainresponse and (d) variation of
incremental stiffness for elastic-perfectly plastic model
