6.4 Multiple Surface Plasticity 111 An alternative approach is to introduce multiple yield surfaces, but to treat each as independent, giving rise to a separate plastic strain component.
Trang 1110 6 Advanced Plasticity Theories
when these two surfaces come in contact with each other The modulus can, for
instance, be specified by an expression of the form,
where as before G PR is the distance from the current stress point to the image
point The form ensures that h 0 H and h G0 h0 This formulation is very
close in concept to bounding surface plasticity
This form of plasticity, however, avoids the problem inherent in the bounding
surface models of ratcheting for small cycles of unloading and reloading The
more sophisticated rules for defining the image point and the translation of the
inner yield surface provide a more realistic way for describing hysteretic
behav-iour and the effect of past loading history
This process can be taken a step further by introducing a set of nesting
sur-faces within the domain between f and 0 F These surfaces can translate 0
and expand or contract due to plastic straining They are capable of encoding in
a more subtle way the details of the past stress history The relative motion
be-tween each adjacent surface is defined by rules similar to (6.6) to ensure that
they remain nested The hardening modulus can be defined by an interpolation
formula such that it is h on the innermost yield surface, and equal to H on the 0
outermost surface For instance, it can be assumed if the stress point is in
con-tact with n surfaces out of a total of N:
The configuration of the nesting surfaces and therefore the subsequent
stiff-ness for a particular stress path depends on the past history of loading The
ma-terial response for any loading history may be studied by following the evolution
of the configurations of the nested surfaces This model possesses a multi-level
memory structure because, for cyclically varying stress, only a certain number of
surfaces undergo translation; the other surfaces may change only isotropically
This approach can be extended to an infinite number of surfaces, although for
practical computations, a finite number is necessary
6.4 Multiple Surface Plasticity
Although it has certain advantages, the translation rule for multiple yield
sur-faces that requires that the sursur-faces remain nested is not strictly necessary (see
Section 6.5) The principal advantage of the nested approach is that this allows
the determination of a single plastic strain component, with its magnitude
estab-lished by one of the above procedures for the hardening modulus
Trang 26.4 Multiple Surface Plasticity 111
An alternative approach is to introduce multiple yield surfaces, but to treat
each as independent, giving rise to a separate plastic strain component The total
strain is the sum of the elastic strain and the plastic strain components:
1
Each yield surface is specified in the form f n ı İij, ij p n 0, where for
simplicity the yield surface depends only on the plastic strains associated
with that surface, and is not coupled to other yield surfaces by dependence
on their plastic strains In principle, non-associated flow can be specified
read-ily by defining plastic potentials distinct from the yield surfaces so that
n
ij ijkl kl
ij n
and the plastic multipliers are eliminated by the consistency conditions for each
of the yield surfaces:
The analysis proceeds exactly as for the single surface model with the
elastic-plastic matrix determined as
Strictly, the summations in the above equations are only for the “active” yield
surfaces, for which f n 0; on the other surfaces, simply, On 0
It can be seen that the multiple surface method is simpler in concept than the
nested surface method It does not involve the plethora of ad hoc rules about
translation and hardening of the inner surface, most of which are introduced
simply to guarantee “nesting” rather than to reproduce any well-defined feature
of material behaviour The multiple surface models do, however, have all the
advantages of nested surface models in modelling hysteresis and stress history
effects An example of this type of model was given by Houlsby (1999)
This approach, too, can be extended to an infinite number of surfaces
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6.5 Remarks on the Intersection of Yield Surfaces
6.5.1 The Non-intersection Condition
The use of kinematic hardening plasticity with multiple yield surfaces has a tory of more than 30 years It has proved a very convenient framework for mod-elling the pre-failure behaviour of soils and other materials, allowing a realistic treatment of issues such as non-linearity at small strain and the effects of recent stress history The growing interest in modelling small strain behaviour of soils has recently resulted in the development of many so-called “bubble” models, such as those described by Stallebrass and Taylor (1997), Kavvadas and Amorosi (1998), Rouainia and Muir Wood (1998), Gajo and Muir Wood (1999), Houlsby (1999), Puzrin and Burland (2000), and Puzrin and Kirshenboim (1999)
his-In all the above models, except Houlsby (1999), the “translation rules” are specified to avoid intersection of the yield surfaces, and it is commonly believed that this non-intersection condition must be met, but some publications express
a contrary view
This subject is discussed here because the non-intersection condition leads to unnecessary complications in kinematic hardening hyperplasticity with multiple yield surfaces As discussed above, the simple translation rules used by Ziegler or Prager correspond to simple forms of Gibbs free energy If non-intersection is to be imposed, much more complex energy expressions are required, in which terms involving cross-coupling between different plastic strain components must appear Puzrin and Houlsby (2001a) argue that the condition is not necessary, but is required only when an incrementally bilinear constitutive law is to be derived Sometimes it is claimed that, even if not strictly necessary, the non-intersection condition should be accepted on pragmatic grounds Incremental bilinearity (and hence non-intersection) certainly offers some advantage in computation The main one is that, if an updated stiffness approach is taken in finite element analysis, the incremental stress-strain relationship is known (for plastic load-ing), reducing the need for iteration At the opposite extreme, if an incremen-
tally non-linear approach (e g as in hypoplastic theories) is used, the
incre-mental stress-strain relationship cannot be determined without prior knowledge
of the path during the increment If intersection of yield surfaces is allowed, an intermediate case occurs: the response is incrementally multilinear (see the discussion below) In practice, this does not prove to be a significant disadvan-tage, since for most relatively smooth stress paths, the incremental plastic re-sponse can be determined in advance for each increment
6.5.2 Example of Intersecting Surfaces
To demonstrate that the non-intersection condition is not strictly necessary, we describe here a model with two yield surfaces that are allowed to intersect It will
Trang 46.5 Remarks on the Intersection of Yield Surfaces 113
be seen that this model poses no theoretical difficulties Consider the plasticity
model with two kinematic hardening yield surfaces in a two-dimensional stress
space, as shown in Figure 6.3 The yield surfaces are
k and k2 are their radii Plastic yielding and hardening are calculated using an
associated flow rule:
Trang 5114 6 Advanced Plasticity Theories
where O 1 and O 2 are non-negative multipliers; 1 ^İ 11,İ12`
T
p2 ^İ p21,İp22`
H are the plastic strain vectors associated with each of the
sur-faces, so that the total plastic strain vector is given by 1 2
Plastic hardening is calculated using Prager’s translation rule (which in this
case is identical to Ziegler’s):
1 1
1 p E
The model defined by Equations (6.14)–(6.17) is a particular case of the
multi-surface model (7.38)–(7.39) which will be derived in Section 7.5 within the
hyperplastic framework
Prager’s and Ziegler’s translation rules are known to violate the
non-intersection condition Consider as an example the case presented in Figure 6.4
During loading, the stress state P was reached, where the two surfaces touch
each other (if they do not touch at only one point, they intersect and the proof is
completed) Next a stress reversal took place and the stress state moved inside
the yield surface f 1 such that the current stress state V was reached, which is
on this yield surface but not on the outer yield surface The next stress
incre-ment dV is such that plastic response of the yield surface f 1 will occur, and
will cause a strain increment 1
p
dH directed along the vector 1 1
prescribed by the associated flow rule (6.15) Then, according to Prager’s
trans-lation rule (6.17), the instantaneous displacement 1
dU of the centre Q of the yield surface f 1 will also be directed along the vector 1 1
F V U fore, if the current stress state V is located so that the angle D between the
There-vectors F and U1 is acute, the instantaneous displacement vector 1
dU will have a component directed along the ray QP In this case, when the stress in-
crement dV takes place, the point P on the yield surface f 1 moves into the
exterior of the yield surface f 2 , and the surfaces intersect
Trang 66.5 Remarks on the Intersection of Yield Surfaces 115
V
Figure 6.4 Example of a violation of the non-intersection condition
6.5.3 What Occurs when the Surfaces Intersect?
There are no significant detrimental effects when yield surface intersect,
pro-vided that the plastic loading and consistency conditions are applied separately
to each yield surface [see, for example, de Borst (1986)] In this case, the
consti-tutive relationship simply becomes multilinear instead of bilinear
Consider, for example, the kinematic hardening model with two yield
sur-faces described by Equations (6.14)–(6.17) The incremental stress-strain
re-sponse of this model is derived by applying consistency conditions f1 0 and
2
0
f separately to each surface as appropriate For this case, the following
incremental relationships can be obtained:
2
1 1
1
, when 02
Trang 7116 6 Advanced Plasticity Theories
and are Macaulay brackets (i e x x x, !0; x 0,xd ) 0
Assuming that the surfaces intersect at the current stress state in Figure 6.5,
four different types of behaviour are encountered, depending on which of the
four possible zones the incremental stress vector is directed into
1 2
2
1 1
2
2 2
2 2
00
1 2
2
1 1
00
Trang 86.6 Alternative Approaches to Material Non-linearity 117
2 2
2
2 2
00
Equations (6.21)–(6.24) represent an example of an incrementally multilinear
constitutive relationship, as opposed to a bilinear one obtained when the
non-intersection condition is satisfied During loading, zones 2 and 3 would be
en-countered only in rather rare circumstances which would involve rather
con-torted stress paths Many other recent developments in generalised plasticity,
hypo-, and hyperplasticity are based on the use of incrementally non-linear and
multilinear constitutive relationships
The main conclusion is that the non-intersection condition is necessary only
when a bilinear constitutive law has to be derived Intersection of yield surfaces,
when treated properly, leads to multilinear constitutive relationships, which are
consistent with recent developments in plasticity theory
Note that we make no case here that every model that allows intersection of
yield surfaces may be theoretically consistent It would be quite possible to
for-mulate such a model so that it was either theoretically unacceptable or produced
unjustifiable results The case we present is simply that intersection of yield
surfaces is allowable and on occasions may offer advantages
6.6 Alternative Approaches to Material Non-linearity
Plasticity theory is not the only method that has been used to model the
irre-versible and non-linear behaviour of rate-independent materials For
complete-ness, two further alternatives should be mentioned
Endochronic theory [Valanis (1975); Bazant (1978)] enjoyed some popularity
at one time, but has now largely fallen into disuse Initially it was an attempt to
model irreversibility within a thermodynamic context and without recourse to
yield surfaces It concentrated instead on the use of an “intrinsic time”, which
was typically identified with some measure of plastic strain Incremental
rela-tions relating stresses, strains, and intrinsic time increment were proposed
Unfortunately, the main purpose of endochronic theory – to avoid yield surfaces
– was the cause of its downfall Real materials that exhibit rate-independent,
irreversible behaviour also exhibit the phenomenon of a yield surface Thus it
became necessary to modify endochronic theory to include yield surfaces
artifi-cially The theories became increasingly contrived, and are now rarely used
Hypoplasticity is closely related to endochronic theory, although it does not
employ an intrinsic time Instead, rate equations are proposed specifying the
Trang 9118 6 Advanced Plasticity Theories
stresses in terms of the strain rates These equations make much use of tensor
analysis to identify the most general forms of first-order (but not necessarily
linear) expressions for stress rate in terms of strain rate For example, Kolymbas
(1977) assumes a direct incrementally non-linear stress-strain relationship:
ij L ijkl kl N ij kl kl
where L ijkland N are linear operators The early theories did not use yield ij
surfaces, but (for the same reasons as encountered by endochronic theory) more
recent theories have become increasingly complex to introduce the
phenome-non of yield surfaces The theories are still popular in some quarters, but in our
view are unlikely to find long-term favour
6.7 Comparison of Advanced Plasticity Models
As seen from the above examples of different plasticity formulations, their
com-mon feature is the existence of an outer or bounding surface F (in soils of-0
ten defined by the degree of material consolidation) In classical plasticity strain
hardening models, this surface is assumed to be a yield surface, containing an
entirely elastic domain To incorporate plastic flow within this surface,
bound-ing surface models, nested surface models, and multiple surface models have
been developed
In bounding surface models, the F 0 surface is treated as a bounding
sur-face, and a loading surface passing through the current stress state is defined
using a specific mapping rule This mapping rule also defines the distance in the
stress space between the stress state and the bounding surface, and the
postu-lated hardening rule depends on this distance The disadvantage of these models
is the unrealistic “ratcheting” behaviour for small unload-reload cycles
In nesting surface models, the stress history of cyclic loading may be
fol-lowed, and the ratcheting problem avoided by the more sophisticated rules for
the evolution of the surfaces Unfortunately, a number of ad hoc assumptions
have to be introduced specifying the motion of the surfaces True multiple
sur-face models (without the nesting requirement) avoid these assumptions, and are
simpler in concept than nested surface models They can accommodate
non-associated flow more easily They too have a disadvantage Since each surface
acts independently, each must be checked for yield, whilst for nested surface
models, it is known that the surfaces are engaged in order from the innermost to
the outermost All multiple surface models can in principle be extended to an
infinite number of surfaces
There is no definitive choice between the more sophisticated plasticity
mod-els In the following, however, we shall develop hyperplasticity versions of
mul-tiple surface models It will be seen that these then lead naturally to a further
extension into models with an infinite number of surfaces
Trang 10
hyper-In an attempt to solve this problem, Iwan (1967) and Mroz (1967) introduced the concept of multiple yield (or loading) surfaces, as discussed in Chapter 6 In multiple yield surface kinematic hardening models, the size of the true linear elastic region can be reduced, even to the limiting case in which it vanishes com-pletely The stress-strain behaviour becomes piecewise linear and can follow more closely the true non-linearity of the material Importantly, the model has
a discrete memory of stress reversals, reflected in the relative configuration of the yield surfaces Generalization of the multiple surface concept to an infinite number of yield surfaces produces models with a continuous field of yield sur-faces These models allow the simulation of the true non-linear stress-strain behaviour and a continuous material memory, and will be the subject of Chap-ter 8
Trang 11120 7 Multisurface Hyperplasticity
7.2 Multiple Internal Variables
For simplicity, in Chapter 4, we considered materials which could be ised by a single kinematic internal variable D , which was in the form of a sec-ijond-order tensor The kinematic internal variable can often be conveniently identified with the plastic strain The significance of the single internal variable
character-is that a single yield surface character-is derived, on which there character-is an abrupt change from elastic to elastic-plastic behaviour The generalisation of the results to some other cases is straightforward; for instance, a scalar internal variable can be obtained simply by dropping the subscripts from the variables D , ij F , and ij F ij
in Chapter 4
The generalisation to some more complex cases is marginally more complex
For instance, N second-order tensor internal variables would mean that the
function for the Gibbs free energy g V D Tij, ij, in Chapter 4 is simply ised to 1
sults for other energy functions and differentials follow a similar pattern When Legendre transformations are made between different functions, the number of possible transformations becomes enormous (for instance, there are 22N possible forms of the energy function) However, it is likely that only a small fraction of the possible forms would be of practical application, and so no sys-tematic presentation of the forms with multiple internal variables is given here
If any of the N internal variables are scalars rather than tensors, then all that
is necessary is to drop the subscripts from the appropriate variables
The main reason for the introduction of multiple internal variables is to allow the definition of models with multiple yield surfaces These can be used for
a variety of purposes, e g.:
x modelling separately compression and shear effects, as may be appropriate
for some granular materials (i e “cone and cap” models);
x modelling anisotropy by using multiple kinematically hardening yield faces;
sur-x modelling the memory of stress reversals; and
x approximation of a smooth transition from elastic to plastic behaviour The last of these purposes is perhaps the most important The use of internal variables (within the thermodynamic framework) is an extremely powerful method for describing the past history of an elastic-plastic material, but suffers from the disadvantage that it inevitably leads to abrupt changes between elastic and elastic-plastic behaviour Although using multiple internal variables allows these changes to be divided into a number of smaller steps, a completely smooth
Trang 127.3 Kinematic Hardening with Multiple Yield Surfaces 121
transition can be achieved only by introducing an infinite number of internal variables Such an idea leads to the concept of an internal function rather than
internal variables The generalisation of the results given in Chapter 4 to internal functions is rather more complex than the generalisations to multiple variables discussed above and will be the subject of Chapter 8
7.3 Kinematic Hardening with Multiple Yield Surfaces
7.3.1 Potential Functions
The model with a single yield surface presented in Section 5.4.3 can be extended
to multiple yield surfaces by modifying the two potential functions that define the constitutive behaviour The specific Gibbs free energy becomes a function of the stress and a finite number of internal variables Dij n, n !1, ,N , where N
is the total number of the yield surfaces We choose the Gibbs free energy in the following form:
we drop the dependence on V in Equation (7.2) ij
7.3.2 Link to Conventional Plasticity
In the conventional formulation of multiple surface kinematic hardening ticity, calculation of incremental stress-strain response requires the equations to
plas-be defined explicitly for all the yield surfaces Then, for each yield surface, the following rules are specified:
... 86. 6 Alternative Approaches to Material Non-linearity 117
2 2
2
2...
Trang 26. 4 Multiple Surface Plasticity 111
An alternative approach is to introduce...
1
, when 02
Trang 71 16 Advanced Plasticity Theories
and are