Fully explicit process with modi®ed stiffness: 1 2 441
3.9 Viscoplasticity ± a generalization .1 General remarks
The purely plastic behaviour of solids postulated in Sec. 3.3 is probably a ®ction as the maximum stress that can be carried is invariably associated with the rate at which this is applied. A purely elasto-plastic behaviour in a uniaxial loading is described in a model of Fig. 3.15(a) in which the plastic strain rate is zero for stresses below yield, that is,
_"p 0 if j ÿ yj < 0 and jj > 0 and _"pis indeterminate when ÿ y 0.
An elasto-viscoplastic material, on the other hand, can be modelled as shown in Fig. 3.15(b), where a dashpot is placed in parallel with the plastic element. Now stresses can exceed yfor strain rates other than zero.
The viscoplastic (or creep) strain rate is now given by a general expression _"vp ÿ y
3:168
78 Inelastic and non-linear materials
where the arbitrary function is such that
The model suggested is, in fact, of a creep-type category described in the previous sections and often is more realistic than that of classical plasticity.
A viscoplastic model for a general stress state is given here and follows precisely the arguments of the plasticity section. In a three-dimensional context becomes a func-tion of the yield condifunc-tion F r; j; de®ned in Eq. (3.44). If this is less than zero, no
`plastic' ¯ow will occur. To include the viscoplastic behaviour we modify Eq. (3.44) as
Fh i ÿ _ 0 3:170
and use Eq. (3.45) to de®ne the plastic strain. Equation (3.175) implies
F
h i 0 if F 4 0
F if F > 0
3:171
and is some `viscosity' parameter. Once again associated or non-associated ¯ows can be invoked, depending on whether F Q or not. Further, any of the yield surfaces described in Sec. 3.3.1 and hardening forms described in Sec. 3.3.3 can be used to de®ne the appropriate ¯ow in detail. For simplicity, F Fmwhere m is a positive power often used to de®ne the viscoplastic rate eects in Eq. (3.170).84
The concept of viscoplasticity in one of its earliest versions was introduced by Bingham in 192285 and a survey of such modelling is given in references 86 and 88.
The computational procedure of using the viscoplastic model can follow any of the general methods described in Sec. 3.4. Early applications commonly used the straight-forward Euler (explicit) method.89ÿ93 The stability requirements for this approach have been considered for several types of yield conditions by Cormeau.94A tangential process can again be used, but unless the viscoplastic ¯ow is associated (F Q), non-symmetric systems of equations have to be solved at each step. Use of an explicit method will yield solution for the associative and non-associative cases and the system matrix remains symmetric. This process is thus similar to that of a modi®ed Newton±
Raphson method (initial stress method) and is quite ecient. Indeed within the stability limit it has been shown that use of an over relaxation method leads to rapid convergence.
(a) σ
Fig. 3.15 (a) Elastoplastic; (b) elasto-viscoplastic; (c) series of elasto-viscoplastic models.
Viscoplasticity ± a generalization 79
3.9.2 Iterative solution
The complete iterative solution scheme for viscoplasticity is identical to that used in plasticity except for the use of Eq. (3.170) instead of Eq. (3.44). To underline this similarity we consider the constitutive model without hardening and use the return map implicit algorithm. The linearized relations are identical except for the treatment of relation (3.175). The form becomes
Dÿ1 Q;rr Q;r
where the discrete residual for Eq. (3.171) is given by rn ÿ Fn 1
tn 3:173
and
0d
dF
Now the equations are almost identical to those of plasticity [see Eq. (3.87)], with dierences appearing only in the 0and 1= t terms.
Again, a consistent tangent can be obtained by elimination of the diand a general iterative scheme is once more available.
Indeed, as expected, t 1 will now correspond to the exact plasticity solution.
This will always be reached by any solution tending to steady state. However, for transient situations this is not the case and use of ®nite values for t will invariably lead to some rate eects being present in the solution.
The viscoplastic laws can easily be generalized to include a series of components, as shown in Fig. 3.15(c). Now we write
_ev _ev1 _ev2 3:179
and again the standard formulation suces. If, as shown in the last element of Fig. 3.15(c), the plastic yield is set to zero, a `pure' creep situation arises in which
¯ow occurs at all stress levels. If a ®nite value is in a term a corresponding rate equation for the associated _j must be used. This is similar to the Koiter treatment for multi-surface plasticity.17;95
The use of the Duvaut and Lions88 approach modi®es the return map algorithm for a rate-independent plasticity solution. Once this solution is available a reduction in the value of is computed to account for rate eects. The interested reader should consult references 14 and 28 for additional information on this approach.
3.9.3 Creep of metals
If an associated form of viscoplasticity using the von Mises yield criterion of Eq. (3.99) is considered the viscoplastic strain rate can be written as
_evp _@jsj
@r _n 3:175
80 Inelastic and non-linear materials
with the rate expressed again as
_ ÿ y
3:176
If y, the yield stress, is set to zero we can write the above as
_evp mn 3:177
and we obtain the well-known Norton±Soderberg creep law. In this, generally the parameter is a function of time, temperature, and the total creep strain (e.g. the analogue to the plastic strain ep). For a survey of such laws the reader can consult specialized references.96;97
An example initially solved using a large number of triangular elements83 is pre-sented in Fig. 3.16, where a much smaller number of isoparametric quadrilaterals are used in a general viscoplastic program.93
Fig. 3.16 Creep in a pressure vessel: (a) mesh end effective stress contours at start of pressurization; (b) effec-tive stress contours 3 h after pressurization.
Viscoplasticity ± a generalization 81
3.9.4 Soil mechanics applications
As we have already mentioned, the viscoplastic model provides a simple and eective tool for the solution of plasticity problems in which transient eects are absent. This includes many classical problems which have been solved in references 93 and 98, and the reader is directed there for details. In this section some problems of soil mechanics are discussed in which the facility of the process for solving non-associated behaviour is demonstrated.99 The whole subject of the behaviour of soils and similar porous media is one in which much yet needs to be done to formulate good constitutive models. For a fuller discussion the reader is referred to texts, conferences, and papers on the subject.100;101
Prescribed
Fig. 3.17 Uniaxial, axisymmetric compression between rough plates: (a) mesh and problem; (b) pressure displacement result; (c) plastic ¯ow velocity patterns.
82 Inelastic and non-linear materials
One particular controversy centres on the `associated' versus `non-associated' nature of soil behaviour. In the example of Fig. 3.17, dealing with an axisymmetric sample, the eect of these dierent assumptions is investigated.86 Here a Mohr±Coulomb law is used to describe the yield surface, and a similar form, but with a dierent friction angle, , is used in the plastic potential, thus reducing the plastic potential to the Tresca form of Fig. 3.8 when 0 and suppressing volumetric strain changes. As can be seen from the results, only moderate changes in collapse load occur, although very appreciable dierences in plastic ¯ow patterns exist.
Figure 3.18 shows a similar study carried out for an embankment. Here, despite quite dierent ¯ow patterns, a prediction of collapse load was almost unaected by the ¯ow rate law assumed.
1 2 3 4
4 5
3 2
10 20 20
30
(b) (a)
1 2 3
1 3 2
10 4 20 40 30
5
Fig. 3.18 Embankment under action of gravity, relative plastic ¯ow velocities at collapse, and effective shear strain rate contours at collapse: (a) associative behaviour; (b) non-associative (zero volume change) behaviour.
Viscoplasticity ± a generalization 83
The non-associative plasticity, in essence caused by frictional behaviour, may lead to non-uniqueness of solution. The equivalent viscoplastic form is, however, always unique and hence viscoplasticity is on occasion used as a regularizing procedure.