Generalized plasticity ± non-associative case

Plastic behaviour characterized by irreversibility of stress paths and the development of permanent strain changes after a stress cycle can be described in a variety of ways.

One form of such description has been given in Sec. 3.3. Another general method is presented here.

3.6.1 Non-associative case ± frictional materials

This approach assumes a priori the existence of a rate process which may be written directly as

_r ˆ D^_e …3:144†

in which the matrix D^depends not only on the stress r and the state of parameters j, but also on the direction of the applied stress (or strain) rate _r (or _e).⁵⁶A slightly less ambitious description arises if we accept the dependence of D^only on two directions

± those of loading and unloading. If in the general stress space we specify a `loading' direction by a unit vector n given at every point (and also depending on the state parameters j), as shown in Fig. 3.9, we can describe plastic loading and unloading by the sign of the projection n^T_r. Thus

n^T_r >0 for loading

<0 for unloading



…3:145†

while n^T_r ˆ 0 is a neutral direction in which only elastic straining occurs. One can now write quite generally that

_r ˆ

D^_L_e for loading

D^_U_e for unloading …3:146†

where the matrices D^_Land D^_Udepend only on the state described by r and j.

68 Inelastic and non-linear materials

The speci®cation of D^_Land D^_Umust be such that in the neutral direction of the stress increment _r the strain rates corresponding to this are equal. Thus we require

_e ˆ …D^_L†^ÿ1_r ˆ D^{ ÿ1}U _r when n^T_r ˆ 0 …3:147†

A general way to achieve this end is to write …D^_L†^ÿ1  D^ÿ1‡ 1

H_Ln_gLn^T and …D^_U†^ÿ1 D^ÿ1‡ 1

H_Un_gUn^T …3:148†

where D is the elastic matrix, n_gLand n_gUare arbitrary unit stress vectors for loading and unloading directions, and H_L and H_Uare appropriate plastic moduli which in general depend on r and j.

The value of the tangent matrices D^_Land D^_Ucan be obtained by direct inversion if H_L=U6ˆ 0, but more generally can be deduced following procedures given in Sect.

3.3.4 or can be written directly using the Sherman±Morrison±Woodbury formula⁵⁷as:

D^_Lˆ D ÿ 1

H_L^ Dn_gLn^TD H^_Lˆ H_L‡ n^TDn_gL …3:149†

This form resembles Eq. (3.72) and indeed its derivation is almost identical. We note further that …D^_L†^ÿ1is now well behaved for H_L zero and a form identical to that of perfect plasticity is represented. Of course, a similar process is used to obtain D^_U.

This simple and general description of generalized plasticity was introduced by MroÂz and Zienkiewicz.^58;59It allows:

1. the full model to be speci®ed by a direct prescription of n, n_gand H for loading and unloading at any point of the stress space;

2. existence of plasticity in both loading and unloading directions;

3. relative simplicity for description of experimental results when these are complex and when the existence of a yield surface of the kind encountered in ideal plasticity is uncertain.

For the above reasons the generalized plasticity forms have proved useful in describing the complex behaviour of soils.^60ÿ64Here other descriptions using various

σ2

σ1

n

σ (unload)

σ (load)

.

.

Fig. 3.9 Loading and unloading directions in stress space.

Generalized plasticity ± non-associative case 69

interpolations of n and moduli form a unique yield surface, known as bounding surface plasticity models, are indeed particular forms of the above generalization and have proved to be useful.⁶⁵

Classical plasticity is indeed a special case of the generalized models. Here the yield surface may be used to de®ne a unit normal vector as

n ˆ 1

‰F^T_;rF_;rŠ¹⁼²F_;r …3:150†

and the plastic potential may be used to de®ne n_gˆ 1

‰Q^T_;rQ_;rŠ¹⁼²Q_;r …3:151†

where once again some care must be exercised in de®ning the matrix notation. Sub-stitution of such values for the unit vectors into Eq. (3.149) will of course retrieve the original form of Eq. (3.72). However, interpretation of generalized plasticity in classical terms is more dicult.

The success of generalized plasticity in practical applications has allowed many com-plex phenomena of soil dynamics to be solved.^66;67We shall refer to such applications later but in Fig. 3.10 we show how complex cyclic response with plastic loading and unloading can be followed.

While we have speci®ed initially the loading and unloading directions in terms of the total stress rate _r this de®nition ceases to apply when strain softening occurs and the plastic modulus H becomes negative. It is therefore more convenient to check the loading or unloading direction by the elastic stress increment _r^e of

100 Mean effective pressure p' (kPa)

Computational model Mean effective pressure p' (kPa) 100

50 0 –50 Deviatoric stress (kPa)–100–10

Axial strain (%) Experimental

–8 –6 –4 –2 0 2

Fig. 3.10 A generalized plasticity model describing a very complex path, and comparison with experimental data. Undrained two-way cyclic loading of Nigata sand.⁶⁸(Note that in an undrained soil test the ¯uid restrains all volumetric strains, and pore pressures develop; see Sec. 19.3.5 of Volume 1).

70 Inelastic and non-linear materials

Eq. (3.65) and to specify

n^T_r^e >0 for loading

<0 for unloading



…3:152†

This, of course, becomes identical to the previous de®nition of loading and unloading in the case of hardening.

3.6.2 Associative case ± J

generalized plasticity

Another modi®cation to the classical rate-independent approach is one in which the transition from an elastic to a fully plastic solution is accomplished with a smooth transition. This approach is useful in improving the match with experimental data for cyclic loading. A particularly simple form applicable to the J₂ model was introduced by Lubliner.^32;33 In this approach, the yield function is modi®ed to a rate form directly and is expressed as

h…F† _F ÿ _ ˆ 0 …3:153†

where h…F† is given by the function

h…F† ˆ F

… ÿ F† ‡ H  …3:154†

in which H ˆ H_i‡ H_k, and ,  are two positive parameters with dimension of stress.

In particular,  is a distance between a limit plastic state and the current radius of the yield surface, and  is a parameter controlling the approach to the limit state with increasing accumulated plastic strain.

On discretization and combination with the return map algorithm a rate-independent process is evident and again only minor modi®cations to the algorithm presented previously is necessary. A full description of the steps involved is given by Auricchio and Taylor.³⁴ Their paper also includes a development for the non-linear kinematic hardening model given in Eq. (3.62). In the case where the yield function is associative (i.e. F ˆ Q) the use of the non-linear kinematic hardening model leads to an unsymmetric tangent sti€ness when used with the return map algorithm. On the other hand, the generalized plasticity model is fully symmetric for this case.

In the next section we present further discussion on use of generalized plasticity to model the behaviour of frictional materials. In general, these involve use of non-associative models where the return map algorithm cannot be used e€ectively.