Some examples of plastic computation

…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.

3.7 Some examples of plastic computation

The ®nite element discretization technique in plasticity problems follows precisely the same procedures as those of corresponding elasticity problems. Any of the elements already discussed can be used for problems in plane stress; however, for plane strain, axisymmetry, and three-dimensional problems it is usually necessary to use elements which perform well in constrained situations such as encountered for near incompressi-bility. For this latter class of problems use of mixed elements is generally recommended,

Some examples of plastic computation 71

although elements and constitutive forms that permit use of reduced integration may also be used.

The use of mixed elements is especially important in metal plasticity as the Huber±

von Mises ¯ow rule does not permit any volume changes. As the extent of plasticity spreads at the collapse load the deformation becomes nearly incompressible, and with conventional (fully integrated) displacement elements the system locks and a true collapse load cannot be obtained.^69;70

Finally, we should remark that the possibility of solving plastic problems is not limited to a displacement and mixed formulation alone. Equilibrium ®elds and, indeed, most of the formulations described in Chapters 11 and 12 of Volume 1 form a suitable vehicle,^71ÿ73 but owing to their convenient and easy interpretation displacement and mixed forms are most commonly used.

3.7.1 Perforated plate ± plane stress solutions

Figure 3.11 shows the con®guration and the division into simple triangular and quadrilateral elements. In this example plane stress conditions are assumed and

(a) (b)

(c) (d)

Fig. 3.11 Perforated plane stress tension strip: mesh used and development of plastic zones at loads of 0.55, 0.66, 0.75, 0.84, 0.92, 0.98, 1.02 times _y. (a) T3 triangles; (b) plastic zone spread; (c) Q4 quadrilaterals; (d) Q9 quadrilaterals.

72 Inelastic and non-linear materials

solution is obtained for both ideal plasticity and strain hardening. This problem was studied experimentally by Theocaris and Marketos⁷⁴ and was ®rst analysed using

®nite element methods by Marcal and King⁷⁵and Zienkiewicz et al.⁴¹(See reference 5 for discussion on these early solutions.) The von Mises criterion is used and, in the case of strain hardening, a constant slope of the uniaxial hardening curve, H, is taken. Data for the problem, from reference 74, are E ˆ 7000 kg/mm², H ˆ 225 kg/mm² and _yˆ 24:3 kg/mm². Poisson's ratio is not given but is here taken as in reference 41 as  ˆ 0:3. To match a con®guration considered in the experimental study a strip with 200 mm width and 360 mm length containing a central hole of 200 mm diameter. Using symmetry only one quadrant is discretized as shown in Fig. 3.11. Displacement boundary restraints are imposed for normal components on symmetry boundaries and the top boundary. Sliding is permitted, to impose the necessary zero tangential traction boundary condition. Loading is applied by a uniform non-zero normal displacement with equal increments. Dis-placement elements of type T3, Q4, and Q9 are used with the same nodal layout.

Results for the three elements are nearly the same, with the extent of plastic zones indicated for various loads in Fig. 3.11 obtained using the Q4 element. The load±deformation characteristics of the problem are shown in Fig. 3.12 and com-pared to experimental results. The strain "_y is the peak value occurring at the hole boundary. This plane stress problem is relatively insensitive to element type and load increment size. Indeed, doubling the number of elements resulted in small changes of all essential quantities.

3.7.2 Perforated plate ± plane strain solutions

The problem described above is now analysed assuming a plane strain situation. Data are the same as for the plane stress case except the lateral boundaries are also restrained to create a zero normal displacement boundary condition. This increases

1.0

0.8

0.6

0.4

0.2

00 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 E (εy/σy)

σmean/σy

T3 – Mises Experimental

Fig. 3.12 Perforated plane stress tension strip: load deformation for strain hardening case (H ˆ 225 kg/mm²).

Some examples of plastic computation 73

the con®nement on the mesh and shows more clearly the locking condition cited previously. In Fig. 3.13 we plot the resultant axial load for each load step in the solution. Figure 3.13(a) shows results for the displacement model using T3, Q4, and Q9 elements and it is evident that the T3 and Q4 elements result in an erroneous increasing resultant load after the fully plastic state has developed. The Q9 element shows a clear limit state and indicates that higher order elements are less prone to locking (even though we have shown that for the fully incompressible state the Q9 displacement element will lock!). Figure 3.13(b) presents the same results for the Q4/1 and Q9/3 mixed elements and both give a clear limit load after the fully plastic state is reached.

2500

2000

1500

1000

500

0

0 5 10 15 20 25 30 35 40 45 50

Step number

Axial load

T3 – Displ Q4 – Displ Q9 – Displ

(a) 2500

2000

1500

1000

500

0

0 5 10 15 20 25 30 35 40 45 50

Step number

Axial load

Q4/1 – Mixed Q9/3 – Mixed

(b)

Fig. 3.13 Limit load behaviour for plane strain perforated strip: (a) displacement (displ.) formulation results;

(b) mixed formulation results.

74 Inelastic and non-linear materials

3.7.3 Steel pressure vessel

This ®nal example, for which test results obtained by Dinno and Gill⁷⁶are available, illustrates a practical application, and the objectives are twofold. First, we show that this problem which can really be described as a thin shell can be adequately represented by a limit number (53) of isoparametric quadratic elements. Indeed, this model simulates well both the overall behaviour and the local stress concentration e€ects [Fig. 3.14(a)]. Second, this problem is loaded by an internal pressure and a solution is performed up to the `collapse' point (where, because there is no hardening, the strains increase without limit) by incrementing the pressure rather than displace-ment. A comparison of calculated and measured de¯ections in Fig. 3.14(b) shows how well the objectives are achieved.