Products: Abaqus/Standard Abaqus/Explicit
This example develops the homogeneous, finite-strain inelastic response of a granular material subject to uniform extension or compression in plane strain.
Results given by Carter et al. (1977) for these cases are used for comparison.
The specimen is initially stress-free and is made of an elastic, perfectly plastic material. The elasticity is linear, with a Young's modulus of 30 MPa and a Poisson's ratio of 0.3. Carter et al. assume that the inelastic response is governed by a Mohr-Coulomb failure surface, defined by the friction angle of the Coulomb line (
30°) and the material's cohesion (c). They also assume that the cohesion is twice the Young's modulus for the extension test and 10% of the Young's modulus in the compression test. The above problem is solved using the Mohr-Coulomb plasticity model in Abaqus with the friction angle and the dilation angle equal to 30°. However, note that this Abaqus Mohr-Coulomb model is not identical to the classical Mohr-Coulomb model used by Carter because it uses a smooth flow potential.
An alternative solution is to use the associated linear Drucker-Prager surface in place of the Mohr-Coulomb surface. In this case it is necessary to relate 
and c to the material constants 
and 
that are used in the Drucker-Prager model. Matching procedures are discussed in “Extended Drucker-Prager models,” Section 23.3.1 of the Abaqus Analysis User's Guide. In this case we select a match appropriate for plane strain conditions:
40°. Using the assumptions of Carter et al., the second equation gives d as 86.47 MPa (
= 120 MPa) for the extension case and d as 4.323 MPa ( = 6 MPa) for the compression case.Uniform extension or compression of the soil sample is specified by displacement boundary conditions since the load-displacement response will be unstable for the extension case.
The results are shown in Figure 1.15.5–1 for extension and in Figure 1.15.5–2 for compression. The solutions for Abaqus/Standard and Abaqus/Explicit are the same. The Drucker-Prager solutions agree well with the results given by Carter et al.; this is to be expected since the Drucker-Prager parameters are matched to the classical Mohr-Coulomb parameters under plane strain conditions. The differences between the Abaqus Mohr-Coulomb solutions and Carter's solutions are due to the fact that the Abaqus Mohr-Coulomb model uses a different flow potential. The Abaqus Mohr-Coulomb model uses a smooth flow potential that matches the classical Mohr-Coulomb surface only at the triaxial extension and compression meridians (not in plane strain).
However, one can also obtain Abaqus Mohr-Coulomb solutions that match Carter's plane strain solutions exactly. As discussed earlier, the classical Mohr-Coloumb model can be matched under plane strain conditions to an associated linear Drucker-Prager model with the flow potential
as computed before. Therefore, we can match the flow potential of the Abaqus Mohr-Coulomb model to that of the Drucker-Prager model. Matching between these two forms of flow potential assumes 
1 and results in 
22° in the Abaqus Mohr-Coulomb model. These Abaqus Mohr-Coulomb solutions are shown in Figure 1.15.5–1 and Figure 1.15.5–2 and match Carter's solutions exactly.Extension case with the Mohr-Coulomb plasticity model ( 30° and 30°) and CPE4 elements.
Extension and compression cases with the linear Drucker-Prager plasticity model and CPE4 elements.
Extension case with the linear Drucker-Prager plasticity model and CPE4I incompatible mode elements.
Compression case with the Mohr-Coulomb plasticity model ( 30° and 30°) and CPE4 elements.
Compression case with the linear Drucker-Prager plasticity model and CPE4 elements.
Extension case with the Mohr-Coulomb plasticity model ( 30° and 22°) and CPE4 elements.
Compression case with the Mohr-Coulomb plasticity model ( 30° and 22°) and CPE4 elements.
Extension and compression cases with the linear Drucker-Prager plasticity model and CPE4R elements.
Extension case with the Mohr-Coulomb plasticity model ( 30° and 30°) and CPE4R elements.
Compression case with the Mohr-Coulomb plasticity model ( 30° and 30°) and CPE4R elements.
Extension case with the Mohr-Coulomb plasticity model ( 30° and 22°) and CPE4R elements.
Compression case with the Mohr-Coulomb plasticity model ( 30° and 22°) and CPE4R elements.
