Triaxial tests on a saturated clay

This example demonstrates the modified Cam-clay plasticity model provided in Abaqus.

This page discusses:

Problem description
Results and discussion
Input files
References
Figures

ProductsAbaqus/Standard

Cam-clay theory provides a reasonable match to the experimentally observed behavior of saturated clays and belongs to the family of critical state plasticity models developed by Roscoe and his colleagues (see Roscoe and Burland—1968—and Schofield and Wroth—1968).

The Cam-clay model in Abaqus permits two extensions of the original Roscoe model: “capping” of the yield ellipse on the wet side of critical state, and consideration of the third stress invariant in the yield function. Both of these extensions to the modified Cam-clay theory are documented in Plasticity for nonmetals and are included in this example.

The general modified Cam-clay yield function (for isotropic plasticity and zero tensile strength) used in Abaqus is

\frac{1}{β^{2}} {(\frac{p}{a} - 1)}^{2} + {(\frac{q}{g M a})}^{2} - 1 = 0,

where the three stress invariants are the equivalent pressure stress given by

p = - \frac{1}{3} trace (σ);

the equivalent shear stress given by

q = \sqrt{\frac{3}{2} S : S},

where $S$ is the deviatoric stress ( $S = σ + p I$ ); and the third stress invariant,

r = {(\frac{9}{2} S : S \cdot S)}^{\frac{1}{3}} .

The other parameters in the function are a, the value of the equivalent pressure stress at critical state; M, a material parameter defining the slope of the critical state lines; $β$ , a “capping” parameter used to provide a different shaped yield ellipse on the wet side of critical state; and g, a function that is dependent on the third stress invariant, used to define different yield surface sizes in compression and extension:

g = \frac{2 K}{1 + K + (1 - K) r^{3} / q^{3}},

where K is a material parameter.

The “standard” Cam-clay yield function has $β = g = $ 1. Including these parameters in the yield surface expression generalizes that expression to allow closer matching of data under various conditions of loading.

Problem description

The material parameters used in this example are as follows:

Elasticity:


Logarithmic bulk modulus, $κ$ :	0.026
Poisson's ratio, $ν$ :	0.3

Plasticity:


Logarithmic hardening modulus, $λ$ :	0.174
Critical state ratio, M:	1.0
Wet cap parameter, $β$ :	0.5
Third stress invariant parameter, K:	0.75
Initial overconsolidation parameter, $a_{0}$ :	58.3 kN/m² (8.455 lb/in²)

The example studies a simple triaxial test: an axisymmetric soil sample contained between two smooth platens, one of which is held fixed and the other of which has prescribed vertical motion, positive for extension and negative for compression. The soil specimen is first loaded by constant pressure. Then the top platen is moved, either downward to test triaxial compression or upward to test triaxial extension. Figure 1 defines the problem geometry. The analyses are meant to simulate drained triaxial tests; therefore, they can be run with the pure displacement elements in Abaqus.

Since the platens are assumed to be smooth and the soil is homogeneous, the stress will be constant throughout the model. For simplicity, large displacement effects are ignored.

Results and discussion

For both cases the initial pressure stress is given via initial conditions, and an initial geostatic step (Geostatic Stress State) is included in which the confining pressure is applied to the outer surface of the specimen. At the start of a soils analysis with initial stresses, Abaqus checks to see that the stress specified does not violate the initial yield surface. If it does, the hardening value (a in the yield surface definition above) is modified to make the yield surface consistent with the stress state. To test this part of the code, in the present example the initial stress state lies within the initial yield surface when the “standard” Cam-clay plasticity theory is used, but it violates the yield criterion with the given initial overconsolidation parameter, $a_{0}$ , when the “capped” plasticity theory is used. The adjustment to the value of $a_{0}$ is shown in Figure 2.

It is recommended that a geostatic procedure always be included at the start of a soils analysis to ensure compatibility between the initially prescribed stress state and the initial loading.

Compression of a drained triaxial specimen

In this case, during the second step of the analysis the top platen is moved down by half the soil sample height. The material response is shown in Figure 3. Depending on the theory used, the soil yields more or less gradually as the displacement increases until critical state is reached (that is, when $q = M p$ : see Figure 2) when the response is perfectly plastic. “Capping” has a strong effect on the material response: for the load path specified (line $O S$ on Figure 2) the “capped” theory predicts that critical state will be reached at a normalized vertical displacement of 0.18, whereas the “standard” Cam-clay theory predicts that critical state will not be reached until the soil sample has been reduced in height by half. It should be emphasized that these results have been obtained with the small-displacement assumption; although the stress-strain response is accurate, the load-displacement response is not because the strains are well beyond the range where linearized strain-displacement relations are reasonable.

Extension of a drained triaxial specimen

In this case, in the second step the top platen is moved vertically upward. This decreases the confining pressure in the soil, so critical state is reached at a lower value of equivalent shear stress than in the compression case. This is clearly seen in Figure 3. Of interest here is the effect of the third stress invariant on the plasticity solution: this dependence is specified via the parameter K (see the Introduction: general for a complete discussion). Since the present case is pure triaxial extension, the critical state condition becomes $q = K M p .$ As seen in Figure 4, this has the effect of lowering the achievable equivalent shear stress states by flattening the yield surface in p–q space. For the load path specified here, the solution follows line $O S$ on Figure 4 for the “standard” Cam-clay theory and line $O S_{1}$ for the case that includes dependence on the third stress invariant.

Input files

triaxialtestclay_cax8r.inp: Compression of a drained triaxial soil specimen modeled with “standard” modified Cam-clay theory. A single CAX8R element is used in this model.
triaxialtestclay_caxa8r1.inp: Same analysis using the CAXA element. A single CAXA8R1 element is used.

Input files for the other cases described in this example are created by including the parameters $β$ and K in the definition of the Cam-clay model.

References

Roscoe, K. H., and J. B. Burland, Stress-strain Behavior of `Wet Clay', Engineering Plasticity, J. Heyman and F. A. Leckie, Editors, Cambridge University Press, 1968.
Schofield, A., and C. P. Wroth, Critical State Soil Mechanics, McGraw-Hill, 1968.

Figures