Maxwell Model: Oedometer Test

Note

To view this project in FLAC3D, use the menu command Help ► Examples…. The project’s main data files are shown at the end of this example.

This example compares numerical and analytical solutions of an oedometer test carried out on a Maxwell substance. In this test, the base of the sample is fixed, lateral deformations are prevented, and a constant vertical load, $P$, is applied at the top of the specimen.

The analytical solution for vertical strain and stresses is

(1) $$\epsilon_{zz}=- {{P} \over {K}} \left( 1 - c e^{-bt} \right)$$

(2) $$\sigma_{zz} = -P$$

(3) $$\sigma_{xx} = \sigma_{yy} = -P \left( 1-a e^{-bt} \right)$$

where $a$, $b$, and $c$ are the three constants:

(4) $$a = 2 {{G} \over {K + 4G/3}}$$

(5) $$b = {{K} \over {\eta}} {{G} \over {K + 4G/3}}$$

(6) $$c = {{4} \over {3}} {{G} \over {K + 4G/3}}$$

$K$ and $G$ are bulk and shear modulus of the substance, and $\eta$ is the viscosity.

The numerical simulation is carried out simultaneously on three samples, represented by one zone, using the Maxwell model, the Burgers model and the Burgers-Mohr model.

Because no value is assigned to the property viscosity-kelvin for the Burgers model and the Maxwell-Mohr model, the Kelvin cell logic is not taken into account by the models. Also, the cohesion property is set to a high value to prevent triggering of the plasticity logic for the Burgers-Mohr model. The initial state is obtained by cycling the model to elastic equilibrium. For the viscous response, velocities are reset to zero, and the initial timestep is set to a small value ($\Delta t$ = 10^-4) compared to the ratio of viscosity over shear modulus ($\eta ^M/G^M$ = 2.0). With the choice of default automatic creep timestep parameter settings used in the example, the timestep increases by a factor of 1.01 when the out-of-balance force ratio is less than 10^-3, until $\Delta t$ = 10^-2. A state of hydrostatic stress is reached at the end of the test. Figure 1 and Figure 2 show the agreement between analytical solutions and numerical predictions for stresses and strains in the three samples.

Figure 1: Oedometer test on a Maxwell substance: analytical and numerical stress values versus time.

Figure 2: Oedometer test on a Maxwell substance: analytical and numerical strain values versus time.

Data File

;------------------------------------------------------------
;       Oedometer test -- Maxwell substance
;-------------------------------------------------------------------
model new
model large-strain off
fish automatic-create off
model title "Oedometer test --- 'Maxwell' substance"
model configure creep
program call 'parameter'
[ini_cons]
; --- model ---
zone create brick size 5 1 1
zone cmodel assign maxwell       range position-x 0 1
zone cmodel assign burgers       range position-x 2 3
zone cmodel assign burgers-mohr  range position-x 4 5

zone null range union position-x 1 2 position-x 3 4
;
zone property density 1 bulk [c_bu]
zone property shear  [c_sh] viscosity  [c_vi ] ...
              range position-x 0 1
zone property shear-maxwell [c_sh] viscosity-maxwell [c_vi]...
              range position-x 2 3
zone property shear-maxwell [c_sh] viscosity-maxwell [c_vi] ...
              cohesion 1e20 tension 1e20 ...
              range position-x 4 5
;
zone gridpoint fix velocity-x
zone gridpoint fix velocity-y
zone gridpoint fix velocity-z range position-z 0
zone face apply stress-normal -1.0 range position-z 1
; --- histories ---
history interval  1
zone history stress-yy position (0.5,0.5,0.5)
zone history stress-yy position (2.5,0.5,0.5)
zone history stress-yy position (4.5,0.5,0.5)
fish history [ana_syy]
zone history stress-zz position (0.5,0.5,0.5)
zone history stress-zz position (2.5,0.5,0.5)
zone history stress-zz position (4.5,0.5,0.5)
fish history [ana_szz]
zone history displacement-z position 0 0 1
zone history displacement-z position 2 0 1
zone history displacement-z position 4 0 1
fish history [ana_ezz]
model history creep time-total
; --- elastic equilibrium ---
model solve ratio 1e-4
; --- reset velocities to zero ---
zone gridpoint initialize velocity (0,0,0)
; --- viscous behaviour ---
model creep timestep starting 1.e-4
model creep timestep minimum  1.e-4 
model creep timestep maximum  1.e-2 
model solve time-total=25.
model save 'maxwell'