Maxwell Model: Oedometer Test

Note

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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^-6) 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 Files

OedometerMaxwell.dat

;------------------------------------------------------------
;       Oedometer test -- Maxwell substance
;-------------------------------------------------------------------
model new
fish automatic-create off
model title "Oedometer test --- 'Maxwell' substance"
model configure creep
model large-strain off

fish define ini_cons
    global c_bu = 1.
    global c_sh = 1.
    global c_vi = 2.
    global c_pr = -1.
    local a1  = c_bu + 4. * c_sh / 3.
    global c_a = 2. * c_sh / a1
    global c_b = c_a * c_bu / (2. * c_vi)
    global c_c = c_a * 2. / 3.
    global c_d = -c_pr / c_bu
end
[ini_cons]

; --- model ---
block create brick 0 1
block zone generate edgelength 1.0
block zone cmodel assign burgers-mohr
block zone property density 1 bulk [c_bu] shear-maxwell [c_sh] cohesion 1E20
block zone property tension 1E20 viscosity-maxwell [c_vi]
block face apply stress 0.0 0.0 [c_pr] 0 0 0 range position-z 1
block gridpoint apply velocity-x 0 range position-x 0 1
block gridpoint apply velocity-y 0 range position-y 0 1
block gridpoint apply velocity-z 0 range position-z 0

; --- elastic equilibrium ---
model solve

; --- histories ---
program call 'analytical.fis'
fish history name 'SXX' sxx
fish history name 'SXX Analytical' ana_sxx
fish history name 'SZZ' szz
fish history name 'SZZ_Analytical' ana_szz
block history name 'Disp-Z' displacement-z position 0.0 0.0 1.0
fish history name 'EXX Analytical' ana_ezz
model history creep time-total
model history timestep

; --- reset velocities to zero ---
block gridpoint initialize velocity (0,0,0)
; --- viscous behaviour ---
model creep timestep starting 1.e-6
model creep timestep minimum  1.e-6 
model creep timestep maximum  1.e-2 
model solve time-total=25.
model save 'maxwell'
program return