Experimental and Theoretical Investigations of the Constitutive Relations of Artificial Frozen Silty Clay
Abstract
:1. Introduction
2. Experimental Conditions
2.1. Experimental Material
2.2. Sample Preparation and Test Apparatus
2.3. Experimental Results and Analysis
3. MDC Model for Frozen Silty Clay
3.1. Constitutive Model and Parameter Determination
3.2. STRAIN-Softening Criterion of the MDC Model
4. SD Model for Frozen Soil
4.1. SD Constitutive Model
- Set the initial value of x = [m, F0] and threshold ε;
- Calculate fi(x) and obtain the vector fk;
- Calculate fi’(x) and obtain the Jacobian matrix Jk;
- Calculate xk+1 by using the iteration form of the Gauss-Newton method;
- Judge the condition ‖xk+1 − xk‖ < ε, if it is not satisfied, repeat step (2) until the condition is satisfied.
4.2. Strain-Softening Criterion of the SD Model
4.3. Model Verification and Analysis
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Notation List
Symbol | Description |
σ1−σ3 | Deviatoric stress |
εa | Axial strain |
a | Parameter related to initial elastic modulus |
b | Parameter related to residual and ultimate strengths |
c | Parameter related to residual and ultimate strengths |
Et | Tangent modulus |
Ei | Initial elastic modulus |
εrs | Calculated initial softening strain |
q | Shear strength |
δe | Positive error correction term |
δb | Length of the confidence interval for parameter b |
δc | Length of the confidence interval for parameter c |
α | Significance level |
Z | Random variable that satisfies normal distribution |
n | Number of samples |
σb | Standard deviation of parameter b |
σc | Standard deviation of parameter c |
δt | Length of the confidence interval of b-c |
εre | Experimental initial softening strain |
σ1ʹ-σ3ʹ | Effective deviatoric stress |
D | Damage variable |
N | Total number of elements |
N(εa) | Number of damaged elements |
f(εa) | Probability density function of the Weibull distribution |
m | Scale parameter of the Weibull distribution |
F0 | Shape parameter of the Weibull distribution |
fi(x) | Residuals between test and simulation results |
x | Parameter set |
xk | Parameter value at k-th iteration |
A | Predicted results calculated by the MDC model |
gi(x) | First order Taylor expansion of fi(x) |
F(x) | Objective function |
Jk | Jacobian matrix |
ε | Error threshold |
εrep | Reference strain |
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Mineral Composition | Quartz | Illite | Halloysite | Kaolinite | Albite | Anorthite | Unknown |
---|---|---|---|---|---|---|---|
Relative content (%) | 39.9 | 10.2 | 14.7 | 5.6 | 13.2 | 12.4 | 4.0 |
No. | Fitting Parameters | No. | Fitting Parameters | ||||||
---|---|---|---|---|---|---|---|---|---|
a | b | c | R2 | a | b | c | R2 | ||
1 | 1.292 | 0.416 | 0.415 | 0.9885 | 15 | 0.421 | 0.191 | 0.191 | 0.9972 |
2 | 0.960 | 0.371 | 0.372 | 0.9939 | 16 | 0.756 | 0.120 | 0.011 | 0.9942 |
3 | 0.690 | 0.244 | 0.244 | 0.9988 | 17 | 0.709 | 0.101 | 0.009 | 0.9976 |
4 | 1.537 | 0.502 | 0.501 | 0.9898 | 18 | 0.486 | 0.238 | 0.260 | 0.9989 |
5 | 1.124 | 0.448 | 0.447 | 0.9970 | 19 | 0.660 | 0.191 | 0.189 | 0.9943 |
6 | 0.796 | 0.288 | 0.288 | 0.9911 | 20 | 0.507 | 0.174 | 0.174 | 0.9976 |
7 | 1.662 | 0.265 | 0.017 | 0.9919 | 21 | 0.279 | 0.097 | 0.065 | 0.9988 |
8 | 1.440 | 0.204 | 0.015 | 0.9859 | 22 | 0.636 | 0.314 | 0.450 | 0.9898 |
9 | 1.139 | 0.565 | 0.914 | 0.9983 | 23 | 0.545 | 0.198 | 0.198 | 0.9907 |
10 | 0.707 | 0.123 | 0.045 | 0.9956 | 24 | 0.481 | 0.152 | 0.152 | 0.9919 |
11 | 0.534 | 0.265 | 0.278 | 0.9963 | 25 | 0.681 | 0.098 | 0.002 | 0.9995 |
12 | 0.368 | 0.179 | 0.181 | 0.9946 | 26 | 0.580 | 0.090 | 0.008 | 0.9983 |
13 | 0.727 | 0.143 | 0.066 | 0.9974 | 27 | 0.386 | 0.209 | 0.209 | 0.9959 |
14 | 0.669 | 0.239 | 0.239 | 0.9973 | – | – | – | – | – |
No. | q(b–c) | δb | δc | 0.5δt | δe | εrs (%) | εre (%) |
---|---|---|---|---|---|---|---|
7 | 0.2514 | 0.0346 | 0.0284 | 0.0315 | 0.0014 | 7.20 | 7~9 |
8 | 0.2504 | 0.0322 | 0.0266 | 0.0294 | 0.0004 | 8.28 | 8~10 |
16 | 0.2514 | 0.0098 | 0.0080 | 0.0089 | 0.0014 | 7.95 | 6~8 |
17 | 0.2550 | 0.0092 | 0.0078 | 0.0087 | 0.0050 | 8.54 | 7~9 |
25 | 0.2533 | 0.0074 | 0.0058 | 0.0071 | 0.0033 | 6.75 | 6~8 |
26 | 0.2492 | 0.0064 | 0.0054 | 0.0059 | 0.0008 | 7.84 | 7~9 |
Fitting Parameters | ||||||||
---|---|---|---|---|---|---|---|---|
a | b | c | F0 | m | εrs (%) | εre (%) | R2 | |
1 | 1.292 | 0.416 | 0.415 | 28.61 | 8.322 | 17.63 | 16~18 | 0.9916 |
4 | 1.537 | 0.502 | 0.501 | 28.20 | 8.012 | 17.17 | 16~18 | 0.9952 |
13 | 0.727 | 0.143 | 0.067 | 27.12 | 7.306 | 14.61 | 14~16 | 0.9986 |
14 | 0.669 | 0.239 | 0.239 | 27.86 | 7.967 | 16.81 | 15~17 | 0.9988 |
19 | 0.660 | 0.182 | 0.181 | 28.01 | 8.043 | 17.36 | 15~17 | 0.9961 |
20 | 0.507 | 0.174 | 0.174 | 28.91 | 8.643 | 17.92 | 16~18 | 0.9982 |
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Li, Z.; Chen, J.; Mao, C. Experimental and Theoretical Investigations of the Constitutive Relations of Artificial Frozen Silty Clay. Materials 2019, 12, 3159. https://doi.org/10.3390/ma12193159
Li Z, Chen J, Mao C. Experimental and Theoretical Investigations of the Constitutive Relations of Artificial Frozen Silty Clay. Materials. 2019; 12(19):3159. https://doi.org/10.3390/ma12193159
Chicago/Turabian StyleLi, Zhiming, Jian Chen, and Chaojun Mao. 2019. "Experimental and Theoretical Investigations of the Constitutive Relations of Artificial Frozen Silty Clay" Materials 12, no. 19: 3159. https://doi.org/10.3390/ma12193159