# Mathematical Models for Stress–Strain Behavior of Nano Magnesia-Cement-Reinforced Seashore Soft Soil

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## Abstract

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## 1. Introduction

## 2. Mathematical Characteristics of Stress–Strain Constitutive Relations of Nmcs

#### 2.1. Materials and Samples

#### 2.2. Mathematical Characteristics of τ-δ Behavior

#### 2.2.1. Strain-Hardening τ-δ Behavior

_{0}. In the plastic stage, the tangent modulus E

_{i}reduces gradually as strain accumulates, leading to a nonlinear τ-δ curve. In the failure stage, the τ-δ curve flattens out and the shear stress reaches the ultimate shear strength τ

_{p}with a corresponding shear strain δ

_{p}. In this stage, the shear stress is mainly contributed by the friction resistance at the failure surface of the soil sample. In sum, the strain-hardening curve includes four mathematical features: through the origin, monotone increasing, convex to τ-axis, and infinite convergence.

#### 2.2.2. Falling τ-δ Behavior

_{p}and shear strength τ

_{p}, respectively. Stage 2 is the falling stage (AB), which evidently shows the falling of shear stress after the soil fails. This falling of shear stress is attributed to the loss of cohesion and the tangent modulus of the τ-δ curve gives a negative value. Stage 3 is the plastic stage (BC) wherein the friction resistance starts to dominate after the loss of cohesion. In this stage, the tangent modulus of the curve approximates the initial elastic modulus at the beginning which subsequently decreases due to the occurrence of plastic shear stress. Stage 4 is the residual shear stress stage (CD) where the tangent modulus approaches zero and the shear stress is equal to residual shear stress. It should be noted that the τ-δ falling curve can be modified by ignoring Stages 2 and 3 (AB and BC), then it returns to the strain-hardening behavior. However, the shear strength in the falling curve refers to the shear stress at the end of Stage 1 which is different from its counterpart in the strain-hardening curve.

#### 2.2.3. S-Type Falling τ-δ Behavior

_{p}and corresponding displacement δ

_{p}at failure occur at the end of Stage 2, i.e., point B. After reaching failure, the falling stage (BC), the frictional plastic stage (CD) and the residual shear stress stage (DE) emerge continuously. These three stages are similar to those in the above falling behavior. Thus, the S-type falling curve can be modified in a similar way as that for the above falling curve. However, the obtained modified curve is still unlike the strain-hardening curve which is convex to τ-axis; namely, an inflection point occurs on the curve, e.g., before this point, the curve is convex to δ-axis, while after this point, it is convex to τ-axis.

#### 2.3. Mathematical Characteristics of σ-δ Behavior

_{p}corresponding to a strain of ε

_{p}. This curve comprises three stages. In the hardening stage (OP), the stress initially increases linearly with strain; the gradient of which represents the initial elastic modulus E

_{0}. This gradient starts to decrease in the later phase of the hardening stage until arriving at the peak, i.e., point P where the brittle failure takes place and the tangent modulus equals to zero. After that the stress reduces monotonically with strain in the softening stage (PM) and remains unchanged when entering the residual stress stage (MN). In sum, the mathematical features of the curve include: through the origin, having extreme value point, and converging at residual stress.

## 3. Established Mathematical Model

#### 3.1. Conventional Shear Stress-Displacement (τ-δ) Models

#### 3.1.1. Hyperbolic Model

_{0}is the nominal elastic modulus (kPa/mm) and τ

_{p}is the shear strength (kPa).

#### 3.1.2. Exponential Model

#### 3.1.3. Power Function Model

#### 3.2. Mathematical REP Model for τ-δ Behavior

_{p}. Hence,

_{0}. Combining with Equation (8) gives

_{0}/τ

_{p}− λk) > 0 if b > 0.

_{p}. Therefore, the new model goes through the origin and has both upper and lower bounds. The coefficients of the second derivative equation are all negative, and the second derivative of the new model is always less than 0. Hence, the REP model is theoretically suitable for the hardening and modified falling curves.

#### 3.3. Mathematical Models for Stress-Displacement (σ-δ) Behavior

_{p}is the peak value of the stress–strain curve (UCS).

_{p}and ε

_{p}are the maximum stress and the corresponding strain, respectively. A and B are the fitting parameters to be determined.

#### 3.4. Application and Analysis

#### 3.4.1. Hardening τ-δ Behavior

_{p}, and lie in the upper part of the measured curve with big difference. In contrast, the power function converges well, and the fitting effect is better than the hyperbolic and exponential functions, although the end of the curve still appears above the measured curve. The REP model also has a good convergence effect and the whole fitted failure stage is very close to that of the measured curve.

#### 3.4.2. Modified Falling τ-δ Behavior

#### 3.4.3. Strong Softening σ-δ Behavior

_{2}SO

_{4}erosion were adopted, Figure 9. As can be seen, the power function model behaves the worst in the modeling of the hump-shaped σ-δ curve. For instance, it is convex at the rising stage of the curve, which is contrary to the characteristic of concave in the measured curve. In addition, it cannot simulate the softening behavior as the measured curve although it converges to an almost constant value in the end. The quadratic function model performs better than the power function model. It is able to simulate the softening behavior and give a peak stress value. However, the shape of the quadratic curve is convex in all stages, which differs with the shape of measured curves. Worse still, there exists a large discrepancy in the maximum stress value and its corresponding strain value between the results of quadratic model and the measured results. This may raise an adverse effect in predicting UCS results. In contrast, the results of proposed REP model agree incredibly well with the measured results. The REP model not only shows consistent shapes, but also predicts a fairly close maximum stress value and a corresponding strain value. This is of great engineering value in predicting the compressive strength.

_{s}is the concentration of sulfuric acid solution.

_{s}into the four parameters of REP model leads to

_{s}), cohesion (c) and internal friction angle (φ), making the new model more suitable for acid erosion environment.

## 4. Mathematical CEL Model for Modified S-Type Falling τ-δ Behavior

#### 4.1. Mathematical CEL Model

_{p}so

_{p}at infinity. Combining the similar items in the first derivative of Equation (29) gives

_{c}, τ’’ > 0 and the curve is convex to the δ-axis; when δ > δ

_{c}, τ’’ < 0 and the curve is convex to the τ-axis.

#### 4.2. Application and Analysis

_{2}SO

_{4}erosion under vertical pressures of 300 kPa and 400 kPa, respectively. The results are shown in Figure 11 and Figure 12.

_{p}than the other models in the failure stage. However, at the initial stage, the CEL curve is slightly lower than the measure curve, while in the failure stage, the difference between the CEL curve and the measured curve is relatively evident.

## 5. Conclusions

- The τ-δ curves with varying cement mixing ratio, Nm mixing ratio and acid erosion concentrations exhibit different behaviors. For example, hardening behavior occurs with low cement mixing ratio; falling behavior takes place after adding Nm and S-type happens with acid erosion environment.
- The proposed REP model is able to satisfy the mathematical characteristics of the stress–strain curves with hardening and modified falling behaviors as well as strong softening behavior. Compared with conventional hyperbolic model, exponential and power function models which easily produce double-“X” discrepancies, the REP model has an evidently higher fitting accuracy in modeling both hardening and modified falling curves. In the modeling of strong softening behavior, the REP model also performs the best. In addition, by introducing an acid erosion factor, the REP model can be further expressed as a sulfuric acid erosion damage model which includes only three typical parameters, i.e., sulfuric acid concentration, cohesion of silty clay, and internal friction angle.
- In the modeling of S-type stress–strain behavior, the conventional models are unable to simulate the characteristic with inflection point and produce even bigger “X” discrepancy than that in the modeling of hardening and falling behavior. In contrast, the proposed CEL model performs much better especially in modeling the range around the inflection point at the early stage.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**Comparison of stress–strain curves: (

**a**) pure water condition (0 mol/L H

_{2}SO

_{4}); (

**b**) 0.08 mol/L H

_{2}SO

_{4}.

**Figure 10.**Comparison of stress–strain curves of different H

_{2}SO

_{4}erosion concentrations with REP model fittings.

H_{2}SO_{4} Erosion Concentrations | Parameter | |||
---|---|---|---|---|

a | b | k | λ | |

0.00 mol/L | 223.23 | 16.62 | 0.08 | 131.20 |

0.02 mol/L | 83.33 | 5.95 | 0.18 | 19.76 |

0.04 mol/L | 100.70 | 7.53 | 0.12 | 48.53 |

0.06 mol/L | 79.73 | 2.36 | 0.25 | 3.60 |

0.08 mol/L | 40.00 | 2.93 | 0.12 | 14.04 |

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**MDPI and ACS Style**

Wang, W.; Fu, Y.; Zhang, C.; Li, N.; Zhou, A.
Mathematical Models for Stress–Strain Behavior of Nano Magnesia-Cement-Reinforced Seashore Soft Soil. *Mathematics* **2020**, *8*, 456.
https://doi.org/10.3390/math8030456

**AMA Style**

Wang W, Fu Y, Zhang C, Li N, Zhou A.
Mathematical Models for Stress–Strain Behavior of Nano Magnesia-Cement-Reinforced Seashore Soft Soil. *Mathematics*. 2020; 8(3):456.
https://doi.org/10.3390/math8030456

**Chicago/Turabian Style**

Wang, Wei, Yong Fu, Chen Zhang, Na Li, and Aizhao Zhou.
2020. "Mathematical Models for Stress–Strain Behavior of Nano Magnesia-Cement-Reinforced Seashore Soft Soil" *Mathematics* 8, no. 3: 456.
https://doi.org/10.3390/math8030456