### 数学代写|概率论代写Probability theory代考|MAST20006

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|概率论代写Probability theory代考|1-D EVP Model by Yin and Graham

Yin and Graham $[122,123,151]$ first clarified the concept of “equivalent time” and gave the physical explanation and mathematic definition of equivalent time. The other important concepts, such as “instant time line” and “reference time line”, were also defined in their researches, as shown in the illustrations in Fig. 1.5.

“Instant time line” or ” $\angle$-line” is used to define the instantaneous vertical strain, which is pure elastic compression. It defines the instant elastic response of soil skeleton due to the variation of effective stress, but not the elastic-plastic response as assumed by Bjerrum [114]. It should be noted that the instant time line in Yin and Graham [122] has the same definition with that by Bjerrum [114], that is, it corresponds to the elastic-plastic deformations. “Reference time line” or ” $\lambda$-line” has zero equivalent time $\left(t_{e}=0\right)$, and it is used as a reference to count the equivalent time. Equivalent time $t_{e}$ has positive value $\left(0<t_{e}<\infty\right)$ below the reference time line, otherwise it is negative $\left(-t_{0}<t_{e}<0\right)$ where $t_{0}$ is a material parameter. The

vertical strain along the reference time line is the elastic-plastic compression. “Limit time line” is defined as a time line that has an infinite equivalent time value and a zero creep rate. Below the limit time line, the soil is considered to be elastic-plastic and time-independent.

The usage of “equivalent time” allows modeling the creep strains in both the overconsolidated and normally consolidated ranges using a constant creep parameter. Furthermore, it implies that the state at any point in Fig. $1.5$ is independent of stress path and loading history. This also implies that the creep rate at point ( $i+1)$ in the figure are the same for any loading path which finally reaches to point $(\mathrm{i}+1)$. For example, the creep rate of soil experienced a loading and unloading path ${\left[\mathrm{i} \rightarrow(\mathrm{i}+2)^{\prime} \rightarrow(\mathrm{i}+2)^{\prime \prime} \rightarrow(\mathrm{i}+1)\right]$ is equal to that in the loading and creep path $} \left.\mathrm{i} \rightarrow(\mathrm{i}+1)^{\prime \prime} \rightarrow(\mathrm{i}+1)\right]$ (see Fig. $\left.1.5\right) .$

## 数学代写|概率论代写Probability theory代考|Development of EVP Models

The 1-D EVP model proposed by Yin and Graham [122, 123, 151] has been verified through numerous consolidation test data [151-154]. Yin et al. [134] developed a three-dimensional EVP model by incorporating a nonlinear function for the creep strain and a new loading surface. This $3-\mathrm{D}$ model is more general than the original EVP model by Yin and Graham [126] and it can be utilized to model the timedependent stress-strain behavior of both overconsolidated and normally clays. In order to consider the creep and swelling simultaneously, Yin and Tong [141] developed the original model while simulating the time-dependent behavior of saturated soils.

Kelln et al. [155] developed an improved EVP model, which incorporates a yield locus into the model based on the creep function, and it maintains good predictive capabilities to the time-dependent behavior under various test conditions. Kelln et al. $[156,157]$ also validated this model by analyzing the geotechnical structures in finite element program. Bodas Freitas et al. [158] proposed a new EVP model considering the isotach viscosity based on the overstress theory [129] and the EVP model by Yin et al. $[126,134,159]$. Zhou et al. [160] further analyzed anisotropy on the basis of the model by Yin [159]. Some researchers investigated different EVP models and adopted them to model the consolidation caused by vertical drains in soil subject to creep $[152,161]$.

These proposed models have their own advantages and disadvantages. In general, the complex model with more parameters can better capture the creep deformation of soft soil. Considering the practical application of engineering, a simple model with sufficient accuracy is preferred to be adopted. Therefore, given a specific soil sample, the most suitable model is required to be selected from the numerous existing models to analyze the deformation behavior.

## 数学代写|概率论代写Probability theory代考|Determination of Model Parameters

Regardless of the time-dependent model utilized in the analysis, the determination of model parameters is the first problem to be solved. At present, several studies demonstrated how to identify the parameters of an EVP model.

Yin and Graham [151] suggested determining parameters by gradually using the curve-fitting technique based on the experimental data at different consolidation stages. They pointed out that the loading increment that is located in the normally consolidated stage can give more information for the determination of creep parameters. The studies by Yin and Yin et al. [134, 159] presented the same way to determine the parameters. They suggested that the reference equivalent time $\left(t_{0}\right)$ can be determined approximately referring to the time at which the excess pore water pressure has been dissipated completely, and then the other two parameters, that is, creep coefficient and creep strain limit, are obtained by fitting the measured data after the primary consolidation stage.

However, Mesri and Vardhanabhuti [162] pointed that it is not the best way to estimate the creep parameters just using the data after complete dissipation of the excess pore water pressure, and the determination of creep parameters should consider the consolidation data during the excess pore water pressures dissipation process. Le et al. [163] developed the trust-region reflective least-squares method to determine all model parameters simultaneously based on the test data from the whole consolidation stage at different loads, and the time parameter $t_{0}$ is assumed the unit minute. Ye et al. [164] used the simplex algorithm with random sampling method to optimize the creep and destructuration parameters of EVP model by Yin et al. $[137,139]$, and the other model parameters in EVP model were suggested to be determined in advance based on test data.

As can be noted, all the previous studies have adopted the deterministic method to determine the model parameters, and they only require the best fitting between the predictions and the observations. The deterministic method ignores the effect of uncertainty in test data and model itself on the model parameters. In Chaps. 4 and 5 , the model parameters will be determined by using the Bayesian probabilistic method, and their uncertainties will also be evaluated at the same time. Moreover, the time parameter as an unknown model parameter with other creep parameters will be determined simultaneously using the whole consolidation data, and no parameter will be determined in advance. Although this study only focused on the parameter determination of 1-D elastic viscoplastic models by Yin et al. [139, 141], the adopted Bayesian probabilistic method can be utilized to analyze any time-dependent models.

## 数学代写|概率论代写Probability theory代考|1-D EVP Model by Yin and Graham

“即时时间线”或“∠-line”用于定义瞬时垂直应变，即纯弹性压缩。它定义了由于有效应力变化引起的土壤骨架的瞬时弹性响应，而不是 Bjerrum [114] 假设的弹塑性响应。需要注意的是，Yin 和 Graham [122] 中的瞬时时间线与 Bjerrum [114] 的定义相同，即对应于弹塑性变形。“参考时间线”或“λ-line”的等效时间为零(吨和=0), 并用作计算等效时间的参考。等效时间吨和有正值(0<吨和<∞)低于参考时间线，否则为负(−吨0<吨和<0)在哪里吨0是材料参数。这

“等效时间”的使用允许使用恒定蠕变参数对过度固结和正常固结范围内的蠕变应变进行建模。此外，这意味着图 1 中任何点的状态。1.5独立于应力路径和加载历史。这也意味着点处的蠕变速率 (一世+1)图中对于任何最终到达点的加载路径都是相同的(一世+1). 例如，土壤的蠕变速率经历了一个加载和卸载路径[一世→(一世+2)′→(一世+2)′′→(一世+1)]$一世s和q在一个l吨○吨H一个吨一世n吨H和l○一个d一世nG一个ndCr和和pp一个吨H$一世→(一世+1)′′→(一世+1)]（见图。1.5).

## 数学代写|概率论代写Probability theory代考|Development of EVP Models

Yin 和 Graham [122, 123, 151] 提出的一维 EVP 模型已经通过大量的固结测试数据 [151-154] 得到验证。尹等人。[134] 通过结合蠕变应变的非线性函数和新的加载表面，开发了一个三维 EVP 模型。这个3−D模型比 Yin 和 Graham [126] 的原始 EVP 模型更通用，它可用于模拟过度固结粘土和正常粘土的时间依赖性应力应变行为。为了同时考虑蠕变和膨胀，Yin 和 Tong [141] 开发了原始模型，同时模拟了饱和土的随时间变化的行为。

## 数学代写|概率论代写Probability theory代考|Determination of Model Parameters

Yin 和 Graham [151] 建议根据不同固结阶段的实验数据逐步使用曲线拟合技术确定参数。他们指出，位于正常固结阶段的加载增量可以为确定蠕变参数提供更多信息。尹和尹等人的研究。[134, 159] 提出了相同的方法来确定参数。他们建议参考等效时间(吨0)可以参考超孔隙水压力完全消散的时间来近似确定，然后通过一次固结阶段后的实测数据拟合得到其他两个参数，即蠕变系数和蠕变应变极限。

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