# Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium
## 2 Methods  
### Deflection Function for an Elastic Compression-Bending Bar within an Elastic Medium  
#### 2.1.1 Basic Model of Viscoelastic Mechanics  
In the southeastern coastal region of China, where the soft soil layer is relatively thick, rock-socketed piles are commonly employed in the design of bridge pile foundations to ensure the stability of the superstructure. Pile lengths often exceed 50 m and can even reach 100 m in some areas. As such, the stability of single piles under these conditions can be approached as a slender bar stability problem. Unlike a compression bar model with a fixed bottom and free top, the pile body is enveloped by a thick layer of soft soil and cannot be simplified as a compression bar stability research model without lateral support. Similar to the support conditions of an elastic foundation beam, the pile side can be transformedinto an elastic foundation support based on the Winkler foundation assumption. In actual working conditions, pile foundations bear not only axial loads transmitted from the bridge superstructure but also additional horizontal stresses caused by lateral deviation of piles, resulting in a state of compression-bending. The constitutive equation of linear viscoelastic materials can be bifurcated into two primary categories: differential and integral types. The differential type is fundamentally premised on the utilization of ideal elements to construct a variety of models. These models subsequently facilitate the derivation of material constitutive equations, which are based on the inherent properties and combinational forms of the elements. This approach offers a level of convenience in resolving certain problems. As per the foundational theory of the differential type, it can be postulated that the one-dimensional differential constitutive equation of any given linear viscoelastic material can be articulated as follows:  
\[P\sigma=Q\epsilon \tag{1}\]  
\[P=\sum_{k=0}^{n}p_{k}\frac{d^{k}}{dt^{k}},\ Q=\sum_{k=0}^{n}q_{k}\frac{d^{k}}{ dt^{k}} \tag{2}\]  
where: the differential operators \(P\) and \(Q\) are shown in Equation (2), where \(p_{k}\) and \(q_{k}\) are constants related to the elastic and viscous properties (\(k\) is only a counting symbol, not the subgrade reaction coefficient mentioned below). The constitutive equation of viscoelastic materials is derived from element models, which are formulated through the integration of elastic (spring) and viscous (dashpot) elements in a multitude of configurations. The spring elements adhere to the principles of [PERSON]'s law. \[\sigma=E\epsilon,\ \tau=G\gamma \tag{3}\]  
where \(\sigma\) and \(\tau\) are the normal stress and shear stress, respectively; \(\epsilon\) and \(\gamma\) are the normal strain and shear strain, respectively; \(E\) and \(G\) are the tensile-compressive elastic modulus and shear elastic modulus. The elements of a viscoelastic model, known as dashpots, adhere to [PERSON]'s law of viscosity, which is a fundamental principle in fluid mechanics as follow:  
\[\sigma=\eta\dot{\epsilon},\ \tau=\eta_{1}\dot{\gamma} \tag{4}\]  
where \(\eta\) and \(\eta_{1}\) is the coefficient of viscosity, \(\dot{\epsilon}\) and \(\dot{\gamma}\) are the normal strain and shear strain of viscosity. Figure 1 shows the mechanical model of Spring and Dashpot element. The Kelvin model is composed of a spring element and a dashpot element in parallel, as is shown in Figure 2. And the Van Der Poel model is formed by a spring element and a Kelvin model in series as is shown in Figure 3.  
Figure 1: Spring elements and dashpot elements. The constitutive equation of Van Der Poel is  
\[\sigma+p_{1}\dot{\sigma}=q_{0}\varepsilon+q_{1}\dot{\varepsilon} \tag{5}\]  
where:  
\[p_{1}=\frac{\eta_{1}}{E_{1}+E_{2}},\ q_{0}=\frac{E_{1}E_{2}}{E_{1}+E_{2}},\ q_{1 }=\frac{E_{2}\eta_{1}}{E_{1}+E_{2}} \tag{6}\]  
Given a constant stress \(\sigma_{0}\), the creep equation of the Van Der Poel model is as follows:  
\[\varepsilon(t)=\frac{\sigma_{0}}{E_{2}}+\frac{\sigma_{0}}{E_{1}}\bigg{(}1-e^{ -\frac{E_{1}}{\eta}t}\bigg{)} \tag{7}\]