# Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium
## 2 Methods
### Deflection Function for an Elastic Compression-Bending Bar in a Viscoelastic Medium  
#### 2.2.1 Mechanical Model of an Elastic Compression-Bending Bar in a Viscoelastic Medium  
The interaction between elastic compression bending bar and the surrounding soil is complex and can be either static or dynamic. To simplify calculations, this interaction is often modeled using an elastic foundation beam. However, this approach has limitations as soil is a discrete, multiphase system with nonlinear and irrecoverable stress-strain characteristics under external forces. As such, modeling soil solely with springs may not accurately capture its behavior. Viscoelastic mechanics can better characterize the creep properties of soft soils by revealing their rheological properties. This typically involves establishing the viscoelastic instantaneous relationship of the material and determining boundary conditions and their solutions. By transforming the parameters characterizing the foundation soil's material properties in the elastic foundation model into time-dependent viscoelastic parameters, the springs in the elastic foundation can be replaced with a suitable viscoelastic model while maintaining the boundary conditions of the elastic foundation model. This allows for the solution of the beam on the elastic foundation to be transformed into a time-dependent viscoelastic solution. In this part, adopts the method of the Winkler foundation hypothesis, selects a suitable viscoelastic constitutive model, and simplifies the soil around the slender bar shown in Figure 5 into discrete viscoelastic element supports. Taking the Van [PERSON] model as an example, as shown in Figure 8.  
Figure 8: Model of elastic compression bending bar in viscoelastic medium. 2.2 A Comprehensive Solution Formula for the Deflection of an Elastic Slender Bar within a Viscoelastic Medium  
In accordance with the principle of elastic-viscoelastic correspondence, the viscoelastic foundation elastic beam problem may be considered as an elastic foundation elastic beam problem with a reaction force coefficient of \(\overline{k}(s,x)\). By analogy, utilizing the m-method to determine the value of the foundation reaction force coefficient, the foundation reaction force coefficient within the phase space formulation is established to be:  
\[\overline{k}(s,x)=A\overline{k}_{0}(s)(L-x) \tag{19}\]  
where: \(\overline{k}\) is reaction force coefficient of viscoelastic medium in kelvin model  
\(A\) is constant  
\(s\) is displacement of elastic bar  
Within Equation (19), \(A\) represents a constant that governs the rate of change for the foundation reaction force coefficient \(\overline{k}(s,x)\) from the ground surface to the depth of the soil. By substituting Equation (19) into Equation (13), the result is obtained:  
\[ \begin{array}{ll}\overline{W}_{nr}&=&\int_{0}^{L}\bigg{(}EI\frac{(2n-1)^{2}(2 r-1)^{2}\pi^{4}}{16\overline{1}^{4}}\cos\frac{(2n-1)\pi x}{2L}\cos\frac{(2r-1) \pi x}{2L}\\ &+A\overline{k}_{0}(s)(L-x)b_{1}\Big{(}1-\cos\frac{(2n-1)\pi x}{2L}\Big{)} \big{(}1-\cos\frac{(2r-1)\pi x}{2L}\Big{)}\\ &-P\frac{(2n-1)(2r-1)\pi^{2}}{4L^{2}}\sin\frac{(2n-1)\pi x}{2L}\sin\frac{(2r- 1)\pi x}{2L}\Big{)}dx,n=1,2,\ldots;r=1,2,\ldots\end{array}  \tag{20}\]  
In the event of an instantaneous increase in the side load of the bar, namely \(q(t,x)=q(x)H(t)\), upon substitution into Equation (14), the result is obtained:  
\[Q_{r}(t)=H(t)b_{1}\int_{0}^{L}q\bigg{(}1-\cos\frac{(2r-1)\pi x}{2L}\bigg{)}dx= H(t)Q_{r} \tag{21}\]  
where: \(W\), \(Q\) and \(H\) are generalized function  
Upon conducting Laplace transform on Equation (21), the result is obtained:  
\[\overline{Q}_{r}=Q_{r}/s \tag{22}\]  
Analogous to Equation (15), the subsequent matrix expression may be derived:  
\[\sum_{r=1}^{i}\sum_{n=1}^{i}\left(\overline{w}_{n}(s)\overline{W}_{nr}- \overline{Q}_{r}\right)=