# Measurement of Relaxation Modulus of Viscoelastic Materials and Design of Testing Device
## 4 Least Square Method  
Previous scholars used a nano-indentation instrument to determine the creep compliance of viscoelastic materials, then used the conversion relationship between creep compliance and relaxation modulus to calculate the relaxation modulus. However, the use of a conversion relationship to calculate the relaxation modulus is not reliable due to the conversion error. Therefore, accurate relaxation modulus can be obtained by directly measuring it rather than indirectly obtaining the value through conversion. In this study, the relaxation modulus was measured using a direct method. The load-displacement relationship of a viscoelastic material can be derived directly from the load-displacement relationship of elastic material under the condition that the contact area between the indenter and the test material does not change. As a result, the relaxation modulus of a viscoelastic material can be obtained. The indenter is gradually pressed down during the test process, that is, the contact area between the indenter and viscoelastic material increases with time. Therefore, the load-displacement relationship of viscoelastic materials cannot be derived directly from the load-displacement relationship of elastic materials. [PERSON] and [PERSON] [24] solved this problem by introducing a generic integral operator. In this study, a series of expressions for linear viscoelasticity of materials were derived based on the Hertz contact theory of linear elastic objects. The load-displacement relationship of a linear elastic material pressed by a spherical indenter can be expressed as follows:  
\[P=\frac{4\sqrt{R}}{3(1-v^{2})}Eh^{\frac{3}{2}} \tag{1}\]  
where \(P\) is the load; \(R\) is the radius of a small ball; \(h\) is the displacement; \(v\) is [PERSON]'s ratio of elastic materials; \(E\) is the elastic modulus of elastic materials. After introducing the generic integral operator, the load-displacement relationship between linear viscoelastic materials and time \(t\) can be expressed as shown below:  
\[P(t)=\frac{4\sqrt{R}}{3(1-v^{2})}{\int_{0}^{t}}E(t-\xi)\frac{dh^{\frac{3}{2}}( \xi)}{d\xi}d\xi \tag{2}\]  
The speed of pressing the small ball into the viscoelastic material is controlled as a uniform linear motion with velocity \(V_{0}\) as follows:  
\[h(t)=V_{0}t \tag{3}\]  
Equation (3) was substituted in Equation (2) to yield:  
\[P(t)=\frac{2V_{0}^{\frac{3}{2}}\sqrt{R}}{1-v^{2}}{\int_{0}^{t}}E(\xi)\sqrt{t- \xi}d\xi \tag{4}\]  
The relaxation modulus equation of viscoelastic materials under the generalized Maxwell model [25] is shown below:  
\[E(t)=E_{0}+\sum_{i=1}^{N}E_{i}e^{-\frac{t}{t_{i}}} \tag{5}\]  
where \(N\) is the number of components; \(E_{0},E_{i}(i=1,2,\ldots,N)\) represents various relaxation modulus parameters; \(\tau_{i}\) represents each relaxation time. Equation (5) was substituted in Equation (4) to yield:  
\[P(t)=\frac{2V_{0}^{\frac{3}{2}}\sqrt{R}}{(1-v^{2})}(\frac{2}{3}E_{0}t^{\frac{3 }{2}}+\sum_{i=1}^{N}{\int_{0}^{t}}E_{i}e^{-\frac{\xi}{t_{i}}}\sqrt{t-\xi}d\xi) \tag{6}\]  
The long expressions can be simplified as follows:  
\[{\int_{0}^{t_{k}}}e^{-\frac{\xi}{t_{i}}}\sqrt{t_{k}-\xi}d\xi=I_{k} \tag{7}\]  
\[\frac{V_{0}^{\frac{3}{2}}\sqrt{R}}{1-v^{2}}=V \tag{8}\]  
where \(t_{k}\) represents each time point \((k=1,2,3,\ldots M)\). \(\tau_{i}(i=1,2,3,\ldots,N-1)\) was defined as equally spaced and can be expressed as shown below:  
\[\lg\tau_{i+1}-\lg\tau_{i}=a(cons\tan t) \tag{9}\]  
The least square method was used in this study. We obtained the load data through experiments. Subsequently, we obtained the relaxation modulus function of the material by minimizing the sum of the squares of the errors between the fitting data and the experimental data. Let the right side of Equation (6) be equal to \(f(t)\) and each time point be \(t_{k}(k=1,2,3,\ldots,M)\).