## 3 The Constitutive Relationship of Cellular Medium
### Phenomenological Constitutive Model
\(\dot{\varepsilon}_{0}\) is a quasi-static strain rate. \(\mathrm{n}(\varepsilon)=\mathrm{a}+\mathrm{b}\varepsilon\) is a linear function of strain, which is the slope of the relationship between the stress and strain rate [83]. a and b are material constants in the ranges of 0-0.0442 and 0.0343-0.145 [83]. [PERSON] [84] considered the influencing factors of temperature and strain rate to establish the uniaxial constitutive relationships of polypropylene foam, polyurethane foam and polystyrene foam according to the work of [PERSON] [83]. They extended the constitutive relationship to triaxial loading and successfully used it in a finite element program. The one-dimensional constitutive model is:  
\[\sigma(\varepsilon)=\sigma_{0}(\varepsilon)\bigg{(}\frac{\dot{\varepsilon}} {\dot{\varepsilon}_{0}}\bigg{)}^{\mathrm{n}(\varepsilon)}\exp\bigg{[}-\frac{ \mathrm{C}_{1}(\mathrm{T}-\mathrm{T}_{\mathrm{r}})}{\mathrm{C}_{2}+\mathrm{T} -\mathrm{T}_{\mathrm{r}}}\bigg{]} \tag{7}\]  
where \(\sigma_{0}(\varepsilon)\) represents the nominal stress response at a constant quasi-static strain rate, \(\mathrm{T}_{\mathrm{r}}\) is the glass transition temperature. The meaning of \(\mathrm{n}(\varepsilon)=\mathrm{a}+\mathrm{b}\varepsilon\) is reflected in Equation (6). C\({}_{1}\) and C\({}_{2}\) are Williams-Landel-Ferry (WLF) function coefficients to be determined from temperature effect experiments. [PERSON] [85; 86] added a modulus function (strain rate dependence) to the [PERSON]'s model [80], and proposed a constitutive model for the stress-strain relationship of cellular medium at low strain rates:  
\[\sigma(\varepsilon)=\bigg{\{}\text{A}\Big{(}1-\text{e}^{-(\text{E}/\text{A}) \varepsilon(1-\varepsilon)^{m}}\Big{)}+\text{B}\bigg{(}\frac{\varepsilon}{1- \varepsilon}\bigg{)}^{n}\bigg{\}}\text{M}\big{(}\varepsilon,\;\dot{\varepsilon }\big{)}. \tag{8}\]  
where the meanings of A, B, E, m and n are reflected in Equation (3) [80] and these parameters can be obtained at a quasi-static strain rate of 0.001 s\({}^{-1}\). They thought the modulus function \(\text{M}\big{(}\varepsilon,\;\dot{\varepsilon}\big{)}\) worked as a scale factor of the shape function linearly varying along the strain axis between a reference strain rate (quasi-static strain rate 0.001 s\({}^{-1}\)) and another strain rate (0.1 s\({}^{-1}\)). And they proposed two modulus functions \(\text{M}\big{(}\varepsilon,\;\dot{\varepsilon}\big{)}\), as shown in Equations (9) [85] and (10) [86]. \[\text{M}\big{(}\varepsilon,\;\dot{\varepsilon}\big{)}=1+(\text{a}+\text{b} \varepsilon)\ln\bigg{(}\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}} \bigg{)} \tag{9}\]  
where a and b are obtained by linearly approximating of the experimental stress-strain data. \(\dot{\varepsilon}_{0}\) is a quasi-static strain rate of 0.001 s\({}^{-1}\). Another modulus function has different expressions of \(\text{n}(\varepsilon)\) depending on three strain regions. A cubic polynomial function of \(\varepsilon\) is used on the transition zone (\(\varepsilon_{1}\leq\varepsilon\leq\varepsilon_{2}\)), which consists of plateau region and densification region adjacent to the critical strain (\(\varepsilon_{1}\leq\varepsilon_{\text{cr}}\leq\varepsilon_{2}\)). Linear functions of \(\text{n}(\varepsilon)\) are used for plateau region (\(\varepsilon\leq\varepsilon_{1}\)) and densification region (\(\varepsilon\geq\varepsilon_{2}\)). c\({}_{1}\), c\({}_{2}\), c\({}_{3}\) and c\({}_{4}\), can be obtained through boundary conditions at \(\varepsilon=\varepsilon_{1}\) and \(\varepsilon=\varepsilon_{2}\). \[\text{M}\big{(}\varepsilon,\;\dot{\varepsilon}\big{)}=1+\text{n}(\varepsilon )\ln\bigg{(}\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}}\bigg{)} \tag{10}\]  
\[\text{n}(\varepsilon)=\left\{ \begin{array}{cc}\text{a}_{1}+\text{b}_{1} \varepsilon&\varepsilon\leq\varepsilon_{1}\\ \text{c}_{1}+\text{c}_{2}\varepsilon+\text{c}_{3}\varepsilon^{2}+\text{c}_{4 }\varepsilon^{3}&\varepsilon_{1}\leq\varepsilon\leq\varepsilon_{2}\\ \text{a}_{2}+\text{b}_{2}\varepsilon&\varepsilon\geq\varepsilon_{2}\end{array} \right.