## 3 The Constitutive Relationship of Cellular Medium
### Phenomenological Constitutive Model  
The phenomenological constitutive model is the mathematical expression between stress and strain obtained by fitting experimental data. It can be used to describe physical phenomena such as linear elastic region, plateau region, and densification region. However, it cannot explain the microstructure deformation and failure mechanism of the foam material, such as the weak part of the material, the stress concentration, and the local deformation [77]. [PERSON] [78] first raised a universal model:  
\[\sigma(\varepsilon)=\varepsilon\mathrm{E_{f}}\psi(\varepsilon). \tag{1}\]  
where \(\mathrm{E_{f}}\) is the apparent Young's modulus, and \(\psi(\varepsilon)=\mathrm{A}\varepsilon^{\mathrm{m}}+\mathrm{B}\varepsilon^{ \mathrm{n}}\) is a dimensionless function of the compressive strain \(\varepsilon\). Parameters A, B, m, and n are curve-fitting constant from a typical logarithmic plot of \(\psi(\varepsilon)\) against \(\varepsilon\). This model was not accurate in describing the densification region. Based on the uniaxial stress-strain test results, [PERSON] and [PERSON] [79] proposed a six-parameter constitutive model of polyurethane foam to fit the linear elasticity, yield plateau, and densification region:  
\[\sigma(\varepsilon)=\mathrm{A}\Big{(}\frac{\mathrm{e}^{\alpha\varepsilon}-1}{ \mathrm{B}+\mathrm{e}^{\beta\varepsilon}}\Big{)}+\mathrm{e}^{\mathrm{C}}( \mathrm{e}^{\gamma\varepsilon}-1) \tag{2}\]  
where parameters A, B, \(\alpha\), and \(\beta\) are constant for a given initial density and strain rate. A is the scaling factor for yield stress; B plays a role of shifting the lower asymptote of the function. And parameters C and \(\gamma\) describe the densification region, varying C implies stretching or shrinking of the curve on the strain axis. [PERSON] [80] improved the curve knee at the elastic and plateau region connection:  
\[\sigma(\varepsilon)=\mathrm{A}\Big{(}1-\mathrm{e}^{-(\mathrm{E}/\mathrm{A}) \varepsilon(1-\varepsilon)^{\mathrm{m}}}\Big{)}+\mathrm{B}\Big{(}\frac{ \varepsilon}{1-\varepsilon}\Big{)}^{\mathrm{m}} \tag{3}\]  
where A and E are the plateau stress and the elastic modulus, exponent m represents the connection between the elastic region and the plateau region. And parameters B and n describe the densification region. [PERSON] [77] proposed Maxwell model:  
\[\sigma(\varepsilon)=\mathrm{e}^{-\frac{\mathrm{k}\varepsilon}{\varepsilon}} \Big{(}-1+\mathrm{e}^{\frac{\mathrm{k}\varepsilon}{\varepsilon}}\Big{)} \mathrm{c}+\big{[}\mathrm{k}_{\mathrm{P}}+\gamma(1-\mathrm{e}^{\varepsilon} )^{\mathrm{n}}\big{]}\varepsilon \tag{4}\]  
where k is equal to the elastic modulus; c is equal to the plateau stress; \(\mathrm{k}_{\mathrm{P}}\) is the slope of the plateau region; and \(\mathrm{k}_{\mathrm{D}}(\varepsilon)=\gamma(1-\mathrm{e}^{\varepsilon})^{ \mathrm{n}}\), which describes the densification region  
[PERSON] [81] proposed a constitutive equation with foam strain rate effect:  
\[\sigma=\mathrm{K}(1-\varepsilon)^{-\mathrm{n}\ \dot{\varepsilon}} \tag{5}\]  
where K is the plateau stress, n and r are constants used to simulate strain dependence and strain rate dependence respectively, typically in the range of \(0\leq\mathrm{n}\leq 4\) and \(0\leq\mathrm{r}\leq 0.3\). The constitutive equation can describe the mechanical behavior well in the strain range of 0.1-0.7, but it cannot fit the linear elasticity and densification region well. [PERSON] [82] considered the influence of the strain rate and proposed a stress-strain framework of cellular medium, as shown in the following formula:  
\[\sigma(\varepsilon)=\mathrm{f}(\varepsilon)\mathrm{M}(\varepsilon,\ \dot{ \varepsilon}) \tag{6}\]  
where \(\mathrm{f}(\varepsilon)=\sum\limits_{\mathrm{i}=1}^{10}\mathrm{A}_{\mathrm{i}} \mathrm{\epsilon}^{\mathrm{n}}\) is the strain function, \(\mathrm{A}_{\mathrm{i}}\)'s are determined using a tenth-order polynomial fit. \(\mathrm{M}(\varepsilon,\ \dot{\varepsilon})=\big{(}\ \dot{\varepsilon}/\ \dot{ \varepsilon}_{0}\big{)}^{\mathrm{n}(\varepsilon)}\) is the modulus function.