## 2 Elastoplastic Constitutive Relationship Theory
### Elastoplastic Stress-Strain Relationship  
According to the classical small-strain elastoplastic theory, the total strain, \(\varepsilon\), is decomposed into an elastic strain part, \(\varepsilon^{\text{e}}\), and a plastic strain part, \(\varepsilon^{\text{p}}\). The elastic strain is calculated using Hooke's Law, while the plastic strain is determined using plasticity theory. When adopting an incremental formulation, we have the following:  
\[\{d\varepsilon\}=\{d\varepsilon^{\text{e}}\}+\{d\varepsilon^{\text{p}}\} \tag{4}\]  
Assuming that the constitutive relationship of the material in the elastic region is given by the following:  
\[\{d\varepsilon^{\text{e}}\}=[C^{\text{e}}]\{d\sigma\} \tag{5}\]  
where \([C^{\text{e}}]\) is the elastic compliance matrix. If the elastic strain increment vector, \(\{d\varepsilon^{\text{e}}\}\), and the stress increment vector, \(\{d\sigma\}\), are principal strain increments and principal stress increments, respectively, then \([C^{\text{e}}]\) is a third-order compliance matrix. When both vectors have dimensions of \(6\times 1\), \([C^{\text{e}}]\) is a sixth-order compliance matrix.  
Since plastic strain increments do not contribute to stress increments, from Equation (5), the relationship between the total stress increment and the total strain increment is given by the following:  
\[\{d\varepsilon-d\varepsilon^{\text{p}}\}=[C^{\text{e}}]\{d\sigma\} \tag{6}\]  
Alternatively, this can be rewritten as follows:  
\[\{d\varepsilon\}=([C^{\text{e}}]+[C^{\text{e}}])\{d\sigma\}=[C^{\text{e}}]\{ d\sigma\} \tag{7}\]  
where \([C^{\text{e}}]\) is the plastic compliance matrix. Using incremental plasticity theory, for non-associated flow rules, we can express the formulation as follows:  
\[\{d\varepsilon^{\text{p}}\}=\Lambda\bigg{\{}\frac{\partial g}{\partial \sigma}\bigg{\}} \tag{8}\]  
where \(\Lambda\) is a scalar factor representing the magnitude of the plastic strain increment, referred to as the plastic multiplier.  
During **plastic loading**, the **consistency condition**\(\text{d}f=0\) must be maintained, leading to the following:  
\[df=\frac{\partial f}{\partial\sigma}:\{d\sigma\}+\frac{\partial f}{\partial H }dH=0 \tag{9}\]  
Since the hardening parameter, \(H\), is often considered a function of plastic strain or accumulated plastic strain, we have the following:  
\[dH=\frac{\partial H}{\partial\varepsilon^{\text{p}}}:\{d\varepsilon^{\text{ p}}\} \tag{10}\]  
Substituting Equation (8) into Equation (10):  
\[dH=\Lambda\bigg{[}\frac{\partial H}{\partial\varepsilon^{\text{p}}}:\bigg{\{} \frac{\partial g}{\partial\sigma}\bigg{\}}\bigg{]} \tag{11}\]Substituting Equation (11) into Equation (9) to obtain \(\Lambda\):  
\[\Lambda=-\frac{\frac{\partial f}{\partial\sigma}:\{d\sigma\}}{\frac{\partial f}{ \partial H}\Big{[}\frac{\partial H}{\partial\sigma^{\prime}}:\Big{\{}\frac{ \partial g}{\partial\sigma}\Big{\}}\Big{]}} \tag{12}\]  
Substitute Equations (8) and (12) into Equation (6):  
\[\{d\epsilon\}=[C^{\epsilon}]\{d\sigma\}+\left(-\frac{\frac{\partial f}{ \partial\sigma}:\{d\sigma\}}{\frac{\partial f}{\partial H}\Big{[}\frac{ \partial H}{\partial\sigma^{\prime}}:\Big{\{}\frac{\partial g}{\partial\sigma} \Big{\}}\Big{]}}\right)\bigg{\{}\frac{\partial g}{\partial\sigma}\Big{\}} \tag{13}\]  
Equation (13) can also be rewritten in terms of the elastoplastic compliance matrix, forming the stress-strain increment equation:  
\[\{d\epsilon\}=\left\{[C^{\epsilon}]+\left(-\frac{\Big{\{}\frac{\partial g}{ \partial\sigma}\Big{\}}\Big{\{}\frac{\partial f}{\partial\sigma}\Big{\}}^{T}}{ \frac{\partial f}{\partial H}\Big{[}\frac{\partial H}{\partial\sigma^{\prime} }:\frac{\partial g}{\partial\sigma}\Big{]}}\right)\right\}\{d\sigma\} \tag{14}\]