## 1 Introduction
### Strength  
Figure 1 describes the strength curve of brittle materials from three aspects: the intact state, damaged state, and fractured state. Different states have their own strength equation which presents  
Figure 1: Strength model of the JH-2 constitutive model.  
Figure 2: Damage model (**a**) and EOS model (**b**) of the JH-2 constitutive model.  
the relation of normalized equivalent stress versus normalized pressure. The analytic function in damaged state can be considered as the general form for the three states, and its normalized equivalent stress is expressed as  
\[\sigma^{*}=\sigma_{i}^{*}-D\Big{(}\sigma_{i}^{*}-\sigma_{f}^{*}\Big{)}=\sigma/ \sigma_{HEL} \tag{1}\]  
where \(\sigma_{i}^{*}\) is the normalized intact equivalent stress, \(\sigma_{f}^{*}\) is the normalized fracture stress, \(D\) is the damage factor (\(0\leq D\leq 1.0\)) and \(\sigma_{HEL}\) is the equivalent stress at the \(HEL\)--the Hugoniot elastic limit, an important concept that stands for the net compressive stress (containing both hydrostatic pressure and deviator stress components) at which a one-dimensional shock wave with uniaxial strain exceeds the elastic limit of the material [4]. Herein, \(\sigma\) is the actual equivalent stress which is calculated by the formula of von Mises stress [36]  
\[\sigma=\sqrt{\frac{1}{2}\Big{[}\big{(}\sigma_{x}-\sigma_{y}\big{)}^{2}+\big{(} \sigma_{x}-\sigma_{z}\big{)}^{2}+\big{(}\sigma_{y}-\sigma_{z}\big{)}^{2}+6 \Big{(}\tau_{xy}^{2}+\tau_{xz}^{2}+\tau_{yz}^{2}\Big{)}\Big{]}} \tag{2}\]  
or  
\[\sigma=\sqrt{\frac{1}{2}\Big{[}\big{(}\sigma_{1}-\sigma_{2}\big{)}^{2}+\big{(} \sigma_{1}-\sigma_{3}\big{)}^{2}+\big{(}\sigma_{2}-\sigma_{3}\big{)}^{2}\Big{]}} \tag{3}\]  
where \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{z}\) are the three normal stresses, \(\tau_{xy}\), \(\tau_{xz}\), and \(\tau_{yz}\) are the three shear stresses, and \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are the three principal stresses.  
The normalized intact strength is given by  
\[\sigma_{i}^{*}=A(P^{*}+T^{*})^{N}\Big{(}1+C\cdot\ln\dot{\varepsilon}^{*}\Big{)} \tag{4}\]  
The normalized fracture strength is given by  
\[\sigma_{f}^{*}=B(P^{*})^{M}\Big{(}1+C\cdot\ln\dot{\varepsilon}^{*}\Big{)}\leq SFMAX \tag{5}\]  
where \(A,B,C,M,N\) are material constants, the normalized pressure is \(P^{*}=P/P_{HEL}\), where \(P\) is the actual hydrostatic pressure, and \(P_{HEL}\) is the hydrostatic pressure at the \(HEL\). The normalized maximum tensile hydrostatic pressure is \(T^{*}=T/P_{HEL}\), where \(T\) is the maximum tensile hydrostatic pressure the material can withstand. The dimensionless strain rate is \(\dot{\varepsilon}^{*}=\dot{\varepsilon}/\dot{\varepsilon}_{0}\), where \(\dot{\varepsilon}\) is the actual equivalent strain rate and \(\dot{\varepsilon}_{0}=1.0s^{-1}\) is the reference strain rate. \(SFMAX\) is the ultimate value of \(\sigma_{f}^{*}\) that provides extra flexibility in defining the fracture strength. The actual equivalent strain rate is analogous to the equivalent stress and is expressed as  
\[\dot{\varepsilon}=\sqrt{\frac{2}{9}\Big{[}\big{(}\dot{\varepsilon}_{x}-\dot{ \varepsilon}_{y}\big{)}^{2}+\big{(}\dot{\varepsilon}_{x}-\dot{\varepsilon}_{ z}\big{)}^{2}+\big{(}\dot{\varepsilon}_{y}-\dot{\varepsilon}_{z}\big{)}^{2}+ \frac{3}{2}\Big{(}\dot{\gamma}_{xy}^{2}+\dot{\gamma}_{xz}^{2}+\dot{\gamma}_{ yz}^{2}\Big{)}\Big{]}} \tag{6}\]  
where \(\dot{\varepsilon}_{x}\), \(\dot{\varepsilon}_{y}\), and \(\dot{\varepsilon}_{z}\) are the three normal strain rates; and \(\dot{\gamma}_{xy}\), \(\gamma_{xz}\), and \(\gamma_{yz}\) are the three engineering shear strain rates.  
In the JH-2 constitutive model, the brittle material begins to soften when damage starts to accumulate (see the damage curve in Figure 1). The softening process can be expressed by Equation (1), which allows for gradual softening of the material under increasing plastic strain, although the softening does not continue when the material is completely damaged (\(D=1\)).