# Unified Analytic Melt-Shear Model in the Limit of Quantum Melting
### Unified Analytic Melt-Shear Thermoelasticity Model  
Unified analytic melt-shear thermoelasticity model describes both the melting temperature (\(T_{m}\)) and cold (\(T=0\)) shear modulus of a solid substance as functions of density (\(\rho\)) at all densities and all temperatures from zero to \(T_{m}\). Such model is required for multiple applications including constitutive modeling for plastic deformations at extremes of temperature and pressure (\(P\)), numerical calculations of elastic and shock wave propagation, and even calculations of the oscillations of low-mass astrophysical objects. The original version of the model was introduced by ourselves in 2003-2004 [4; 5] and is based on the analytic model for the Gruneisen parameter,  
\[\gamma(\rho)=\frac{1}{2}+\frac{\gamma_{1}}{\rho^{1/3}}+\frac{\gamma_{2}}{\rho^ {q}}, \tag{1}\]  
where the parameters \(\gamma_{1}\), \(\gamma_{2}\), and \(q\) can be determined from ambient data by solving a system of three algebraic equations [4; 5]. Equation (1) in conjunction with  
\[\frac{d\ln G(\rho,0)}{d\ln\rho}=2\,\gamma(\rho)+\frac{1}{3}, \tag{2}\]  
and  
\[\frac{d\ln T_{m}(\rho)}{d\ln\rho}\equiv 2\bigg{(}\gamma(\rho)-\frac{1}{3} \bigg{)} \tag{3}\]  
yields  
\[G(\rho,0)=G(\rho_{ref1})\bigg{(}\frac{\rho}{\rho_{ref1}}\bigg{)}^{4/3}\exp \Bigg{\{}6\gamma_{1}\left(\frac{1}{\rho_{ref1}^{1/3}}-\frac{1}{\rho^{1/3}} \right)+\frac{2\gamma_{2}}{q}\,\left(\frac{1}{\rho_{ref1}^{q}}-\frac{1}{\rho^ {q}}\right)\Bigg{\}}, \tag{4}\]  
and  
\[T_{m}(\rho)=T_{m}(\rho_{ref2})\bigg{(}\frac{\rho}{\rho_{ref2}}\bigg{)}^{1/3} \exp\Bigg{\{}6\gamma_{1}\left(\frac{1}{\rho_{ref2}^{1/3}}-\frac{1}{\rho^{1/3} }\right)+\frac{2\gamma_{2}}{q}\,\left(\frac{1}{\rho_{ref2}^{q}}-\frac{1}{\rho ^{q}}\right)\Bigg{\}} \tag{5}\]  
where two \(\rho_{ref}\)s are conveniently chosen reference densities, usually those that correspond to \(T=0\), \(P=0\) for \(G(\rho,0)\) and to the ambient melting point \(T_{m}(P=0)\) for \(T_{m}(\rho)\). Then \(G(\rho,T)\) is given by the Preston-Wallace [6] model:  
\[G(\rho,T)=G(\rho,0)\bigg{(}1-\beta\,\frac{T}{T_{m}(\rho)}\bigg{)}\,. \tag{6}\]The thermoelastic softening parameter \(\beta\) may be density dependent. Note that when \(\beta\approx\) const, i.e., density-independent, \(T_{m}\) and \(G\) along the solidus approximately satisfy the relation [4; 7; 8]  
\[\frac{G(\rho,T_{m}(\rho))}{\rho\,T_{m}(\rho)}=\frac{G(\rho_{ref},T_{m}(\rho_{ref }))}{\rho_{ref}\,T_{m}(\rho_{ref})}\,, \tag{7}\]  
for arbitrary \(\rho_{ref}\). The model can be constructed for any substance, and is applicable at any \(\rho\) and any \(T,0\leq T\leq T_{m}\). A table containing model parameters for 28 elemental solids and 4 alloys can be found in Ref. [9]. For polymorphic (multi-phase) materials, the \(\gamma_{1}\)-\(\gamma_{2}\)-\(q\) values for the ambient solid structure may differ significantly from those for the high-\(P\) structure(s), and therefore the model may fail in its description of those high-\(P\) structure(s). The case of a multi-phase solid was considered in detail in Ref. [10]. In this case, the above equations are replaced with  
\[\frac{d\ln G(\rho,0)}{d\ln\rho}=2\,\gamma_{G}(\rho)+\frac{1}{3}, \tag{8}\]  
and  
\[\frac{d\ln T_{m}(\rho)}{d\ln\rho}\equiv 2\bigg{(}\gamma_{T_{m}}(\rho)-\frac{1} {3}\bigg{)}, \tag{9}\]  
where  
\[\gamma_{G}(\rho)=\frac{1}{2}+\frac{\gamma_{1}}{\rho^{1/3}}+\frac{\gamma_{2}}{ \rho^{\prime 2}} \tag{10}\]  
and  
\[\frac{d\ln G(\rho,0)}{d\ln\rho}=2\,\gamma(\rho)+\frac{1}{3}, \tag{2}\]0  
Hence, in contrast to the previous case where a set of parameters \((\gamma_{1},\,\gamma_{2},\,q)\) describes both \(G\) and \(T_{m}\), for a polymorphic material they are described by their own individual sets: \((\gamma_{1},\,\gamma_{2},\,q_{2})\) for \(G\) and \((\gamma_{1},\,\gamma_{3},\,q_{3})\) for \(T_{m}\). In this case the only common parameters of \(G\) and \(T_{m}\) is [10]  
\[\frac{d\ln G(\rho,0)}{d\ln\rho}=2\,\gamma(\rho)+\frac{1}{3}, \tag{2}\]1  
where \(Z\) is the atomic number of the material.