Previous work has shown that the one-dimensional (1D) inviscid compressible flow (Euler) equations admit a wide variety of scale-invariant solutions (including the famous Noh, Sedov, and Guderley shock solutions) when the included equation of state (EOS) closure model assumes a certain scale-invariant form. However, this scale-invariant EOS class does not include even simple models used for shock compression of crystalline solids, including many broadly applicable representations of Mie-Gr\"uneisen EOS. Intuitively, this incompatibility naturally arises from the presence of multiple dimensional scales in the Mie-Gr\"uneisen EOS, which are otherwise absent from scale-invariant models that feature only dimensionless parameters (such as the adiabatic index in the ideal gas EOS). The current work extends previous efforts intended to rectify this inconsistency, by using a scale-invariant EOS model to approximate a Mie- Gr\"uneisen EOS form. To this end, the adiabatic bulk modulus for the Mie-Gr\"uneisen EOS is constructed, and its key features are used to motivate the selection of a scale-invariant approximation form. The remaining surrogate model parameters are selected through enforcement of the Rankine-Hugoniot jump conditions for an infinitely strong shock in a Mie-Gr\"uneisen material. Finally, the approximate EOS is used in conjunction with the 1D inviscid Euler equations to calculate a semi-analytical, Guderley-like imploding shock solution in a metal sphere, and to determine if and when the solution may be valid for the underlying Mie-Gr\"uneisen EOS.

翻译：先前的工作已经显示, 单维( 1D) 隐隐隐压缩流( Euler) 方程式在包含的州( EOS) 关闭模型的方程式以某种规模异化形式出现时, 单维(1 D) 隐隐隐压缩流( Euler) 方程式会接受多种规模反差解决方案( 包括著名的 Noh、 Sedov 和 Guderley 休克解决方案) 。 然而, 这种规模反差类并不包括晶体固体冲击压缩所用的简单模型, 包括许多广泛适用的 Mie- Gr\\" uneisen EOS 参数。 直观地, 这种不相容自然产生于Mie- Gr\\ URsen EOS 的多维度级级计算( 包括著名的 Neh- gr\ " ), 等离异性OS- 的硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性软性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性