The Quarterly Journal of Mechanics and Applied Mathematics, Volume 71 (1) – Feb 1, 2018

21 pages

/lp/ou_press/scaling-in-cavity-expansion-equations-using-the-isovector-method-55VDfkAs3e

- Publisher
- Oxford University Press
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- Published by Oxford University Press 2017. This work is written by a US Government employee and is in the public domain in the US.
- ISSN
- 0033-5614
- eISSN
- 1464-3855
- D.O.I.
- 10.1093/qjmam/hbx023
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- See Article on Publisher Site

Summary Cavity-expansion approximations are widely-used in the study of penetration mechanics and indentation phenomena. We apply the isovector method to a well-established model in the literature for elastic-plastic cavity-expansion to systematically demonstrate the existence of Lie symmetries corresponding to scale-invariant solutions. We use the symmetries obtained from the equations of motion to determine compatible auxiliary conditions describing the cavity wall trajectory and the elastic-plastic material interface. The admissible conditions are then compared with specific similarity solutions in the literature. 1. Introduction Cavity formation occurs in gases, liquids and solids due to intense loading by internal pressure. Many direct examples of cavity formation are presented by detonation of high explosives in various media, for example, underwater explosions are described by Cole and Weller (1) and detonation of high-explosives in ductile metals are discussed by Hill (2). The development of cavity-expansion models starting with the experimental work of Bishop et al. (3) is summarized by Hopkins (4). In particular, spherical cavity-expansion models in purely elastic materials were proposed by Blake (5) and in elastic-plastic materials by Hunter and Crozier (6). The use of cavity-expansion approximations to describe subsurface indentation due to high-velocity impact of rigid projectiles in penetration mechanics has been developed by Forrestal et al. (7, 8). The review by Backman and Goldsmith (9) discusses many of the analytical and experimental results describing the interaction between penetrators and target materials. The existence of group-invariant solutions in the form of similarity solutions for related problems in gas dynamics have been intensively studied leading to many well-known exact solutions, for example, Sedov–von Neumann–Taylor blast wave (10, 11), Guderley (12), and Noh (13). More recently, Ovsiannikov (14), Axford (15), Axford and Holm (16), Holm (17), Coggeshall (18) and Ramsey and Baty (19) have applied more unified approaches to classify the group-invariant solutions admitted for general equations of state. However, while group-theoretic techniques have been used widely in the aforementioned areas of gas dynamics, it appears they have been applied less frequently to closely related problems in solid mechanics. Motivated by this observation, the aim of this research is to investigate the existence of a class of group-invariant solutions to a well-established model for elastic-plastic cavity-expansion. In particular, constructing scaling invariant solutions serves three objectives. First, new exact solutions have immediate application to hydrocode verification. Second, this analysis is an initial step toward the future investigation of scaling criteria that can be used to guide physical experiments. Finally, this study is the primary step toward broader study of related models characterizing more general material response. Hence, the main contribution of this article is to systematically derive a similarity solution for a basic class of spherical cavity-expansion models, which provides a foundation for other related work. The application of group invariance or symmetry analysis to the study of solutions to differential equations dates back to Lie (20). These techniques received renewed interest by Birkhoff (21), Ovsiannikov (14), Olver (22), and Bluman et al. (23). The isovector method or the method of differential forms proposed by Estabrook and Harrison (24, 25, 26) and further developed by Edelen (27, 28) is an alternative formulation of more traditional group invariance methods. The foundation for the isovector method is due to Cartan’s work in geometric analysis of partial differential equations (PDEs). A modern account of his work can be found in Ivey and Landsberg (29). The isovector method requires reformulating the original system of PDEs as an equivalent system of differential forms. The usual infinitesimal representation of the associated invariance group is then re-interpreted more concisely through its natural association with a vector field called the isovector field. The isovector field can be used to generate the classical invariance groups describing the symmetries of the original PDE. However, although the isovector method may offer some computational advantages in many cases, it should be emphasized from the outset that identical results can be obtained using other group-theoretic approaches. This remainder of this study is divided into several sections as follows. In section 2 a well-established model for one-dimensional (1D) spherically symmetric elastic-plastic cavity-expansion is reviewed. In section 3 we recast the equations of motion for the elastic-plastic cavity-expansion model as an equivalent exterior differential system, which provides a more concise setting to analyze its associated Lie symmetries. In section 3.2 we apply the isovector method to demonstrate the existence of various Lie symmetries corresponding to scale-invariant solutions of the dynamics cavity-expansion problem. Conclusions drawn from this analysis are discussed in section 4. 2. Spherically symmetric cavity-expansion In this section we review a well-established model for elastic-plastic cavity-expansion in arbitrary material. The analytical study of dynamic expanding cavity problems in plastic materials was initiated by Bishop et al. (3), cf. Hill (30). Blake (5) investigated the spherically symmetric problem with purely elastic response and Hunter and Crozier (6) investigated the corresponding problem with elastic-plastic response. The review by Hopkins (4) covers much of the theoretical developments through 1960. More recently, Forrestal and Luk (7, 8) have developed expanding cavity models in compressible, elastic-plastic material in connection with penetration mechanics. Following (7), we assume that an internally pressurized cavity of initial radius zero is driven radially outward with a constant velocity $$c_{w}$$. Hence, the cavity wall trajectory can be written $$r_w(t)=c_{w}t$$, where $$r_w$$ denotes the radial potion of the cavity wall and $$t$$ denotes elapsed time. As the cavity expands, we assume that a spherical shell immediately surrounding the cavity deforms plastically. Outside the plastic region we assume that a second spherical shell undergoes elastic deformation. The elastic-plastic boundary is determined by a moving material interface. The medium within the unbounded region surrounding plastic and elastic sub-regions is assumed to be stress-free or undisturbed. Assuming the cavity expands uniformly in the radial direction, we can approximate the domain by the cross section depicted in Fig. 1. Fig. 1. View largeDownload slide Cross section of the elastic-plastic domain Fig. 1. View largeDownload slide Cross section of the elastic-plastic domain 2.1 Plastic deformation 1$$D$$ spherically symmetric expansion in the plastic region governed by conservation of mass and momentum written in Eulerian coordinates reduces to \begin{eqnarray} -\frac{D\rho}{Dt}&=&\rho\left(\frac{\partial v}{\partial r} + \frac{2v}{r}\right),\\ \end{eqnarray} (2.1) \begin{eqnarray} -\rho\frac{Dv}{Dt}&=&\frac{\partial \sigma_{r}}{\partial r} + \frac{2}{r}(\sigma_{r} -\sigma_{\theta}), \end{eqnarray} (2.2) where $$r$$ denotes the coordinate along the radial direction with respect to the origin, $$t$$ again denotes elapsed time, $$\rho$$ denotes the density, $$v$$ the velocity, $$\sigma_r$$ denotes the radial stress and $$D / Dt:= \partial / \partial t + v \partial / \partial r$$ denotes the material derivative. Equations (2.1)–(2.2) of motion describing isentropic expansion are supplemented by a material model for strength in the form of two constitutive equations. The first constitutive equation describes compressibility in terms of a linear pressure-volumetric strain relation \begin{eqnarray} p=K\eta, \qquad \eta = \left(1-\frac{\rho_0}{\rho}\right), \end{eqnarray} (2.3) where $$p$$ denotes pressure, $$K$$ is the bulk modulus, $$\eta$$ is the volumetric strain and $$\rho_0$$ denotes the density in the undeformed state. Recall that pressure is defined as the trace of the stress; see for example, Fung (31) \begin{eqnarray} p=\frac13(\sigma_{r}+\sigma_{\theta}+\sigma_{\phi}), \end{eqnarray} (2.4) where $$\sigma_{r}$$, $$\sigma_{\theta}$$ and $$\sigma_{\phi}$$ are the radial, hoop (tangential) and meridional meridional stress components, respectively. As a consequence of spherical symmetry \begin{eqnarray} \sigma_{\phi}=\sigma_{\theta}. \end{eqnarray} (2.5) The second constitutive equation describes the material strength or yield properties and we assume is given by the Tresca criterion \begin{eqnarray} \sigma_{r}-\sigma_{\theta} = Y, \end{eqnarray} (2.6) where the constant $$Y$$ denotes the yield or flow stress. In (2.1)–(2.2), $$\rho$$ and $$\sigma_{\theta}$$ can be eliminated through the constitutive relations (2.3) and (2.6), which yields the closed system \begin{eqnarray} \frac{D\sigma_{r}}{Dt}& = &-(\gamma - \sigma_r)\left(\frac{\partial v}{\partial r} + \frac{2}{r}v\right)\! ,\\ \end{eqnarray} (2.7) \begin{eqnarray} \rho_0K\frac{Dv}{Dt} &=& -(\gamma - \sigma_r)\left(\frac{\partial \sigma_{r}}{\partial r} + \frac{2}{r}Y\right)\!, \end{eqnarray} (2.8) where we have introduced a new constant $$\gamma = K + 2/ 3 Y$$. 2.2 Elastic deformation Motion in the elastic region is governed by the linear elastic wave equation, cf. Fung (31) \begin{equation} \frac{\partial^2 u}{\partial r^2} + \frac{2}{r}\frac{\partial u}{\partial r} - \frac{2u}{r^2}=\frac{1}{c_d^2}\frac{\partial^2 u}{\partial t^2}, \end{equation} (2.9) where $$u$$ denotes radial displacement. Equation (2.9) is obtained from (2.2) by treating the elastic region as incompressible and neglecting the convective term. Using Hooke’s law, the components of the stress are reexpressed through the radial displacement (strain) through the standard relations, for example, Fung (31), \begin{eqnarray} \sigma_{r}& =& -\frac{E}{(1+\nu)(1-2\nu)}\Big((1-\nu)\frac{\partial u}{\partial r} + 2\nu\frac{u}{r}\Big),\\ \end{eqnarray} (2.10) \begin{eqnarray} \sigma_{\theta} &= &-\frac{E}{(1+\nu)(1-2\nu)}\Big(\nu\frac{\partial u}{\partial r} + \frac{u}{r}\Big), \end{eqnarray} (2.11) where the constant $$E$$ denotes Young’s modulus, the constant $$\nu$$ denotes Poisson’s ratio. Finally, the wave propagation speed $$c_d$$ in (2.9) is given by \begin{equation} c_d^2 = \frac{E(1-\nu)}{\rho_0(1+\nu)(1-2\nu)}. \end{equation} (2.12) Additionally, after we neglect the convective term, there is no distinction in (2.9) between the Lagrangian and Eulerian coordinates and the relationship between particle velocity and displacement is then given by \begin{equation} \frac{\partial u}{\partial t} = v\left(1 - \frac{\partial u}{\partial r}\right). \end{equation} (2.13) 2.3 Interface conditions The well-known Hugoniot matching conditions expressing conservation of mass and momentum across the material interface (see for example, Courant and Friedrichs (32)), are assumed along elastic-plastic interface and are given by the relations \begin{eqnarray} \rho_1(v_1 -c_{i}) - \rho_2(v_2 - c_{i})& =& 0, \\ \end{eqnarray} (2.14) \begin{eqnarray} \sigma_{2r} - \sigma_{1r}& =&\frac{\rho_1}{K}(c_{i}-v_1)(v_2-v_1), \end{eqnarray} (2.15) where $$\sigma_{1r}$$ and $$\sigma_{2r}$$ denote the radial components of the stress, $$\rho_1$$ and $$\rho_2$$ denote the densities, $$v_1$$ and $$v_2$$ are the velocities, in the elastic and plastic regions respectively. The velocity of the elastic-plastic interface is denoted $$c_{i}$$. (2.15) is obtained from the more familiar expression for conservation of momentum \begin{equation} \sigma_{1r} - \sigma_{2r}= \rho_1v_1(c_{i}-v_1) - \rho_2v_2(c_{i}-v_2) \end{equation} (2.16) by imposing (2.14). The radial stresses along the interface computed from (2.6) are given by \begin{eqnarray} \sigma_{1r} &=& 2Y/3 + K(1-\rho_0/\rho_1),\\ \end{eqnarray} (2.17) \begin{eqnarray} \sigma_{2r} &=& 2Y/3 + K(1-\rho_0/\rho_2). \end{eqnarray} (2.18) From (2.15), we can then write \begin{equation} \rho_1 = K\frac{(\sigma_{2r}-\sigma_{1r})}{(v_2 - v_1) (c_{i}- v_1)}, \end{equation} (2.19) which combined with (2.15) gives \begin{equation} \rho_2= K\frac{(\sigma_{2r}-\sigma_{1r})}{(v_2 - v_1) (c_{i}- v_2)}. \end{equation} (2.20) Equating (2.17) and (2.18) through the yield strength we obtain \begin{equation} \rho_1(\rho_2\sigma_{2r} + K\rho_0) = \rho_2(\rho_1\sigma_{1r} + K\rho_0). \end{equation} (2.21) Substituting the new expressions for $$\rho_1$$ and $$\rho_2$$ into (2.21) yields \begin{equation} (\sigma_{2r}-\sigma_{1r})^2 = \rho_0(v_2 - v_1)^2. \end{equation} (2.22) Finally, assuming the radial stress is independent of the velocity, we conclude that no jumps are present in the radial stress and particle velocity at the elastic-plastic interface, that is, \begin{eqnarray} v_1&=&v_2,\\ \end{eqnarray} (2.23) \begin{eqnarray} \sigma_{1r}&=&\sigma_{2r}. \end{eqnarray} (2.24) 3. Symmetry analysis In this section, we apply the isovector method of Harrison and Estabrook (24) to systemically identify symmetries corresponding to scale-invariant solutions admitted by the elastic-plastic equations in section 2. Similarity solutions have been studied extensively in gas dynamics, see Birkhoff (21), Sedov (10), Stanyukovich (33) and Barenblatt (34) and the references therein. We also note that some symmetry properties of a different system from the theory of plasticity are considered by Senashov et al. (35, 36). The isovector method is but one of several related techniques available to identify Lie symmetries of differential equations Ovsiannikov (14), Olver (22), Cantwell (37) and Bluman et al. (23). In the isovector method of Harrison and Estabrook (24), also called the method of differential forms, the infinitesimal representations of the associated invariance groups are interpreted more precisely as isovector fields. The motivation behind this generalization rests upon placing the study of symmetries in the setting of differential geometry where the dual relationship the tangent vector field to cotangent vector field provides a more natural way to identify invariants or coordinate-free geometric features of certain classes of solutions to the original PDEs. A comprehensive development of the isovector method in the framework of exterior analysis is outside the scope of this article. Symmetry analysis via the method of differential forms can be found in Estabrook and Harrison (24), Edelen (28) and Stephani (38). Differential forms are discussed in complete detail by Flanders (39). In-depth treatment of exterior analysis of PDEs can be found in Bryant et al. (40), Ivey and Landsberg (29) and Edelen (27). 3.1 Exterior differential system In this section, we apply the necessary tools from exterior analysis as a convenient theoretical framework to systematically identify scale-invariant properties of solutions to the cavity-expansion model in section 2. Systems of PDEs defined on an $$n$$-dimensional differentiable manifold can be reformulated locally as an equivalent exterior differential systems (EDS) on an extended differentiable manifold. A system of PDEs is transformed into an equivalent EDS through a general procedure by replacing all second-order and higher equations by systems of first-order equations by introducing the necessary number of additional variables. The resulting system of differential forms is then determined by inspection from this first-order system. We illustrate these ideas below to obtain an equivalent EDS for the elastic-plastic equations of motion reviewed in section 2. The required objects for symmetry analysis in this setting are differential forms denoted $$\omega$$ and vector fields denoted $$V$$ defined in an $$n$$-dimensional differentiable manifold. A 1-form maps a vector field into function called the contraction or interior product. In an $$n$$-dimensional differentiable manifold there exists $$n$$-independent 1-forms denoted $${\mathrm{d}} \omega^i$$. The contraction denoted $$\lrcorner$$ can be represented in coordinate form as \begin{equation} V \lrcorner \omega = v^i\omega^i. \end{equation} (3.1) The wedge or exterior product is denoted $$\wedge$$ and is defined between two independent 1-forms and defines a 2-form. The exterior product is linear in each argument and anti-symmetric, that is, \begin{equation} \omega_j \wedge \omega_k = -\omega_k \wedge \omega_j, \end{equation} (3.2) and \begin{equation} \omega_j \wedge \omega_j = 0. \end{equation} (3.3) 3.1.1 Plastic equations The first-order equations (2.1)–(2.2) describing motion in the plastic region are can be cast as an EDS composed of a pair of 2-forms \begin{eqnarray} \omega_1 &=&-{\mathrm{d}} \sigma_r \wedge {\mathrm{d}} r + v {\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t \nonumber\\ &&+\, (\gamma - \sigma_r)\left( {\mathrm{d}} v \wedge {\mathrm{d}} t + \frac{2v}{r} {\mathrm{d}} r \wedge {\mathrm{d}} t\right),\\ \end{eqnarray} (3.4) \begin{eqnarray} \omega_2 &=&\rho_0 K (-{\mathrm{d}} v \wedge {\mathrm{d}} r + v {\mathrm{d}} v \wedge {\mathrm{d}} t)\nonumber\\ &&+\, (\gamma - \sigma_r) \left( {\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t + \frac{2Y}{r} {\mathrm{d}} r \wedge {\mathrm{d}} t \right). \end{eqnarray} (3.5) Equations (2.1)–(2.2) can be recovered from the differential forms in (3.4)–(3.5) in two steps. First we impose the appropriate compatibility requirements on $$\sigma_r$$ and $$v$$ in terms of the independent variables $$r$$ and $$t$$, which amounts to expanding their corresponding total differentials in (3.4)–(3.5), we obtain \begin{eqnarray} \omega_1 &=& -\Big(\frac{\partial \sigma_r}{\partial r} {\mathrm{d}} r + \frac{\partial \sigma_r}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} r + v\Big(\frac{\partial \sigma_r}{\partial r} {\mathrm{d}} r + \frac{\partial \sigma_r}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} t \nonumber\\ &&+\, (\gamma - \sigma_r)\Big\{ \Big(\frac{\partial v}{\partial r} {\mathrm{d}} r + \frac{\partial v}{\partial t} {\mathrm{d}} t\Big)\wedge {\mathrm{d}} t + \frac{2v}{r} {\mathrm{d}} r \wedge {\mathrm{d}} t\Big\} \nonumber \\ &=& \Big\{\frac{\partial \sigma_r}{\partial t} + v \frac{\partial \sigma_r}{\partial r} + (\gamma - \sigma_r) \Big(\frac{\partial v}{\partial r} + \frac{2v}{r} \Big) \Big\} {\mathrm{d}} r \wedge {\mathrm{d}} t, \\ \end{eqnarray} (3.6) \begin{eqnarray} \omega_2 &=&\rho_0 K \Big\{-\Big(\frac{\partial v}{\partial r} {\mathrm{d}} r + \frac{\partial v}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} r + v \Big(\frac{\partial v}{\partial r} {\mathrm{d}} r + \frac{\partial v}{\partial t} {\mathrm{d}} t \Big) \wedge {\mathrm{d}} t\Big\} \nonumber \\ && + \,(\gamma - \sigma_r) \Big\{ \Big(\frac{\partial \sigma_r}{\partial r} {\mathrm{d}} r + \frac{\partial \sigma_r}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} t + \frac{2Y}{r} {\mathrm{d}} r \wedge {\mathrm{d}} t \Big\} \nonumber\\ &=&\Big\{ \rho_0 K \Big( \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial r} \Big) + (\gamma - \sigma_r) \Big( \frac{\partial \sigma_r}{\partial r} + \frac{2Y}{r}\Big) \Big\} {\mathrm{d}} r \wedge {\mathrm{d}} t . \end{eqnarray} (3.7) Second, the previous forms are evaluated at zero to recover the original PDEs in (2.1)–(2.2) from (3.6)–(3.7) \begin{eqnarray} \Big\{\Big(\frac{\partial \sigma_r}{\partial t} + v \frac{\partial \sigma_r}{\partial r} + (\gamma - \sigma_r) \Big(\frac{\partial v}{\partial r} + \frac{2v}{r} \Big) \Big\} {\mathrm{d}} r \wedge {\mathrm{d}} t &=& 0,\\ \end{eqnarray} (3.8) \begin{eqnarray} \Big\{ \rho_0 K \Big( \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial r} \Big) + (\gamma - \sigma_r) \Big( \frac{\partial \sigma_r}{\partial r} + \frac{2Y}{r}\Big) \Big\} {\mathrm{d}} r \wedge {\mathrm{d}} t &=&0. \end{eqnarray} (3.9) Equations (3.4)–(3.5) form a differential ideal $$I=\{\omega_1,\omega_2\}$$. The ideal $$I$$ should be closed with respect to exterior differentiation, that is, \begin{equation} {\mathrm{d}} I \subset I, \end{equation} (3.10) where $${\mathrm{d}}(\cdot)$$ is a differential operator called the exterior derivative. In the case of functions (0-forms) the exterior derivative coincides with the total derivative, that is, \begin{equation} {\mathrm{d}} f = \sum\frac{\partial f}{\partial x^i}{\mathrm{d}} x^i. \end{equation} (3.11) In the case of 1-forms denoted $$\omega_j$$ and $$\omega_k$$ and a scalar function denoted $$f$$, the exterior derivative has the properties \begin{eqnarray} {\mathrm{d}}({\mathrm{d}} \omega_j) &=& 0,\\ \end{eqnarray} (3.12) \begin{eqnarray} {\mathrm{d}}(\omega_j + \omega_k) &=& {\mathrm{d}} \omega_j + {\mathrm{d}} \omega_k,\\ \end{eqnarray} (3.13) \begin{eqnarray} {\mathrm{d}} (\omega_j \wedge \omega_k)& =& {\mathrm{d}} \omega_j \wedge \omega_k - \omega_j \wedge {\mathrm{d}} \omega_k,\\ \end{eqnarray} (3.14) \begin{eqnarray} {\mathrm{d}}(f\omega_j) &=& {\mathrm{d}} f \wedge \omega_j + f{\mathrm{d}}\omega_j. \end{eqnarray} (3.15) We verify (3.10) by demonstrating the exterior derivative of any element of the ideal $$I$$ can be written as a form linear in $$\omega_1$$ and $$\omega_2$$. Hence, the ideal $$I$$ is seen to be closed by writing \begin{eqnarray} {\mathrm{d}} \omega_1 &=& \frac{4}{\gamma-\sigma_r}\Big(\omega_1 \wedge {\mathrm{d}} \sigma_r + \omega_2 \wedge {\mathrm{d}} v + \frac{2v}{r} \omega_1 \wedge {\mathrm{d}} t - \frac{Y}{r \rho_0K}\omega_2 \wedge {\mathrm{d}} t \Big),\\ \end{eqnarray} (3.16) \begin{eqnarray} {\mathrm{d}} \omega_2 &=& \frac{-2Y}{r}({\mathrm{d}} \omega_1 \wedge {\mathrm{d}} t). \end{eqnarray} (3.17) 3.1.2 Elastic equations Proceeding analogously to section 3.1.1, first we reduce the second-order differential equation (2.9) to a system of first-order equations \begin{eqnarray} w &=& \frac{\partial u}{\partial r},\\ \end{eqnarray} (3.18) \begin{eqnarray} z &=& \frac{\partial u}{\partial t},\\ \end{eqnarray} (3.19) \begin{eqnarray} \frac{\partial w}{\partial r} - \frac{1}{c_p^2}\frac{\partial z}{\partial t} + \Big(\frac{2w}{r} - \frac{2u}{r^2}\Big) &=& 0, \end{eqnarray} (3.20) where the new variables $$w$$ and $$z$$ correspond to strain and velocity, respectively. The system of equations in (3.18)–(3.20) can be reduced to the equivalent EDS \begin{eqnarray} \omega_3 &=& {\mathrm{d}} u \wedge {\mathrm{d}} t - w {\mathrm{d}} r \wedge {\mathrm{d}} t,\\ \end{eqnarray} (3.21) \begin{eqnarray} \omega_4 & =& {\mathrm{d}} u \wedge {\mathrm{d}} r + z {\mathrm{d}} r \wedge {\mathrm{d}} t,\\ \end{eqnarray} (3.22) \begin{eqnarray} \omega_5 &=&{\mathrm{d}} w \wedge {\mathrm{d}} t + \frac{1}{c_p^2} {\mathrm{d}} z \wedge {\mathrm{d}} r + \left( \frac{2w}{r} - \frac{2u}{r^2} \right) {\mathrm{d}} r \wedge {\mathrm{d}} t . \end{eqnarray} (3.23) We outline an equivalence between the equations of motion cast as familiar PDEs (3.18)–(3.20) and the EDS (3.21)–(3.23). Again, we enforce the required compatibility condition for the dependent variables $$u,w$$ and $$z$$ in terms of the independent variables $$r$$ and $$t$$ by expanding their corresponding total differentials to obtain \begin{eqnarray} \omega_3 &=& \Big(\frac{\partial u}{\partial r} {\mathrm{d}} r + \frac{\partial u}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} t - w {\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber\\ &=&\Big(\frac{\partial u}{\partial r} - w \Big) {\mathrm{d}} r \wedge {\mathrm{d}} t,\\ \end{eqnarray} (3.24) \begin{eqnarray} \omega_4 &=& \Big(\frac{\partial u}{\partial r} {\mathrm{d}} r + \frac{\partial u}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} r + z {\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber\\ &=& \Big(-\frac{\partial u}{\partial t} + z\Big) {\mathrm{d}} r \wedge {\mathrm{d}} t, \\ \end{eqnarray} (3.25) \begin{eqnarray} \omega_5 &=& \Big( \frac{\partial w}{\partial r} {\mathrm{d}} r + \frac{\partial w}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} t + \frac{1}{c_p^2} \Big(\frac{\partial z}{\partial r} {\mathrm{d}} r + \frac{\partial z}{\partial t} {\mathrm{d}} t\Big) \wedge {\mathrm{d}} r \nonumber\\ && +\, \Big( \frac{2w}{r} - \frac{2u}{r^2} \Big) {\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber \\ &=&\Big\{ \frac{\partial w}{\partial r} -\frac{1}{c_p^2}\frac{\partial z}{\partial t} + \Big( \frac{2w}{r} - \frac{2u}{r^2} \Big)\Big\} {\mathrm{d}} r \wedge {\mathrm{d}} t. \end{eqnarray} (3.26) The original PDEs in (3.18)–(3.20) are recovered from the previous forms by setting them to zero \begin{eqnarray} \Big(\frac{\partial u}{\partial r} - w \Big) {\mathrm{d}} r \wedge {\mathrm{d}} t&=&0,\\ \end{eqnarray} (3.27) \begin{eqnarray} \Big(-\frac{\partial u}{\partial t} + z\Big) {\mathrm{d}} r \wedge {\mathrm{d}} t &=&0,\\ \end{eqnarray} (3.28) \begin{eqnarray} \Big\{ \frac{\partial w}{\partial r} -\frac{1}{c_p^2} \Big(\frac{\partial z}{\partial t} + \Big( \frac{2w}{r} - \frac{2u}{r^2} \Big)\Big\} {\mathrm{d}} r \wedge {\mathrm{d}} t&=&0. \end{eqnarray} (3.29) Similarly, to the exterior differential system in the plastic region. Differential forms (3.21)–(3.23) form a closed ideal as seen from \begin{eqnarray} {\mathrm{d}}\omega_3 &=& -\omega_5 \wedge {\mathrm{d}} r,\\ \end{eqnarray} (3.30) \begin{eqnarray} {\mathrm{d}} \omega_4 &=& c_p^2 \omega_5 \wedge {\mathrm{d}} t,\\ \end{eqnarray} (3.31) \begin{eqnarray} {\mathrm{d}} \omega_5 &=& \frac{2}{r} \omega_3 \wedge {\mathrm{d}} r - \frac{2}{r^2} \omega_4 \wedge {\mathrm{d}} t.\end{eqnarray} (3.32) 3.2 Scaling symmetry To motivate the use of the method of differential forms to identify symmetries, we recall several key ideas behind Lie group analysis. Scaling symmetries, considered as a subclass of the larger collections local Lie symmetries or Lie point transformations, are determined by dilation or contraction of the original variables. To illustrate, scaling transformations corresponding to the variables in (2.7)–(2.8) are given by \begin{equation} \tilde{t} = s_1 t,\quad \tilde{r} = s_2 r, \quad \tilde{v} = s_3 v, \quad \tilde{\sigma}_r = s_4 \sigma_r. \end{equation} (3.33) Next suppose $$F$$ is a function of the form $$F(t,r,v,\sigma_r).$$ A function $$F$$ is invariant under the transformations in (3.33), if \begin{equation} F(\tilde{t},\tilde{r},\tilde{v},\tilde{\sigma}_r) = F(t,r,v,\sigma_r). \end{equation} (3.34) In this case $$F$$ is said to possess a scaling symmetry determined by the transformations in (3.33). Additionally, if $$F$$ corresponds to a smooth parameterized map of the variable $$t$$ and if there exists a value of $$s_1$$ corresponding to the identity of the scaling law for $$t$$ defined by (3.34), then the left-hand side of (3.34) can be expanded as a Taylor series about the identity, that is, $$s_1 = 1$$ \begin{eqnarray} F(\tilde{t},r,v,\sigma_r) &=& F(t,r,v,\sigma_r) + (s_1 -1)\frac{\partial F}{\partial s_1}\Big\rvert_{s_1 = 1} \nonumber\\ && +\, \frac{(s_1-1)^2}{2}\frac{\partial^2 F}{\partial s_1^2}\Big\rvert_{s_1 = 1} + \cdots \nonumber \\ &=& F(t,r,v,\sigma_r) + (s_1 -1)\frac{\partial F}{\partial \tilde{t}}\frac{\partial \tilde{t}}{\partial s_1}\Big\rvert_{s_1 = 1} \nonumber\\ &&+\, \frac{(s_1-1)^2}{2}\frac{\partial}{\partial s_1}\Big(\frac{\partial F}{\partial \tilde{t}}\frac{\partial \tilde{t}}{\partial s_1}\Big)\Big\rvert_{s_1 = 1} + \cdots \nonumber\\ &=& F(t,r,v,\sigma_r) + (s_1 -1)t\frac{\partial F}{\partial t}\Big\rvert_{s_1 = 1}\nonumber\\ && +\, \frac{(s_1-1)^2}{2}t\frac{\partial}{\partial t}\Big(t\frac{\partial F}{\partial t}\Big)\Big\rvert_{s_1 = 1} + \dots \nonumber\\ &=& F(t,r,v,\sigma_r) + (s_1 -1)X^{(t)}F\Big\rvert_{s_1 = 1} \nonumber\\ && +\, \frac{(s_1-1)^2}{2}X^{(t)}\Big(X^{(t)}F\Big)\Big\rvert_{s_1 = 1} + \cdots, \end{eqnarray} (3.35) where $$X^{(t)}:=t\partial / \partial t$$. Therefore the invariance condition for scaling $$t$$ in (3.34) can be re-characterized by the condition $$X^{(t)}F = 0$$. Similar infinitesimal representations of the invariance condition can be obtained for the scaling transformations in the remaining variables. Invariance according to (3.34) can then be restated equivalently in terms of the condition \begin{equation} X^{(sc)}F = 0, \end{equation} (3.36) provided $$F=0$$, and where $$X^{(sc)}$$ is a differential operator given by a linear combination of the form \begin{equation} {X^{(sc)}}:= a_1 t\frac{\partial}{\partial t} + a_2r\frac{\partial}{\partial r} + a_3v\frac{\partial}{\partial v}+ a_4\sigma_r\frac{\partial}{\partial \sigma_r}. \end{equation} (3.37) The linear differential operator $$X^{(sc)}$$ is called the group generator corresponding to the local Lie group of scaling transformations in (3.33). The final step is then to determine any restrictions on the set of coefficients $$a_1,a_2,a_3$$ and $$a_4$$ admitted by (3.36). In the case where $$F$$ represents a first-order PDE of the form \begin{equation} F(t,r,v,\sigma_r, \tfrac{\partial v}{\partial r}, \tfrac{\partial v}{\partial t}, \tfrac{\partial \sigma_r}{\partial r}, \tfrac{\partial \sigma_{r}}{\partial t})=0 \end{equation} (3.38) just as in (2.7) or (2.8). Therefore the condition for invariance in (3.36) must be achieved in a larger space of variables, which includes the first derivatives of the dependent variables. Before transitioning back to the setting of differential forms, we outline several connections between the group generator $$X^{(sc)}$$ in (3.37) and Lie derivatives, which are associated with a vector field $$V$$. In the case of ordinary functions (0-forms), the Lie derivative corresponds to the familiar directional derivative \begin{equation} \mathscr{L}_Vf = \sum v^i\frac{\partial f}{\partial x^i}, \end{equation} (3.39) where $$v^i$$ denote the components of a vector field $$V$$. The action of the Lie derivative with respect to forms is defined in terms of the contraction operation \begin{equation} \mathscr{L}_V \omega = V\lrcorner {\mathrm{d}} \omega + d(V\lrcorner \omega). \end{equation} (3.40) The Lie derivative acts on forms according to the following rules \begin{eqnarray} \mathscr{L}_V[{\mathrm{d}}(\omega)]& =& {\mathrm{d}}(\mathscr{L}_V[\omega])\\ \end{eqnarray} (3.41) \begin{eqnarray} \mathscr{L}_V(\omega_j \wedge \omega_k)& =& \mathscr{L}_V\omega_j \wedge \omega_k + \omega_i \wedge \mathscr{L}_V \omega_k. \end{eqnarray} (3.42) In the method of differential forms, symmetries of a closed ideal $$I={\omega_1, \omega_2, \dots, \omega_N}$$ of differential forms are classified in terms of invariance with respect to action of the Lie derivative, which can be expressed \begin{equation} \mathscr{L}_V I \subset I. \end{equation} (3.43) In other words, the Lie derivatives of the differential forms in $$I$$ vanish, if $$\omega_j=0$$ for $$j=1,2,\dots,N$$. This condition is clearly satisfied if $$\mathscr{L}_V \omega_j = \lambda_k\omega_k$$ for $$\omega_k \in I$$. Hence, statement (3.43) is the analog of (3.36). Moreover, by setting \begin{equation} v^{(t)}=a_1t, \quad v^{(r)}=a_2r, \quad v^{(v)}=a_3v, \quad v^{(\sigma_r)}=a_4\sigma_r, \end{equation} (3.44) we see that the group generator in (3.37) is equivalent to (3.39). To stress this connection between the classical group generator and the vector field attached to the Lie derivative, we will denote the Lie derivative corresponding to the infinitesimal representation of the Lie symmetry by \begin{equation} \mathscr{L}_{X^{(sc)}} := \mathscr{L}_V. \end{equation} (3.45) More complete analysis of these connections can be found in Olver (22) and Stephani (38). 3.3 Plastic constraints We proceed by determining the group generator of the form \begin{equation} \mathscr{L}_{X^{(sc)}}:= a_1 t\frac{\partial}{\partial t} + a_2r\frac{\partial}{\partial r} + a_3v\frac{\partial}{\partial v}+ a_4\sigma_r\frac{\partial}{\partial \sigma_r}, \end{equation} (3.46) corresponding to a Lie group of scaling symmetries. The existence of scaling symmetries may be reduced to satisfying the condition (3.43), which is written \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_1] &=&\lambda_1\omega_1 + \lambda_2\omega_2, \\ \end{eqnarray} (3.47) \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_2]& =&\lambda_3 \omega_1 + \lambda_4 \omega_2, \end{eqnarray} (3.48) where the Lagrange multipliers $$\lambda_1, \lambda_2, \lambda_3$$ and $$\lambda_4$$ are unknown functions, in this case, 0-forms determined below. First we compute the action of the Lie derivative on (3.4) \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_1]&=& -{\mathrm{d}}\mathscr{L}_{X^{(sc)}}[\sigma_r]\wedge {\mathrm{d}} r -{\mathrm{d}} \sigma_r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \nonumber\\ &&+\, \mathscr{L}_{X^{(sc)}}[v] {\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t + v\big( {\mathrm{d}}\mathscr{L}_{X^{(sc)}}[\sigma_r]\wedge {\mathrm{d}} t + {\mathrm{d}} \sigma_r \wedge {\mathrm{d}}\mathscr{L}_{X^{(sc)}} [t]\big) \nonumber\\ &&+\,\mathscr{L}_{X^{(sc)}}[\gamma - \sigma_r] \Big( {\mathrm{d}} v \wedge {\mathrm{d}} t + \frac{2v}{r} {\mathrm{d}} r \wedge {\mathrm{d}} t \Big) \nonumber\\ &&+\, (\gamma - \sigma_r) \Big\{ {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[v] \wedge {\mathrm{d}} t + {\mathrm{d}} v \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t] \nonumber\\ &&+\, \frac{2}{r} \mathscr{L}_{X^{(sc)}}[v] {\mathrm{d}} r \wedge {\mathrm{d}} t + 2v \mathscr{L}_{X^{(sc)}} \Big[ \frac{1}{r} \Big] {\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber\\ &&+\, \frac{2v}{r} \Big( {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \wedge {\mathrm{d}} t + {\mathrm{d}} r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t] \Big)\Big\} \nonumber\\ &=& -(a_2 + a_4){\mathrm{d}} \sigma_r \wedge {\mathrm{d}} r +v(a_1+a_3+a_4){\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t \nonumber\\ &&+\, \big\{(\gamma-\sigma_r)(a_1+a_3) - \sigma_ra_4\big\} \big({\mathrm{d}} v \wedge {\mathrm{d}} t + \frac{2v}{r}{\mathrm{d}} r \wedge {\mathrm{d}} t\big). \end{eqnarray} (3.49) Equating the coefficients of the five unique 2-forms resulting from (3.47) it can be read off that $$\lambda_2$$ must vanish, hence \begin{eqnarray} ({\mathrm{d}} \sigma_r \wedge {\mathrm{d}} r)&:& a_2 + a_4 = \lambda_1,\\ \end{eqnarray} (3.50) \begin{eqnarray} ({\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t)&:& (a_1 + a_3 + a_4)v = \lambda_1 v,\\ \end{eqnarray} (3.51) \begin{eqnarray} ({\mathrm{d}} v \wedge {\mathrm{d}} t)&:& (\gamma - \sigma_r)(a_1 + a_3) - \sigma_ra_4 = \lambda_1(\gamma - \sigma_r),\\ \end{eqnarray} (3.52) \begin{eqnarray} ({\mathrm{d}} r \wedge {\mathrm{d}} t)&:&\big( (\gamma-\sigma_r) (a_1 + a_3) - \sigma_r a_4\big)v =\lambda_1(\gamma-\sigma_r)v. \end{eqnarray} (3.53) Eliminating $$\lambda_1$$ yields \begin{equation} a_3 = a_2 - a_1. \end{equation} (3.54) Finally, the condition for $$({\mathrm{d}} v \wedge {\mathrm{d}} t)$$ then implies $$a_4 = 0$$. Note that (3.54) simply amounts to dimensional consistency between the radial velocity and the space and time coordinates. Next, we expand (3.48) starting with the action of the Lie derivative on (3.5) \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_2]&=& \rho_0K \big\{-{\mathrm{d}} \mathscr{L}_{X^{(sc)}}[v] \wedge {\mathrm{d}} r - {\mathrm{d}} v \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}} [r] \nonumber\\ && + \,\mathscr{L}_{X^{(sc)}}[v]{\mathrm{d}} v \wedge {\mathrm{d}} t \nonumber\\ &&+\, v({\mathrm{d}} \mathscr{L}_{X^{(sc)}}[v] \wedge {\mathrm{d}} t + {\mathrm{d}} v \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}} [t]) \big\} \nonumber\\ &&+\, \mathscr{L}_{X^{(sc)}}[\gamma - \sigma_r]\left( {\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t - \frac{2Y}{r} {\mathrm{d}} r \wedge {\mathrm{d}} t \right) \nonumber\\ && +\, (\gamma - \sigma_r) \Big\{ {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[\sigma_r] \wedge {\mathrm{d}} t + {\mathrm{d}} \sigma_r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t] \nonumber\\ && -\,2Y\mathscr{L}_{X^{(sc)}}\left[\frac{1}{r}\right] {\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber\\ && - \,\frac{2Y}{r} \Big({\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \wedge {\mathrm{d}} t + {\mathrm{d}} r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}} [t]\Big) \Big\} \nonumber\\ &=&\rho_0K \Big\{-(a_2 + a_3) {\mathrm{d}} v \wedge {\mathrm{d}} r + v(a_1 + 2a_3) {\mathrm{d}} v \wedge {\mathrm{d}} t\Big\} \nonumber\\ &&+\,\Big\{(\gamma-\sigma_r)(a_1 + a_4) - a_4 \sigma_r\Big\} {\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t \nonumber\\ &&-\,\frac{2Y}{r}\Big\{(\gamma -\sigma_r)a_1 - \sigma_ra_4\Big\}{\mathrm{d}} r \wedge {\mathrm{d}} t. \end{eqnarray} (3.55) Again, equating the coefficients of the five unique 2-forms resulting from (3.48), we have immediately that $$\lambda_3=0$$ and \begin{eqnarray} ({\mathrm{d}} v \wedge {\mathrm{d}} r)&:&\rho_0 K (a_2 + a_3) = \lambda_4\rho_0 K\\ \end{eqnarray} (3.56) \begin{eqnarray} ({\mathrm{d}} v \wedge {\mathrm{d}} t)&:& \rho_0 K (a_1+2a_3)v = \lambda_4\rho_0Kv\\ \end{eqnarray} (3.57) \begin{eqnarray} ({\mathrm{d}} \sigma_r \wedge {\mathrm{d}} t)&:& (\gamma - \sigma_r) (a_1 + a_4) - \sigma_ra_4 = \lambda_4(\gamma - \sigma_r)\\ \end{eqnarray} (3.58) \begin{eqnarray} ({\mathrm{d}} r \wedge {\mathrm{d}} t)&:&-\frac{2Y}{r}\left((\gamma-\sigma_r)a_1 - \sigma_ra_4\right) =-\lambda_4\frac{2Y}{r}(\gamma-\sigma_r). \end{eqnarray} (3.59) Eliminating $$\lambda_4$$ yields \begin{equation} a_3 = a_2 - a_1. \end{equation} (3.60) This constraint together with the previous constraint in (3.54) imply that \begin{equation} a_1 = a_2, \end{equation} (3.61) which also implies that $$a_3 = 0$$. Therefore, the group generator in (3.46) in the plastic region becomes \begin{equation} \mathscr{L}_{X^{(sc)}}^{(p)}:= a_1\left(t\frac{\partial}{\partial t} + r\frac{\partial}{\partial r}\right), \end{equation} (3.62) which indicates that (2.7)–(2.8) admit at most a one-parameter family of scaling transformations. Additionally, the group generator in (3.62) corresponds to a similarity variable which agrees with the similarity transformation used by Forrestal and Luk (7) acquired via ansatz and dimensional considerations. The similarity variable $$\xi$$ is determined by the solution to the characteristic equations associated with (3.62) \begin{equation} Xf = 0, \end{equation} (3.63) where $$f$$ is any function with the same symmetry as the PDEs (2.7)–(2.8). The characteristic equations can be written in the form \begin{equation} \frac{{\mathrm{d}} r}{r} = \frac{{\mathrm{d}} t}{t}. \end{equation} (3.64) Equation (3.64) can be solved to obtain the similarity transformation \begin{equation} \xi= \frac{r}{\alpha t}, \end{equation} (3.65) where $$\alpha=\text{const}$$. 3.4 Elastic constraints Following the process described in section 3.3, we begin by assuming the group generator is represented as \begin{equation} \mathscr{L}_{X^{(sc)}}:= a_1t\frac{\partial}{\partial t} + a_2r\frac{\partial}{\partial r} + a_6u\frac{\partial}{\partial u} + a_7w\frac{\partial}{\partial w} + a_8z\frac{\partial}{\partial z}. \end{equation} (3.66) Again, we proceed by the method of unknown Lagrange multipliers to determine the relationships between the coefficients in (3.66) from \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_3] &=& \lambda_1\omega_1 + \lambda_2\omega_2 + \lambda_3\omega_3,\\ \end{eqnarray} (3.67) \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_4] &=& \lambda_4\omega_1 + \lambda_5\omega_2 + \lambda_6\omega_3,\\ \end{eqnarray} (3.68) \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_5] &=& \lambda_7\omega_1 + \lambda_8\omega_2 + \lambda_9\omega_3, \end{eqnarray} (3.69) where $$\lambda_1, \lambda_2, \dots, \lambda_9$$ are unknown 0-forms. First we compute the action of the Lie derivative on the set of 2-forms in (3.21)–(3.23). Starting with (3.21), \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_3] &=& {\mathrm{d}}\mathscr{L}_{X^{(sc)}}[u] \wedge {\mathrm{d}} t + {\mathrm{d}} u \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t] \nonumber\\ &&-\,\mathscr{L}_{X^{(sc)}}[w] {\mathrm{d}} r \wedge {\mathrm{d}} t - w \big( {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \wedge {\mathrm{d}} t + {\mathrm{d}} r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}} [t] \big) \nonumber\\ &=&(a_1 + a_6){\mathrm{d}} u \wedge {\mathrm{d}} t \nonumber \\ && -\, a_7 w {\mathrm{d}} r \wedge {\mathrm{d}} t - (a_1 + a_2)w{\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber \\ &=&(a_1 + a_6){\mathrm{d}} u \wedge {\mathrm{d}} t - (a_1 + a_2 + a_7)w{\mathrm{d}} r \wedge {\mathrm{d}} t. \end{eqnarray} (3.70) Similarly for (3.22), \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_4] &=& -{\mathrm{d}} \mathscr{L}_{X^{(sc)}}[u] \wedge {\mathrm{d}} r - {\mathrm{d}} u \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \nonumber \\ &&-\, \mathscr{L}_{X^{(sc)}}[z] {\mathrm{d}} r \wedge {\mathrm{d}} t - z \big( {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \wedge {\mathrm{d}} t + {\mathrm{d}} r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}} [t] \big) \nonumber\\ &=&-(a_2 + a_6){\mathrm{d}} u \wedge {\mathrm{d}} r \nonumber\\ &&- \,a_8 z {\mathrm{d}} r \wedge {\mathrm{d}} t - (a_1 + a_2)z{\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber\\ &=&-(a_2 + a_6){\mathrm{d}} u \wedge {\mathrm{d}} r - (a_1 + a_2 + a_8)z{\mathrm{d}} r \wedge {\mathrm{d}} t. \end{eqnarray} (3.71) Finally for (3.23), \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_5] &=& {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[w] \wedge {\mathrm{d}} t + {\mathrm{d}} w \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t] \nonumber\\ &&+\, \frac{1}{c_d^2}({\mathrm{d}}\mathscr{L}_{X^{(sc)}}[z] \wedge {\mathrm{d}} r + {\mathrm{d}} z \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r]) \nonumber\\ &&+\,\mathscr{L}_{X^{(sc)}}\Big[\frac{2w}{r} - \frac{2u}{r^2}\Big] {\mathrm{d}} r\wedge {\mathrm{d}} t\nonumber\\ && +\, \Big(\frac{2w}{r} -\frac{2u}{r^2}\Big)({\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \wedge {\mathrm{d}} t + {\mathrm{d}} r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t])\nonumber \\ &=& (a_1 + a_7) {\mathrm{d}} w \wedge {\mathrm{d}} t + \frac{1}{c_d^2}(a_2 + a_8) {\mathrm{d}} z \wedge {\mathrm{d}} r \nonumber\\ &&+\,\Big[(-a_2 + a_7)\frac{2w}{r} + (2a_2 -a_6)\frac{2u}{r^2}\Big]{\mathrm{d}} r\wedge {\mathrm{d}} t \nonumber\\ &&+\, \Big(\frac{2w}{r} - \frac{2u}{r^2}\Big)(a_1 + a_2){\mathrm{d}} r \wedge {\mathrm{d}} t \nonumber\\ &=& (a_1 + a_7) {\mathrm{d}} w \wedge {\mathrm{d}} t + \frac{1}{c_d^2}(a_2 + a_8) {\mathrm{d}} z \wedge {\mathrm{d}} r \nonumber\\ &&+\,\Big[(a_1 + a_7)\frac{2w}{r} + (-a_1 + a_2 -a_6)\frac{2u}{r^2}\Big]{\mathrm{d}} r\wedge {\mathrm{d}} t. \end{eqnarray} (3.72) Equating coefficients corresponding to the distinct 2-forms in (3.67)–(3.69), from (3.67) we can immediately read off $$\lambda_2=\lambda_3 = 0$$, leaving \begin{eqnarray} ({\mathrm{d}} u \wedge {\mathrm{d}} t)&:& \lambda_1 = a_1 + a_6,\\ \end{eqnarray} (3.73) \begin{eqnarray} ({\mathrm{d}} r \wedge {\mathrm{d}} t)&:& -(a_1 + a_2 + a_7)w = -\lambda_1w. \end{eqnarray} (3.74) Eliminating $$\lambda_1$$ yields \begin{equation} a_7 = a_6 - a_2. \end{equation} (3.75) Next, from (3.68) we see that $$\lambda_4 = \lambda_6 = 0$$, which leaves \begin{eqnarray} ({\mathrm{d}} u \wedge {\mathrm{d}} r)&:& \lambda_5 = a_2 + a_6,\\ \end{eqnarray} (3.76) \begin{eqnarray} ({\mathrm{d}} r \wedge {\mathrm{d}} t)&:& -(a_1 + a_2 + a_8)z = \lambda_5z. \end{eqnarray} (3.77) Similarly, eliminating $$\lambda_5$$ leaves \begin{equation} a_8 = a_6 - a_1. \end{equation} (3.78) Note that constraints (3.75) and (3.78) correspond to dimensional consistency for strain and particle velocity respectively. From (3.69), \begin{eqnarray} ({\mathrm{d}} w \wedge {\mathrm{d}} t)&:& \lambda_9 = a_1 + a_7,\\ \end{eqnarray} (3.79) \begin{eqnarray} ({\mathrm{d}} z \wedge {\mathrm{d}} r)&:& \lambda_9 = a_2 + a_8,\\ \end{eqnarray} (3.80) \begin{eqnarray} ({\mathrm{d}} r \wedge {\mathrm{d}} t)&:& \lambda_9\left(\frac{2w}{r} - \frac{2u}{r^2}\right) = (a_1 + a_7) \frac{2w}{r} + (-a_1 + a_2 -a_6)\frac{2u}{r^2}. \end{eqnarray} (3.81) Eliminating $$\lambda_9$$, these constraints yield the remaining constraint \begin{eqnarray} a_1 + a_7 = a_2 + a_8, \end{eqnarray} (3.82) which after applying the previous constraints, further reduces to \begin{eqnarray} a_1 &=& a_2,\\ \end{eqnarray} (3.83) \begin{eqnarray} a_7 &=& a_8,\\ \end{eqnarray} (3.84) \begin{eqnarray} a_6 &=& a_1 + a_8. \end{eqnarray} (3.85) Hence, the group generator in (3.66) becomes \begin{equation} \mathscr{L}_{X^{(sc)}}:= a_1\Big(t\frac{\partial}{\partial t} + r\frac{\partial}{\partial r}\Big) + (a_1 + a_8)u\frac{\partial}{\partial u} + a_8\Big(w\frac{\partial}{\partial w} + z\frac{\partial}{\partial z}\Big). \end{equation} (3.86) The remaining constraints are determined from relating the displacement $$u$$ to the velocity $$v$$ using (2.13). Writing (2.13) as a 2-form, \begin{equation} \omega_6 = -{\mathrm{d}} u \wedge {\mathrm{d}} r - v({\mathrm{d}} r \wedge {\mathrm{d}} t - {\mathrm{d}} u \wedge {\mathrm{d}} t). \end{equation} (3.87) Applying the Lie derivative to (3.87) we obtain \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[\omega_6]& =& -{\mathrm{d}} \mathscr{L}_{X^{(sc)}}[u] \wedge {\mathrm{d}} r -{\mathrm{d}} u \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \nonumber\\ &&-\,\mathscr{L}_{X^{(sc)}}[v]({\mathrm{d}} r \wedge {\mathrm{d}} t - {\mathrm{d}} u \wedge {\mathrm{d}} t) \nonumber\\ &&-\,v({\mathrm{d}} \mathscr{L}_{X^{(sc)}}[r] \wedge {\mathrm{d}} t + {\mathrm{d}} r \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t] \nonumber\\ && -\, {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[u] \wedge {\mathrm{d}} t - {\mathrm{d}} u \wedge {\mathrm{d}} \mathscr{L}_{X^{(sc)}}[t] ) \nonumber\\ &=& -(2a_1+a_8){\mathrm{d}} u \wedge {\mathrm{d}} r \nonumber\\ &&-\,v(a_1 {\mathrm{d}} r \wedge {\mathrm{d}} t + a_1{\mathrm{d}} r \wedge {\mathrm{d}} t - (a_1+a_8) {\mathrm{d}} u \wedge {\mathrm{d}} t - a_1 {\mathrm{d}} u \wedge {\mathrm{d}} t) \nonumber\\ &=& -(2a_1+ a_8){\mathrm{d}} u \wedge {\mathrm{d}} r \nonumber\\ &&-\,v\big\{ 2a_1{\mathrm{d}} r \wedge {\mathrm{d}} t - (2a_1 + a_8){\mathrm{d}} u \wedge {\mathrm{d}} t\big\}. \end{eqnarray} (3.88) Equating the coefficients corresponding to the three distinct 2-forms in \begin{equation} \mathscr{L}_{X^{(sc)}}[\omega_6] = \lambda_{10} \omega_6 \end{equation} (3.89) yields \begin{eqnarray} ({\mathrm{d}} u \wedge {\mathrm{d}} r)&:& \qquad \qquad \lambda_{10} = 2a_1 + a_8, \qquad \qquad\\ \end{eqnarray} (3.90) \begin{eqnarray} ({\mathrm{d}} r \wedge {\mathrm{d}} t)&:& \qquad \qquad \lambda_{10} = 2a_1, \qquad \qquad\\ \end{eqnarray} (3.91) \begin{eqnarray} ({\mathrm{d}} u \wedge {\mathrm{d}} t)&:& \qquad \qquad \lambda_{10} = 2a_1 + a_8, \qquad \qquad \end{eqnarray} (3.92) which reduces to \begin{equation} a_8 = 0. \end{equation} (3.93) The scaling group generator in (3.86) in the elastic region simplifies to \begin{equation} \mathscr{L}_{X^{(sc)}}^{(e)}:= a_1\Big(t\frac{\partial}{\partial t} + r\frac{\partial}{\partial r} + u\frac{\partial}{\partial u}\Big). \end{equation} (3.94) The similarity transformations corresponding to (3.94) are determined from the characteristic equations derived from $$\mathscr{L}_{X^{(sc)}}^{(e)}f$$, where $$f$$ is any function with the same symmetry as the PDE (2.9) \begin{equation} \frac{{\mathrm{d}} t}{t}=\frac{{\mathrm{d}} r}{r}=\frac{{\mathrm{d}} u}{u}. \end{equation} (3.95) From (3.95), we obtain the similarity transformation in the plastic region in (3.65) and obtain a second similarity variable given by \begin{equation} \eta = \frac{u}{\beta t}. \end{equation} (3.96) for some constant $$\beta$$. It is also important to note that although the results presented here could have also been obtained equivalently using a classical scaling argument, in which case solutions of the power law-type are taken as an ansatz, for example, see Sedov (10). The advantage of applying the isovector method, along with any other group-theoretic program is to demonstrate that the obtained similarity variables are the only scaling invariant transformations admitted by the current model. In section 4, we also discuss several additional advantages behind this approach from a modeling perspective. 3.5 Interface conditions Compatibility of the interface conditions in (2.14)–(2.15) with the group generator in (3.62) and (3.86) is formulated as \begin{eqnarray} \mathscr{L}^{(p)}_{X^{(sc)}}\big[\rho_1(v_1 - c_{i}) - \rho_2(v_2-c_{i})\big] &=& 0,\\ \end{eqnarray} (3.97) \begin{eqnarray} \mathscr{L}^{(p)}_{X^{(sc)}}\big[\sigma_{2r} - \sigma_{1r} + \rho_1(v_2-v_1 )(v_1 - c_{i})\big] &=& 0, \end{eqnarray} (3.98) if (2.14) and (2.15) hold. From (3.97) we have $$\partial c_{i} / \partial t = 0$$, that is $$c_{i}$$ is a constant. Similarly, (3.98) also yields $$\partial c_{i} / \partial t = 0$$ or $$v_2 = v_1$$. We observe that the analogous result obtained by replacing the group generator (3.62) above with the group generator in the elastic region (3.86), (3.97)–(3.98) produce identical constraints on the interface velocity. Therefore, the interface conditions in (2.14)–(2.15) are invariant with respect both group generators, and hence compatible with the corresponding scaling transformations at the interface. 3.6 Wall trajectory Next we ensure compatibility between the boundary conditions describing the cavity wall trajectory with the group generator in (3.62) in the plastic region. We can state generic boundary conditions for the cavity wall trajectory as \begin{equation} r_w=a(t). \end{equation} (3.99) Compatibility with (3.62) is formulated as \begin{equation} \mathscr{L}_{X^{(sc)}}[r_w - a(t) ] = 0, \end{equation} (3.100) if (3.99) holds. Then constraint (3.100) yields \begin{eqnarray} \mathscr{L}_{X^{(sc)}}[ r_w - a(t) ]= a_1\left( r_w -t\frac{{\mathrm{d}} a}{{\mathrm{d}} t} \right). \end{eqnarray} (3.101) This implies \begin{equation} r_w -t\frac{{\mathrm{d}} a}{{\mathrm{d}} t} = 0. \end{equation} (3.102) Under the assumption that $$r_w=a$$, we have \begin{equation} a -t\frac{{\mathrm{d}} a}{{\mathrm{d}} t}=0. \end{equation} (3.103) Hence, (3.100) implies $${\mathrm{d}} a / {\mathrm{d}} t = c_w \cdot t$$, where $$c_w$$ is the constant of integration. Therefore the only admissible class of boundary conditions of the form (3.99) compatible with the scale-invariant transformations admitted by (2.7)–(2.8) is the constant velocity wall trajectory. This is in agreement with boundary conditions used in connection with cavity dynamics, for example, Forrestal and Luk (7) and Hunter and Crozier (6). 4. Conclusions We have systematically derived a scaling invariant solution to the spherical cavity expansion model described in section 2 using the isovector method of Estabrook and Harrison (24). We also verified that the obtained isovector field is compatible with the auxiliary boundary and interface conditions assumed along the cavity wall and elastic-plastic interface respectively. The resulting similarity transformations can be utilized for the purposes of constructing exact solutions, which have immediate application to hydrocode verification. It is important to note that, while we have a number of known exact solutions for related models in gas dynamics, far fewer solutions exist for models describing elastic-plastic cavity formation in solids. Additionally, the utility of these models to the study penetration mechanics is described in detail in, for example, Forrestal and Luk (8). It is important to note that for the SCE model described in section 2, which is supplemented by an explicit material response model in the form of the plastic flow criterion and pressure volumetric-strain relation in (2.3) and (2.5) respectively, the obtained results could have been determined equivalently by a conventional scaling argument, in which case solutions of the power law-type are used as an ansatz, as in for example Sedov (10). However, the advantage of the current approach is that the isovector (or equivalently other group-theoretic) approaches can also be used under much more general assumptions regarding the constitutive relations describing material strength and other constitutive relations. Hence, this effort constitutes an important initial step toward the major aim of determining other admissible classes of material response models that admit scaling invariance. In particular, the ultimate aim is to utilize the group-theoretic approach applied in this work as a unifying framework to guide the selection of more general constitutive models that also admit scaling invariance, which from a modeling perspective has implications toward the design and evaluation of laboratory-based experiments and simulations. Additionally, under the restriction that the constitutive relations must be compatible with a specified local Lie group of scalings, it is also possible to derive functional restrictions for admissible constitutive relationships, for example, for the yield criterion. An illustration of this approach for related problems in gas dynamics was used in Ovsiannikov (14) Axford and Holm (16) and Axford (15) to derive restrictions on the existence of scaling invariant transformations for various classes of the supplementary equation of state. Acknowledgements This work was completed under the auspices of the United States Department of Energy by Los Alamos National Security, LLC, at Los Alamos National Laboratory under contract DE-AC52-06NA25396. The authors thank J. Guzik, B. K. Harrison, R. Macek, J. Hogan and W. Patterson for valuable insights on these topics. References 1. Cole R. H. and Weller R. Underwater explosions, Physics Today 1 ( 1948) 35– 35. Google Scholar CrossRef Search ADS 2. Hill R. Cavitation and the influence of headshape in attack of thick targets by non-deforming projectiles, J. Mech. Phys. Solids. 28 ( 1980) 249– 263. Google Scholar CrossRef Search ADS 3. Bishop R. Hill R. and Mott N. The theory of indentation and hardness tests, Proc. Phys. Soc. 57 ( 1945) 147. Google Scholar CrossRef Search ADS 4. Hopkins H. Dynamic expansion of spherical cavities in metals, Progress in Solid Mechanics (ed. Hill R. and Sneddon I. North-Holland Publishing Company, Amsterdam 1960). 5. 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This work is written by a US Government employee and is in the public domain in the US.

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