A new tri-axial pressure-based constitutive expression has been found using Cauchy's stress tensor. This stress state emphasizes pressure and shear stress. The description is a pressure plus an effective shear stress allowing for a constitutive law based on atomic solid-state phase changes in crystalline cells due to pressure plus shear-based dislocation motion commonly associated with plasticity. Pressure has a new role in the material's constitutive response as it is separated from plasticity. The thermo-mechanical system describes third-order Gibbs’ expressions without specific volume restrictions placed upon the material. Isothermally, the ratio of heat to shear work in elastic copper is shown to approach zero at a very low temperature and become larger than one as temperature approaches melting. Wave compression models investigated are elastic and plastic: in fully elastic materials, the planar wave is restricted by Poisson's effect although plastic shear changes this constraint. Plastic deformation, dominated by dissipative shear stresses in uniaxial strain, heats the material while excluding phase changes from hydrostatic pressures. The material properties *per se* across Hugoniot shocks are described with entropy concepts. Shock waves are exceedingly complex since the constitutive laws are linked at extreme temperatures, pressures, and shear stresses. Isothermal, isentropic, isochoric, and iso-shear conditions are used throughout with Jacobian algebra.

## I. INTRODUCTION

Longitudinal, propagating, planar, compression waves in solids are launched by high-powered, ablative, pulsed lasers impacting solids and liquids^{1–6} and many other methods.^{7–14} The compression waves generated are often called *P* waves. These pulses at elevated temperatures and pressures are known to form new phases and transform materials.^{15–22} Pressures achieved by materials in laser compression, ramp compression, single shock, and double shocks in solid materials can approach 10 TPa. High pressures promote solid-state phase changes that are often unexpected, for example, hydrogen and deuterium become metals,^{23,24} common metals become insulators and insulators become metals,^{25,26} electrons released by over-lapping atomic orbitals form electride phases^{16,27,28} with the itinerant electrons residing in the interstices of the crystalline structures, room temperature superconductors have been discovered at very high static pressures,^{29,30} etc. High pressures also create phases seen in the Earth's interior materials.^{31–33} Extreme pressures and temperatures also form phases of metastable materials that exhibit unusual properties when returned to atmospheric conditions. Diamonds, especially industrial nano-diamonds,^{34,35} and martensitic steels^{36} are two well-known metastable materials invaluable in today's technologies.

Compressive waves, at much lower pressures, are ideal for determining material properties from wave time-of-flight measurements and scattering in geophysics. It allows for structure determination^{37,38} of the interior of the Earth. Seismic mappings of our planet at very sophisticated levels are obtained using the material's prescribed pressure and temperature dependence^{39} for the scattering and reflection of longitudinal and transverse waves: these methods prescribe a Helmholtz inversion which determines internal structures. Compression and shear waves are also extensively used in bio-medical ultrasound, which is based on density changes for propagation, scattering, and reflection measurements. These data determine a numerically inverted Helmholtz equation for the visualization of biological structures.^{40–42} Ultrasound propagation and absorption in tissues and understanding the biological structures observed using pulsed waves are important medical advances for detecting breast and other cancers. An understanding of longitudinal wave propagation in pressurized and sheared solids is of critical importance for solving many problems.

Most analytic methods for describing ramp compression, single shock, and multiple shocks emphasize pressure as the basic mechanical constituent of compressed material. The Rankine–Hugoniot expressions^{43–45} are used to determine the pressures from shock measurements of Lagrangian and Eulerian velocities in liquid materials. Yet, the most recent experimental irreversible material properties are shear driven as they describe rate-dependent plasticity^{46} from dislocation motion and twinning. Estimates of the temperature of laser compressed materials are an important objective of this paper.

Analytically, density functional theories and Monte Carlo calculations of realistic, complex crystal structures^{47–50} are widely used to describe atomic arrangements in solids^{51} at high temperatures and pressures. These calculations, although complex, predict structures observed at extreme pressures and temperatures.

In what follows below, a uniaxial strain system is used to describe stresses and strains with separate volumetric compression and attendant shear driven shape changes that will restrict the constitutive laws for materials. This is achieved with both the stress and strain tensors using octahedral components as is described in detail in Sec. II. The normal compressive displacements and the shear strains in a uniaxial strain system will be thermodynamically described with pressures and shear stresses, respectively. Thermodynamics is fully developed and applied to the material's constitutive responses. Jacobian algebra is used to describe isothermal, isochoric, entropic, and iso-shear constraints on physical properties.

## II. STRESS TENSORS FOR PLANAR, LONGITUDINAL WAVES

The direction of propagation and the two axes in the perpendicular plane are principal stress directions for planar, longitudinal, and compression stresses in isotropic solid materials. Figure 1(a) shows a compressive pulse propagating in the 1 direction with a planar wave front in the 2–3 plane. The material displacement for stress compression is in the direction of propagation.^{52–54} If the material is isotropic, then axes 2 and 3 are equivalent with equal stresses; if the material was textured or a single crystal,^{55} then axes 2 and 3 are not necessarily equivalent but can always be rotated to achieve principal stresses and coordinates.

Equation (1) is the stress tensor, $\sigma ~$, with the 1 axis aligned to the propagation direction and the perpendicular stresses along the 2 and 3 axes as shown in Fig. 1(b),

for isotropic material: $\sigma 22=\sigma 33$ and that assumption will be continued throughout the remainder of this paper.

Material constraints generate normal stresses in directions 2 and 3 for a propagating planar wave because the normal strains are zero; the material is doubly constrained. The perpendicular strains are zero because the wave is planar. The stresses are due to constraints from Poisson's effect. In Eq. (1) and shown in Figs. 1(c) and 1(d), the largest shear stresses are the deviatoric stresses or ½ the differences between the principal stresses; the shear on the 1 face is zero for all materials since it is a principal stress direction, while shear stresses seen in Figs. 1(c) and 1(d) are maximum on planes at 45° to the direction of wave propagation. The stresses in Fig. 1(d) are

as is well known. Figures 1(a)–1(d) are in the same stress state with (c) and (d) displaying maximum shear stresses. The state of stress in Figs. 1(c) and 1(d) is given by the second stress tensor in Eq. (3),

In what follows below, a stress tensor rotation separates the pressure and the volume changes from the shear shape changes as seen on the far-right side of Eq. (3). All the diagonal normal stress terms are equal.

The stress tensor on the right of (3) can be obtained by a complex rotation. At first, it may seem to complicate the stress state, but it is intended to separate deformation into pressure-induced phenomena and plastic deformation instigated by shear stresses. Pressure is envisioned as producing solid-state phase changes with atomic rearrangements to different crystal structures or even phase changes from a solid to a liquid. These changes are first order and have latent heat with the transformation. The shear stress driven changes are quite different and are thought of as coming from slip dislocation motion, long-range structures such as shear driven deformation twins, stacking fault motions, or shear band propagation. Dislocation motion and shear phenomena are highly irreversible which generate local heating in the lattice. Although, shear stresses in a material may also be elastic and reversible, provided a shear yield stress, is not exceeded. Finally, both pressure and shear under isentropic constraints will induce temperature changes in the material and also temperature from heat through irreversible processes.

Figure 2(a) shows the rotation of the stress tensor; it has been rotated about an axis in the plane perpendicular to the direction of propagation through an Eulerian rotation angle $\psi $. This rotation involves all three coordinates. The stress tensor rotation is intended to establish all three normal stresses having identical values: the 1 axis is rotated through $\psi $ so the $1\u2032$ axis is equally distanced to both the 2 and 3 axes as seen in Figs. 2(a) and 2(b). $\psi $ is chosen so that the normal stress on the $1\u2032$ face (the rotated face) is equal to 1/3 of the trace of the stress tensor in Eq. (3). Symmetry dictates that the normal stresses on the $2\u2032$ and $3\u2032$ faces are equal for isotropic materials so the three normal stresses are the same; it then follows that in choosing the angle $\psi $ for the $1\u2032$ face, as given above, implies that all normal stresses are equal [see Fig. 2(c)]. Thus, the normal stresses can be considered as a “Pressure”, *p* since the three normal stresses are equal; the stress state, however, has shear stress, so it is not a hydrostatic pressure state which does not include shear stresses. There are always shear stresses in solid materials with the stress state in Eq. (1).

The stress state shown on the far-right side of expression (3) is of particular interest here as it has a representation of equal normal stresses as noted above. This stress state is found using Eqs. (4)–(6) below. The tensorial notation and the direction cosine matrix for the rotated coordinates are

The direction cosine matrix for the coordinate system shown in Fig. 2(b) is

Equation (4) may be solved for $sin2\psi $ by substituting $cos2\psi =1\u2212sin2\psi $ on the right and solving for $\psi $ gives $\psi =54.74\xb0$. The shear stress tensor elements in the rotated coordinate system can now be found as

The stress state in Eq. (1) and the state of stress for material in longitudinal wave compression come from a uniaxial compressive strain with the $\sigma 22$ stress from Poisson's effect. The stress tensor seen in Fig. 2(c) and Eq. (3) on the far right is especially relevant as it determines an energy balance, which is of major importance in rapidly propagating longitudinal waves in solids. The stress pulses are generated by laser ablation of materials and from shocks in longitudinal compression.

## III. ENERGY BALANCES IN PLANAR LONGITUDINAL WAVES

The incremental energy balance per unit mass from the 1st law of thermodynamics for the stress state shown on the far-right side of expression (3) is

The first term on the left, *du*, is the incremental internal energy of the system per unit mass. *T* is the temperature; *S* is the entropy per unit mass; *v* is the specific volume, i.e., volume per unit mass; $\sigma 1\u20321\u2032$ is the normal stress on the $1\u2032$ plane; and $\epsilon 1\u20321\u2032$, $\epsilon 2\u20322\u2032$and $\epsilon 3\u20323\u2032$are the normal strains on the $1\u2032,2\u2032,and3\u2032$ faces, respectively. $\tau 1\u20322\u2032$is the shear stress on the $1\u2032$ face in the $2\u2032$ direction as shown in Fig. 2(c). The symmetry of $2\u2032and3\u2032$ faces dictates that two of the shears are the same. $\gamma 1\u20322\u2032$ is the corresponding engineering shear strain. $\tau 2\u20323\u2032$, is the shear stress on the $2\u2032and3\u2032$ planes, as seen in the figure, and is zero for isotropic materials and not included in Eq. (7). $TdS$ is the incremental heat added to the system, $v\sigma i\u2032i\u2032d\epsilon i\u2032i\u2032$ is the incremental mechanical work per unit mass done on the $i\u2032$ surface due to the normal stress, $\sigma i\u2032i\u2032$, and $v\tau i\u2032j\u2032d\gamma i\u2032j\u2032$ is the incremental mechanical work per unit mass due to shear stress, $\tau i\u2032j\u2032$. So, Eq. (7) becomes

The trace of the incremental normal strains in Eq. (8) is well known,

So, with this substitution and $\sigma 1\u20321\u2032=\u2212p$, we have from Eq. (8),

The shear strain–volume^{56,57} is defined incrementally using the symbol $\gamma ij_$,

The strain–volume definition is incremental and places no restriction on *v* as it is transformed from the internal energy, *u*, to the Gibbs’ like free energy, *g*. *g* is the free energy at constant stress,

The effective shear strain–volume and the effective shear stress are used in what follows:

From Eq. (13), the incremental complimentary shear energy term on the right is now replaced with effective shears so the Gibbs’ like function is

### A. Thermodynamic property definitions

In what follows below, the physical thermodynamic properties are defined following very closely the new pioneering description^{58} of plane stress thermodynamics. Plane shear stress analysis was applied to the deformation and heat capacities of clays but these relations apply generally to many other materials. The following thermodynamics applies to longitudinal waves in uniaxial strain. From Eq. (15), we define heat capacity at a constant pressure and shear stress as

The volumetric, thermal expansion coefficient,

The isobaric, shear thermal expansion coefficient,

The isothermal, constant shear stress compressibility,

The isobaric, isothermal shear compliance,

The isothermal, constant shear stress, shear strain pressure interaction,

These six physical properties describe the thermodynamic system for the longitudinal compressive stress pulse shown in Figs. 1(a) and 2(c). They are organized into a Jacobian format in Table I for seeing interactions and for using in calculating other physical properties with different constraints. The constant entropy properties have already been described in the literature for a plane shear stress system with pressure and shear variables.^{58}

f . | $\u2202f\u2202T|p,\tau eff$ . | $\u2202f\u2202p|T,\tau eff$ . | $\u2202f\u2202\tau eff|p,T$ . |
---|---|---|---|

Gibbs Function, g | −S | v | −2γ_{eff} |

Temperature, T | 1 | 0 | 0 |

Pressure, p | 0 | 1 | 0 |

Shear stress, τ_{eff} | 0 | 0 | 1 |

System entropy, S | $Cp,\tau eff/T$ | −vβ | 2vα |

Specific volume, v | vβ | $\u2212v\kappa T,\tau eff$ | −2vχ |

Effective engineering shear strain, γ_{eff} | vα | −vχ | vλ_{T,p} |

Internal energy, u | $Cp,\tau eff\u2212pv\beta +2\tau v\alpha $ | $\u2212Tv\beta +pv\kappa T,\tau eff+2\tau effv\lambda T,p$ | 2Tvα + 2pvχ + 2τ_{eff}vλ_{T,p} |

f . | $\u2202f\u2202T|p,\tau eff$ . | $\u2202f\u2202p|T,\tau eff$ . | $\u2202f\u2202\tau eff|p,T$ . |
---|---|---|---|

Gibbs Function, g | −S | v | −2γ_{eff} |

Temperature, T | 1 | 0 | 0 |

Pressure, p | 0 | 1 | 0 |

Shear stress, τ_{eff} | 0 | 0 | 1 |

System entropy, S | $Cp,\tau eff/T$ | −vβ | 2vα |

Specific volume, v | vβ | $\u2212v\kappa T,\tau eff$ | −2vχ |

Effective engineering shear strain, γ_{eff} | vα | −vχ | vλ_{T,p} |

Internal energy, u | $Cp,\tau eff\u2212pv\beta +2\tau v\alpha $ | $\u2212Tv\beta +pv\kappa T,\tau eff+2\tau effv\lambda T,p$ | 2Tvα + 2pvχ + 2τ_{eff}vλ_{T,p} |

Table II displays the third-order derivatives of the Gibbs function. These derivatives are important in establishing connections among the six physical properties in Table I. All the physical properties are not independent but are related through third-order derivatives. The table is unique because of Eq. (11). The specific volume has not been assumed constant, as would ordinarily be the case if energy per unit volume were used rather than energy per unit mass, as is done here. Analyses based on strain enthalpy or Helmholtz energies in third-order are also generally not restricted by a specific volume. In Table II’s case, third-order derivatives per unit mass properties are used. If one were to use third-order Gibbs functions using energy per unit volume, then the third-order expressions are generally restricted to a specific volume being held constant and differ from Table II. The equations just below Table II contain *v*; if *v* was equal to a reference state or (say) 1, then expressions for energy per unit volume in the third-order derivatives would be identical between volume and mass. If constant volume restrictions are applied to materials at extreme pressures, it might be unwise since atoms can enter or leave the system without penalty.

Physical property . | $\u2202f\u2202T|p,\tau eff$ . | $\u2202f\u2202p|T,\tau eff$ . | $\u2202f\u2202\tau eff|p,T$ . |
---|---|---|---|

$Cp,\tau eff/T$ | c_{000} | c_{100} | 2c_{200} |

vβ | −c_{100} | −c_{110} | −c_{210} |

vα | c_{200} | c_{210} | c_{220} |

$\u2212v\kappa T,\tau eff$ | c_{110} | c_{111} | c_{211} |

vχ | c_{210} | c_{211} | c_{221} |

vλ_{T,p} | c_{220} | c_{221} | c_{222} |

Physical property . | $\u2202f\u2202T|p,\tau eff$ . | $\u2202f\u2202p|T,\tau eff$ . | $\u2202f\u2202\tau eff|p,T$ . |
---|---|---|---|

$Cp,\tau eff/T$ | c_{000} | c_{100} | 2c_{200} |

vβ | −c_{100} | −c_{110} | −c_{210} |

vα | c_{200} | c_{210} | c_{220} |

$\u2212v\kappa T,\tau eff$ | c_{110} | c_{111} | c_{211} |

vχ | c_{210} | c_{211} | c_{221} |

vλ_{T,p} | c_{220} | c_{221} | c_{222} |

### B. An application of thermodynamics used to find the heat generated from mechanical shear work. Establishing the ratio of incremental heat to the shear work on an isobaric, isotherm

The section directly below is an example of the use of thermodynamic variables defined above. The example is finding the ratio of the heat to the shear work isothermally. It also establishes that the shear thermal expansion coefficient is very different from a normal strain thermal expansion coefficient. Assume the system is elastic. The ratio will be evaluated from experimental data,

$J(S,T,p)$ is the Jacobian read from Table I into the determinant as shown in Eq. (22). For linear materials, the shear thermal expansion coefficient $\alpha $ is proportional to $\tau eff$ and the temperature dependence of $\lambda T,p$ is as seen in Fig. 3(a) and displayed in Eq. (23).

For linear material,

The thermal derivative of the natural log of the shear compliance is compared to the change of the natural log of *T*. This ratio of energy terms is seen in the expressions in Eqs. (22)–(24). An estimate of the value for copper can be made from experimental shear modulus data^{59,60} as shown in Fig. 4(a). The data, in a form that is useful for Eq. (24), are given in Fig. 4(b). The ratio of heat to work has a value of 1.2 × 10^{−3} at *T *= 30 K and a value of 2.2 at *T *= 1 100 K close to the melting point. The ratio should be zero at *T *= 0 from the third law of thermodynamics and is seen to increase sharply with *T*. If the material were an ideal gas, then the absolute ratio would be 1 and temperature independent. So, shear heat in the solid, isothermally, is quite different from gases and fluids as the copper data goes from 0 as $T\u21920$ to a number that is larger than 1 near melting temperatures. Equations (22)–(24) are again revisited for plastic wave propagation in Sec. IV A 1 and the supplemental material without restriction to elastic behavior.

## IV. STRAIN TENSORS FOR PLANAR, LONGITUDINAL WAVES: UNIAXIAL STRAIN

The strain of the material is only in the direction of the particle motion in Fig. 1(a). The strain state is uniaxial. The uniaxial strain tensor is rotated below using the same rotation seen in Eq. (3). The 1, 2, and 3 axes are still the principal coordinates for longitudinal wave propagation. An octahedron with all eight planes—each making an angle of 54.74° to the 1, 2, and 3 coordinates—defines the octahedral^{61,62} stress/strain system. The octahedral rotation at first seems to complicate the strains, but this rotation allows the materials’ properties between volumetric deformation and shear deformation to be fully established. Consider the strain tensors,

The strain $\epsilon 11$ is obtained from the deformation of the sample as it is compressed in the 1 direction. The strain octahedron is, therefore, compressed in the 1 direction and constrained in the 2–3 plane. Equation (25) in the center and on the right establishes the system's volume changes and shape changes. The left tensor is heterogeneous while the right is homogeneous in normal strains. Shape and volumetric strain changes are related by

The relationship in (26) places a significant restriction on deformation processes in compressed solid materials. Although Eq. (26) was found for a solid, it also is a restriction on a liquid: the shape and volumetric fluid changes are related to the uniaxial strain of longitudinally compressed materials. Equation (26) shows that the ratio of the volumetric strains to the effective engineering shear strain is exactly 1:2. Tensor shear strains are ½ of the engineering shear strains at 1:1.

The specimen's thickness vs time is shown in Fig. 5. The compressive strain, $\epsilon 11$, is found with reference to an instantaneous gage length: as a true strain,

where *h* is the final thickness of the specimen and $h0$ is the initial thickness in Eq. (27). The $\u2113n$ argument is from the evaluation of the integral's limits. Expression (27) is the true strain in the 1 direction. The experimental data accompanying compression gives compressive volume changes vs pressure, while the accommodating shear strains are from (25) or (26). This system is not reversible as shear which accompanies pressure changes is typically irreversible. For example, in Ref. 4, with both the front and back surfaces of the deuterium specimen measured during compression with x-ray radiography allows for both *h*_{0} and *h* to be measured, $\epsilon 11$ is estimated from their data as *h*_{0 }= 300 *μ*m ; *h *= 140 *μ*m; and $\epsilon 11=\u22120.76$. These measurements provide the starting and final compressions of the thickness ratio in Eq. (27). Figure 5 is a schematic of the time dependence of the specimen thickness vs time,

The shear deformation is used below to determine the heating in a compressed specimen.

### A. Irreversible shear deformation

The thermodynamic system described above is generally considered elastic while plastic deformation in the shear system is typically irreversible. The material's response to plastic shear may be treated as a fictitious material that is “elastic” but non-reversible. The response is non-reversible so loading must always increase the shear stress or remain constant, while stresses must never unload. The shear compliance in Table I may be used with this restriction to determine the extent of plastic deformation within the sheared material. Below, the plastic energy is directly converted into heat without the aid of Table I.

The shear deformation contributes to temperature changes. The temperature of compressed samples^{45} is frequently measured using a Streak Optical Pyrometer, SOP. Figure 6 is a schematic of the shear stress vs the shear strain. The shear system below is treated by considering an elastic-plastic, shear strain-hardening material;^{46,62} shear hardening occurs in material deformed in uniaxial stress but maybe less so in uniaxial strain. Strain hardening is from dislocation interactions while strain-rate sensitivity is from dislocation mobility, both contribute to $\tau eff$ (see Refs. 46 and 62–64). The strain-rate is elevated from 10^{−4}/s rates (typical in engineering compression tests) to rates as high as 10^{+6}/s—as measured in laser compression experiments; positive changes in strain-rates typically elevate $\tau eff$.

Balancing enthalpy for the heat energy to the irreversible plasticity gives

where $\Delta T$ is the increase in temperature due to plastic work. The temperature *T** is an increase over the initial temperature, *T _{i}*. Plastic deformation is converted into heat and results in a temperature,

*T**, given in Eq. (30),

The total plastic strain in Fig. 6 and Eqs. (29) and (30) is the material's shear deformation from Eqs. (14) and (26)–(28).

An estimate of Eq. (29) obtained from Fig. 6 (ignoring the small elastic shear) is found by summing the two triangles. $\lambda T$ is defined in Fig. 6 and $\tau eff$ is in Eq. (14),

The plastic strain-rate sensitivity is included in $\tau eff$ and $\tau y$ since both of these quantities are rate-sensitive, responding to dislocation behavior. Furthermore, in expression (31), the plastic strain $\gamma eff$ is the true plastic strain; in all cases, it is equal to $\u2113n(hh0)$ or $\u2113n(\rho 0\rho )$ from (28). Knowledge of $\sigma 22$ is needed for determining $\tau eff$ in Eqs. (6) and (14); unfortunately, $\sigma 22$ remains unmeasured.

#### 1. Application to ramp-compressed aluminum

The temperature of a specimen that has been ramp-compressed can be estimated from Eqs. (30) and (31). The estimate is found by using data previously reported for aluminum and the standard room temperature and atmospheric pressure properties of aluminum. See Ref. 15 for the data of aluminum at pressures of 475 GPa. $\sigma 22$ in Eqs. (6) and (14) is found using Poisson's effect. The stress $\sigma 11$ is directly measured using Rankine–Hugoniot equations evaluated with a velocity interferometric system for any reflector, VISAR measurement. We have measured $\sigma 11$ at 475 GPa. The strain in the 2 and 3 directions are zero, with Poisson's ratio, $\upsilon $, taken as 0.34 for aluminum (at atmospheric pressure and temperature) so $\sigma 22=\upsilon /(1\u2212\upsilon )\sigma 11=245GPa$. The effective shear stress with $\sigma 11and\sigma 22$ is found from Eqs. (6) and (14) to be about 74.5 GPa. The value for heat capacity is taken at atmospheric pressure and temperature. The specific volume is from the density estimated at 6.9 g/cm^{3} at the highest pressure. Equation (30) may now be evaluated from Fig. 6 with $\tau y=0.T\u2217$ at this pressure is estimated to be about 11 300 K using the equations and values cited. The large temperature increase is from the effective shear strain taken from the volume change and the very large shear stress. As the material heats and yields, the temperature will increase and reduce the constraining stress through $\upsilon $. This effect will help to equilibrate the stresses and reduce the temperature from plasticity's contribution. Although the shear stress is significantly smaller than the pressure term, it is still a very large value. This temperature is between the Hugoniot and isentropic temperatures. The model also does not account for thermal emissivity. The estimated temperature is larger than expected.

### B. Pressures in the octahedral system from compressive strains

The isothermal compressibility is defined in Eq. (19). This expression may be integrated using the elastic constitutive law from Refs. 65–67,

where *B* is the bulk modulus; *ρ* is the density; *C*_{0} is a constant determined (say) at room temperature and atmospheric pressure; *m* is an empirical constant determined from the measurements of specific volume and compressibility at selected temperatures; and *m* is the Anderson–Grüneisen parameter. Equation (32) may be integrated on an isotherm. Separate the terms, integrate, and apply *p *= 0 with *v *= *v*_{0} and *B *= *B*_{0} to determine the constant of integration. *B*_{0} is the bulk modulus at zero pressure and *T *= 298 K. We find

Equation (33) applied to copper with *B*_{0 }= 123 GPa, *ρ*_{0 }= 8.936 g/cm^{3}, and *m *= 5.22 is given in Fig. 7. Generally, Eq. (33) is considered as an equation of state; here, shear stress is also a state variable; so (33) is an expression restricted to an isotherm with constant shear stress at zero shear stress.

### C. Linear elastic wave speeds

Material with small stresses will behave elastically and follow Hooke's law relating the stresses to the strains. Taking $\epsilon 11$ to be the normal strain in the direction of wave propagation as seen in Fig. 1(b), we have

where *E* is Young's Modulus and $\upsilon $ is Poisson's ratio. In isotropic solids, $\sigma 22=\sigma 33$ and the strains in the 2 and 3 directions are zero for planar compressive stresses, as seen in Fig. 1(b). Hooke's law in the 2 direction yields

The compressional wave speed in the solid follows directly from Eq. (36),

This speed is well known^{52–55} in the literature. It not only includes $\kappa S,\tau eff$ but also Poisson's ratio. $\upsilon $ makes it clear that pressure and shear energies are included in the compressional wave propagation system.

### D. Pressure relation to stress σ_{11} and Poisson's ratio

Linear elastic materials as described in Eqs. (34) and (35) are doubly constrained. The constraints, through Hooke's law, establish a unique relation between *p* and $\sigma 11$ that is only dependent on $\upsilon $, as seen in Eqs. (4), (34), and (35). The expression is

Figure 8 is a plot of $p/\sigma 11$ vs Poisson's ratio. Auxetic solids^{68,69} are materials with $\upsilon $ less than 0 and are on the left side of Fig. 8. Materials with $\upsilon =1/2$ are solids, but these materials are “fluid-like” since $p/\sigma 11=\u22121$ and normal stresses are the same in all directions.

The shear stress, pressure and normal stress, $\sigma 11$, have a simple relation from the constraint, $\sigma 22$,

with $\upsilon =1/2$, hydrostatic pressure is found as noted above, plus from (38b), we find $\tau eff=0$; with $\upsilon =\u22121$, a pure shear stress is found with $p=0$ and $\tau eff=\sigma 11/2$; and with no constraint, $\upsilon =0$ and $\tau eff=\sigma 11/3$, the octahedral shear stress.

### E. Relating density to uniaxial strain

The experimental data from the sample's compression is related to the change in density. The compression strain from the change in thickness vs time and with $\epsilon 22$ and $\epsilon 33$ zero in a planar wave also describes the change in density. If *h* is the final thickness of the specimen and $h0$ is the initial thickness in Fig. 5, then using Eq. (26) on the isotherm, we find

The density of compression is from the 1 coordinate system while the pressure is from the $1\u2032$ coordinate system; the 1 coordinate supports stress, not pressure. $\sigma 11$ stress is proportional to both pressure and shear, as noted earlier. Measurements using VISAR on the front and back surface positions of transparent specimens or from independent measurements on both the front and back surfaces of opaque specimens as seen in Eq. (28) is the uniaxial strain in the material as it is compressed. $B0andm$ should reflect isentropic conditions in expression (39). For example, in Refs. 24 and 70, the front and back surfaces of the deuterium specimen are measured before it becomes opaque during compression with two VISAR traces from the back and front coated surface in a transparent sample. Or in Ref. 71 that reports the density change in LiF from $\epsilon 11=\u22121.06$, as estimated with Eq. (39) on the left, using the density of compression data for LiF, gives an increase in density as determined from the VISAR measurements of shock velocity.

## V. MATERIAL PROPERTIES ACROSS SHOCKS WITH HUGONIOT RESTRICTIONS

The longitudinal wave generates stress in the direction of propagation. The materials in ambient conditions before and after the shock are in very different thermodynamic states. The waves and shocks are so fast that external entropy does not have adequate time to enter the system and the shock is considered isentropic plus entropy modifications by Hugoniot expressions. However, the temperature changes and separating the system into pressure and shear systems aid in describing the material changes *per se* for thermal changes. The change in entropy in the temperature system is followed by an investigation of the pressure system, and then, the shear system. Of course, both pressure and shear are changing simultaneously, but their contributions to temperature across the shock front are very dissimilar. Consider the system's entropy,

where $\Delta S$ for weak shocks is near zero because the process is isentropic across the shock; for strong shocks, $\Delta S$ has been estimated.^{72} The first term on the right in Eq. (41) is well known; the Debye or Einstein heat capacity or even a Dulong–Petit heat capacity on an isobar describes a positive function of temperature. The second term in Eq. (41) decreases the entropy as pressure increases for all materials that have positive thermal expansion coefficients. The third term may be integrated with the aid of Eq. (23) for linear elastic materials. It yields

Equation (41), after rearranging terms, thus yields

As the pressure is increased and a strong shock is initiated, the left side of Eq. (43) increases; the right side's increase is shared between the entropy due to temperature in the first term and shear stress plus *T* in the second terms on the right. The overall entropy is changed in the system since the shock process generates entropy. The entropy contribution for each independent variable is reflected in (43). If shear were ignored, then the pressure would increase only in terms on the left and *T* as seen in the first term on the right in expression (43). As already explained, both *p* and $\tau eff$ are linear in the stress $\sigma 11$; the shear term in Eq. (43) is squared in stress, while the left side's second term is proportional to $\sigma 11$.

The Hugoniot energy conservation across the shock involves Eulerian and Lagrangian velocities when applied to either a solid or a fluid. Conservation of mass and conservation of momentum laws plus the internal energy of the solid or the fluid are also needed. The thermodynamic internal energy for solids is listed on the bottom line in Table I for pressure and shear variables; for fluids, there are no shear terms. However, the entropy as discussed above changes *T* of the material while the system's entropy is increased by the shock and/or is nearly constant for very weak shocks.

## VI. CONCLUSIONS

A full equilibrium thermodynamic description of a compressed solid has been developed and exploited for *P* waves. The state of stress includes both the pressure and shear components. This stress state, thus, has two mechanical state variables. Many descriptions of solids rely only on pressure as the main mechanical state variable; most engineering systems emphasize shear stress as the key mechanical variable in solids and ignore pressure. Both pressure and shear are described in Table I and Ref. 58. Pressure and shear are related to the stresses, $\sigma 11and\sigma 22$. The constitutive laws are unchanged by coordinate rotations but easier to understand by our stress rotation: the pressure system no longer includes shear phenomena and the shear system excludes pressure effects.

Material deformation beyond the yield stress is mostly due to shear deformation in unconstrained samples. For example, in an engineering tensile test, which is unconstrained and in tension, the sample significantly elongates due to plastic deformation without any significant changes in stress or volume, i.e., the specific volume is nearly constant during plastic deformation. The same processes are present in engineering compression tests. Plasticity often changes the shape of the material without elevating the stress level or specific volume. In unconstrained samples, specimens always significantly change shape during plastic deformation. The “effective” Poisson's ratio approaches ½ in the compressed, constrained, plastically deformed samples with significant shear deformation. It implies that $\sigma 22\u21d2\sigma 11$ and the solid is in a more “fluid-like” condition.

The stress tensor rotation proposed is unique but not unknown;^{61,62} the ramp compression and shock wave communities do not generally consider pressure and shear at the same time. Pressure and shear stress are fully developed here with thermodynamic descriptions. Our tensor rotation assures that all the normal stresses are the same on each of the rotated stress cube's faces as seen in Fig. 2(c) and are, therefore, describable as pressure. This stress state also includes shear stresses.

Elastic thermodynamic experimental descriptions of solids are sparse in the literature. For example, the measurements of temperature changes due to isentropic elastic stresses are predicted in the supplemental material and extensively in the literature. Has it been systematically measured? Temperature measurements from plastic shear are large and very important; especially, at high stresses as described in Refs. 24, 70 and 71. Additional equations are proposed in the supplemental material and should help for comparison to SOP temperature measurements. The total shear strains are twice the normal strains in this uniaxial strain system. Pressure, plus shear stress, and *υ* plus shear strain are all related through a biaxial stress state and uniaxial strain state.

The literature often considers (incorrectly) that elastic adiabatic shear compliances are equal to isothermal shear compliances, but that is not in agreement with equation (S-6) developed here, except at zero shear stress. If a solid were assumed to have zero shear interactions, then shear stresses could never result in any temperature changes. We have not found, in the literature for solids, the experimental measurements of temperature changes with fully elastic thermodynamic conditions for compressive waves in isotropic solids. The temperature changes in pressure waves are well known analytically; *T* changes have been less frequently measured to our knowledge. The temperature changes, especially in fully elastic shear waves, are small but measurable [see Eq. (S-4)]. Laplace established adiabatic conditions on Newton's isothermal sound wave speeds in gases to bring theory and experiment into an agreement.

Most researchers would refer to Eq. (33) as an equation of state. The stress state proposed and described here contains shear; so the concept of a universal equation of state has two mechanical state variables. Shear stress plays an important role in the stress tensors seen in Eq. (3)—it is just a part of the physical description of a uniaxial strain compressed solid.

Third-order violations of the elastic Gibbs’ function are removed with the use of strain volumes in Eq. (11). Thus, the thermodynamic system used here keeps track of atoms when they enter or leave the system. At very high pressures, as measured in recent shock waves in solids where atoms and electrons are entering and leaving the system, the use of strain volumes energetically accounts for these atoms and electrons. Constant volume systems do not keep track of the atoms; constant mass systems, not using strain volumes, may have inconsistencies between the Gibbs’ function and the first law of thermodynamics.

## SUPPLEMENTARY MATERIAL

See the supplementary material for three additional properties: (1) changes in temperature in an elastic longitudinal wave, restricted to adiabatic conditions; (2) adiabatic compressibility for a constant shear stress, elastic pressure wave; and (3) changes in the compressive wave speed with the latent heat of phase changes.^{73,74 }

## ACKNOWLEDGMENTS

S.J.B. would like to thank S. M. Gracewski for helpful discussions. He would also like to thank R. Perucchio for support and encouragement and the LLE staff for interactions. For the other authors, this material is partially based on a work supported by the Department of Energy National Nuclear Security Administration under Award No. DENA0003856, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflict of interest to disclose.

### Authors Contributions

All authors contributed equally to this work.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

_{3}perovskite at high pressures: Equation of state, structure, and melting transition

*Properties of Copper and Copper Alloys at Cryogenic Temperatures*, Monograph 177 (National Institute Standard Technology, 1992), pp. 6–11, 7–31, and 7–32.