discontinuous across the front. Kinetic equations (182) and (187) therefore
reduce as follows, with $\bar{\pi}^{\alpha}_{\rm D}$, $\pi^{\alpha}_{{\rm
D}k}$ in (III.3), (III.3):

$\displaystyle\bar{\pi}^{1}_{\rm D}=0,\qquad\pi^{1}_{{\rm D}1}=0.$ (213)

For each decrement of $J$, (213) are solved simultaneously for $\bar{D}$ and
$D_{1}$, affecting $P$ and $U$ in Hugoniot equation (208). Damage can be
nonzero, so the generalized Finsler metrics of Sec. III.5 are non-trivial.
Here, $\bar{g}_{ij}=\delta_{ij},\bar{G}^{\alpha}_{IJ}=\delta_{IJ}$, with
$\hat{g}^{i}_{j}$ and $(\hat{G}^{\alpha})^{I}_{J}$ in (200)–(203). Isotropic
matrix damage gives (202), anisotropic fiber damage (203). Determinants and
their derivatives, (204) and (III.5), enter (III.2) and (213). For the present
loading and material symmetries, with (200), $\hat{g}^{i}_{j}$ and
$(\hat{G}^{\alpha})^{I}_{J}$ do not affect $J$ or ${\rm tr}\,\tilde{\bf C}$.
Finsler or osculating Riemannian metrics enter the analysis only through
(III.2) and (213).

Properties for the ECF and tissue phase used in calculations are in Table 1.
Experimental data on hydrated muscle (e.g., [112]) furnish properties of the
mixture as a whole, not the isolated $\alpha=1$ phase. Given (209), the
mixture density, isentropic bulk modulus, bulk sound speed, volumetric thermal
expansion coefficient, specific heat, and Grüneisen parameter are,
respectively,

$\displaystyle\rho_{0}=\sum_{\alpha}n_{0}^{\alpha}\rho_{{\rm
R}0}^{\alpha},\quad B_{\eta}=\sum_{\alpha}n_{0}^{\alpha}B^{\alpha}_{\eta},$
(214) $\displaystyle c_{\rm B}=\sqrt{(B_{\eta}+p_{0})/\rho_{0}},\quad
A=\sum_{\alpha}n_{0}^{\alpha}A^{\alpha},$ (215) $\displaystyle\rho_{0}c_{\rm
p}=\sum_{\alpha}n_{0}^{\alpha}\rho_{{\rm R}0}^{\alpha}c_{\rm
p}^{\alpha}=\rho_{0}c_{\epsilon}(1+A\gamma_{0}\theta_{0}),$ (216)
$\displaystyle\gamma_{0}={AB_{\eta}}/({\rho_{0}c_{\rm
p}})={AB_{\theta}}/({\rho_{0}c_{\epsilon}}),$ (217)

where $c_{\rm p}^{\alpha}$ is specific heat at constant pressure of phase
$\alpha$. Given properties of the mixture [108, 112] and ECF ($\alpha=2$),
(214)–(217) are inverted and solved for thermoelastic properties of the tissue
phase ($\alpha=1$). Ultimately, experimental values [112] of $\rho_{0}$ and
$c_{\rm B}$ yield tissue density and bulk modulus. Nonlinear bulk stiffening
parameters $B^{1}_{\eta{\rm p}}$ and $k^{1}_{\rm V}$ are fit to the shock
Hugoniot data [112].

(

Figure 2:  Model results and/or shock compression data for (a) Hugoniot stress
vs. mass density, (b) shock velocity vs. particle velocity, and (c)
temperature. Data in (a,b) are for water [106] human blood [111], and porcine
skeletal muscle [112].

a)
b)
(c)

(

All static and dynamic shear properties are not fully established from Ref.
[112], so order-of-magnitude estimates are used based on literature values
[102, 114, 9, 115, 23, 19] for skeletal, and in some cases cardiac, muscle. A
standard scalar measure [88] of shear stress $\tau$ of the mixture for
uniaxial shock compression is the first of

$\displaystyle\tau={\textstyle{\frac{3}{4}}}(P-p);\quad\tau=-{\textstyle{\frac{3}{4}}}\sum_{\alpha}[(\sigma^{\alpha})^{1}_{1}+p^{\alpha}].$
(218)

If a material is isotropic, $\tau$ is half the von Mises stress under uniaxial
strain. The second expression in (218) specializes the first. Both phases
$\alpha=1,2$ contribute to $p$ via each EOS; only the tissue phase contributes
to $\tau$ via deviatoric matrix and fiber, elastic and viscoelastic, stresses.
Low-rate data [102, 114, 9, 115, 23, 19] suggest $\mu^{1}_{\rm S}$ and
$\mu^{1}_{1}$ should be in the kPa range, with fiber exponential stiffening
$k^{1}_{1}$ on the order of 10. Define the sums of glassy viscoelastic
stiffening factors $\beta^{1}_{\rm S}=\sum_{m}\beta^{1}_{{\rm S}m}$ and
${\beta}^{1}_{{\Phi}1}=\sum_{n}{\beta}^{1}_{{\Phi}1,n}$. Low- to moderate-rate
data on cardiac tissue [19] suggest values up to the order of $10^{3}$.
Dynamic compression data on porcine muscle [114] show von Mises stresses in
the MPa range for strain rates on the order of $10^{3}$/s. Extrapolating,
$\tau$ is anticipated up to the order of 10 MPa for shock loading, where
strain rates appear on the order of $10^{5}$/s given rise times under 1 $\mu$s
[112]. Accordingly, $\beta^{1}_{\rm S}$ and ${\beta}^{1}_{{\Phi}1}$ are
estimated for shock compression as $10^{5}$, probing very high-frequency
modes.

For fracture cohesive energy in (III.2), the usual phase-field description is
invoked for matrix and fibers: $\bar{E}^{1}_{\rm
C}=\bar{\Upsilon}^{1}/\bar{l}^{1}=\bar{\mathsf{J}}^{1}_{\rm C}/(2\bar{l}^{1})$
and ${E}^{1}_{{\rm C}1}={\Upsilon}^{1}_{1}/{l}_{1}={\mathsf{J}}^{1}_{{\rm
C}1}/(2{l}^{1}_{1})$. Toughness ${\mathsf{J}}_{\rm C}$ of muscle is known only
for the whole tissue [116], so here the simplest physical choice
$n_{0}^{1}\bar{\mathsf{J}}^{1}_{\rm C}=n_{0}^{1}{\mathsf{J}}^{1}_{{\rm
C}1}=\frac{1}{2}{\mathsf{J}}_{\rm C}$ is used. Similarly, length constants for
each mechanism are both set equal to a value calibrated later for modeling
tensile damage: $\bar{l}^{1}=l^{1}_{1}=l^{1}$, on the order of 10-15 single-
fiber diameters [112]. Values are about $20\times$ those used elsewhere for
modeling skin [59]. Standard phase-field choices
$\bar{\vartheta}^{1}=\vartheta^{1}_{1}=\vartheta^{1}=2$ [26, 59] are used for
degradation functions (165) and (III.2). Regarding generalized Finsler
metrics, $\bar{r}^{1}=\tilde{r}^{1}_{1}=2$ is adopted from prior work on skin
[59], and remanent microstructure strain factors are set equal:
$\bar{\epsilon}^{1}=n_{0}^{1}\bar{\kappa}^{1}/\bar{r}^{1}=\tilde{\epsilon}^{1}_{1}=n_{0}^{1}\tilde{\kappa}^{1}_{1}/\tilde{r}^{1}_{1}=\epsilon^{1}_{\rm
r}$. Experimental data on vascular tissue [117] furnish $\epsilon_{\rm
r}=0.2$, set positive (dilative) here, as in other soft tissues [18, 59].
Under uniaxial-stress compression [117], axial shortening is overcompensated
by radial and circumferential expansion: the arterial wall tissue is
residually stretched. Arterial data [117] suggest $\epsilon^{1}_{\rm r}$ is
higher (lower) for tissues with more collagen (less elastin), but experimental
data on skeletal muscle components do not exist to justify different choices
of $\epsilon^{1}_{\rm r}$ for matrix and fibers.

Shown in Fig. 2(a) for skeletal muscle is mixture Hugoniot stress $P$ versus
mixture density ratio $\rho/\rho_{0}$. Experimental data on muscle [112],
blood [111], and water [106] are shown for comparison, along with model
predictions for the ECF in isolation. The mixture theory captures most of the
shock data well, exceptions being several anomalous points in the domain
$\rho_{0}/\rho\in[0.82,0.87]$. Similar statements apply for mixture
$\mathcal{U}$ versus $\upsilon$ data [112] and model results in Fig. 2(b).
Muscle is stiffer than blood, ECF, and water. Hugoniot $\theta$ predictions in
Fig. 2(c) show a substantial temperature rise, with higher temperatures in
muscle than ECF in isolation. This could cause burn damage, not modeled here.

(

Figure 3:  Predictions for parallel and transverse fiber orientations and
Finsler metrics: (a) shear stress vs. mass density and (b) damage. Model
results for Finsler and Euclidean metrics and parallel orientation: (c) damage
and (d) shear stress.

a)
b)
c)
d)

(

(

(

Results in Fig. 2 are for shock compression parallel to the fiber direction
${\bm{\iota}}$. Shear stress of the mixture (supplied only by the tissue
phase) $\tau$ is predicted in Fig. 3(a) for tissue shocked parallel and
transverse to the fiber direction. Contributions of matrix and fiber
deviatoric stresses are delineated; these simply sum to give $\tau$. For
parallel compression, matrix and fibers contribute similarly in magnitude. For
transverse compression, fibers support less load, and $\tau$ is lower. In both
arrangements, $\tau$ is at most on the order of $10^{-2}P$, so orientation
does not discernibly influence the Hugoniot stress that is dominated by
$p=P-\frac{4}{3}\tau$. For $\rho_{0}/\rho\lesssim 0.95$, damage in matrix and
fibers causes a reduction in strength, leading to reduced $\tau$ at high
compressions. Order parameters $\bar{D}$ (matrix) and $D_{1}$ (muscle fibers)
at Hugoniot states are shown in Fig. 3(b). For parallel loading, degradation
occurs similarly for matrix and fibers. For transverse loading, less
degradation occurs in the fibers; their strain energy is lower in this
arrangement, giving smaller elastic driving force in $\pi^{1}_{{\rm D}1}$.
Shock-recovered samples [5] show microstructure changes indicative of damage
in fibers (myofibrils) and slippage at cellular interfaces, implying matrix
damage. Other experiments, including microscopy and histology after static
crushing of muscle, show shear-induced damage in fibers, interfaces, and
extracellular matrix [118, 119]. Predictions agree with these trends.

Model results for muscle in Figs. 2 and 3(a,b) use the generalized Finsler
metric with remnant strain $\epsilon^{1}_{\rm r}=0.2$ (Table 1). Predictions
in Fig. 3(c,d) compare aforementioned results for damage and $\tau$ with those
obtained when $\epsilon^{1}_{\rm r}=0$, producing Euclidean metrics. Recall
$\epsilon^{1}_{\rm r}>0$ depicts dilatation of the material manifold
$\mathfrak{M}^{\alpha}=\mathfrak{M}^{1}$ as tearing commences and internal
surfaces enlarge. Under shock compression, shear-induced dilatation can occur
in solid phases as fracture surfaces slide and open [88]. As a result, area of
free surfaces increases, leading to an increase in total fracture surface
energy in the model at fixed damage order parameters. As corroborated by Fig.
3(c), damage is suppressed (i.e., more diffuse tearing and rupture) at large
deformation when a Finsler metric is used relative to a Euclidean metric.
Higher energetic cost of fracture in (III.2) for the former
($\hat{G}^{\alpha}>1$) explains this. Conversely, with higher values of
$\bar{D}$ and $D_{1}$, $\tau$ decays more rapidly with increasing compression
in Fig. 3(d) when a Euclidean metric ($\hat{G}^{\alpha}=1$) is used.

For shock stability, $P>0$, ${\rm d}P/{\rm d}J<0$, and ${\rm d}^{2}P/{\rm
d}J^{2}>0$ [72]. These, and complementary conditions along isentropes plus
$\partial P/\partial\eta>0$ [71], were verified for $J\in[0.7,1]$ for all
cases in Fig. 3. Damage reduces the tangent shear modulus, but this is more
than offset by increasing tangent elastic bulk and shear moduli under
compression.

###  IV.2  Static and dynamic uniaxial stress response

Liver. The theory is implemented to model uniaxial-stress compression of
bovine liver across a wide range of strain rates as studied experimentally
[120], demonstrating efficacy of the model’s viscoelastic and damage kinetics.
Liver parenchyma is comprised of cells (hepatocytes), blood vessels
(sinusoids), lymphatic vessels, bile ducts, and fibrous extracellular matrix
(ECM). The organ is encased in a membrane (peritoneum) and connective tissue
(Glisson’s capsule). In vivo, the liver is internally pressurized, expanded,
and perfused with blood, with a fluid volume fraction on the order of 0.5 [7].
Most experimental characterizations, including those modeled here [120],
consider excised samples of the parenchyma, initially at ambient pressure
(i.e., not perfused), excluding the peritoneum, Glisson’s capsule, and major
vessels and ligaments. In these cases, initial blood volume is substantially
lower, and the response is usually isotropic.

The material is depicted as a mixture of two phases ($N=2$): a solid tissue
phase ($\alpha=1$) and a fluid phase ($\alpha=2$) consisting of blood. The EOS
used in Sec. IV.1 is reinvoked, with properties in Table 1, any differences
between bovine and human blood ignored. In the non-perfused state, the initial
fluid fraction is $n_{0}^{2}=0.12$ [121], the solid fraction $n_{0}^{1}=0.88$.
Effects of intracellular and extracellular fluids other than blood are
encompassed by the EOS of the first phase, with free energy of (III.2) and
(129). In addition, free energy of the solid phase ($\alpha=1$) consists of
matrix deviatoric elastic ($\Psi^{1}_{\rm S}$) and viscoelastic
($\Psi^{1}_{\Gamma}$) terms, fiber elastic ($\Psi^{1}_{{\rm F}1}$) and
viscoelastic ($\Psi^{1}_{{\Phi}1}$) terms, and damage to matrix and fibers
($\Psi^{1}_{\rm D}$). A single fiber family is sufficient ($k=1$), fully
dispersed with $\kappa^{\alpha}_{k}\rightarrow\kappa^{1}_{1}=\frac{1}{3}$ in
(140) for isotropy. Damage order parameters for matrix and fibers,
$\bar{D}^{\alpha}\rightarrow\bar{D}^{1}=\bar{D}$ and
$D^{\alpha}_{k}\rightarrow D^{1}_{1}=D_{1}$, reduce deviatoric stress in
(III.2) and furnish surface energy in (III.2). For loading rates up to the
order of $10^{3}$/s and viscosities $\hat{B}^{\alpha},\hat{\mu}^{\alpha}$ in
Table 1, viscous stress from blood should not exceed tens of Pa. This is
negligible relative to total stresses in the kPa to MPa range [120], and thus
ignored. For compression, cavitation damage in the fluid is irrelevant.

The sample is a cylindrical annulus [120], deformed uniformly in the
longitudinal (i.e., axial) direction to a stretch of
$F^{1}_{1}(t)=\lambda(t)=1-\dot{\epsilon}t\leq 1$ at constant “engineering
strain” rate of $\dot{\epsilon}$. A Cartesian coordinate frame defines the
axial 1-direction and orthogonal (radial) 2- and 3-directions. Longitudinal
velocity history $(\upsilon^{\alpha})^{1}(t)$ and axial deformation gradient
$(F^{\alpha})^{1}_{1}(t)$ are identical in each phase. In the initial state,
partial pressure in each phase is equilibrated to reference ambient pressure:
$p^{\alpha}_{0}=n_{0}^{\alpha}p^{\alpha}_{{\rm R}0}$ with $p^{\alpha}_{{\rm
R}0}=1$ atm. At low rates ($\dot{\epsilon}\lesssim 10^{2}$/s), isothermal
conditions apply: $\theta^{\alpha}=\theta_{0}=310$ K. For high rates
($\dot{\epsilon}\gtrsim 10^{2}$/s), macroscopically adiabatic conditions
apply: ${\bf q}^{\alpha}={\bf 0}$. Interphase mass transfer is excluded:
$\hat{c}^{\alpha}=0$.

Two different boundary conditions are considered for transverse (i.e., radial)
stress and deformation. First is a “drained” condition, whereby each phase
expands or contracts independently to maintain equilibrium with external
atmosphere:
$(\sigma^{\alpha})^{2}_{2}=(\sigma^{\alpha})^{3}_{3}=p^{\alpha}_{0}$.
Transverse velocities $(\upsilon^{1})^{2}=(\upsilon^{1})^{3}$ and
$(\upsilon^{2})^{2}=(\upsilon^{2})^{3}$ are not necessarily equal in each
phase, so transverse diffusion velocities
$(\mu^{\alpha})^{2},(\mu^{\alpha})^{3}$ need not vanish. Hydraulic
conductivity is assumed infinite as a limiting case, so
$\lambda^{\alpha\beta}\rightarrow 0$ and ${\bf h}^{\alpha}\rightarrow{\bf 0}$
in (190). For the drained case, each $\alpha$ likewise maintains its own
temperature $\theta^{\alpha}(t)$, with $\omega^{\alpha\beta}\rightarrow 0$ in
(III.3) as a similarly limiting case, so $\epsilon^{\alpha}\rightarrow 0$.
Temperatures are updated by solving (111) separately for $\alpha=1,2$.

Second is an “undrained” or “tied” condition, whereby each phase expands or
contracts radially with the same transverse velocity history. All diffusion
velocities vanish: ${\bm{\mu}}^{\alpha}={\bf 0}$. Transverse deformation is
obtained by equilibrating the total transverse stress of (57) to atmospheric
pressure:
$\sigma^{2}_{2}=\sum_{\alpha=1,2}(\sigma^{\alpha})^{2}_{2}=p^{\alpha}_{{\rm
R}0}=1$ atm. Consistently, for high-rate loading, each phase has the same
temperature: $\theta^{1}(t)=\theta^{2}(t)=\theta(t)$, updated by integrating
the sum of (111) over $\alpha=1,2$. Undrained conditions are consistent with
limiting very high, yet still finite,
$\lambda^{\alpha\beta}\tilde{\rightarrow}\infty$ and
$\omega^{\alpha\beta}\tilde{\rightarrow}\infty$, so ${\bf
h}^{\alpha}\rightarrow{\bf 0}$ and $\epsilon^{\alpha}\rightarrow 0$.

Axial deformation $\lambda(t)$ is imposed over small time steps $\Delta t$.
Thermomechanical responses of each phase and the mixture as a whole are
obtained by solution and integration of the constitutive (i.e., stress-
deformation-temperature) equations, (111) if high-rate loading, and kinetic
laws for viscoelasticity and damage. For viscoelasticity, two relaxation modes
are sufficient for the deviatoric matrix ($m=1,2$ in (144)) and one for fiber
($n=1$ in (155)) contributions to stress, with volumetric (pressure)
contributions omitted for reasons explained in Sec. IV.1. The algorithm of
Refs. [98, 19] is used to solve (147) and (157). Damage is absent in the fluid
and spatially homogeneous in the solid: $\nabla\bar{D}=\nabla D_{1}={\bf 0}$,
ensuring stress fields are homogeneous within each phase, consistent with
momentum conservation in the absence of acceleration waves. Gradient energies
in (III.2) and conjugate forces $\bar{\bm{\zeta}}^{\alpha}_{\rm
D},{\bm{\zeta}}^{\alpha}_{{\rm D}k}$ in (180), (III.3) vanish. Order
parameters and dissipated energies are obtained by integrating (182), (183),
(187), and (III.3) over the load history with nonzero fracture viscosities
$\bar{\nu}^{1}_{\rm D}$ and $\nu^{1}_{{\rm D}1}$.

Table 2: Physical properties or model parameters ($\alpha=k=1$) for bovine liver ($n_{0}^{1}=0.88$) and rabbit muscle ($n_{0}^{1}=0.9$) Property | Bovine Liver | Rabbit Muscle
---|---|---
$\rho^{\alpha}_{{\rm R}0}$ [g/cm3] | 1.06 | 1.10
$B^{\alpha}_{\eta}$ [GPa] | 2.67 | 3.28
$c^{\alpha}_{\epsilon}$ [J/g$\cdot$K] | 3.51 | 3.25
$\gamma_{0}^{\alpha}$ [-] | 0.114 | 0.313
$B^{\alpha}_{\eta{\rm p}}$ [-] | 8.0 | 8.0
$k^{\alpha}_{\rm V}$ [-] | 6.0 | 6.0
$\mu^{\alpha}_{\rm S}$ [kPa] | 1.0 | 1.0
$\mu^{\alpha}_{k}$ [kPa] | 100 | 600
$k^{\alpha}_{k}$ [-] | $1.0\times 10^{-6}$ | 2.1
$\beta^{\alpha}_{{\rm S1}}$ [-] | 20 | 900
$\beta^{\alpha}_{{\rm S2}}$ [-] | 150 | $\ldots$
$\beta^{\alpha}_{\Phi k,1}$ [-] | 1.0 | 0.1
$\tau^{\alpha}_{{\rm S1}}$ [s] | 0.05 | 0.05
$\tau^{\alpha}_{{\rm S2}}$ [s] | $1.0\times 10^{-3}$ | $\ldots$
$\tau^{\alpha}_{\Phi k,1}$ [s] | $1.0\times 10^{-3}$ | 0.05
$\hat{\nu}^{\alpha}_{\rm D}$ [s] | 0.05 | 0
${\mathsf{J}}_{\rm C}$ [kJ/m2] | 0.08 | 0.84
$l^{\alpha}$ [mm] | 1.00 | 0.88
$\vartheta^{\alpha}$ [-] | 2.0 | 2.0
${\epsilon}^{\alpha}_{\rm r}$ [-] | 0.2 | 0.2
$\bar{r}^{\alpha}=\tilde{r}^{\alpha}_{k}$ [-] | 2.0 | 2.0

Properties for the isolated solid tissue phase ($\alpha=1$) of bovine liver
are given in Table 2. EOS properties, namely $\rho_{{\rm R}0}^{\alpha}$,
$B^{\alpha}_{\eta}$, $\gamma_{0}^{\alpha}$, and $c^{\alpha}_{\epsilon}$, are
calculated from mixture rules in (214)–(217) using known values for the fluid
($\alpha=2$) phase (i.e., blood in Table 1), $n_{0}^{1}=0.88$ [121], and
available properties for the liver as a whole (solid + fluid) [122, 123, 108].
Bulk nonlinear stiffening coefficients $k^{\alpha}_{\rm V}$ and
$B^{\alpha}_{\eta{\rm p}}=B^{\alpha}_{\theta{\rm p}}$ are assumed identical to
those of skeletal muscle in Table 1 since high-pressure data are not available
for their determination. Values are inconsequential for pressures obtained
here under uniaxial-stress compression, wherein $|J^{\alpha}-1|<10^{-4}$.
Shear moduli $\mu^{\alpha}_{\rm S}$ and $\mu^{\alpha}_{k}$, stiffening
$k^{\alpha}_{k}$, viscoelastic strength factors $\beta^{\alpha}_{{\rm S}m}$
and $\beta^{\alpha}_{{\Phi}k,n}$, and relaxation times $\tau^{\alpha}_{{\rm
S}m}$ and $\tau^{\alpha}_{{\Phi}k,n}$ for $\alpha=k=1$, $m=1,2$ and $n=1$ are
fit to data [120] in Fig. 4(a,b) at rates $\dot{\epsilon}=0.01$/s,
$\dot{\epsilon}=10$/s, and $\dot{\epsilon}=2000$/s. Total first Piola-
Kirchhoff or “engineering” stress magnitude for this purpose, noting
$J^{1}\approx 1$ and $(\sigma^{\alpha})^{1}_{1}\leq-
n_{0}^{\alpha}p^{\alpha}_{{\rm R}0}$, is measured relative to ambient pressure
$p^{\alpha}_{{\rm R}0}$:

$P=\frac{J^{1}|(\sigma^{1}_{1}+p^{1}_{{\rm
R}0})|}{(F^{1})^{1}_{1}}=\frac{J^{1}}{\lambda}\bigr{\lvert}\sum_{\alpha}[(\sigma^{\alpha})^{1}_{1}+n_{0}^{\alpha}p^{\alpha}_{{\rm
R}0}]\bigr{\rvert}.$ (219)

Fracture toughness of the mixture, ${\mathsf{J}}_{\rm C}$, is obtained from
Ref. [124], presumed similar for porcine and bovine liver. Procedures of Sec.
IV.1 give $n_{0}^{1}\bar{\mathsf{J}}^{1}_{\rm
C}=n_{0}^{1}{\mathsf{J}}^{1}_{{\rm C}1}=\frac{1}{2}{\mathsf{J}}_{\rm C}$.
Length constants for matrix and fibers are set equal to the value in Table 2
to best represent data in Fig. 4(a,b): $\bar{l}^{1}=l^{1}_{1}=l^{1}=1$ mm.
Recalling $\bar{E}^{1}_{\rm
C}=\bar{\Upsilon}^{1}/\bar{l}^{1}=\bar{\mathsf{J}}^{1}_{\rm C}/(2\bar{l}^{1})$
and ${E}^{1}_{{\rm C}1}={\Upsilon}^{1}_{1}/{l}_{1}={\mathsf{J}}^{1}_{{\rm
C}1}/(2{l}^{1}_{1})$, for homogeneous damage, (III.2) and evolution of order
parameters depend only on the ratio of toughness to length (i.e., cohesive
energies $\bar{E}^{1}_{\rm C}$, ${E}^{1}_{{\rm C}1}$) and not toughness and
length independently. If gradient regularization lengths $\bar{l}^{1}_{\rm
R},l^{1}_{{\rm R}1}$ must be chosen in (III.2) based on mesh size constraints
rather than physical observations (e.g., the smaller value of 0.1 mm used for
arterial rupture in Ref. [26]), $\bar{\alpha}^{1}=\bar{l}^{1}_{\rm
R}/\bar{l}^{1}$ and $\alpha^{1}_{1}=l^{1}_{{\rm R}1}/l^{1}_{1}$ can be invoked
independently without affecting the cohesive energy. Standard values
$\bar{\vartheta}^{1}=\vartheta^{1}_{1}=\vartheta^{1}=2$ [26, 59] enter (165)
and (III.2). The same rate dependence of damage, $\hat{\nu}^{\alpha}_{\rm D}$,
normalized by cohesive energy $E^{\alpha}_{\rm C}=\bar{E}^{\alpha}_{\rm
C}=E^{\alpha}_{{\rm C}k}$ and with units of time, is used for matrix and
fibers ($\alpha=k=1$): $\bar{\nu}_{\rm D}^{\alpha}=\nu^{\alpha}_{{\rm
D}k}=E^{\alpha}_{\rm C}\hat{\nu}^{\alpha}_{\rm D}$. The value in Table 2
produces credible results in the context of Fig. 4(c). The same parameters for
generalized Finsler metrics ${\bf G}^{\alpha}$ are used for liver and muscle,
explained in Sec. IV.1.

(

Figure 4:  Model results and experimental data [120] for bovine liver
compressed to stretch $\lambda$: (a) axial stress $P$ at strain rate
$\dot{\epsilon}=2000$/s, (b) $P$ at $\dot{\epsilon}=0.01$/s and
$\dot{\epsilon}=10$/s, and (c) predicted matrix and fiber damage $\bar{D}$ and
$D_{1}$ at all three rates

a)
b)
(c)

(

The high-rate response of compressed liver is shown in Fig. 4(a). Model
results assume adiabatic conditions, but predicted temperature change is
negligible. Damage order parameters attained small maxima at high rates (i.e.,
$\bar{D}\lesssim 0.03$ and $D_{1}\lesssim 0.003$ in Fig. 4(c)) due to the
viscosity $\hat{\nu}^{\alpha}_{\rm D}$ preventing notable degradation over the
brief time period of deformation. This is consistent with data [120] that show
continuously increasing stiffness at this loading rate, with no evidence of
material failure. Differences between drained and undrained conditions are
indiscernible because the solid tissue is nearly incompressible.

Low- and moderate-rate stress histories are reported in Fig. 4(b). Model
results are isothermal and for drained conditions only; results for undrained,
not shown, are nearly identical. Fits to data are reasonable but not as close
as those for high-rate loading in Fig. 4(a). At low rates, data show reduction
in the degree of stiffening at large compression, indicative of strength
degradation [2, 11, 125]. This phenomenon is captured by damage kinetics, Fig.
4(c). Damage increases monotonically with compressive strain $1-\lambda$, and
at moderate- to high-rates is more severe in matrix than fibers. Histological
analysis after dynamic blunt impact [126] showed fractures in liver parenchyma
avoid fibers and interlobular septa and more often propagate along interfaces,
consistent with higher levels of “matrix” damage $\bar{D}$ relative to fiber
damage $D_{1}$ in Fig. 4(c). Conversely, at the lowest strain rate (0.01/s),
fiber damage overtakes matrix damage at large compression ($\lambda\lesssim
0.77$) , and is more pronounced due to longer load times for relaxation to
ensue.

Skeletal muscle. The theory is now implemented to study uniaxial-stress
tensile behavior of rabbit skeletal muscle at low and moderate strain rates,
with and without activation from electrical stimulation. Model results seek to
depict experiments reported in Refs. [127, 23].

Calculations proceed in the same manner as just described for modeling liver,
with a few exceptions. First, tension is modeled rather than compression, with
stretch ratio $F^{1}_{1}(t)=\lambda(t)=1+\dot{\epsilon}t\geq 1$ at two rates
[127]: $\dot{\epsilon}=0.17$/s and $\dot{\epsilon}=15$/s. Engineering tensile
stress is $P$ of (219), where now $(\sigma^{\alpha})^{1}_{1}\geq-
n_{0}^{\alpha}p^{\alpha}_{{\rm R}0}$. Both drained and undrained conditions
are considered, all isothermal. Second, the data do not indicate any
consistent rate sensitivity of damage or failure stretch [23], so equilibrium
equations (213) used in Sec. IV.1 for porcine muscle still apply. These
correspond to (182) and (187) with null viscosities $\bar{\nu}_{\rm
D}^{\alpha}=\nu^{\alpha}_{{\rm D}k}\rightarrow 0$, giving zero dissipation in
(183) and (III.3). Lastly, active tension, irrelevant for liver, is considered
for muscle. The form of free energy $\Psi^{\alpha}_{\rm A}$ and Cauchy stress
term ${\bm{\sigma}}^{\alpha}_{\rm A}$ in (163) and (164) are adapted directly
from Ref. [23] since they reproduce the over-stress from activation recorded
in isometric experiments [127]:

$\displaystyle\Psi^{1}_{\rm A}={\textstyle{\frac{2}{3}}}\Delta_{\rm A}\mu_{\rm
A}(\lambda_{{\rm A}1}-\lambda_{{\rm A}0})[\bar{\Lambda}^{p_{\rm A}}/p_{\rm
A}-\bar{\Lambda}^{r_{\rm A}}/r_{\rm A}],$ (220)
$\displaystyle{\bm{\sigma}}^{1}_{\rm
A}={\textstyle{\frac{2}{3}}}\frac{\rho^{\alpha}}{\rho^{\alpha}_{{\rm
R}0}}\frac{\Delta_{\rm A}\mu_{\rm A}}{\lambda^{1}_{1}}\bar{\Lambda}^{p_{\rm
A}-1}[1-\bar{\Lambda}^{q_{\rm A}-1}]\tilde{\bf h}^{1}_{1},$ (221)
$\displaystyle\bar{\Lambda}=\begin{cases}&\frac{\lambda^{1}_{1}-\lambda_{{\rm
A}0}}{\lambda_{{\rm A}1}-\lambda_{{\rm
A}0}}\quad\forall\,\lambda^{1}_{1}\in(\lambda_{{\rm A}0},\lambda_{{\rm
A}1}),\\\ &0\qquad\qquad\qquad(\text{otherwise}).\end{cases}$ (222)

Active tension vanishes for $\bar{\Lambda}$ outside domain $[0,1]$. Recall
fiber stretch obeys $(\lambda^{\alpha}_{k})^{2}={I^{\alpha}_{k}}={\tilde{\bf
C}^{\alpha}:{\bf H}^{\alpha}_{k}}$ with $\alpha=k=1$. Parameters are $\mu_{\rm
A}$ (stress units) and dimensionless set $(\lambda_{{\rm A}0},\lambda_{{\rm
A}1},p_{\rm A},q_{\rm A})$, with $r_{\rm A}=p_{\rm A}+q_{\rm A}-1$. A
dimensionless internal variable for activation is
$\\{\Delta_{k}^{\alpha}(t)\\}\rightarrow\Delta_{1}^{1}(t)=\Delta_{\rm A}$.
Only discrete states are considered: $\Delta_{\rm A}=1$ in the fully active
state and $\Delta_{\rm A}=0$ in the fully passive state. Transient switching
between states and partial activation are not addressed here or in the
experiments [127, 23]. Energy $\chi^{\alpha}_{k}$ in (163) and kinetic law
(176) need not be prescribed; no contribution to dissipation arises since
$D_{t}^{1}\Delta_{\rm A}=0$ in (178). The $\frac{2}{3}$ in (220) is omitted in
Ref. [23] where compressibility is ignored.

As prescribed in Sec. IV.1 for porcine muscle, rabbit muscle consists of solid
tissue $\alpha=1$ and ECF ($\alpha=2$), where parameters for ECF are in Table
1. The initial solid volume fraction remains $n_{0}^{1}=0.9$. Parameters for
rabbit skeletal muscle are compiled in Table 2. Thermophysical properties
entering the EOS are identical to those for porcine tissue in Table 1. Shear
properties $\mu^{\alpha}_{\rm S}$, $\mu^{\alpha}_{k}$, and $k^{\alpha}_{k}$
are calibrated to the data [127, 23], with $k=1$ sufficient. Fibers are fully
aligned, $\kappa^{\alpha}_{k}\rightarrow\kappa^{1}_{1}=0$ in (140), giving
anisotropic response. The glassy viscoelastic assumption used for modeling
shocks in Sec. IV.1 is inappropriate for low and moderate strain rates.
Instead, viscoelastic strength factors $\beta^{\alpha}_{{\rm S}m}$ and
$\beta^{\alpha}_{{\Phi}k,n}$ and relaxation times $\tau^{\alpha}_{{\rm S}m}$
and $\tau^{\alpha}_{{\Phi}k,n}$ are fit to experimental data; here, a single
mode suffices: $m=n=1$. The activation parameters in (220) are verbatim from
Ref. [23]: $\mu_{\rm A}=962$ kPa, $\lambda_{{\rm A}0}=0.9$, $\lambda_{{\rm
A}1}=1.32$, $p_{\rm A}=1.65$, and $q_{\rm A}=2.0$. Assumptions for properties
modulating matrix and fiber damage are the same as those explained for porcine
tissue in Sec. IV.1, with matching toughness $\mathsf{J}_{\rm C}$ [116].
Length $l^{1}=\bar{l}^{1}=l^{1}_{1}$ provides cohesive energies $E_{\rm
C}=\bar{E}_{\rm C}=E_{{\rm C}1}$ in (III.2). The value in Table 2 best fits
softening and failure in test data [127, 23] at the lowest loading rate.
Finsler metric parameters in Table 2 match those of porcine muscle in Table 1;
none are adjusted when fitting the data.

For tensile loading, cavitation of the fluid ($\alpha=2$) is not impossible.
Calibration of the model for water under isentropic expansion to its 8.7 MPa
cavitation stress [128] gives $\bar{E}^{\alpha}_{{\rm C}}=0.1818$ MPa. Use of
the same cohesive energy for ECF gives a cavitation stress of 8.9 MPa, and
$J^{\alpha}\gtrsim 1.001$ ($\alpha=2$) is needed to initiate detectable damage
$\bar{D}^{\alpha}$. Such expansion is never achieved in the present modeling
of muscle response: damage in the ECF is negligible here.

(

Figure 5:  Model results and experimental data [127, 23] for rabbit skeletal
muscle to tensile stretch $\lambda$ at rates of $\dot{\epsilon}=0.17$/s and
$\dot{\epsilon}=15$/s: (a) axial stress $P$ in active state ($\Delta_{\rm
A}=1$), (b) $P$ in passive state ($\Delta_{\rm A}=0$), and (c) predicted
matrix and fiber damage $\bar{D}$ and $D_{1}$, active state (passive nearly
identical)

a)
b)
(c)

(

Model outcomes and experimental stress-stretch data are compared in Fig. 5(a)
for active states and Fig. 5(b) for passive states, at engineering strain
rates of $\dot{\epsilon}=0.17$/s and $\dot{\epsilon}=15$/s. The fiber
direction ${\bm{\iota}}={\bm{\iota}}^{1}_{1}$ is aligned with the direction of
elongation. Model results in Fig. 5 are for drained lateral boundaries;
predictions for undrained conditions are nearly indiscernible from drained and
thus not shown. For active and passive states, the material is stiffer at the
higher rate, with larger peak (failure) stress. Predicted failure stretch is
similar at both strain rates. The material supports larger $P$ in the active
state over domain $1\leq\lambda\lesssim 1.32$, including an initial stress of
$P=0.192$ MPa at $\lambda(t=0)=1$ that matches experiments.

The model closely depicts the majority of data points [127, 23], an exception
under-prediction of large $\lambda$-$P$ data at $\dot{\epsilon}=15$/s for the
passive state in Fig. 5(b). An over-prediction of peak stress for active
loading at the higher rate was obtained from a phenomenological model [23].
That model, however, contained five adjustable parameters, whereas only the
length parameter $l^{1}$ was adjusted for damage modeling in application of
the present theory. More elaborate coupling among viscoelastic and damage
kinetics, with more parameters, could improve agreement, but such an exercise
is unjustified for closer fitting of relatively few data points. Unlike
results for compression in Sec. IV.1 where the matrix and fibers supported
similar stress, here, under tensile loading, the fibers bear the majority of
the load $P$, with the ratio of fiber to matrix stress increasing as stretch
increases and rate decreases. The correspondingly larger strain energy in the
fibers provides a larger driving force for fiber damage $D_{1}$ than matrix
damage $\bar{D}$, which is nearly negligible at $\dot{\epsilon}=0.17$/s, as
shown in Fig. 5(c). As discussed in Ref. [129], under tensile loading at low
rates, damage to muscle fibers is prominently observed over delamination and
damage to the endomysium (i.e., the matrix including connective tissue in
which fibers are embedded). The current predictions concur with these
observations. At the higher rate of $\dot{\epsilon}=15$/s, viscoelastic energy
of the matrix is sufficient to induce matrix damage, though it remains less
severe than fiber damage for $\lambda\gtrsim 1.23$, which is nearly the same
at both rates. The damage model is decoupled from $\Psi_{\rm A}$, so order
parameters $\bar{D}$ and $D_{1}$ have indistinguishable histories for active
versus passive states.

###  IV.3  Shock evolution

Analytical solution. Growth and decay of planar shock waves are studied, with
shock fronts treated as singular surfaces per theory of Sect. II.2. An
analytical solution is derived for solid-fluid mixtures with viscoelastic and
damage mechanisms, extending Ref. [74] that considered nonlinear elastic
solid-fluid mixtures without internal variables and Ref. [72] that considered
fluids with internal state variables. To streamline notation, let internal
variables
$\\{{\bm{\xi}}^{\alpha}\\}\rightarrow({\bm{\mathsf{a}}}^{\alpha},{\bm{\mathsf{b}}}^{\alpha})$.
Generic internal variable class ${\bm{\mathsf{a}}}^{\alpha}$ obeys kinetic
laws of form (92) specialized to

$D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}=D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}({\bf
F}^{\alpha},{\eta}^{\alpha}({\bf
F}^{\alpha},{\theta}^{\alpha},{\bm{\mathsf{a}}}^{\alpha},{\bm{\mathsf{b}}}^{\alpha}),{\bm{\mathsf{a}}}^{\alpha},{\bm{\mathsf{b}}}^{\alpha}).$
(223)

Generic class ${\bm{\mathsf{b}}}^{\alpha}$ obeys equilibrium conditions of
form

${\bm{\pi}}^{\alpha}_{\mathsf{b}}=\rho^{\alpha}\partial
u^{\alpha}/\partial{\bm{\mathsf{b}}}^{\alpha}={\bf 0}.$ (224)

Type ${\bm{\mathsf{a}}}^{\alpha}$ include dissipative variables for
viscoelasticity and order parameters for rate-dependent fracture;
${\bm{\mathsf{b}}}^{\alpha}$ include rate-insensitive order parameter(s). Now
excluding gradient regularization and explicit ${\bf X}^{\alpha}$-dependence,

$u^{\alpha}=u^{\alpha}({\bf
F}^{\alpha},\eta^{\alpha},{\bm{\mathsf{a}}}^{\alpha},{\bm{\mathsf{b}}}^{\alpha}),\,\theta^{\alpha}=\theta^{\alpha}({\bf
F}^{\alpha},\eta^{\alpha},{\bm{\mathsf{a}}}^{\alpha},{\bm{\mathsf{b}}}^{\alpha})$
(225)

are internal energy and temperature of (102) and (103). Unless
$\llbracket{\bm{\xi}}^{\alpha}\rrbracket={\bf 0}$, gradient regularization
yields infinite energy density in the shock front as its width approaches
zero. It also introduces complexity in Rankine-Hugoniot equations via
${\bm{\zeta}}^{\alpha}$, precluding analytical solutions without undue
assumptions on shock structure [87] not used here.

From (200), $\hat{g}=\hat{G}^{\alpha}\Rightarrow g/G^{\alpha}=1$. Then 1-D
kinematics, continuum balance laws, and entropy production are, with
$\hat{c}^{\alpha}=0$, ${\bf q}^{\alpha}={\bf 0}$, $r^{\alpha}=0$,
$h^{\alpha}={\bf h}^{\alpha}\cdot{\bf n}$, and $P^{\alpha}=-t^{\alpha}$,

$\displaystyle F^{\alpha}=\partial\chi^{\alpha}/\partial
X^{\alpha}=J^{\alpha},\qquad
D_{t}^{\alpha}J^{\alpha}=\partial\upsilon^{\alpha}/\partial X^{\alpha},$ (226)
$\displaystyle\rho_{0}^{\alpha}=J^{\alpha}\rho^{\alpha},\quad\rho_{0}^{\alpha}D_{t}^{\alpha}\upsilon^{\alpha}=-(\partial
P^{\alpha}/\partial X^{\alpha})+J^{\alpha}h^{\alpha},$ (227)
$\displaystyle\rho_{0}^{\alpha}D_{t}^{\alpha}u^{\alpha}=-P^{\alpha}(\partial\upsilon^{\alpha}/\partial
X^{\alpha})+J^{\alpha}\epsilon^{\alpha},$ (228)
$\displaystyle\rho_{0}^{\alpha}\theta^{\alpha}D_{t}^{\alpha}\eta^{\alpha}=J^{\alpha}(\epsilon^{\alpha}-{\bm{\pi}}^{\alpha}_{\mathsf{a}}\cdot
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}),$ (229)
$\displaystyle\sum_{\alpha}({J^{\alpha}}/{\theta^{\alpha}})(\epsilon^{\alpha}-{\bm{\pi}}^{\alpha}_{\mathsf{a}}\cdot
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}+c^{\alpha}\theta^{\alpha}\eta^{\alpha})\geq
0.$ (230)

A single shock propagates in the ($x,X^{\alpha}$)-direction at Lagrangian
speed $\mathcal{U}^{\alpha}$. Ahead of the shock front $\Sigma^{\alpha}$, each
phase $\alpha$ obeys equilibrium and uniformity conditions:

$\displaystyle
J^{\alpha+}=1,\quad\upsilon^{\alpha+}=0,\quad\theta^{\alpha+}=\theta_{0},\quad\eta^{\alpha+}={\rm
constant}$ $\displaystyle\Rightarrow\,\rho^{\alpha+}=\rho_{0}^{\alpha},\quad
h^{\alpha+}=0,\quad\epsilon^{\alpha+}=0,\quad{\mathcal{U}^{\alpha}}={\mathcal{U}};$
(231)
$\displaystyle{\bm{\mathsf{a}}}^{\alpha+}={\bm{\mathsf{a}}}^{\alpha}_{0}={\rm
constant},\quad{\bm{\mathsf{b}}}^{\alpha+}={\bm{\mathsf{b}}}^{\alpha}_{0}={\rm
constant}.$ (232)

Rankine-Hugoniot equations ($P^{\alpha}\\!>0\\!\Leftrightarrow\\!{\rm
compression}$) are

$\displaystyle\llbracket\upsilon^{\alpha}\rrbracket=-{\mathcal{U}}\llbracket
J^{\alpha}\rrbracket,\qquad\llbracket
P^{\alpha}\rrbracket=-\rho_{0}^{\alpha}{\mathcal{U}}^{2}\llbracket
J^{\alpha}\rrbracket,$ (233) $\displaystyle\rho_{0}^{\alpha}\llbracket
u^{\alpha}\rrbracket=-\langle P^{\alpha}\rangle\llbracket
J\rrbracket,\qquad\sum_{\alpha}\rho_{0}^{\alpha}\llbracket\eta^{\alpha}\rrbracket\geq
0.$ (234)

To avoid infinite dissipation in the shock front [72, 109, 110],

$\displaystyle\llbracket{\bm{\mathsf{a}}}^{\alpha}\rrbracket={\bf
0}\,\Rightarrow\,{\bm{\mathsf{a}}}^{\alpha-}={\bm{\mathsf{a}}}^{\alpha}_{0}.$
(235)

Jumps $\llbracket{\bm{\mathsf{b}}}^{\alpha}\rrbracket$ in non-dissipative
variables can be nonzero across $\Sigma^{\alpha}$ so long as (224) holds.
However, it is assumed that (224) can be solved, at least implicitly, at any
$(X^{\alpha},t)$ with the first of each of (225), (227), and then via (235),

${\bm{\mathsf{b}}}^{\alpha}=\bar{\bm{\mathsf{b}}}^{\alpha}(J^{\alpha},\eta^{\alpha},{\bm{\mathsf{a}}}^{\alpha}),\,\,\,{\bm{\mathsf{b}}}^{\alpha-}=\bar{\bm{\mathsf{b}}}^{\alpha-}(J^{\alpha-},\eta^{\alpha-},{\bm{\mathsf{a}}}^{\alpha}_{0}).$
(236)

Now with (235) and (236) at a given $(\cdot)^{+}$ state, the first of (234)
can be written
$H(J^{\alpha-},\eta^{\alpha-},\bar{\bm{\mathsf{b}}}^{\alpha-}(J^{\alpha-},\eta^{\alpha-}))=0$.
Again, at least implicitly, this can be solved for entropy along the Hugoniot
and then the other state variables:

$\displaystyle\llbracket\eta^{\alpha}\rrbracket=\eta^{\alpha}_{\rm
H}(\llbracket
J^{\alpha}\rrbracket),\quad\llbracket{\bm{\mathsf{b}}}^{\alpha}\rrbracket={\bm{\mathsf{b}}}^{\alpha}_{\rm
H}(\llbracket J^{\alpha}\rrbracket),$ (237) $\displaystyle\llbracket
P^{\alpha}\rrbracket=P^{\alpha}_{\rm H}(\llbracket
J^{\alpha}\rrbracket),\quad\llbracket u^{\alpha}\rrbracket=u^{\alpha}_{\rm
H}(\llbracket J^{\alpha}\rrbracket).$ (238)

Hugoniot states do not depend explicitly on $h^{\alpha-}$ or
$\epsilon^{\alpha-}$. From (104) with ${\bf F}^{\alpha}={\rm
diag}(J^{\alpha},1,1)$, note, then define

$\displaystyle P^{\alpha}=-\rho^{\alpha}_{0}\partial u^{\alpha}/\partial
J^{\alpha},\quad\theta^{\alpha}=\partial u^{\alpha}/\partial\eta^{\alpha};$
(239) $\displaystyle{\mathsf{C}}^{\alpha}=-\partial P^{\alpha}/\partial
J^{\alpha}=\rho^{\alpha}_{0}\partial^{2}u^{\alpha}/\partial(J^{\alpha})^{2},$
(240) $\displaystyle{\mathsf{G}}^{\alpha}=-\partial
P^{\alpha}/\partial\eta^{\alpha}=-\rho^{\alpha}_{0}J^{\alpha}\theta^{\alpha}(\gamma^{\alpha})^{1}_{1},$
(241) $\displaystyle{\bm{\mathsf{A}}}^{\alpha}=-\partial
P^{\alpha}/\partial{\bm{\mathsf{a}}}^{\alpha},\quad{\bm{\mathsf{B}}}^{\alpha}=-\partial
P^{\alpha}/\partial{\bm{\mathsf{b}}}^{\alpha},$ (242)
$\displaystyle{\bm{\mathsf{b}}}^{\prime\alpha}=\partial\bar{\bm{\mathsf{b}}}^{\alpha}/\partial
J^{\alpha},\quad{\bm{\mathsf{b}}}^{\prime\prime\alpha}=\partial^{2}\bar{\bm{\mathsf{b}}}^{\alpha}/(\partial
J^{\alpha})^{2},$ (243)
$\displaystyle\hat{\mathsf{C}}^{\alpha}={\mathsf{C}}^{\alpha}+{\bm{\mathsf{B}}}^{\alpha}\cdot{\bm{\mathsf{b}}}^{\prime\alpha},\quad{\bm{\mathsf{B}}}^{\prime\alpha}=\partial{\bm{\mathsf{B}}}^{\alpha}/\partial
J^{\alpha},$ (244)
$\displaystyle{\mathsf{C}^{\prime}}^{\alpha}=\partial{\mathsf{C}}^{\alpha}/\partial
J^{\alpha}=\rho^{\alpha}_{0}\partial^{3}u^{\alpha}/\partial(J^{\alpha})^{3},$
(245)
$\displaystyle\hat{\mathsf{C}}^{\prime\alpha}={\mathsf{C}}^{\prime\alpha}+{\bm{\mathsf{B}}}^{\prime\alpha}\cdot{\bm{\mathsf{b}}}^{\prime\alpha}+{\bm{\mathsf{B}}}^{\alpha}\cdot{\bm{\mathsf{b}}}^{\prime\prime\alpha},$
(246)
$\displaystyle\hat{\mathsf{G}}^{\alpha}={\mathsf{G}}^{\alpha}+{\bm{\mathsf{B}}}^{\alpha}\cdot{\bm{\mathsf{b}}}_{\eta}^{\alpha}.$
(247)

Weak shocks are analyzed in the limit $\llbracket
J^{\alpha}\rrbracket\rightarrow 0$. Applying theorems [71] relating isentropic
and tangent moduli, (237) and (238) are expanded from a $(\cdot)^{+}$ state:

$\displaystyle\llbracket
P^{\alpha}\rrbracket=-\hat{\mathsf{C}}^{\alpha+}\llbracket
J^{\alpha}\rrbracket-{\textstyle{\frac{1}{2}}}\hat{\mathsf{C}}^{\prime\alpha+}\llbracket
J^{\alpha}\rrbracket^{2}+O(\llbracket J^{\alpha}\rrbracket^{3}),$ (248)
$\displaystyle\llbracket\rho_{0}^{\alpha}u^{\alpha}\rrbracket=-P^{\alpha+}\llbracket
J^{\alpha}\rrbracket+{\textstyle{\frac{1}{2}}}\hat{\mathsf{C}}^{\alpha+}\llbracket
J^{\alpha}\rrbracket^{2}+O(\llbracket J^{\alpha}\rrbracket^{3}),$ (249)
$\displaystyle\llbracket\rho_{0}^{\alpha}\theta^{\alpha}\rrbracket=\hat{\mathsf{G}}^{\alpha+}\llbracket
J^{\alpha}\rrbracket+{\textstyle{\frac{1}{2}}}(\textstyle{\frac{\partial\hat{\mathsf{G}}}{\partial
J}})^{\alpha+}\llbracket J^{\alpha}\rrbracket^{2}+O(\llbracket
J^{\alpha}\rrbracket^{3}),$ (250)
$\displaystyle\llbracket{\bm{\mathsf{b}}}^{\alpha}\rrbracket={\bm{\mathsf{b}}}^{\prime\alpha+}\llbracket
J^{\alpha}\rrbracket+{\textstyle{\frac{1}{2}}}{\bm{\mathsf{b}}}^{\prime\prime\alpha+}\llbracket
J^{\alpha}\rrbracket^{2}+O(\llbracket J^{\alpha}\rrbracket^{3}),$ (251)
$\displaystyle\llbracket\eta^{\alpha}\rrbracket={\textstyle{\frac{1}{12}}}\\{\hat{\mathsf{C}}^{\prime\alpha+}/(\rho_{0}^{\alpha}\theta_{0})\\}\llbracket
J^{\alpha}\rrbracket^{3}+O(\llbracket J^{\alpha}\rrbracket^{4}).$ (252)

From (252), $\llbracket\eta^{\alpha}\rrbracket$ is of order three in
$\llbracket J^{\alpha}\rrbracket$, and $\llbracket J^{\alpha}\rrbracket\leq 0$
when $\hat{\mathsf{C}}^{\prime\alpha+}<0$ to satisfy the second of (234) for a
single $\alpha$. Interactions ${\bf h}^{\alpha}$ and $\epsilon^{\alpha}$ are
respectively odd and even functions of $\upsilon^{\beta}$ [73, 74], so from
(IV.3), (233), and (250),

$\displaystyle h^{\alpha-}=-{\mathcal{U}}\sum_{\beta}(\partial
h^{\alpha}/\partial\upsilon^{\beta})^{+}\llbracket
J^{\beta}\rrbracket+O(\llbracket J^{\beta}\rrbracket^{2}),$ (253)
$\displaystyle\epsilon^{\alpha-}=\sum_{\beta}(\partial\epsilon^{\alpha}/\partial\theta^{\beta})^{+}(\hat{\mathsf{G}}^{\beta}/\rho^{\beta}_{0})^{+}\llbracket
J^{\beta}\rrbracket+O(\llbracket J^{\beta}\rrbracket^{2}).$ (254)

From (234) and (248), $\mathcal{U}^{\beta}$ approaches the sound speed:

$\displaystyle({\mathcal{U}}^{\beta})^{2}=({\mathcal{C}}^{\beta})^{2}+{\textstyle{\frac{1}{2}}}(\hat{\mathsf{C}}^{\prime\beta}/\rho^{\beta}_{0})^{+}\llbracket
J^{\beta}\rrbracket+O(\llbracket J^{\beta}\rrbracket^{2}),$
$\displaystyle\mathcal{C}^{\beta}=\sqrt{\hat{\mathsf{C}}^{\beta+}/\rho_{0}^{\beta}}.$
(255)

Considering $\alpha\neq\beta$ and
${\mathcal{C}}^{\alpha}\neq{\mathcal{C}}^{\beta}$, (IV.3) implies [73, 74]

$\displaystyle\\{({\mathcal{C}}^{\alpha})^{2}-({\mathcal{C}}^{\beta})^{2}\\}\llbracket
J^{\beta}\rrbracket+O(\llbracket J^{\alpha}\rrbracket\llbracket
J^{\beta}\rrbracket)+O(\llbracket J^{\beta}\rrbracket^{2})=0$
$\displaystyle\Rightarrow\,{\mathcal{U}}^{2}=({\mathcal{C}}^{\alpha})^{2}+O(\llbracket
J^{\alpha}\rrbracket)\quad({\rm for\,one}\,\llbracket J^{\alpha}\rrbracket\neq
0),$ $\displaystyle\quad\llbracket
J^{\beta}\rrbracket=0\,\,(\forall\,\beta=1,2,\ldots,\alpha-1,\alpha+1,\ldots,N).$
(256)

Each distinct wave speed ${\mathcal{C}}^{\alpha}$ corresponds to isolated jump
$\llbracket J^{\alpha}\rrbracket$. From (248)–(252), a weak shock in phase
$\alpha$ does not induce jumps $\llbracket P^{\beta}\rrbracket$, $\llbracket
u^{\beta}\rrbracket$, $\llbracket\theta^{\beta}\rrbracket$, etc. in other
phases $\beta$. Since $\llbracket J^{\beta}\rrbracket=0$ for
$\beta\neq\alpha$, (253) and (254) become

$\displaystyle h^{\alpha-}=-{\mathcal{C}}^{\alpha}(\partial
h^{\alpha}/\partial\upsilon^{\alpha})^{+}\llbracket
J^{\alpha}\rrbracket+O(\llbracket J^{\alpha}\rrbracket^{2}),$ (257)
$\displaystyle\epsilon^{\alpha-}=(\partial\epsilon^{\alpha}/\partial\theta^{\alpha})^{+}(\hat{\mathsf{G}}^{\alpha}/\rho^{\alpha}_{0})^{+}\llbracket
J^{\alpha}\rrbracket+O(\llbracket J^{\alpha}\rrbracket^{2}),$ (258)

that also apply for $h^{\beta-},\epsilon^{\beta-}$, $\beta\neq\alpha$ with
$(\partial h^{\alpha}/\partial\upsilon^{\alpha})^{+}\rightarrow(\partial
h^{\beta}/\partial\upsilon^{\alpha})^{+}$ and
$(\partial\epsilon^{\alpha}/\partial\theta^{\alpha})^{+}\rightarrow(\partial\epsilon^{\beta}/\partial\theta^{\alpha})^{+}$
[73].

Resuming analysis of the nonlinear (i.e., strong-shock) regime, denote by
$f^{\alpha}$ any of ($J^{\alpha}$, $P^{\alpha}$, $\eta^{\alpha}$,
$\theta^{\alpha}$, $\rho^{\alpha}$, $\upsilon^{\alpha}$, $u^{\alpha}$,
${\bm{\mathsf{b}}}^{\alpha}$). Recall that across surface $\Sigma^{\alpha}$,
$f^{\alpha}$, $D_{t}^{\alpha}f^{\alpha}$, and
$\nabla^{\alpha}_{0}f^{\alpha}=\partial f^{\alpha}/\partial X^{\alpha}$ can be
discontinuous. From (235), ${\bm{\mathsf{a}}}^{\alpha}$ is continuous;
however, $D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}$ and
$\nabla^{\alpha}_{0}{\bm{\mathsf{a}}}^{\alpha}$ need not be so. Applying (42)
with the last of (IV.3) gives [72]

$\displaystyle\delta_{t}\llbracket f^{\alpha}\rrbracket=\llbracket
D_{t}^{\alpha}f^{\alpha}\rrbracket+{\mathcal{U}}\llbracket\nabla_{0}^{\alpha}f^{\alpha}\rrbracket,$
(259) $\displaystyle\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket=-{\mathcal{U}}\llbracket\nabla_{0}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket.$
(260)

Recall (9) in 1-D is
$D_{t}^{\alpha}J^{\alpha}=\nabla_{0}^{\alpha}\upsilon^{\alpha}$ via (226).
Using (259) on $\llbracket J^{\alpha}\rrbracket$ and
$\llbracket\upsilon^{\alpha}\rrbracket$ and (42) on ${\mathcal{U}}$ in the
first of (233) gives

$\displaystyle 2{\mathcal{U}}\delta_{t}\llbracket
J^{\alpha}\rrbracket+\llbracket
J^{\alpha}\rrbracket\delta_{t}{\mathcal{U}}={\mathcal{U}}^{2}\llbracket\nabla_{0}^{\alpha}J^{\alpha}\rrbracket-\llbracket
D_{t}^{\alpha}\upsilon^{\alpha}\rrbracket.$ (261)

From (227), (229), (IV.3), (233), and (239)–(242),

$\displaystyle\rho_{0}^{\alpha}\llbracket
D_{t}^{\alpha}\upsilon^{\alpha}\rrbracket={\mathsf{C}}^{\alpha-}\llbracket\nabla_{0}^{\alpha}J^{\alpha}\rrbracket+{\mathsf{G}}^{\alpha-}\llbracket\nabla_{0}^{\alpha}\eta^{\alpha}\rrbracket+J^{\alpha-}\llbracket
h^{\alpha}\rrbracket$
$\displaystyle\qquad\qquad\quad\,\,+{\bm{\mathsf{A}}}^{\alpha-}\cdot\llbracket\nabla_{0}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket+{\bm{\mathsf{B}}}^{\alpha-}\cdot\llbracket\nabla_{0}^{\alpha}{\bm{\mathsf{b}}}^{\alpha}\rrbracket,$
(262) $\displaystyle\rho_{0}^{\alpha}\theta^{\alpha-}\llbracket
D_{t}^{\alpha}\eta^{\alpha}\rrbracket=J^{\alpha-}(\llbracket\epsilon^{\alpha}\rrbracket-{\bm{\pi}}^{\alpha-}_{\mathsf{a}}\cdot\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket),$ (263)

where ${\bm{\pi}}^{\alpha}_{\mathsf{a}}=\rho^{\alpha}\partial
u^{\alpha}/\partial{\bm{\mathsf{a}}}^{\alpha}$. With
$f^{\alpha}\rightarrow\eta^{\alpha}$, putting (263) into (259) gives
$\llbracket\nabla_{0}^{\alpha}\eta^{\alpha}\rrbracket$ in terms of
$\delta_{t}\llbracket\eta^{\alpha}\rrbracket$ and jumps on the right side of
(263). This $\llbracket\nabla_{0}^{\alpha}\eta^{\alpha}\rrbracket$ is
substituted for the second term on the right in (IV.3). The fourth term on the
right of (IV.3) is
$-\frac{1}{\mathcal{U}}{\bm{\mathsf{A}}}^{\alpha-}\cdot\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket$ via (260). Using this
result, (243), and (263), the fifth term includes

$\displaystyle\llbracket\nabla_{0}^{\alpha}{\bm{\mathsf{b}}}^{\alpha}\rrbracket={\bm{\mathsf{b}}}^{\prime\alpha-}\llbracket\nabla_{0}^{\alpha}J^{\alpha}\rrbracket-(1/{\mathcal{U}})\,{{\bm{\mathsf{b}}}_{\mathsf{a}}^{\alpha-}}\cdot\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket$
$\displaystyle+\frac{{\bm{\mathsf{b}}}_{\eta}^{\alpha-}}{\mathcal{U}}\biggr{\\{}\delta_{t}\llbracket\eta^{\alpha}\rrbracket-\frac{J^{\alpha-}(\llbracket\epsilon^{\alpha}\rrbracket-{\bm{\pi}}^{\alpha-}_{\mathsf{a}}\cdot\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket)}{\rho_{0}^{\alpha}\theta^{\alpha-}}\biggr{\\}},$
(264)
$\displaystyle{\bm{\mathsf{b}}}_{\mathsf{a}}^{\alpha}=\partial\bar{\bm{\mathsf{b}}}^{\alpha}/\partial{\bm{\mathsf{a}}}^{\alpha},\qquad{\bm{\mathsf{b}}}_{\eta}^{\alpha}=\partial\bar{\bm{\mathsf{b}}}^{\alpha}/\partial\eta^{\alpha}.$
(265)

Putting (IV.3) into (IV.3), the latter is inserted into (261):

$\displaystyle\delta_{t}\llbracket
J^{\alpha}\rrbracket=\frac{1}{2{\mathcal{U}}}\biggr{\\{}\biggr{(}{\mathcal{U}}^{2}-\frac{\hat{\mathsf{C}}^{\alpha-}}{\rho_{0}^{\alpha}}\biggr{)}\llbracket\nabla_{0}^{\alpha}J^{\alpha}\rrbracket-\frac{\hat{\mathsf{G}}^{\alpha-}}{\rho_{0}^{\alpha}{\mathcal{U}}}\delta_{t}\llbracket\eta^{\alpha}\rrbracket$
$\displaystyle\qquad\qquad-\delta_{t}{\mathcal{U}}\llbracket
J^{\alpha}\rrbracket-\frac{J^{\alpha-}}{\rho_{0}^{\alpha}}\llbracket
h^{\alpha}\rrbracket+\frac{J^{\alpha-}\hat{\mathsf{G}}^{\alpha-}}{(\rho^{\alpha}_{0})^{2}{\mathcal{U}}\theta^{\alpha-}}\llbracket\epsilon^{\alpha}\rrbracket$
$\displaystyle\qquad\qquad+{\bm{\mathsf{L}}}^{\alpha-}\cdot\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket\biggr{\\}},$ (266)
$\displaystyle{\bm{\mathsf{L}}}^{\alpha}=\frac{1}{\rho_{0}^{\alpha}{\mathcal{U}}}\biggr{\\{}{\bm{\mathsf{A}}}^{\alpha}+{\bm{\mathsf{B}}}^{\alpha}\cdot{\bm{\mathsf{b}}}_{\mathsf{a}}^{\alpha}-\frac{J^{\alpha}\hat{\mathsf{G}}^{\alpha}}{\rho_{0}^{\alpha}\theta^{\alpha}}{\bm{\pi}}^{\alpha}_{\mathsf{a}}\biggr{\\}}.$
(267)

From (224), (233), (234), (239), (244), (260), and (247),

$\displaystyle\rho_{0}^{\alpha}\delta_{t}\llbracket
u^{\alpha}\rrbracket=-(P^{\alpha-}+\rho_{0}^{\alpha}{\mathcal{U}}^{2}\llbracket
J^{\alpha}\rrbracket)\delta_{t}\llbracket J^{\alpha}\rrbracket$
$\displaystyle\qquad\qquad\qquad-\llbracket
J^{\alpha}\rrbracket\delta_{t}P^{\alpha-}-\rho_{0}^{\alpha}{\mathcal{U}}\llbracket
J^{\alpha}\rrbracket^{2}\delta_{t}{\mathcal{U}},$ (268)
$\displaystyle\rho_{0}^{\alpha}\delta_{t}\llbracket
u^{\alpha}\rrbracket=-P^{\alpha-}\delta_{t}\llbracket
J^{\alpha}\rrbracket+\rho_{0}^{\alpha}\theta^{\alpha-}\delta_{t}\llbracket\eta^{\alpha}\rrbracket,$
(269) $\displaystyle\delta_{t}P^{\alpha-}=\delta_{t}\llbracket
P^{\alpha}\rrbracket=-\hat{C}^{\alpha-}\delta_{t}\llbracket
J^{\alpha}\rrbracket-\hat{G}^{\alpha-}\delta_{t}\llbracket\eta^{\alpha}\rrbracket.$
(270)

Using (IV.3)–(270) to eliminate $\delta_{t}\llbracket u^{\alpha}\rrbracket$
and $\delta_{t}P^{\alpha-}$, then differentiating (233) produces the
following:

$\displaystyle\rho_{0}^{\alpha}\llbracket
J^{\alpha}\rrbracket^{2}{\mathcal{U}}\delta_{t}{\mathcal{U}}={\mathsf{G}}^{\alpha-}\llbracket
J^{\alpha}\rrbracket(1-1/\hat{\zeta}^{\alpha})\delta_{t}\llbracket\eta^{\alpha}\rrbracket$
$\displaystyle\qquad\qquad\qquad\quad+\hat{\mathsf{C}}^{\alpha-}(1-\hat{\xi}^{\alpha})\llbracket
J^{\alpha}\rrbracket,$ (271)
$\displaystyle\hat{\xi}^{\alpha}=\rho_{0}^{\alpha}{\mathcal{U}}^{2}/\hat{\mathsf{C}}^{\alpha-},\quad\hat{\zeta}^{\alpha}=\hat{\mathsf{G}}^{\alpha-}\llbracket
J^{\alpha}\rrbracket/(\rho^{\alpha}_{0}\theta^{\alpha-});$ (272)
$\displaystyle 2\rho_{0}^{\alpha}\llbracket
J^{\alpha}\rrbracket{\mathcal{U}}\delta_{t}{\mathcal{U}}=\hat{\mathsf{C}}^{\alpha-}(1-\hat{\xi}^{\alpha})\delta_{t}\llbracket
J^{\alpha}\rrbracket+\hat{G}^{\alpha-}\delta_{t}\llbracket\eta^{\alpha}\rrbracket.$
(273)

Solving (IV.3) and (273) for $\delta_{t}{\mathcal{U}}$ and
$\delta_{t}\llbracket\eta^{\alpha}\rrbracket$, then insertion in (IV.3) with
(IV.3) and (270) yields the fully nonlinear shock evolution equations for
$\delta_{t}\llbracket J^{\alpha}\rrbracket$ and $\delta_{t}\llbracket
P^{\alpha}\rrbracket$:

$\displaystyle\delta_{t}{\mathcal{U}}={\mathcal{U}}\frac{(1-\hat{\xi}^{\alpha})}{(2-\hat{\zeta}^{\alpha})\hat{\xi}^{\alpha}\llbracket
J^{\alpha}\rrbracket}\delta_{t}\llbracket J^{\alpha}\rrbracket,$ (274)
$\displaystyle\delta_{t}\llbracket\eta^{\alpha}\rrbracket=\frac{\hat{\mathsf{C}}^{\alpha-}}{\hat{\mathsf{G}}^{\alpha-}}\frac{\hat{\zeta}^{\alpha}(1-\hat{\xi}^{\alpha})}{(2-\hat{\zeta}^{\alpha})}\delta_{t}\llbracket
J^{\alpha}\rrbracket,$ (275) $\displaystyle\delta_{t}\llbracket
J^{\alpha}\rrbracket={\mathcal{U}}\frac{(1-\hat{\xi}^{\alpha})(2-\hat{\zeta}^{\alpha})\\{\Lambda^{\alpha}-(\nabla_{0}^{\alpha}J^{\alpha})^{-}\\}}{(3\hat{\xi}^{\alpha}+1)-\hat{\zeta}^{\alpha}(3\hat{\xi}^{\alpha}-1)},$
(276) $\displaystyle\delta_{t}\llbracket
P^{\alpha}\rrbracket=-\hat{\mathsf{C}}^{\alpha-}{\mathcal{U}}\frac{\\{3-\hat{\xi}^{\alpha}(1+\hat{\zeta}^{\alpha})\\}\\{\Lambda^{\alpha}-(\nabla_{0}^{\alpha}J^{\alpha})^{-}\\}}{(3\hat{\xi}^{\alpha}+1)-\hat{\zeta}^{\alpha}(3\hat{\xi}^{\alpha}-1)},$
(277) $\displaystyle\Lambda^{\alpha}=\frac{1+\llbracket
J^{\alpha}\rrbracket}{(1-\hat{\xi}^{\alpha})\hat{\mathsf{C}}^{\alpha-}}\bigr{\\{}\frac{\rho_{0}^{\alpha}}{J^{\alpha-}}[{\bm{\mathsf{L}}}^{\alpha-}\cdot(D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha})^{-}]$
$\displaystyle\qquad\qquad\qquad\quad-h^{\alpha-}+[{\hat{\mathsf{G}}^{\alpha-}}/({\rho_{0}^{\alpha}{\mathcal{U}}\theta^{\alpha-}})]\epsilon^{\alpha-}\bigr{\\}}.$
(278)

When $(\nabla_{0}^{\alpha}J^{\alpha})^{-}$ equals a critical strain gradient
$\Lambda^{\alpha}$ (a function of $(\cdot)^{-}$ conditions immediately behind
the wave front), (274)–(277) vanish so the shock is steady.

Preceding derivations are for phase $\alpha$. Now let $\mathcal{U}$ for phase
$\alpha$ be imposed simultaneously on $\beta\neq\alpha$. The trivial solution
to (233) is $\llbracket J^{\beta}\rrbracket=0$. Noting (273) and (275) still
apply with $\alpha\rightarrow\beta$, substitution of (274) into the former
gives a non-trivial solution (i.e., shock interaction law):

$\displaystyle\delta_{t}\llbracket
J^{\beta}\rrbracket=\frac{(2-\hat{\zeta}^{\beta})\hat{\xi}^{\beta}(1-\hat{\xi}^{\alpha})}{(2-\hat{\zeta}^{\alpha})\hat{\xi}^{\alpha}(1-\hat{\xi}^{\beta})}\frac{\llbracket
J^{\beta}\rrbracket}{\llbracket J^{\alpha}\rrbracket}\delta_{t}\llbracket
J^{\alpha}\rrbracket.$ (279)

In the weak limit,
$\mathcal{U}={\mathcal{U}}^{\alpha}\rightarrow{\mathcal{C}}^{\alpha}={\rm
constant}$ and shock evolution depends only on $(\cdot)^{+}$ states. As
$\llbracket J^{\alpha}\rrbracket\rightarrow 0$, from (233), (234), (248),
(250), (253), (254), (276), and (IV.3),

$\displaystyle\hat{\xi}^{\alpha}=1-{\textstyle{\frac{1}{2}}}(\hat{\mathsf{C}}^{\prime\alpha+}/\hat{\mathsf{C}}^{\alpha+})\llbracket
J^{\alpha}\rrbracket+O(\llbracket J^{\alpha}\rrbracket^{2}),$ (280)
$\displaystyle\hat{\zeta}^{\alpha}=\\{\hat{\mathsf{G}}^{\alpha+}/(\rho_{0}^{\alpha}\theta_{0})\\}\llbracket
J^{\alpha}\rrbracket+O(\llbracket J^{\alpha}\rrbracket^{2}),$ (281)
$\displaystyle\Lambda^{\alpha}=-4\,{\mathcal{C}}^{\alpha}\rho_{0}^{\alpha}\omega^{\alpha}/\hat{\mathsf{C}}^{\prime\alpha+}+O(\llbracket
J^{\alpha}\rrbracket),$ (282)
$\displaystyle\omega^{\alpha}=-\frac{1}{2\rho_{0}^{\alpha}}\biggr{\\{}\frac{1}{({\mathcal{C}}^{\alpha})^{2}}\biggr{[}{\bm{\mathsf{A}}}^{\alpha+}+{\bm{\mathsf{B}}}^{\alpha+}\cdot{\bm{\mathsf{b}}}_{\mathsf{a}}^{\alpha+}-\frac{\hat{\mathsf{G}}^{\alpha+}}{\rho_{0}^{\alpha}\theta_{0}}{\bm{\pi}}^{\alpha+}_{\mathsf{a}}\biggr{]}$
$\displaystyle\qquad\qquad\quad\cdot\left[\biggr{(}\frac{\partial(D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha})}{\partial
J^{\alpha}}\biggr{)}^{+}+\biggr{(}\frac{\partial(D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha})}{\partial{\bm{\mathsf{b}}}^{\alpha}}\cdot{\bm{\mathsf{b}}}^{\prime\alpha}\biggr{)}^{+}\right]$
$\displaystyle\qquad\quad+\left(\frac{\partial
h^{\alpha}}{\partial\upsilon^{\alpha}}\right)^{+}+\frac{1}{\theta_{0}}\left(\frac{\hat{\mathsf{G}}^{\alpha+}}{\rho_{0}^{\alpha}{\mathcal{C}}^{\alpha}}\right)^{2}\left(\frac{\partial\epsilon^{\alpha}}{\partial\theta^{\alpha}}\right)^{+}\biggr{\\}},$
(283) $\displaystyle\delta_{t}\llbracket
J^{\alpha}\rrbracket=-\omega^{\alpha}\llbracket
J^{\alpha}\rrbracket+O(\llbracket
J^{\alpha}\rrbracket\llbracket\nabla^{\alpha}_{0}J^{\alpha}\rrbracket;\llbracket
J^{\alpha}\rrbracket^{2}).$ (284)

Omitting higher-order products on the right of (284) [73],

$\displaystyle\llbracket J^{\alpha}\rrbracket(t)=\Delta
J^{\alpha}_{0}\exp(-\omega^{\alpha}t),\,\,\,\Delta J^{\alpha}_{0}=\llbracket
J^{\alpha}\rrbracket(t=0).$ (285)

For small $\llbracket J^{\alpha}\rrbracket$, if
$\nabla_{0}^{\alpha}J^{\alpha}$ remains negligible behind wave front
$\Sigma^{\alpha}$, shock amplitude evolves at a rate determined by
$\omega^{\alpha}={\rm constant}$. Jumps $\llbracket P^{\alpha}\rrbracket$,
$\llbracket u^{\alpha}\rrbracket$, $\llbracket\theta^{\alpha}\rrbracket$, and
$\llbracket{\bm{\mathsf{b}}}^{\alpha}\rrbracket$ evolve proportionally via
(248)–(251); $\llbracket\eta^{\alpha}\rrbracket\rightarrow 0$ by (252).

Derivations of Sec. IV.3 apply trivially for a single-phase material
($\alpha=N=1$, $h^{\alpha}\rightarrow 0$, $\epsilon^{\alpha}\rightarrow 0$).
They can also describe a shock moving with velocity $\mathcal{U}$ through the
mixture as a whole per Sec. II.3. Assuming $\upsilon^{\alpha}=\upsilon$ and
$\theta^{\alpha}=\theta$ for all constituents, then $J^{\alpha}=J$, diffusion
velocities ${\bm{\mu}}^{\alpha}={\bm{0}}$, and thus $h^{\alpha}=0$ and
$\epsilon^{\alpha}=0$. Then from Sec. II.3, mixture quantities include
$\psi=u-\theta\eta$ and

$\displaystyle
P=\sum_{\alpha}P^{\alpha},\quad\rho_{0}=\sum_{\alpha}n_{0}^{\alpha}\rho^{\alpha}_{{\rm
R}0},\quad\rho_{0}u=\sum_{\alpha}\rho_{0}^{\alpha}u^{\alpha},$
$\displaystyle\hat{\mathsf{C}}=\sum_{\alpha}\hat{\mathsf{C}}^{\alpha},\quad\hat{\mathsf{G}}=\sum_{\alpha}\frac{\partial
P^{\alpha}}{\partial\eta},\quad\eta=-\frac{\partial\psi}{\partial\theta}.$
(286)

Since $u^{\alpha}$ is independent of
$({\bm{\mathsf{a}}}^{\beta},{\bm{\mathsf{b}}}^{\beta})\,\forall\,\beta\neq\alpha$
per (225),

$\displaystyle{\bm{\mathsf{L}}}\cdot\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}\rrbracket=\sum_{\alpha}{\bm{\mathsf{L}}}^{\alpha}\cdot\llbracket
D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha}\rrbracket.$ (287)

Biologic tissue. Quantities entering $\Lambda^{\alpha}$ and $\omega^{\alpha}$
of (282) and (IV.3) are evaluated for constitutive frameworks of Sec. III.
First, consider an ideal gas. Variables
(${\bm{\mathsf{a}}}^{\alpha},{\bm{\mathsf{b}}}^{\alpha}$) are irrelevant,
$\rho_{0}^{\alpha}=n_{0}^{\alpha}\rho_{{\rm R}0}^{\alpha}$, and (125) and
(127) give

$\displaystyle P^{\alpha+}=n_{0}^{\alpha}p_{{\rm
R}0}^{\alpha},\quad\hat{\mathsf{C}}^{\alpha+}=n_{0}^{\alpha}p_{{\rm
R}0}^{\alpha}(1+\gamma_{0}^{\alpha}),$ (288)
$\displaystyle\hat{\mathsf{C}}^{\prime\alpha+}=-n_{0}^{\alpha}p_{{\rm
R}0}^{\alpha}(1+\gamma_{0}^{\alpha})(2+\gamma_{0}^{\alpha}),$ (289)
$\displaystyle\hat{\mathsf{G}}^{\alpha+}=-\rho_{0}^{\alpha}\theta_{0}\gamma^{\alpha}_{0},\quad{\mathcal{C}}^{\alpha}=[n_{0}^{\alpha}p_{{\rm
R}0}^{\alpha}(1+\gamma_{0}^{\alpha})/\rho^{\alpha}_{0}]^{1/2},$ (290)
$\displaystyle(\partial\hat{\mathsf{G}}^{\alpha}/\partial
J^{\alpha})^{+}=-\hat{\mathsf{G}}^{\alpha+}(1+\gamma^{\alpha}_{0}).$ (291)

Next, consider a compressible fluid obeying the EOS in (III.2)–(135). With
null cavitation for compression, (${\bm{\mathsf{a}}}^{\alpha}$,
${\bm{\mathsf{b}}}^{\alpha}$) are again irrelevant,
$\rho_{0}^{\alpha}=n_{0}^{\alpha}\rho_{{\rm R}0}^{\alpha}$, (291) holds, and

$\displaystyle P^{\alpha+}=n_{0}^{\alpha}p_{{\rm
R}0}^{\alpha},\quad\hat{\mathsf{C}}^{\alpha+}=n_{0}^{\alpha}(p_{{\rm
R}0}^{\alpha}+B_{\eta}^{\alpha}),$ (292)
$\displaystyle\hat{\mathsf{C}}^{\prime\alpha+}=-n_{0}^{\alpha}\\{2p^{\alpha}_{{\rm
R}0}+B^{\alpha}_{\eta}(1+B_{\eta{\rm p}}^{\alpha})\\},$ (293)
$\displaystyle\hat{\mathsf{G}}^{\alpha+}=-\rho_{0}^{\alpha}\theta_{0}\gamma^{\alpha}_{0},\quad{\mathcal{C}}^{\alpha}=[n_{0}^{\alpha}(p_{{\rm
R}0}^{\alpha}+B_{\eta}^{\alpha})/\rho^{\alpha}_{0}]^{1/2}.$ (294)

Last, a solid tissue with EOS (III.2)–(135), viscoelastic matrix, viscoelastic
fibers, matrix- and fiber-damage is addressed. Bulk and shear viscoelasticity
and fiber family $k=1$ furnish internal variables
${\bm{\mathsf{a}}}^{\alpha}\rightarrow\\{{\bm{\Gamma}}^{\alpha}_{{\rm
V}l},{\bm{\Gamma}}^{\alpha}_{{\rm S}m},{\bm{\Gamma}}^{\alpha}_{{\Phi}k,n}\\}$
with initial conditions $\\{{\bm{\Gamma}}^{\alpha+}_{{\rm
V}l}={\bm{\Gamma}}^{\alpha+}_{{\rm
S}m}={\bm{\Gamma}}^{\alpha+}_{{\Phi}k,n}={\bf 0}\\}$. Rate-insensitive damage
supplies order parameters
${\bm{\mathsf{b}}}^{\alpha}\rightarrow\\{\bar{D}^{\alpha},D^{\alpha}_{k}\\}$
with initial conditions $\\{\bar{D}^{\alpha+}=D^{\alpha+}_{k}=0\\}$. Fibers
are either aligned, $\kappa^{\alpha}_{k}=0$ parallel to ($x,X^{\alpha}$), or
isotropic, $\kappa^{\alpha}_{k}=\frac{1}{3}$. Constant active tension is
permitted, affecting energy, stress $(\sigma_{\rm A})^{1}_{1}$, and stiffness
via (163) and (164); since $\Delta^{\alpha}_{k}={\rm constant}$, this does not
affect dissipation nor need enter ${\bm{\mathsf{a}}}^{\alpha}$. As usual,
$\rho_{0}^{\alpha}=n_{0}^{\alpha}\rho_{{\rm R}0}^{\alpha}$ and
$({\mathcal{C}}^{\alpha})^{2}={\hat{\mathsf{C}}^{\alpha+}/\rho^{\alpha}_{0}}$.
Lengthy, yet routine, derivations yield

$\displaystyle P^{\alpha+}=n_{0}^{\alpha}\\{p^{\alpha}_{{\rm
R}0}-(\sigma^{\alpha}_{\rm
A})^{1+}_{1}\\},\quad{\mathsf{G}}^{\alpha+}=-\rho_{0}^{\alpha}\theta_{0}\gamma^{\alpha}_{0};$
(295) $\displaystyle(\partial\hat{\mathsf{G}}^{\alpha}/\partial
J^{\alpha})^{+}=-\hat{\mathsf{G}}^{\alpha+}(1+\gamma^{\alpha}_{0}),$ (296)
$\displaystyle\hat{\mathsf{C}}^{\alpha+}={\mathsf{C}}^{\alpha+},\quad\hat{\mathsf{C}}^{\prime\alpha+}={\mathsf{C}}^{\prime\alpha+},\quad\hat{\mathsf{G}}^{\alpha+}={\mathsf{G}}^{\alpha+},$
(297) $\displaystyle{\mathsf{C}}^{\alpha+}={\mathsf{C}}^{\alpha+}_{\rm
V}+{\mathsf{C}}^{\alpha+}_{\rm
S}+{\mathsf{C}}^{\alpha+}_{\Gamma}+{\mathsf{C}}^{\alpha+}_{\rm
A}+{\mathsf{C}}^{\alpha+}_{\rm F}+{\mathsf{C}}^{\alpha+}_{\Phi},$ (298)
$\displaystyle{\mathsf{C}}^{\prime\alpha+}={\mathsf{C}}^{\prime\alpha+}_{\rm
V}+{\mathsf{C}}^{\prime\alpha+}_{\rm
S}+{\mathsf{C}}^{\prime\alpha+}_{\Gamma}+{\mathsf{C}}^{\prime\alpha+}_{\rm
A}+{\mathsf{C}}^{\prime\alpha+}_{\rm F}+{\mathsf{C}}^{\prime\alpha+}_{\Phi};$
(299) $\displaystyle{\mathsf{C}}^{\alpha+}_{\rm V}=n_{0}^{\alpha}(p_{{\rm
R}0}^{\alpha}+B_{\eta}^{\alpha}),\quad{\mathsf{C}}^{\alpha+}_{\rm
S}={\textstyle{\frac{4}{3}}}n_{0}^{\alpha}\mu_{\rm S}^{\alpha},$ (300)
$\displaystyle{\mathsf{C}}^{\alpha+}_{\Gamma}=n_{0}^{\alpha}\bigr{(}B_{\theta}^{\alpha}\sum_{l}\beta^{\alpha}_{{\rm
V}l}+{\textstyle{\frac{4}{3}}}\mu^{\alpha}_{\rm S}\sum_{m}\beta^{\alpha}_{{\rm
S}m}\bigr{)},$ (301) $\displaystyle{\mathsf{C}}^{\alpha+}_{\rm
A}=n^{\alpha}_{0}(\partial^{2}\Psi^{\alpha}_{{\rm
A}k}/\partial(J^{\alpha})^{2})^{+}\qquad(k=1),$ (302)
$\displaystyle{\mathsf{C}}^{\alpha+}_{\rm
F}=\begin{cases}\frac{8}{9}n_{0}^{\alpha}\mu^{\alpha}_{k}&(\kappa^{\alpha}_{k}=0),\\\
0&(\kappa^{\alpha}_{k}=\frac{1}{3}),\\\ \end{cases}$ (303)
$\displaystyle{\mathsf{C}}^{\alpha+}_{\Phi}=\begin{cases}\frac{8}{9}n_{0}^{\alpha}\mu^{\alpha}_{k}\sum_{n}\beta^{\alpha}_{\Phi
k,n}&(\kappa^{\alpha}_{k}=0),\\\ 0&(\kappa^{\alpha}_{k}=\frac{1}{3});\\\
\end{cases}$ (304) $\displaystyle{\mathsf{C}}^{\prime\alpha+}_{\rm
V}=-n_{0}^{\alpha}\\{2p^{\alpha}_{{\rm R}0}+B^{\alpha}_{\eta}(1+B_{\eta{\rm
p}}^{\alpha})\\},$ (305) $\displaystyle{\mathsf{C}}^{\prime\alpha+}_{\rm
S}=-{\textstyle{\frac{28}{9}}}n_{0}^{\alpha}\mu_{\rm S}^{\alpha},$ (306)
$\displaystyle{\mathsf{C}}^{\prime\alpha+}_{\Gamma}=-3n_{0}^{\alpha}\bigr{(}B_{\theta}^{\alpha}\sum_{l}\beta^{\alpha}_{{\rm
V}l}+{\textstyle{\frac{28}{27}}}\mu^{\alpha}_{\rm
S}\sum_{m}\beta^{\alpha}_{{\rm S}m}\bigr{)},$ (307)
$\displaystyle{\mathsf{C}}^{\prime\alpha+}_{\rm
A}=n^{\alpha}_{0}(\partial^{3}\Psi^{\alpha}_{{\rm
A}k}/\partial(J^{\alpha})^{3})^{+}\qquad(k=1),$ (308)
$\displaystyle{\mathsf{C}}^{\prime\alpha+}_{\rm
F}=\begin{cases}\frac{32}{27}n_{0}^{\alpha}\mu^{\alpha}_{k}&(\kappa^{\alpha}_{k}=0),\\\
0&(\kappa^{\alpha}_{k}=\frac{1}{3}),\\\ \end{cases}$ (309)
$\displaystyle{\mathsf{C}}^{\prime\alpha+}_{\Phi}=\begin{cases}\frac{32}{27}n_{0}^{\alpha}\mu^{\alpha}_{k}\sum_{n}\beta^{\alpha}_{\Phi
k,n}&(\kappa^{\alpha}_{k}=0),\\\ 0&(\kappa^{\alpha}_{k}=\frac{1}{3});\\\
\end{cases}$ (310) $\displaystyle{\bm{\mathsf{B}}}^{\alpha+}\rightarrow{\bf
0},\quad{\bm{\mathsf{b}}}^{\alpha+}_{\mathsf{a}}\rightarrow{\bf
0},\quad{\bm{\mathsf{b}}}^{\prime\alpha+}\rightarrow{\bf
0},\quad{\bm{\mathsf{b}}}^{\alpha+}_{\eta}\rightarrow{\bf 0},$ (311)
$\displaystyle{\bm{\mathsf{A}}}^{\alpha+}\rightarrow-
n_{0}^{\alpha}\biggr{\\{}\frac{\partial{\bf Q}^{\alpha}_{{\rm V}l}}{\partial
J^{\alpha}},\frac{\partial{\bf Q}^{\alpha}_{{\rm S}m}}{\partial
J^{\alpha}},\frac{\partial{\bf Q}^{\alpha}_{\Phi k,n}}{\partial
J^{\alpha}}\biggr{\\}}^{+},$ (312)
$\displaystyle\biggr{(}\frac{\partial(D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha})}{\partial
J^{\alpha}}\biggr{)}^{+}\rightarrow\biggr{\\{}\frac{\partial{\bf
Q}^{\alpha}_{{\rm V}l}/\partial J^{\alpha}}{\beta^{\alpha}_{{\rm
V}l}B_{\theta}^{\alpha}\tau^{\alpha}_{{\rm V}l}},\frac{\partial{\bf
Q}^{\alpha}_{{\rm S}m}/\partial J^{\alpha}}{\beta^{\alpha}_{{\rm S}m}\mu_{\rm
S}^{\alpha}\tau^{\alpha}_{{\rm S}m}},\cdots$ $\displaystyle(\partial{\bf
Q}^{\alpha}_{\Phi k,n}/\partial J^{\alpha})/(\beta^{\alpha}_{\Phi
k,n}\mu_{k}^{\alpha}\tau^{\alpha}_{\Phi
k,n})\bigr{\\}},\quad\,{\bm{\pi}}^{\alpha+}_{\mathsf{a}}\rightarrow{\bf
0},\quad$ (313)
$\displaystyle{\bm{\mathsf{A}}}^{\alpha+}\cdot\bigr{(}{\partial(D_{t}^{\alpha}{\bm{\mathsf{a}}}^{\alpha})}/{\partial
J^{\alpha}}\bigr{)}^{+}\rightarrow{\mathcal{A}}^{\alpha}_{\Gamma{\rm
V}}+{\mathcal{A}}^{\alpha}_{\Gamma{\rm S}}+{\mathcal{A}}^{\alpha}_{\Phi},$
(314) $\displaystyle{\mathcal{A}}^{\alpha}_{\Gamma{\rm
V}}=-3n_{0}^{\alpha}B_{\theta}^{\alpha}\sum_{l}\frac{\beta^{\alpha}_{{\rm
V}l}}{\tau^{\alpha}_{{\rm V}l}},\quad{\mathcal{A}}^{\alpha}_{\Gamma{\rm
S}}=-{\textstyle{\frac{8}{3}}}n_{0}^{\alpha}\mu^{\alpha}_{\rm
S}\sum_{m}\frac{\beta^{\alpha}_{{\rm S}m}}{\tau^{\alpha}_{{\rm S}m}},$
$\displaystyle{\mathcal{A}}^{\alpha}_{\Phi}=\begin{cases}-\frac{32}{27}n_{0}^{\alpha}\mu^{\alpha}_{k}\sum_{n}{\beta^{\alpha}_{\Phi
k,n}}/{\tau^{\alpha}_{\Phi k,n}}&(\kappa_{k}^{\alpha}=0),\\\
0&(\kappa_{k}^{\alpha}=\frac{1}{3}).\end{cases}$ (315)

Table 3: Shock evolution parameters for rabbit muscle (ECF and solid, $n_{0}^{1}=0.9$), bovine liver (blood and solid, $n_{0}^{1}=0.88$) and canine lung (air and solid, $n_{0}^{1}=0.336$). “Mixture” invokes same shock simultaneously to each phase. $(\Delta P)^{\alpha}_{0}$, $(\Delta\theta)^{\alpha}_{0}$, $(\Delta\bar{D})^{\alpha}_{0}$, and $(\Delta D_{1})^{\alpha}_{0}$ are initial stress, temperature, and matrix/fiber damage jumps for initial strain change $\Delta J^{\alpha}_{0}=\llbracket J^{\alpha}\rrbracket(t=0)=-0.1$. Property or |  | Muscle |  |  | Liver |  |  | Lung |
---|---|---|---|---|---|---|---|---|---
model prediction | ECF | Solid | Mixture | Blood | Solid | Mixture | Air | Solid | Mixture
$\rho^{\alpha}_{0}$ [g/cm3] | 0.103 | 0.990 | 1.093 | 0.127 | 0.933 | 1.060 | 7.56$\times 10^{-4}$ | 0.337 | 0.338
$\hat{\mathsf{C}}^{\alpha+}$ [GPa] | 0.220 | 2.954 | 3.174 | 0.317 | 2.350 | 2.667 | 9.42$\times 10^{-5}$ | 9.26$\times 10^{-5}$ | 1.87$\times 10^{-4}$
$\hat{\mathsf{C}}^{\prime\alpha+}$ [GPa] | -1.751 | -26.57 | -28.32 | -4.118 | -21.15 | -25.27 | -2.26$\times 10^{-4}$ | -2.42$\times 10^{-4}$ | -4.68$\times 10^{-4}$
${\mathcal{C}}^{\alpha}$ [km/s] | 1.462 | 1.727 | 1.704 | 1.578 | 1.587 | 1.586 | 0.353 | 1.66$\times 10^{-2}$ | 2.35$\times 10^{-2}$
$\hat{\mathsf{G}}^{\alpha+}$ [g$\cdot$K/cm3] | -4.215 | -96.06 | -99.15 | -6.309 | -32.97 | -39.31 | -9.38$\times 10^{-2}$ | -11.91 | -11.95
${\omega}^{\alpha}$ [1/s] | 8.69$\times 10^{9}$ | 9.05$\times 10^{8}$ | 7.01$\times 10^{-3}$ | 1.89$\times 10^{7}$ | 2.57$\times 10^{6}$ | 6.62$\times 10^{-2}$ | 2.97$\times 10^{4}$ | 1.13$\times 10^{2}$ | 2.95$\times 10^{-2}$
${\Lambda}^{\alpha}$ [1/m] | 2.99$\times 10^{6}$ | 2.33$\times 10^{5}$ | 1.84$\times 10^{-6}$ | 3.68$\times 10^{3}$ | 7.21$\times 10^{2}$ | 1.76$\times 10^{-5}$ | 1.40$\times 10^{2}$ | 1.04$\times 10^{1}$ | 2.01$\times 10^{-3}$
$(\Delta P)^{\alpha}_{0}$ [MPa] | 30.8 | 428 | 459 | 52.3 | 341 | 393 | 1.06$\times 10^{-2}$ | 1.05$\times 10^{-2}$ | 2.10$\times 10^{-2}$
$(\Delta\theta)^{\alpha}_{0}$ [K] | 4.32 | 10.3 | 9.66 | 5.25 | 3.73 | 3.92 | 13.3 | 3.73 | 3.74
$(\Delta\bar{D})^{\alpha}_{0}$ [-] matrix | $\ldots$ | 0.045 | 0.045 | $\ldots$ | 0.100 | 0.100 | $\ldots$ | 5.03$\times 10^{-3}$ | 5.03$\times 10^{-3}$
$(\Delta D_{1})^{\alpha}_{0}$ [-] fibers | $\ldots$ | 0.022 | 0.022 | $\ldots$ | 0 | 0 | $\ldots$ | 0 | 0

For rate-insensitive damage, from $\bar{\pi}^{\alpha}_{{\rm D}}=0$ and
$\pi^{\alpha}_{{\rm D}k}=0$ via (224) and using (251) and (311) with
$\bar{\vartheta}^{\alpha}=\vartheta^{\alpha}_{k}=2$,

$\displaystyle\bar{D}^{\alpha-}=\frac{1}{n_{0}^{\alpha}\bar{E}_{\rm
C}^{\alpha}}({\mathsf{C}}^{\alpha+}_{\rm
S}+{\mathsf{C}}^{\alpha+}_{\Gamma})\llbracket
J^{\alpha}\rrbracket^{2}+O(\llbracket J^{\alpha}\rrbracket^{3}),$ (316)
$\displaystyle{D}^{\alpha-}_{k}=\frac{1}{n_{0}^{\alpha}{E}_{{\rm
C}k}^{\alpha}}({\mathsf{C}}^{\alpha+}_{\rm
F}+{\mathsf{C}}^{\alpha+}_{\Phi})\llbracket
J^{\alpha}\rrbracket^{2}+O(\llbracket J^{\alpha}\rrbracket^{3}).$ (317)

Notice ${D}^{\alpha-}_{k}\rightarrow 0$ in (317) if
$\kappa^{\alpha}_{k}=\frac{1}{3}$. To at least $O(\llbracket
J^{\alpha}\rrbracket^{2})$, (316) and (317) are unaffected by Finsler versus
Euclidean metrics $({\bf g},{\bf G}^{\alpha})$ for current prescriptions
$\bar{r}^{\alpha}\geq 2$ and $\tilde{r}^{\alpha}_{k}\geq 2$. From
${\bm{\mathsf{b}}}^{\alpha+}\rightarrow{\bf 0}$ and (311), rate-insensitive
fractures do not affect weak-shock evolution (282)–(285).

If fractures are rate dependent, then
${\bm{\mathsf{b}}}^{\alpha}\rightarrow{\bf 0}$ and
${\bm{\mathsf{a}}}^{\alpha}\rightarrow\\{{\bm{\Gamma}}^{\alpha}_{{\rm
V}l},{\bm{\Gamma}}^{\alpha}_{{\rm
S}m},{\bm{\Gamma}}^{\alpha}_{{\Phi}k,n},\bar{D}^{\alpha},D^{\alpha}_{k}\\}$.
Then (235) yields $\bar{D}^{\alpha-}=0$ and $D^{\alpha-}_{k}=0$ in lieu of
(316) and (317). From (182) and (187), damage kinetics do not contribute to
${\bm{\mathsf{A}}}^{\alpha+}$ or ${\bm{\pi}}^{\alpha+}_{\mathsf{a}}$ nor
$\Lambda^{\alpha}$ or $\omega^{\alpha}$ in (282), (IV.3); (312)–(314) are
unchanged. Importantly, damage, regardless of rate (in)dependence, can still
affect strong-shock evolution in (274)–(IV.3).

Phase interactions affect $\Lambda^{\alpha}$ and $\omega^{\alpha}$, from
(190)–(III.3),

$\displaystyle\biggr{(}\frac{\partial
h^{\alpha}}{\partial\upsilon^{\alpha}}\biggr{)}^{+}\\!=-\sum_{\beta\neq\alpha}\bar{\lambda}^{\alpha\beta+},\,\biggr{(}\frac{\partial\epsilon^{\alpha}}{\partial\theta^{\alpha}}\biggr{)}^{+}\\!=-\sum_{\beta\neq\alpha}\bar{\omega}^{\alpha\beta+}\\!.$
(318)

Consider now a two-phase mixture of solid ($\alpha=1\rightarrow{\rm s}$) and
fluid ($\alpha=2\rightarrow{\rm f}$). Adopting physics in Refs. [48, 51],

$\displaystyle\bar{\lambda}^{12+}=(n^{\rm f}_{0})^{2}\hat{\mu}^{\rm
f}/\Xi,\qquad\bar{\omega}^{12+}=\alpha_{\rm v}{\kappa}^{\rm fs},$ (319)

with fluid viscosity $\hat{\mu}^{\rm f}$, system permeability $\Xi$,
interfacial area per unit volume $\alpha_{\rm v}$, and heat transfer
coefficient ${\kappa}^{\rm fs}$. Although macroscopic Newtonian viscosity and
Fourier conduction are excluded for singular shocks [109, 110], microscopic
$h^{\alpha}$ and $\epsilon^{\alpha}$ include viscosity and heat transfer.

Recall from (IV.3) that the weak-shock solution (285) for a multi-phase
material corresponds to strain jump $\Delta J^{\alpha}_{0}$ and resulting
discontinuities in $P^{\alpha}$ and $\theta^{\alpha}$ applied as a loading
condition for one phase $\alpha$, with all other phase
$\beta=1,2,\ldots\alpha-1,\alpha+1,\ldots,N$ witnessing no discontinuities in
$J^{\beta}$, $P^{\beta}$, or $\theta^{\beta}$. This shock moves through all
phases at speed ${\mathcal{C}}^{\alpha}$; phase interactions $h^{\alpha}$ and
$\epsilon^{\alpha}$ induce decay in amplitude $\llbracket
J^{\alpha}\rrbracket(t)$ so long as $\bar{\lambda}^{\alpha\beta}>0$ and
$\bar{\omega}^{\alpha\beta}>0$. Velocities $\upsilon^{\beta}$ and temperatures
$\theta^{\beta}$ ($\beta\neq\alpha$) can evolve continuously in space-time
behind the wave front from such interactions, their values indeterminate.

In contrast, if the mixture is idealized as homogeneous with matching
$\upsilon^{\alpha}$ and $\theta^{\alpha}$, then $h^{\alpha}$ and
$\epsilon^{\alpha}$ do not explicitly affect shock evolution. In this “tied”
case, $\Delta J^{\alpha}_{0}$ is applied simultaneously at $t=0$ to all phases
$\alpha=1,\ldots,N$ as a loading condition. Speed
$\mathcal{C}=(\hat{\mathsf{C}}/\rho_{0})^{1/2}$ results from stiffness and
density of the whole mixture in (IV.3), and

$\displaystyle\hat{\mathsf{C}}^{\prime+}=\sum_{\alpha}\hat{\mathsf{C}}^{\prime\alpha+},\quad\hat{\mathsf{G}}^{+}=-\rho_{0}\theta_{0}\frac{\sum_{\alpha}\gamma_{0}^{\alpha}\rho_{0}^{\alpha}c_{\epsilon}^{\alpha}}{\sum_{\alpha}\rho_{0}^{\alpha}c_{\epsilon}^{\alpha}}.$
(320)

Predictions. The analytical solution for weak shock evolution, (282)–(285), is
applied to three biologic systems at $\theta_{0}=310\,$K, each comprised of
one solid tissue phase and one fluid: skeletal muscle with interstitial fluid,
liver with blood, and lung with air. Properties, $\omega^{\alpha}$, and
$\Lambda^{\alpha}$ are given in Table 3 for each component, and for the
homogeneous idealization of (IV.3), (287), and (320) labeled “Mixture”. The
lower four rows contain initial jumps in stress (248), temperature (250),
matrix- (316), and fiber-damage (317), each to $O(\llbracket
J^{\alpha}\rrbracket^{2}$) for initial strain jump $\Delta
J^{\alpha}_{0}=-0.1$. If damage is rate-dependent, its jumps are zero instead.
Normalized exponential decay of $\llbracket P^{\alpha}\rrbracket$ and
$\llbracket\theta^{\alpha}\rrbracket$ arising from (285) is shown for each
component in Fig. 6. When mixtures are shocked uniformly, decay from
viscoelastic dissipation alone manifests over much larger distances (not
shown).

(

Figure 6:  Predicted ratio of jump in shock stress $(\Delta P)^{\alpha}$ or
temperature $(\Delta\theta)^{\alpha}$ normalized by initial magnitude $(\Delta
P)^{\alpha}_{0}$ or $(\Delta\theta)^{\alpha}_{0}$, for shock amplitude $\Delta
J^{\alpha}_{0}$ applied individually to phase $\alpha$, vs. propagation
distance $X^{\alpha}$ in the weak shock limit: (a) rabbit muscle, (b) bovine
liver, and (c) canine lung. Wave speed is $\mathcal{C}^{\alpha}$; different
scales (i.e., $\mu$m, mm, or m) used for $X^{\alpha}$.

a)
b)
(c)

(

Constitutive and metric parameters are those for rabbit skeletal muscle and
bovine lung of Sec. IV.2, Table 1 (ECF, blood) and Table 2 (solid phases). For
muscle, active tension from (221) affects initial stress but does not
appreciably change tabulated results. Air is modeled as an ideal gas with
${\mathfrak{R}}^{\alpha}=287\,$J/kg$\cdot$K and $\gamma_{0}^{\alpha}=0.4$
[74]. Viscosity for air in (319) is $\hat{\mu}^{\rm f}=18.3\,\mu{\rm
Pa}\cdot{\rm s}$ [48]. Solid properties are for canine lung [95, 6, 74] with
bulk and shear viscoelasticity ($l=m=1$), isotropic fibers
($\kappa^{\alpha}_{k}=\frac{1}{3}$), solid fraction $n_{0}^{1}=0.336$,
$B^{\alpha}_{\eta}=164\,$kPa, $\mu_{\rm S}^{\alpha}=2.98\,$kPa,
$\beta^{\alpha}_{{\rm V}l}=0.009$, $\beta^{\alpha}_{{\rm S}m}=1.5$,
$\tau^{\alpha}_{{\rm V}l}=\tau^{\alpha}_{{\rm S}m}=0.5\,$s. Values
$\gamma_{0}^{\alpha}=0.114$ and $\bar{E}_{\rm C}^{\alpha}=22.7\,$kPa,
unmeasured for lung parenchyma, are borrowed from liver parenchyma. This value
of $\gamma_{0}^{\alpha}$ for lung is $\approx 2\times$ that of Ref. [74], with
the latter estimate $10^{3}\times$ that of classical thermodynamics [130]
using the low $B^{\alpha}_{\theta}$ of the highly porous structure.

In (319), $\Xi=6.7\times 10^{-18}{\rm m}^{2}$ for muscle [131], $\Xi=1.5\times
10^{-14}{\rm m}^{2}$ for liver [8], and $\Xi=1.83\times 10^{-10}{\rm m}^{2}$
for lung [48]. Regarding heat transfer, $\kappa^{\rm fs}=6\,$W/m${}^{2}\cdot$K
for muscle and liver [132], and $\kappa^{\rm fs}=41.2\,$W/m${}^{2}\cdot$K for
lung [133]. From idealized microstructure geometries [131, 121, 1], a contact
area estimate for muscle is $\alpha_{\rm v}=\pi/R_{0}$ with $R_{0}=30\,\mu$m
the fiber radius [112], for liver $\alpha_{\rm v}=2n_{0}^{\rm f}/R_{0}$ with
$R_{0}=4\,\mu$m the capillary radius [132], and for lung $\alpha_{\rm
v}=\pi/(2R_{0})$ with $R_{0}=30\,\mu$m the alveolar radius [95].

From Table 3 and Fig. 6, when $\Delta J^{\alpha}_{0}$ is applied to one phase
alone, then $|\Lambda^{\alpha}|$ and $|\omega^{\alpha}|$ are large, with
$\omega^{\alpha}>0$ for transient decay. For ${\rm d}P^{\alpha}_{\rm H}/{\rm
d}\llbracket J^{\alpha}\rrbracket\approx-\hat{\mathsf{C}}^{\alpha-}<0$ and
since $\Lambda^{\alpha}>0$, a negative stress gradient $\nabla_{0}P^{\alpha-}$
is needed for a steady shock: $P^{\alpha}$ should decrease steeply as
$\Sigma^{\alpha}$ is approached from the $(\cdot)^{-}$ side. For muscle and
liver, $(\partial h^{\alpha}/\partial\upsilon^{\alpha})^{+}$ dominates
$\omega^{\alpha}$ due to relatively large $\Xi$ and $\hat{\mu}^{\rm f}$ in
(319). For lung, both $(\partial h^{\alpha}/\partial\upsilon^{\alpha})^{+}$
and $(\partial\epsilon^{\alpha}/\partial\theta^{\alpha})^{+}$ have significant
influence on shock decay, the latter due to a comparatively low
${\mathcal{C}}^{\alpha}$. Viscoelastic dissipation embodied in
${\mathcal{A}}_{\Gamma{\rm V}}$, ${\mathcal{A}}_{\Gamma{\rm S}}$, and
${\mathcal{A}}_{\Phi}$ has relatively small effects, negligible compared to
interphase drag and heat exchange. The near-incompressibility of these
tissues, also typical of soft polymers at high rates [134], leads to small
glassy shear moduli relative to bulk moduli, the latter (bulk) modeled here
having little or no viscoelastic relaxation.

From Fig. 6, decay distance is shortest in muscle and longest in lung. In each
system, a shock applied to the fluid decays over a much shorter distance than
one applied to the solid. From Table 3, stress rises $(\Delta P)^{\alpha}_{0}$
are highest in solid phases of muscle and liver having largest tangent moduli.
Temperature $(\Delta\theta)^{\alpha}_{0}$ is highest in air in the lung.
Damage is greatest in liver and smallest in lung; fiber damage is absent in
both organs in the weak-shock limit as fibers are isotropic. In anisotropic
muscle, fiber damage $(\Delta{D}_{k})^{\alpha}_{0}$ is
$\approx\frac{1}{2}\times$ matrix damage $(\Delta\bar{D})^{\alpha}_{0}$, in
contrast to (strong-shock) Hugoniot solutions of Fig. 3(b,c) whereby these
damage mechanisms evolve more similarly.

When $\Delta J^{\alpha}_{0}$ is applied to both phases at once and the
homogeneous idealization is invoked, $|\Lambda^{\alpha}|$ and
$|\omega^{\alpha}|$ are deemed small, with contributions only from
viscoelasticity. For these conditions, weak shocks should be steady with near-
uniform stress-strain states trailing their wave fronts. Weak-shock
experiments are needed to confirm this prediction; known data [112] are for
strong shocks.

##  V  Concluding remarks

A theoretical framework is posited for modeling multi-phase soft biologic
materials over wide ranges of loading rate and pressure. Results for uniaxial-
stress and shock loading agree with experimental data on biologic fluids,
skeletal muscle, and liver. Damage, represented by order parameters for matrix
and fiber fractures, is found to be rate insensitive in muscle but rate
dependent in liver. An analytical solution has been derived for shock
evolution including phase interactions, viscoelasticity, and tissue damage.
Predictions for weak shock decay reveal dominance of interphase momentum and
energy exchange over viscoelastic dissipation and damage/fracture kinetics.

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