Following the approach of continuum mechanics, we replace the discrete molecular structure of materials by a continuously distributed set of material points which undergo mappings from the initial (reference), Ω_{0}, to current, Ω, configuration: **x**↦**y**(**x**). The deformation in the vicinity of the material points is described by the deformation gradient **F**=Grad**y**(**x**).

In what follows we use the Lagrangean description with respect to the initial or reference configuration and define the local mass balance in the form

$$ \frac{d\rho}{dt}=\text{Div}\mathbf{s}+\xi, $$

(1)

where *ρ* is the referential (Lagrangean) mass density; **s** is the referential mass flux; *ξ* is the referential mass source (sink); and Div**s**=*∂*
*s*
_{
i
}/*∂*
*x*
_{
i
} in Cartesian coordinates.

*We further assume that failure and, consequently, mass flow are highly localized and the momenta and energy balance equations can be written in the standard form without adding momenta and energy due to the mass alterations.*

In view of the assumption above, we write momenta and energy balance equations in the following forms accordingly

$$ \frac{d(\rho\mathbf{v})}{dt}=\text{Div}\mathbf{P}+\rho\mathbf{b},\quad\mathbf{P}\mathbf{F}^{\mathrm{T}}=\mathbf{F}\mathbf{P}^{\mathrm{T}}, $$

(2)

and

$$ \frac{d(\rho e)}{dt}=\mathbf{P}:\dot{\mathbf{F}}+\rho r-\text{Div}\mathbf{q}, $$

(3)

where \(\mathbf {v}=\dot {\mathbf {y}}\) is the velocity of a material point; **b** is the body force per unit mass; **P** is the first Piola-Kirchhoff stress and (Div**P**)_{
i
}=*∂*
*P*
_{
ij
}/*∂*
*x*
_{
j
}; *e* is the specific internal energy per unit mass; *r* is the specific heat source per unit mass; and **q** is the referential heat flux.

Entropy inequality reads

$$ \frac{d(\rho\eta)}{dt}\geq\frac{1}{T}(\rho r-\text{Div}\mathbf{q})+\frac{1}{T^{2}}\mathbf{q}\cdot\text{Grad}T, $$

(4)

where *T* is the absolute temperature.

Substitution of (*ρ*
*r*−Div**q**) from (3) to (4) yields

$$ \rho\dot{\eta}+\dot{\rho}\eta\geq\frac{1}{T}(\rho\dot{e}+\dot{\rho}e-\mathbf{P}:\dot{\mathbf{F}})+\frac{1}{T^{2}}\mathbf{q}\cdot\text{Grad}T, $$

(5)

or, written in terms of the internal dissipation,

$$ D_{\text{int}}=\mathbf{P}:\dot{\mathbf{F}}-\rho(\dot{e}-T\dot{\eta})-\dot{\rho}(e-T\eta)-\frac{1}{T}\mathbf{q}\cdot\text{Grad}T\geq0. $$

(6)

We introduce the specific Helmholtz free energy per unit mass

and, consequently, we have

$$ e=w+T\eta,\quad\dot{e}=\dot{w}+\dot{T}\eta+T\dot{\eta}. $$

(8)

Substituting (8) in (6) we get

$$ D_{\text{int}}=\mathbf{P}:\dot{\mathbf{F}}-\rho(\dot{w}+\dot{T}\eta)-\dot{\rho}w-\frac{1}{T}\mathbf{q}\cdot\text{Grad}T\geq0. $$

(9)

Then, we calculate the Helmholtz free energy increment

$$ \dot{w}=\frac{\partial w}{\partial\mathbf{F}}:\dot{\mathbf{F}}+\frac{\partial w}{\partial T}\dot{T}, $$

(10)

and substitute it in (9) as follows

$$ D_{\text{int}}=\left(\mathbf{P}-\rho\frac{\partial w}{\partial\mathbf{F}}\right):\dot{\mathbf{F}}-\rho\left(\frac{\partial w}{\partial T}+\eta\right)\dot{T}-\dot{\rho}w-\frac{1}{T}\mathbf{q}\cdot\text{Grad}T\geq0. $$

(11)

The Coleman-Noll procedure suggests the following choice of the constitutive laws

$$ \mathbf{P}=\rho\frac{\partial w}{\partial\mathbf{F}},\quad\eta=-\frac{\partial w}{\partial T}. $$

(12)

and, consequently, the dissipation inequality reduces to

$$ D_{\text{int}}=-\dot{\rho}w-\frac{1}{T}\mathbf{q}\cdot\text{Grad}T\geq0. $$

(13)

*We further note that the process of the bond breakage is very fast as compared to the dynamic deformation process and the mass density changes in time as a step function. So, strictly speaking, the density rate should be presented by the Dirac delta in time. We will not consider the super fast transition to failure, which is of no interest on its own, and assume that the densities before and after failure are constants and, consequently,*

$$\dot{\rho}=\text{Div}\mathbf{s}+\xi=0, $$

or

$$ \text{Div}\mathbf{s}+\xi=0. $$

(14)

Then, the dissipation inequality reduces to

$$ D_{\text{int}}=-\frac{1}{T}\mathbf{q}\cdot\text{Grad}T\geq0, $$

(15)

which is obeyed because the heat flows in the direction of the lower temperature.

It remains to settle the boundary and initial conditions.

Natural boundary conditions for zero mass flux represent the mass balance on the boundary *∂*Ω_{0}

$$ \mathbf{s}\cdot\mathbf{n}=0, $$

(16)

where **n** is the unit outward normal to the boundary in the reference configuration.

Natural boundary conditions for given traction \(\bar {\mathbf {t}}\) represent the linear momentum balance on the boundary *∂*Ω_{0}

$$ \mathbf{P}\mathbf{n}=\bar{\mathbf{t}}, $$

(17)

or, alternatively, the essential boundary conditions for placements can be prescribed on *∂*Ω_{0}

$$ \mathbf{y}=\bar{\mathbf{y}}. $$

(18)

Initial conditions in Ω_{0} complete the formulation of the coupled mass-flow-elastic initial boundary value problem

$$ \mathbf{y}(t=0)=\mathbf{y}_{0},\quad\mathbf{v}(t=0)=\mathbf{v}_{0}. $$

(19)

**Remark** The fact that we ignore the process of the transition to failure and use (14) instead of (1) might be difficult to comprehend at first glance. To ease the comprehension the reader might find it useful to consider the analogy between fracture and the buckling process in thin-walled structure. The pre-buckled and post-buckled states of a structure are usually analyzed by using a time-independent approach. The very process of the fast dynamic transition to the buckled state is of no interest and it is normally ignored in analysis by dropping the inertia terms from the momentum balance equation: \(\frac {d(\rho \mathbf {v})}{dt}=\text {Div}\mathbf {P}+\rho \mathbf {b}=\mathbf {0}\) or Div**P**+*ρ*
**b**=**0**. By analogy with the buckling analysis we are only interested in the pre-cracked and post-cracked states while the transition (bond rupture) process can be ignored. The latter is the reason why the mass balance can be written in the simplified form: \(\frac {d\rho }{dt}=\text {Div}\mathbf {s}+\xi =0\) or Div**s**+*ξ*=0. It is also important to emphasize that the proposed simplification does not affect the natural boundary condition (16). This boundary condition is the expression of the mass balance on the boundary, which is obtained by using the standard Cauchy tetrahedron argument.