2.4 Axioms of Conservation

The governing equations of continuum mechanics are given by the axioms of conservation of mass, linear and angular momentum, and energy. These axioms produce differential equations that need to be solved for the material response. However, on their own, these equations are insufficient to solve for all the unknowns in an analysis, therefore we must also provide equations of state that describe the behavior of specific classes of materials, such as elastic solids, viscoelastic solids, viscous fluids, etc. These equations of state are formulated as constitutive models, which may not be chosen arbitrarily: they must be constrained by the axiom of entropy inequality. In this section, we summarize all the axioms of conservation for a pure substance, as well as the axiom of entropy inequality.

Axiom of Mass Balance

The axiom of mass balance simply states that mass is conserved, or equivalently, its time rate of change is zero. Normally, we formulate this axiom in a volumetric domain (such as a fixed control volume with permeable boundary, a material region with impermeable boundary, or a material region with permeable boundary), then convert it to its differential form. In a spatial (Eulerian) frame, the differential statement of the axiom of mass balance takes the form

\[ \begin{equation} \frac{\partial\rho}{\partial t}+\divg\left(\rho\mathbf{v}\right)=0\,,\label{eq:axiom-mass-balance} \end{equation} \]

where \(\rho\) is the mass density of the matter and \(\mathbf{v}\) is its velocity vector. Since \(\divg\left(\rho\mathbf{v}\right)=\grad\rho\cdot\mathbf{v}+\rho\divg\mathbf{v}\), we may also write this relation in the form

\[ \begin{equation} \frac{D\rho}{Dt}+\rho\divg\mathbf{v}=0\,,\label{eq:mass-balance-redux} \end{equation} \]

where

\[ \begin{equation} \frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+\grad\rho\cdot\mathbf{v}\label{eq:material-derivative-density} \end{equation} \]

is the material time derivative of the mass density in the spatial frame.

Axioms of Momentum Balance

The axioms of linear and angular momentum balance are also known as Newton's second law of motion. The axiom of linear momentum balance states that the time rate of change of linear momentum is equal to the sum of all surface and body forces. The differential statement of the axiom of mass balance in a spatial frame is given by

\[ \begin{equation} \rho\frac{D\mathbf{v}}{Dt}=\divg\boldsymbol{\sigma}+\rho\mathbf{b}\,,\label{eq:axiom-linear-momentum} \end{equation} \]

where \(\boldsymbol{\sigma}\) is the Cauchy stress tensor in the material, and \(\mathbf{b}\) is the specific body force vector (body force per mass), which is typically prescribed. Note that \(D\mathbf{v}/Dt\) is the acceleration of the matter at every point in the continuum, as evaluated in the spatial frame.

The axiom of angular momentum balance states that the time rate of change of angular momentum is equal to the sum of all surface and body moments. The angular momentum represents the moment of the linear momentum about a fixed point (such as the origin) in an inertial frame. In a non-polar medium, we assume that moments are only produced by surface and body forces. In that case, the axiom of angular momentum balance is satisfied by the linear momentum balance and by letting the Cauchy stress be symmetric,

\[ \begin{equation} \boldsymbol{\sigma}^{T}=\boldsymbol{\sigma}\,.\label{eq:axiom-angular-momentum} \end{equation} \]

Axiom of Energy Balance

The axiom of energy balance is also known as the first law of thermodynamics. The time rate of change of internal and kinetic energy is balanced by the rate of work done by body and surface forces, the rate of heat flow into the medium, and rate of heat supply from external sources. Its differential statement takes the form

\[ \begin{equation} \rho\frac{D\varepsilon}{Dt}=\boldsymbol{\sigma}:\mathbf{L}-\divg\mathbf{q}+\rho r\,,\label{eq:axiom-energy-balance} \end{equation} \]

where \(\varepsilon\) is the specific internal energy (internal energy per mass), \(\mathbf{L}=\grad\mathbf{v}\) is the velocity gradient, \(\mathbf{q}\) is the heat flux in the medium (units of power per area), and \(r\) is the heat supply per unit mass from external sources that are not modeled explicitly (such as microwave heating in a framework that does not model microwave radiation explicitly, or Joule heating in electrical conductors, in a framework that does not model frictional interactions between electrons and atomic nuclei explicitly).

For non-polar media, since the Cauchy stress is symmetric according to eq.\eqref{eq:axiom-angular-momentum}, it follows that \(\boldsymbol{\sigma}:\mathbf{L}=\boldsymbol{\sigma}:\mathbf{D}\), where \(\mathbf{D}=\frac{1}{2}\left(\mathbf{L}+\mathbf{L}^{T}\right)\) is the symmetric part of the velocity gradient, also known as the rate of deformation tensor.

Axiom of Entropy Inequality

The axioms of conservation presented above have introduced several functions of state that describe the behavior of specific materials, such as the Cauchy stress \(\boldsymbol{\sigma}\), the specific internal energy \(\varepsilon\), and the heat flux \(\mathbf{q}\). These axioms of conservation have a total of ten scalar unknowns (six compoments of \(\boldsymbol{\sigma}\), three components of \(\mathbf{q}\), and \(\varepsilon\)), whereas the equations have five scalar components (three components of the vectorial linear momentum balance, and one component for each of the scalar mass and energy balance equations). Therefore, there are too few equations to solve for all the unknowns. Consequently, constitutive models must be provided for these functions of state, which depend on a user-specified set of state variables (such as strain for elastic solids, rate of deformation for viscous fluids, temperature for heat transfer, etc.).

Constitutive relations must be formulated in a manner that does not violate the axiom of entropy inequality, also known as the second law of thermodynamics. The second law of thermodynamics stipulates that the time rate of change of entropy in a medium is greater than the rate of entropy supply into the medium by various sources of heat. In a spatial frame, the differential statement of the axiom of entropy inequality is given by

\[ \begin{equation} \rho\frac{D\eta}{Dt}+\divg\frac{\mathbf{q}}{\theta}-\rho\frac{r}{\theta}\geqslant0\,,\label{eq:axiom-entropy-inequality} \end{equation} \]

where \(\eta\) is the specific entropy (entropy per unit mass) of the matter. Entropy is a measure of disorder in a medium. It represents another function of state.

Since Eq.\eqref{eq:axiom-entropy-inequality} does not include all of the functions of state, we may combine it with the energy balance to recover \(\boldsymbol{\sigma}\) and \(\varepsilon\) in the entropy inequality. This may be achived by multiplying the entropy inequality in \eqref{eq:axiom-entropy-inequality} by the absolute temperature \(\theta\) (recalling that \(\theta\) is always positive) and adding it to the energy balance in \eqref{eq:axiom-energy-balance} to produce

\[ -\rho\left(\frac{D\varepsilon}{Dt}-\theta\frac{D\eta}{Dt}\right)+\boldsymbol{\sigma}:\mathbf{D}-\frac{1}{\theta}\mathbf{q}\cdot\grad\theta\geqslant0\,. \]

The term in parentheses may be rearranged as

\[ \frac{D\varepsilon}{Dt}-\theta\frac{D\eta}{Dt}=\frac{D}{Dt}\left(\varepsilon-\theta\eta\right)+\eta\frac{D\theta}{Dt}\,. \]

We now introduce the specific Helmholtz free energy, \(\psi\), which relates the specific internal energy and entropy according to

\[ \begin{equation} \psi=\varepsilon-\theta\eta\,.\label{eq:specific-free-energy} \end{equation} \]

We may say that the free energy is the part of the internal energy which is available to produce useful work. Substituting this expression above produces the form of the Clausius-Duhem inequality that involves all the independent functions of state appearing in these axioms, (\(\psi\), \(\eta\), \(\boldsymbol{\sigma}\) and \(\mathbf{q}\)),

\[ \begin{equation} -\rho\left(\dot{\psi}+\eta\dot{\theta}\right)+\boldsymbol{\sigma}:\mathbf{D}-\frac{1}{\theta}\mathbf{q}\cdot\grad\theta\geqslant0\,,\label{eq:Clausius-Duhem-inequality} \end{equation} \]

where we have substituted \(D\left(\cdot\right)/Dt\) with the \(\left(\dot{}\right)\) operator, for notational simplicity.

It is important to understant that all functions of state are non-observable properties of matter: No instruments exist to measure the complete state of stress inside a material, nor the heat flux, the internal energy, or the entropy. However, we can formulate constitutive models that relate these functions of state to observable state variables, then calculate their value for a specific state. Observable state variables are all based on measurements of space (length, area and volume) and time. For example, we can measure (observe) displacement, velocity and acceleration of matter at various spatial locations, from which we can calculate their spatial gradients and treat these as observable state variables. From measurements of length in calibrated materials, we can derive measurements of force (using a load cell) and temperature (using a thermometer).

To better understand the choice of state variables relevant to mechanics, we proceed to describe the kinematics of a continuum in the next section.