As noted in Sect. 10.6, the first law of thermodynamics states that the rate of change of the (total) energy *E* must balance the rate at which work is done on the body plus the rate at which heat is added to the body. Here, let us derive a differential equation for energy balance (similar to the differential equations for mass balance and linear momentum balance in Sects. 8.1 and 8.2 of Chap. 8) for the special case of no mechanical work (e.g., a rigid solid); once done, we then list the more general thermomechanical energy equation.

In contrast to Sect.

10.5, let the energy per unit mass

*e* be given by the internal energy

*u* alone (i.e., no kinetic or potential energy). Moreover, let the heat be added to the body via two means: through the surface, via a heat flux vector

**q** defined per unit area, and volumetrically, via a heat source (scalar)

*q*_{ s } defined per unit mass. Note, too, that convention defines

**q** positive outward, and being an arbitrary vector, we need

**q** ·

**n**, where

**n** is an outward unit normal vector to the surface through which the flux occurs. Because

**q** ·

**n** acts at each point over a differential area and

*q*_{ s } acts at each point over a differential volume and because the first law requires the balance of total energy, we have

$\frac{d}{dt}{\displaystyle \int u\rho d\overline{)v}=-{\displaystyle \int q\cdot \mathit{n}da+}{\displaystyle \int \rho {q}_{s}d\overline{)v}}}\text{.}$

(A10.1)

The minus sign in the flux term accounts for our desire to quantify heat

*addition* to the body. Recall from Chap.

7 that the divergence theorem allows us to convert a surface integral to a volume integral, namely

$-{\displaystyle \int \mathit{q}\cdot \mathit{n}da=-{\displaystyle \int \nabla \cdot \mathit{q}d\overline{)v}}}\text{.}$

(A10.2)

Now, if we can exchange the order of the time differentiation and the volume integration in Eq. (

A10.1), then we can collect all terms into a single integral and thereby obtain our governing differential equation (cf. Sect.

8.1). As in Sect.

8.1, therefore, we seek to relate

$d\overline{)v}$ to

$d\overline{)V}$ the original differential volume that is independent of time.

Employing the same arguments as in Sect.

8.1, let the original and current differential volumes both be cuboidal; hence,

$d\overline{)v}=\frac{\partial x}{\partial X}\frac{\partial y}{\partial Y}\frac{\partial z}{\partial Z}d\overline{)V}$

(A10.3)

and

$\frac{d}{dt}{\displaystyle \int u\rho}d\overline{)v}={\displaystyle \int \frac{d}{dt}}\left(u\rho \frac{\partial x}{\partial X}\frac{\partial y}{\partial Y}\frac{\partial z}{\partial Z}\right)d\overline{)V}\text{.}$

(A10.4)

Using the product rule for the differentiation and exploiting results from Sect.

8.1, this equation can be written as

$\int \left[\frac{du}{dt}\rho +u\left(\frac{d\rho}{dt}+\rho \nabla \cdot \mathit{v}\right)\right]}\frac{\partial x}{\partial X}\frac{\partial y}{\partial Y}\frac{\partial z}{\partial Z}d\overline{)V$

(A10.5)

and, consequently, using Eq. (

A10.3), Eq. (

A10.1) becomes

$\int \left[\frac{du}{dt}\rho +u\left(\frac{d\rho}{dt}+\rho \nabla \cdot \mathit{v}\right)\right]}d\overline{)v}={\displaystyle \int \left(-\nabla \cdot \mathit{q}+\rho {q}_{s}\right)d\overline{)v}\text{,}$

(A10.6)

or

$\int \left[\frac{du}{dt}\rho +u\left(\frac{d\rho}{dt}+\rho \nabla \cdot \mathit{v}\right)+\nabla \cdot \mathit{q}-\rho {q}_{s}\right]}d\overline{)v}=0\text{,$

(A10.7)

which must hold for

*all* arbitrary domains (volumes). This can be satisfied if the integrand is always zero; that is,

$$ \frac{du}{dt}\rho +u\left(\frac{d\rho }{dt}+\rho \nabla \cdot \boldsymbol{v}\right)=-\nabla \cdot \boldsymbol{q}+\rho {q}_s $$

(A10.8)

where we recall from mass balance (Eq.

8.11) that the second term on the left-hand side must be zero. Thus, our differential equation for energy balance, in the absence of mechanical work terms, is

$$\rho\frac{du}{dt} = - \nabla \cdot {\boldsymbol{q}} + {pq}_{s}\cdot $$

(A10.9)

This equation is similar to the general equations of linear momentum balance in Sect.

8.2, in that we have not specified particular material behaviors (i.e., constitutive relations).

The most commonly used constitutive equation for heat flux, however, is “Fourier’s law,” which states that

**q** is proportional to the temperature gradient, namely

$$ \boldsymbol{q}=-k\nabla T, $$

(A10.10)

where

*k* is a material constant called the thermal conductivity. Typical values for soft tissues are on the order of

*k =* 4.76 mW/cm°C for normal human aorta and

*k* = 4.85 mW/cm°C for a fibrous atherosclerotic plaque, both measured at 35 °C. A commonly assumed constitutive relation for the internal energy, in the absence of deformation, is

*u = c*_{ v }*T*, where

*c*_{ v } is the specific heat, at a constant volume, and

*T* is the absolute temperature. Hence, in this special case, Eq. (

A10.9) becomes

$$ \rho \frac{d}{dt}\left({c}_vT\right)=-\nabla \cdot \left(-k\nabla T\right)+\rho {q}_s $$

(A10.11)

or, for constant

*c*_{ v } and

*k*, the famous heat diffusion equation

$$ \rho {c}_v\frac{dT}{dt}=k{\nabla}^2T+\rho {q}_s\to \frac{dT}{dt}=\alpha {\nabla}^2T+\frac{q_s}{c_v}, $$

(A10.12)

where

*α = k/ρc*_{ v } is the so-called thermal diffusivity (a material property). This equation is widely studied in applied mathematics and allows one to determine temperature fields

*T*(

*x*,

*y*,

*z*,

*t*) in terms of the material parameters (

*ρ*,

*c*_{ v },

*k*) and the heat supply

*q*_{ s }*.* Microwave energy is a prime source of

*q*_{ s }*.*Finally, note that in the case of mechanical work, Eq. (

A10.9) can be shown to become (Humphrey

2002)

$$ \rho \frac{du}{dt}={\sigma}_{xx}{D}_{xx}+{\sigma}_{yy}{D}_{yy}+{\sigma}_{zz}{D}_{zz}+{\sigma}_{xy}{D}_{xy}+\cdots +{\sigma}_{zx}{D}_{zx}-\nabla \cdot \boldsymbol{q}+\rho {q}_s, $$

(A10.13)

which couples mass, linear momentum, and energy balance equations, thus resulting in a formidable system of five partial differential equations. The solution of such coupled problems is important in biomechanics, but this is the topic of advanced courses. We merely introduce a few coupled problems in Chap.

11.