The equations that govern the thermomechanics of a solid include the balance laws for mass, momentum, and energy. Kinematic equations and constitutive relations are needed to complete the system of equations. Physical restrictions on the form of the constitutive relations are imposed by an entropy inequality that expresses the second law of thermodynamics in mathematical form.
The balance laws express the idea that the rate of change of a quantity (mass, momentum, energy) in a volume must arise from three causes:
Let Ω be the body (an open subset of Euclidean space) and let be its surface (the boundary of Ω).
Let the motion of material points in the body be described by the map
where is the position of a point in the initial configuration and is the location of the same point in the deformed configuration.
Recall that the deformation gradient () is given by
Let be a physical quantity that is flowing through the body. Let be sources on the surface of the body and let be sources inside the body. Let be the outward unit normal to the surface . Let be the velocity of the physical particles that carry the physical quantity that is flowing. Also, let the speed at which the bounding surface is moving be un (in the direction ).
Then, balance laws can be expressed in the general form
Note that the functions , , and can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with.
It can be shown that the balance laws of mass, momentum, and energy can be written as
In the above equations is the mass density (current), is the material time derivative of ρ, is the particle velocity, is the material time derivative of , is the Cauchy stress tensor, is the body force density, is the internal energy per unit mass, is the material time derivative of e, is the heat flux vector, and is an energy source per unit mass.
With respect to the reference configuration, the balance laws can be written as
In the above, is the first Piola-Kirchhoff stress tensor, and ρ0 is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by
We can alternatively define the nominal stress tensor which is the transpose of the first Piola-Kirchhoff stress tensor such that
Then the balance laws become
Keep in mind that:
The gradient and divergence operators are defined such that
where is a vector field, is a second-order tensor field, and are the components of an orthonormal basis in the current configuration. Also,
where is a vector field, is a second-order tensor field, and are the components of an orthonormal basis in the reference configuration. The inner product is defined as
The Clausius-Duhem inequality can be used to express the second law of thermodynamics for elastic-plastic materials. This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved.
Just like in the balance laws in the previous section, we assume that there is a flux of a quantity, a source of the quantity, and an internal density of the quantity per unit mass. The quantity of interest in this case is the entropy. Thus, we assume that there is an entropy flux, an entropy source, and an internal entropy density per unit mass (η) in the region of interest.
Let Ω be such a region and let be its boundary. Then the second law of thermodynamics states that the rate of increase of η in this region is greater than or equal to the sum of that supplied to Ω (as a flux or from internal sources) and the change of the internal entropy density due to material flowing in and out of the region.
Let move with a velocity un and let particles inside Ω have velocities . Let be the unit outward normal to the surface . Let ρ be the density of matter in the region, be the entropy flux at the surface, and r be the entropy source per unit mass. Then the entropy inequality may be written as
The scalar entropy flux can be related to the vector flux at the surface by the relation . Under the assumption of incrementally isothermal conditions, we have
where is the heat flux vector, s is a energy source per unit mass, and T is the absolute temperature of a material point at at time t.
We then have the Clausius-Duhem inequality in integral form:
We can show that the entropy inequality may be written in differential form as
In terms of the Cauchy stress and the internal energy, the Clausius-Duhem inequality may be written as