1.5.5 Energy balance

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The conservation of energy implied by the first law of thermodynamics states that the time rate of change of kinetic energy and internal energy for a fixed body of material is equal to the sum of the rate of work done by the surface and body forces. This can be expressed as

where

is the current mass density,

is the velocity field vector,

is the internal energy per unit mass,

is the surface traction vector,

is the body force vector, and

is the normal direction vector on boundary .

Using Gauss' theorem and the identity that on the boundary , the first term of the right-hand side of Equation 1.5.5–1 can be rewritten as

where we have used the fact that

is symmetric, and we also know (see “Equilibrium and virtual work,” Section 1.5.1) that

where

is the strain rate tensor (see “Rate of deformation and strain increment,” Section 1.4.3). Substituting Equation 1.5.5–2 into Equation 1.5.5–1 yields

From Cauchy's equation of motion we have

Substituting this into Equation 1.5.5–3 gives

From this we get the energy equation

Integrating this equation we find

where

is the energy at time

. To make the energy balance (Equation 1.5.5–1) more convenient to use, we integrate it in time:

where

defined as the rate of work done to the body by external forces and contact friction forces between the contact surfaces.

, the kinetic energy, is given by

and

is defined as

To track physically distinguishable engineering phenomena more narrowly, we introduce decompositions of the stress, strain, and tractions.

We can split the traction, , into the surface distributed load, , the solid infinite element radiation traction, , and the frictional traction, . Then can be written as

where

is the rate of work done to the body by external forces,

is the rate of energy dissipated by the damping effect of solid medium infinite elements, and

is the rate of energy dissipated by contact friction forces between the contact surfaces. An energy balance for the entire model can then be written as

For convenience, the dissipated portions of the internal energy are split off:

where

is the stress derived from the user-specified constitutive equation, without viscous dissipation effects included;

is the elastic stress;

is the viscous stress (defined for bulk viscosity, material damping, and dashpots);

is the energy dissipated by viscous effects; and

is the remaining energy, which we continue to call the internal energy. If we introduce the strain decomposition,

(where

, and

are elastic, plastic, and creep strain rates, respectively), the internal energy,

, can be expressed as

where

is the applied elastic strain energy,

is the energy dissipated by plasticity, and

is the energy dissipated by time-dependent deformation (creep, swelling, and viscoelasticity).

If damage occurs in the material, not all of the applied elastic strain energy is recoverable. At any given time, the stress, , can be expressed in terms of the “undamaged” stress, , and the continuum damage parameter, :