Energy balance

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The conservation of energy can be expressed as

\begin{matrix} (1) & \frac{d}{d t} \int_{V} (\frac{1}{2} ρ v \cdot v + ρ U) d V = \int_{S} v \cdot t d S + \int_{V} f \cdot v d V, \end{matrix}

where

$ρ$: is the current mass density,
$v$: is the velocity field vector,
U: is the internal energy per unit mass,
$t$: is the surface traction vector,
$f$: is the body force vector, and
$n$: is the normal direction vector on boundary S.

Using Gauss' theorem and the identity that $t = σ \cdot n$ on the boundary S, the first term of the right-hand side of Equation 1 can be rewritten as

\begin{matrix} (2) & \begin{matrix} \int_{S} v \cdot t d S & ​ = \int_{V} (\frac{\partial}{\partial x}) \cdot (v \cdot σ) d V \\ = \int_{V} [(\frac{\partial}{\partial x} \cdot σ) \cdot v + \frac{\partial v}{\partial x} : σ] d V \\ = \int_{V} [(\frac{\partial}{\partial x} \cdot σ) \cdot v + \dot{ε} : σ] d V, \end{matrix} \end{matrix}

where we have used the fact that $σ$ is symmetric, and we also know (see Equilibrium and virtual work) that

\frac{\partial v}{\partial x} : σ = \dot{ε} : σ,

where $\dot{ε}$ is the strain rate tensor (see Rate of deformation and strain increment). Substituting Equation 2 into Equation 1 yields

\begin{matrix} (3) & \frac{d}{d t} \int_{V} (\frac{1}{2} ρ v \cdot v + ρ U) d V = \int_{V} [(\frac{\partial}{\partial x} \cdot σ + f) \cdot v + σ : \dot{ε}] d V . \end{matrix}

From Cauchy's equation of motion we have

\frac{\partial}{\partial x} \cdot σ + f = ρ \frac{d v}{d t} .

Substituting this into Equation 3 gives

\begin{matrix} \frac{d}{d t} \int_{V} (\frac{1}{2} ρ v \cdot v + ρ U) d V & = \int_{V} (ρ \frac{d v}{d t} \cdot v + σ : \dot{ε}) d V \\ = \int_{V} [\frac{d}{d t} (\frac{1}{2} ρ v \cdot v) + σ : \dot{ε}] d V . \end{matrix}

From this we get the energy equation

ρ \frac{d U}{d t} = σ : \dot{ε} .

Integrating this equation we find

\int_{0}^{t} (\int_{V} σ : \dot{ε} d V) d t = \int_{V} ρ U d V + U_{0},

where $U_{0}$ is the energy at time $0$ . To make the energy balance (Equation 1) more convenient to use, we integrate it in time:

\int_{V} \frac{1}{2} ρ v \cdot v d V + \int_{V} ρ U d V = \int_{0}^{t} {\dot{E}}_{W F} d τ + constant,

E_{K} + E_{U} = \int_{0}^{t} {\dot{E}}_{W F} d τ + constant,

where

{\dot{E}}_{W F} = \int_{S} v \cdot t d S + \int_{V} f \cdot v d V,

defined as the rate of work done to the body by external forces and contact friction forces between the contact surfaces. $E_{K}$ , the kinetic energy, is given by

E_{K} = \int_{V} \frac{1}{2} ρ v \cdot v d V,

and $E_{U}$ is defined as

E_{U} = \int_{V} ρ U d V = \int_{0}^{t} (\int_{V} σ : \dot{ε} d V) d τ - U_{0} .

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

We can split the traction, $t$ , into the surface distributed load, $t^{l}$ , the solid infinite element radiation traction, $t^{q b}$ , and the frictional traction, $t^{f}$ . Then ${\dot{E}}_{W F}$ can be written as

{\dot{E}}_{W F} = (\int_{S} v \cdot t^{l} d S + \int_{V} f \cdot v d V) - (- \int_{S} v \cdot t^{f} d S) - (- \int_{S} v \cdot t^{q b} d S) \equiv {\dot{E}}_{W} - {\dot{E}}_{F} - {\dot{E}}_{Q B},

where ${\dot{E}}_{W}$ is the rate of work done to the body by external forces, ${\dot{E}}_{Q B}$ is the rate of energy dissipated by the damping effect of solid medium infinite elements, and ${\dot{E}}_{F}$ 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

\begin{matrix} (4) & E_{U} + E_{K} + E_{F} - E_{W} - E_{Q B} = constant . \end{matrix}

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

\begin{matrix} E_{U} & = \int_{0}^{t} (\int_{V} σ : \dot{ε} d V) d τ = \int_{0}^{t} [\int_{V} (σ^{c} + σ^{v}) : \dot{ε} d V] d τ \\ = \int_{0}^{t} (\int_{V} σ^{c} : \dot{ε} d V) d τ + \int_{0}^{t} (\int_{V} σ^{v} : \dot{ε} d V) d τ \\ ​ \equiv E_{I} + E_{V}, \end{matrix}

where $σ^{c}$ is the stress derived from the user-specified constitutive equation, without viscous dissipation effects included; $σ^{e l}$ is the elastic stress; $σ^{v}$ is the viscous stress (defined for bulk viscosity, material damping, and dashpots); $E_{V}$ is the energy dissipated by viscous effects; and $E_{I}$ is the remaining energy, which we continue to call the internal energy. If we introduce the strain decomposition, $\dot{ε} = {\dot{ε}}^{e l} + {\dot{ε}}^{p l} + {\dot{ε}}^{c r}$ (where ${\dot{ε}}^{e l}$ , ${\dot{ε}}^{p l}$ , and ${\dot{ε}}^{c r}$ are elastic, plastic, and creep strain rates, respectively), the internal energy, $E_{I}$ , can be expressed as

\begin{matrix} (5) & \begin{matrix} E_{I} & = \int_{0}^{t} (\int_{V} σ^{c} : \dot{ε} d V) d τ \\ = \int_{0}^{t} (\int_{V} σ^{c} : {\dot{ε}}^{e l} d V) d τ + \int_{0}^{t} (\int_{V} σ^{c} : {\dot{ε}}^{p l} d V) d τ + \int_{0}^{t} (\int_{V} σ^{c} : {\dot{ε}}^{c r} d V) d τ \\ ​ = E_{S} + E_{P} + E_{C}, \end{matrix} \end{matrix}

where $E_{S}$ is the applied elastic strain energy, $E_{P}$ is the energy dissipated by plasticity, and $E_{C}$ 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, $σ^{c}$ , can be expressed in terms of the “undamaged” stress, $σ^{u}$ , and the continuum damage parameter, d:

σ^{c} = (1 - d) σ^{u} .

The damage parameter, d, starts at zero (undamaged material) and increases to a maximum value of no more than one (fully damaged material). Hence, we can write

E_{S} = \int_{0}^{t} (\int_{V} (1 - d) σ^{u} : {\dot{ε}}^{e l} d V) d τ .

We assume that, upon unloading, the damage parameter remains fixed at the value attained at time t. Therefore, the recoverable strain energy is equal to

E_{E} = \int_{0}^{t} (\int_{V} (1 - d_{t}) σ^{u} : {\dot{ε}}^{e l} d V) d τ = \int_{0}^{t} (\int_{V} \frac{(1 - d_{t})}{(1 - d)} σ^{c} : {\dot{ε}}^{e l} d V) d τ,

and the energy dissipated through damage is equal to

E_{D} = \int_{0}^{t} (\int_{V} (d_{t} - d) σ^{u} : {\dot{ε}}^{e l} d V) d τ = \int_{0}^{t} (\int_{V} \frac{(d_{t} - d)}{(1 - d)} σ^{c} : {\dot{ε}}^{e l} d V) d τ .

If we define

f^{u} (ε^{e l}) = \int_{0}^{t} σ^{u} : {\dot{ε}}^{e l} d τ

as the undamaged elastic energy function, we can write

E_{E} = \int_{0}^{t} (\int_{V} (1 - d_{t}) {\dot{f}}^{u} d V) d τ

and

E_{D} = \int_{0}^{t} (\int_{V} (d_{t} - d) {\dot{f}}^{u} d V) d τ .

Interchanging the integrals yields

E_{E} = \int_{V} (\int_{0}^{t} (1 - d_{t}) {\dot{f}}^{u} d τ) d V = \int_{V} ((1 - d_{t}) f^{u}) d V

and

E_{D} = \int_{V} (\int_{0}^{t} (d_{t} - d) {\dot{f}}^{u} d τ) d V = \int_{V} [{(d_{t} - d) f^{u} |}_{0}^{t} + \int_{0}^{t} \dot{d} f^{u} d τ] d V .

The first term in the last expression vanishes, since at time t, $d = d_{t}$ and at time zero, $f^{u} = 0$ . If we now define the damage strain energy function

f^{c} = (1 - d) f^{u},

then

E_{D} = \int_{V} (\int_{0}^{t} \frac{\dot{d}}{1 - d} f^{c} d τ) d V = \int_{0}^{t} \int_{V} \frac{\dot{d}}{1 - d} f^{c} d V d τ .

For a linear elastic energy function

f^{c} = \frac{1}{2} σ^{u} : ε^{e l},

and, hence,

E_{D} = \int_{0}^{t} \int_{V} \frac{\dot{d}}{2 (1 - d)} σ^{u} : ε^{e l} d V d τ .