Energy balance

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.

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The conservation of energy can be expressed as

(1)ddtV(12ρvv+ρU)dV=SvtdS+VfvdV,

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=σn on the boundary S, the first term of the right-hand side of Equation 1 can be rewritten as

(2)SvtdS=V(x)(vσ)dV=V[(xσ)v+vx:σ]dV=V[(xσ)v+ε˙:σ]dV,

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

vx:σ=ε˙:σ,

where ε˙ is the strain rate tensor (see Rate of deformation and strain increment). Substituting Equation 2 into Equation 1 yields

(3)ddtV(12ρvv+ρU)dV=V[(xσ+f)v+σ:ε˙]dV.

From Cauchy's equation of motion we have

xσ+f=ρdvdt.

Substituting this into Equation 3 gives

ddtV(12ρvv+ρU)dV=V(ρdvdtv+σ:ε˙)dV=V[ddt(12ρvv)+σ:ε˙]dV.

From this we get the energy equation

ρdUdt=σ:ε˙.

Integrating this equation we find

0t(Vσ:ε˙dV)dt=VρUdV+U0,

where U0 is the energy at time 0. To make the energy balance (Equation 1) more convenient to use, we integrate it in time:

V12ρvvdV+VρUdV=0tE˙WFdτ+constant,

or

EK+EU=0tE˙WFdτ+constant,

where

E˙WF=SvtdS+VfvdV,

defined as the rate of work done to the body by external forces and contact friction forces between the contact surfaces. EK, the kinetic energy, is given by

EK=V12ρvvdV,

and EU is defined as

EU=VρUdV=0t(Vσ:ε˙dV)dτ-U0.

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, tl, the solid infinite element radiation traction, tqb, and the frictional traction, tf. Then E˙WF can be written as

E˙WF=(SvtldS+VfvdV)-(-SvtfdS)-(-SvtqbdS)E˙W-E˙F-E˙QB,

where E˙W is the rate of work done to the body by external forces, E˙QB is the rate of energy dissipated by the damping effect of solid medium infinite elements, and 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

(4)EU+EK+EF-EW-EQB=constant.

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

EU=0t(Vσ:ε˙dV)dτ=0t[V(σc+σv):ε˙dV]dτ=0t(Vσc:ε˙dV)dτ+0t(Vσv:ε˙dV)dτEI+EV,

where σc is the stress derived from the user-specified constitutive equation, without viscous dissipation effects included; σel is the elastic stress; σv is the viscous stress (defined for bulk viscosity, material damping, and dashpots); EV is the energy dissipated by viscous effects; and EI is the remaining energy, which we continue to call the internal energy. If we introduce the strain decomposition, ε˙=ε˙el+ε˙pl+ε˙cr (where ε˙el, ε˙pl, and ε˙cr are elastic, plastic, and creep strain rates, respectively), the internal energy, EI, can be expressed as

(5)EI=0t(Vσc:ε˙dV)dτ=0t(Vσc:ε˙eldV)dτ+0t(Vσc:ε˙pldV)dτ+0t(Vσc:ε˙crdV)dτ=ES+EP+EC,

where ES is the applied elastic strain energy, EP is the energy dissipated by plasticity, and EC 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, σc, can be expressed in terms of the “undamaged” stress, σu, and the continuum damage parameter, d:

σc=(1-d)σu.

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

ES=0t(V(1-d)σu:ε˙eldV)dτ.

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

EE=0t(V(1-dt)σu:ε˙eldV)dτ=0t(V(1-dt)(1-d)σc:ε˙eldV)dτ,

and the energy dissipated through damage is equal to

ED=0t(V(dt-d)σu:ε˙eldV)dτ=0t(V(dt-d)(1-d)σc:ε˙eldV)dτ.

If we define

fu(εel)=0tσu:ε˙eldτ

as the undamaged elastic energy function, we can write

EE=0t(V(1-dt)f˙udV)dτ

and

ED=0t(V(dt-d)f˙udV)dτ.

Interchanging the integrals yields

EE=V(0t(1-dt)f˙udτ)dV=V((1-dt)fu)dV

and

ED=V(0t(dt-d)f˙udτ)dV=V[(dt-d)fu|0t+0td˙fudτ]dV.

The first term in the last expression vanishes, since at time t, d=dt and at time zero, fu=0. If we now define the damage strain energy function

fc=(1-d)fu,

then

ED=V(0td˙1-dfcdτ)dV=0tVd˙1-dfcdVdτ.

For a linear elastic energy function

fc=12σu:εel,

and, hence,

ED=0tVd˙2(1-d)σu:εeldVdτ.