where D indicates the intrinsic local dissipation. Assuming the body a thermodynamically isolated system, the following insolation condition holds ZV P dV =0. (6) Substitution of (1) into (5) leads to D=σ: ˙ε− ∂ψ ∂ε : ˙ε− ∂ψ ∂d ˙d− ∂ψ ∂κ ˙κ− ∂ψ ∂¯κ ¯˙κ+P ≥0 in V, (7) which holds for any, reversible or irreversible, deformation mechanism. Following standard procedures, it is possible to recognize the following state laws: σ= ∂φ ∂ε =(1−d)2C0 : ε, Y :=− ∂ψ ∂d =(1−d)ε : C0 : ε (8a,b) χ:= ∂ψ ∂κ =hl κ, χnl := ∂ψ ∂¯κ =hnl ¯κ=hnl R(κ), (8c,d) so that the explicit form of the dissipation function reads D=Y ˙d−χ ˙κ−χnl ¯˙κ+P ≥0 in V. (9) The thermodynamic force Y, conjugated to the damage d, represents the energy release for unitary increment of damage. χand χnl are the thermodynamic forces associated to the local κ and nonlocal ¯κ internal variables, respectively. The dissipative irreversible mechanism associated to the damage is governed by the local fluxes ˙d, ˙κ. The dissipation Dcan therefore alternatively be written as the bilinear form D=Y ˙d−X˙κ≥0 in V, (10) where X represents the equivalent nonlocal force that is associated to the increment of the local variable κ. By comparing eqs. (9) and (10), we can specify P as P =χ ˙κ+χnl ¯˙κ−X˙κ in V. (11) The imposition of the insulation condition (6) leads to ZV χ ˙κ+χnl ¯˙κ−X˙κ¡dV =0 (12) for each dissipative mechanism and then for each ˙κ. It can be shown that the following (Green) identity holds for the operator R ZV χnl R(˙κ)dV =Z V R∗(χnl)(˙κ)dV, (13) where R∗ is the adjoint operator of Rdefined by R∗(χnl)¬ ¬ x =Z V 1 Vr(y) α(||x−y||)χnl(y)dV(y). (14) By considering identity (13), eq. (12) turns out to be ZV ¢R∗(χnl)+χ−X£˙κdV =0. (15)
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