ABSTRACT
Where we introduce the notion of Eshelby stress in elasticity and its direct generalizations, and witness its first applications to materially inhomogeneous bodies.
3.1.1 Quasistatic Eshelby Stress
For the sake of simplicity we consider the case of the quasistatics of materially inhomogeneous purely elastic bodies (no dissipation of any kind, inertial effects neglected). Then the balance of linear (physical) momentum in the Piola-Kirchhoff formulation reads:
divR W
W WT f 0 T F
F X+ = = ∂ ∂
= ( )ρ0 , , ; . (3.1)
On applying F to the right of the first equation and noting that
R RWT F T F T F T F( ) = ( ) − ∇( ) = ( ) − ∇ −. . . . finh , (3.2)
since
∇ = ∂ ∂
∇( ) + ∂∂ = − ∂ ∂
W W W W F
F X
f XF
. , : fixed
− ∂ ∂ W X F fixed
, (3.3)
we obtain the fully material equilibrium equation
div ext inhRb f f 0+ + = , (3.4)
wherein we have set
b T F f f F: . , : .= − = −W R1 0ext ρ (3.5)
The fully material stress tensor b is called the quasistatic Eshelby stress tensor in honor of J.D. Eshelby (cf. Maugin and Trimarco, 1992), who previously called it the elastic energy-momentum tensor or Maxwell stress tensor (but this denomination is somewhat misleading; see further developments in the following). This material tensor has no specific symmetry. However, on account of the local balance of angular momentum (2.82), we easily show that it satisfies the following symmetry condition with respect to the CauchyGreen (or deformed) material metric C:
C b C b b C. . . .= ( ) =T T (3.6)
The material forces fext and finh are the externally applied material force (obtained by the negative of the pull back of the prescribed body force) and the material force of inhomogeneity. According to its definition in (3.3), the latter captures the explicit dependence of the function W on the material point (e.g., the dependence of the elasticity coefficients on X). At all regular material points X, (3.4) is an identity deduced from the basic balance law in (3.1). In field theory (see Section 4.2) this is called a conservation law, but here it is in fact a nonconservation because of the lack of invariance of the physical system under material translation. In the absence of body force and material inhomogeneity, (3.4) reduces to a strict conservation law. In the case where (3.1) and (3.4) hold good, the preceding manipulation is equivalent to writing the following identity:
div div ext inhR RT f F b f f 0+( ) + + +( ) =ρ0 . , (3.7)
which may be referred to as the Ericksen identity of elasticity, as Ericksen (1977) gave it in the special case (no body force, no elastic inhomogeneity):
div divR RT F b 0( ) + ( ) =. . (3.8)
In field theory (Section 4.2), these are none other than special cases of the celebrated Noether’s identity.
