Correction to: Eshelby force and power for uniform bodies
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CORRECTION
Mawafag F. Alhasadi · Marcelo Epstein · Salvatore Federico
Correction to: Eshelby force and power for uniform bodies
Received: 11 August 2020 / Accepted: 15 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Correction to: Acta Mech 230, 1663-1684 (2019) https://doi.org/10.1007/s00707-018-2353-6 In the original paper, we erroneously used the total potential energy (Helmholtz free energy of the material plus potential energy of the external forces) in the entropy inequality. This is because we incautiously merged two sets of calculations: in the first set, W denoted the total potential energy and, in the second, W denoted the Helmholtz free energy. This confusion caused the appearance of incorrect terms, featuring the kinetic energy density K and its time derivative K˙ . Thus, we must correct the text between Eqs. (43) and (45) in Sect. 2.4 on the Lagrangian for a thermoelastic body, Remark 3 and the entire Sect. 4.6 on the entropy inequality [note that, in the corrected version of Sect. 4.6, there are no Eqs. (147)–(153)]. In order to reduce the number of corrections to a minimum, we renamed the total energy and the Helmholtz free energy, but not the potential energy of the external forces, as per the scheme below:
Total potential energy Helmholtz free energy Potential energy of the external forces
Original paper
This erratum
W Wel Wext
Wtot W Wext
Moreover, in Sect. 2.3 on the material balance equations, when writing the most general form of the balance of energy and of the entropy inequality, we erroneously omitted the non-compliant surface contribution to the internal energy. This had no consequences in the later calculations because we neglected all non-compliant terms. However, for the sake of completeness and consistency, we felt that it was necessary to amend these equations. Finally, we take this chance to provide some other minor corrections. Note that, although the most serious corrections are to Sects. 2.4 and 4.6, we report the corrections to Sect. 2.3 first, in order to maintain the order in the original paper. The original article can be found online at https://doi.org/10.1007/s00707-018-2353-6. M. F. Alhasadi · M. Epstein · S. Federico (B) Department of Mechanical and Manufacturing Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada E-mail: [email protected]
M. F. Alhasadi et al.
Corrections to Section 2.3 In Sect. 2.3, Eqs. (25)–(30), as well as Eq. (33), should have featured the non-compliant surface contribution to the internal energy, which we denote U, and should have read: ˙ + H + U + z | φ) ˙ ∂t ( 21 ρ R φ˙ . φ˙ + ρ R E) = ( 21 φ˙ . φ˙ + E + f | φ (25 corr.) R R ˙ [φ˙ (T + A) + (E + 21 φ˙ . φ)M + U + Q] N , + ∂R
∂t
1
2 ρR
φ˙ . φ˙ + ρ R E =
˙ + H + U + z | φ ˙ + φ˙ . φ˙ + E + f | φ 1 ˙ + Div φ˙ (T + A) + (E + φ˙ . φ)M +U + Q ,
(26 corr.)
˙ + Grad E | M + Div ( Q + U). ρ R E˙ = H + U + (T + A) : F
(27 corr.)
N ,
(28 corr.)
1 2
2
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