TY - JOUR
T1 - Corrigendum to “Macroscopic constitutive relations for elastomers reinforced with short aligned fibers
T2 - Instabilities and post-bifurcation response”(Journal of the Mechanics and Physics of Solids (2016) 91 (240–264) (S0022509615301605)(10.1016/j.jmps.2016.02.028)
AU - Avazmohammadi, Reza
AU - Ponte Castañeda, Pedro
PY - 2017/12
Y1 - 2017/12
N2 - The authors regret that some of the results presented in Fig. 12 of the above-referenced paper are incorrect. More specifically, the results for the five traces of the modulus tensor Lc, associated with the post-bifurcation solution (PBS), need to be corrected inside the ‘relaxed’ region (identified by α > αcr and [Formula Presented]), while the corresponding results outside the relaxed region remain the same. Thus, Fig. 1 in this corrigendum shows a revised version of Fig. 12 with the corrected PBS results. In particular, we observe from Fig. 1(a) that the component [Formula Presented] is zero inside the relaxed region which is consistent with the convex hull construction for the relaxed stored-energy function [Formula Presented] (see Fig. 8 in the original paper). In addition, the corresponding results for the principal solution (PS) are included in Fig. 1 as they appear in Fig. 12 in the original paper, except for the result for the trace [Formula Presented] in Fig. 1(b), which is shown only up to [Formula Presented]. This is because this trace becomes imaginary for some α > αcr as the product [Formula Presented] becomes negative. Moreover, we find from the corrected results that the PBS is rank-one convex inside the relaxed region—and not strongly elliptic as incorrectly stated in the discussion of Fig. 12, as well as in the context of Eq. (64), in the original paper. In other words, the PBS solution is rank-one convex, but not strictly so. Thus, the condition (B.3) for the corrected moduli traces inside the relaxed region in Fig. 1 is satisfied with the strict inequality ( > ) replaced by relaxed inequality ( ≥ ), as in expression (11) of the original paper. More precisely, the only non-zero coefficients in the quartic polynomial associated with the condition (B.3) are [Formula Presented] and [Formula Presented] which are both positive inside the relaxed region. Therefore, the quartic polynomial remains non-negative within the entire relaxed region, implying that the PBS is rank-one convex in this region. However, the polynomial can become zero when [Formula Presented] implying that the PBS is not strongly elliptic inside the relaxed region. This correction is inconsequential for the rest of the results presented in the paper. Thus, the stored-energy function [Formula Presented] is still rank-one convex and polyconvex, and therefore corresponds to the quasiconvexification of the stored-energy function [Formula Presented] (as discussed in the context of Eqs. (64)–(66) in the original paper).
AB - The authors regret that some of the results presented in Fig. 12 of the above-referenced paper are incorrect. More specifically, the results for the five traces of the modulus tensor Lc, associated with the post-bifurcation solution (PBS), need to be corrected inside the ‘relaxed’ region (identified by α > αcr and [Formula Presented]), while the corresponding results outside the relaxed region remain the same. Thus, Fig. 1 in this corrigendum shows a revised version of Fig. 12 with the corrected PBS results. In particular, we observe from Fig. 1(a) that the component [Formula Presented] is zero inside the relaxed region which is consistent with the convex hull construction for the relaxed stored-energy function [Formula Presented] (see Fig. 8 in the original paper). In addition, the corresponding results for the principal solution (PS) are included in Fig. 1 as they appear in Fig. 12 in the original paper, except for the result for the trace [Formula Presented] in Fig. 1(b), which is shown only up to [Formula Presented]. This is because this trace becomes imaginary for some α > αcr as the product [Formula Presented] becomes negative. Moreover, we find from the corrected results that the PBS is rank-one convex inside the relaxed region—and not strongly elliptic as incorrectly stated in the discussion of Fig. 12, as well as in the context of Eq. (64), in the original paper. In other words, the PBS solution is rank-one convex, but not strictly so. Thus, the condition (B.3) for the corrected moduli traces inside the relaxed region in Fig. 1 is satisfied with the strict inequality ( > ) replaced by relaxed inequality ( ≥ ), as in expression (11) of the original paper. More precisely, the only non-zero coefficients in the quartic polynomial associated with the condition (B.3) are [Formula Presented] and [Formula Presented] which are both positive inside the relaxed region. Therefore, the quartic polynomial remains non-negative within the entire relaxed region, implying that the PBS is rank-one convex in this region. However, the polynomial can become zero when [Formula Presented] implying that the PBS is not strongly elliptic inside the relaxed region. This correction is inconsequential for the rest of the results presented in the paper. Thus, the stored-energy function [Formula Presented] is still rank-one convex and polyconvex, and therefore corresponds to the quasiconvexification of the stored-energy function [Formula Presented] (as discussed in the context of Eqs. (64)–(66) in the original paper).
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U2 - 10.1016/j.jmps.2017.08.010
DO - 10.1016/j.jmps.2017.08.010
M3 - Comment/debate
AN - SCOPUS:85028871368
VL - 109
SP - 198
EP - 199
JO - Journal of the Mechanics and Physics of Solids
JF - Journal of the Mechanics and Physics of Solids
SN - 0022-5096
ER -