Correction to: Nonlocal effects on curved double-walled carbon nanotubes based on nonlocal theory (Archive of Applied Mechanics, (2025), 95, 2, (57), 10.1007/s00419-025-02762-2)

In this article, multiple errors have occurred and corrected as described below. In Eq. 9, Dtt2 has been incorrectly written and must be corrected as Dtt2. The corrected equation is given below. (Formula presented.) On page 5, line 5, the terms involving Dtt1 and Dtt2 are incorrectly written and must be corrected to Dtt1 and Dtt2. In Eq. (26) and Eq. (28), the terms Dtt1 and Dtt2 are incorrectly written. All occurrences must be corrected to Dtt1 and Dtt2. The corrected equations are given below. (Formula presented.) (Formula presented.) In Eq. 9, Dtt2 has been incorrectly written and must be corrected as Dtt2. The corrected equation is given below. On page 5, line 5, the terms involving Dtt1 and Dtt2 are incorrectly written and must be corrected to Dtt1 and Dtt2. In Eq. (26) and Eq. (28), the terms Dtt1 and Dtt2 are incorrectly written. All occurrences must be corrected to Dtt1 and Dtt2. The corrected equations are given below. In matrix A (Eq. (29), page 6), the terms involving Dtt1 and Dtt2 are incorrectly written. All occurrences must be corrected to Dtt1 and Dtt2. On page 7, the termsDtt1, Dtt2,Itt1 and Itt2 are incorrectly written. All occurrences must be corrected to Dtt1, Dtt2,Itt1, andItt2. The simulations were performed with an outer radius of r1=0.7nm, an inner radius of r2=0.35nm, and a tube thickness of 0.34nm [46–48]. A concentrated load of P=10-11N is applied to the nanotube. Both tubes are assumed to have the same Young’s modulus, E=1TPa, while the shear modulus G is calculated as G=E2(1+v), with a Poisson’s ratio of v=0.3. It has been reported by several authors that the value of nonlocal parameter γ typically ranges from 0 to 2 nm to capture the small-scale effects in the buckling analysis of CNTs [40, 42, 67]. Herein, γ is set as γ=5×10-14m. The bending rigidities along the binormal direction are expressed as (Dnn)1=E(Inn)1 and (Dnn)2=E(Inn)2, where (Inn)1=πr14/4 and (Inn)2=πr24/4 represent the moments of inertia in the inner and outer tubes, respectively. Similarly, the bending rigidities along the tangent direction are (Dtt)1=G(Itt)1 and (Dtt)2=G(Itt)2, with (Itt)1=πD14/32 and (Itt)2=πD24/32, where D1=2r1 and D2=2r2. The van der Waals interaction coefficient c, which quantifies the coupling between the inner and outer tubes, is derived from the interlayer energy potential, as discussed in [64–66]. The original article has been corrected.