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Elastic rods with incompatible strain: macroscopic versus microscopic buckling. (English) Zbl 1442.74075

Summary: We consider the buckling of a long prismatic elastic solid under the combined effect of a pre-stress that is inhomogeneous in the cross-section, and of a prescribed displacement of its endpoints. A linear bifurcation analysis is carried out using different structural models (namely a double beam, a rectangular thin plate, and a hyper-elastic prismatic solid in 3-d): it yields the buckling mode and the wavenumber \(q_{\operatorname{c}}\) that are first encountered when the end-to-end displacement is progressively decreased with fixed pre-stress. For all three structural models, we find a transition from a long-wavelength \((q_{\operatorname{c}}=0)\) to a short-wavelength first buckling mode \((q_{\operatorname{c}}\neq 0)\) when the inhomogeneous pre-stress is increased past a critical value. A method for calculating the critical inhomogeneous pre-stress is proposed based on a small-wavenumber expansion of the buckling mode. Overall, our findings explain the formation of multiple perversions in elastomer strips, as well as the large variations in the number of perversions as a function of pre-stress and cross-sectional geometry, as reported by J. Liu et al. [“Structural transition from helices to hemihelices”, PLoS ONE 9, No. 4, Article ID e93183 (2014; doi:10.1371/journal.pone.0093183)].

MSC:

74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)

Software:

SLEPc
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References:

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