Minimization of Shear Energy in Two Dimensional Continua with Two Orthogonal Families of Inextensible Fibers: The Case of Standard Bias Extension Test

Abstract

Gradient elasticity • Mechanics of woven fabrics • Inextensible fibers • Non-local integral equations • Extensional bias test Mathematics Subject Classification 74B20 • 74K99 • 74E99 Actually we call standard bias test the extension test where used rectangular specimens (i) have length exactly three times longer than their width, (ii) are clamped at their shorter sides and where (iii) a relative displacement (compatible with the presence of inextensible 1. vanishing displacement of the line Σ 1 , 2. imposed displacement u 0 = u 0 (D 1 + D 2) of the line Σ 2. 10 ∪ 23

Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence

“…The field μ (ξ) is piecewise C 2 , and the field ϑ (ξ) is piecewise C 1 with possible discontinuities in the point ξ = 2. The pairs of fields (μ, ϑ), together with the conditions (16), verify the local condition…”

“…For more details, the reader is referred to [16]. The reader is warned: no attempt is made here to model damage and failure phenomena.…”

“…3 and equal to the identity on Δ 00 and a translation in Δ 33 (for the proof the reader is referred to [16]). …”

“…Moreover, the continuity conditions in P i = (1, 1) and P f = (3, 3) of the placement field can be written as (see again [16]):…”

“…These cylinders form an array of elastic pivots, whose elastic energy determines the shear deformation energy of the pantographic sheet, while the bending energy of fibers will produce a second-gradient effect in the continuum model deformation energy (see [19,34,35,83] or [16,17]). This system has a microstructure which seems suitably described by the aforementioned continuum models when secondgradient deformation energies are taken into account considered: In particular, the deformation energy has to account for the so-called geodesic bending [81].…”