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This paper considers whether non-trivial identities are satisfied by certain ‘plactic-like’ monoids that, like the plactic monoid, are closely connected with combinatorics. New results show that the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy identities. The existing state of knowledge is discussed for the plactic and Bell monoids.

{We consider a velocity tracking problem for the Navier-Stokes equations in a 2D bounded domain. The control acts on the boundary through an injection-suction device, and the flow is allowed to slip against the surface wall. We study the well-posedness of the state equations, linearized state equations, and adjoint equations. In addition, we show the existence of an optimal solution and establish the first order optimality condition.}

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Orthotropic stiffness components of Pinus pinaster Ait. wood are simultaneously determined by means of a heterogeneous plate bending test. The proposed inverse identification approach couples full-field slope measurements provided by deflectometry with the virtual fields methods. Wooden plates oriented in the longitudinal–radial and longitudinal–tangential material planes were manufactured. A procedure was implemented to allow suitable specular reflective coating of the wooden plates, required in the deflectometry technique. Reconstructed curvature fields, applied load and plate dimensions were input in virtual fields methods for material parameter identification, assuming Kirchhoff–Love classical plate theory. Several virtual fields and load cases were analysed to address the identifiability of the method. The values of the orthotropic elastic constants obtained from the proposed approach were found in good agreement with regard to reference ones for the same species and determined from classical tensile, compression and shear mechanical tests.

The paper presents a nonlinear analytical solution for the prediction of the full-range debonding response of mechanically-anchored FRP composites from the substrate. The nonlinear analytical approach predicts, for any monotonic loading history or bonded length the relative displacements (or slips) between materials, the strains in the FRP composite, the bond stresses within the interface and the stresses developed in the substrate. The load-slip responses FRP-to-substrate interfaces with a short and a long bonded lengths are motive of analysis and discussion. The solutions obtained from the proposed approach are also compared with other experimental results found in the literature.

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