By Yury Vetyukov
This ebook provides a hybrid method of the mechanics of skinny our bodies. Classical theories of rods, plates and shells with limited shear are in response to asymptotic splitting of the equations and boundary stipulations of third-dimensional elasticity. The asymptotic options turn into actual because the thickness decreases, and the three-d fields of stresses and displacements may be decided. The research contains essentially vital results of electromechanical coupling and fabric inhomogeneity. The extension to the geometrically nonlinear diversity makes use of the direct technique in line with the main of digital paintings. Vibrations and buckling of pre-stressed buildings are studied with assistance from linearized incremental formulations, and direct tensor calculus rounds out the record of analytical strategies used in the course of the book.
A novel conception of thin-walled rods of open profile is in this case built from the versions of rods and shells, and often utilized equations are confirmed to be asymptotically particular. The impact of pre-stresses at the torsional stiffness is proven to be an important for buckling research. Novel finite aspect schemes for classical rod and shell constructions are offered with a entire dialogue concerning the theoretical foundation, computational elements and implementation information. Analytical conclusions and closed-form strategies of specific difficulties are demonstrated opposed to numerical effects. the vast majority of the simulations have been played within the Wolfram Mathematica atmosphere, and the compact resource code is equipped as a considerable and quintessential a part of the book.
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Additional info for Nonlinear Mechanics of Thin-Walled Structures: Asymptotics, Direct Approach and Numerical Analysis
2. As we have seen, the practical application of the approach requires certain threedimensional analysis to be performed prior to the solution of the dimensionally reduced problem, which provides the strain energy function (stiffness coefficients) for purely elastic structures or the enthalpy function for electromechanically coupled ones. Moreover, this three-dimensional analysis allows recovering the stressed state of the actual continuous body from the computed force factors of the reduced model; for combined presentations of both the direct and the three-dimensional analyses see comprehensive works by Antman , Basar and Krätzig , Chró´scielewski et al.
Consistently bridging the gap between appropriate three-dimensional and reduced models is the central point of the hybrid approach (see Sect. 3), which is used throughout this book. 4 Asymptotic Analysis The technique of formal series expansions applied to the solution of differential equations of mechanics of three-dimensional continuum is broadly discussed in the literature. , Andrianov et al. , Nayfeh . An important role is played by the asymptotic methods in fluid mechanics [137, 157].
10) 0 The principal term σt in the tangential stress component remains yet undetermined. 11) = 0. 12) for a straight rod the asymptotics looks slightly different. 11) we arrive at M = −Qn ; (. ) ≡ ∂s (. ). 14) The third step completes the procedure. We collect the terms of the order λ0 and integrate them over the thickness. 16) (ft + 2ατ + nα∂nτ + ∂s σt ) dn = 0. 17) we arrive at qn + Qn − αQt = 0, qt + Qt + αQn = 0. 19) Q = Qn n + Qt t, Eqs. 18) are rewritten in the known vectorial form of the equations of equilibrium of forces and moments in the rod [6, 172]: Q + q = 0, M + t × Q = 0.
Nonlinear Mechanics of Thin-Walled Structures: Asymptotics, Direct Approach and Numerical Analysis by Yury Vetyukov