Second-Order Structured Deformations: Relaxation, Integral Representation and Applications
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Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications.
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- 1.Acharya, A., Fressengeas, C.: Continuum mechanics of the interaction of phase boundaries and dislocations in solids. Differential Geometry and Continuum Mechanics, Vol. 137 (Eds. Chen G.Q , Grinfeld M. and Knops R.J.) Springer Proceedings in Mathematics & Statistics, Berlin, 125–168, 2015Google Scholar
- 2.Agrawal V., Dayal K.: Dynamic phase-field model for structural transformations and twinning: regularized interfaces with transparent prescription of complex kinetics and nucleation. Part I: formulation and one-dimensional characterization. J. Mech. Phys. Solids, 85, 270–290 (2015)ADSMathSciNetCrossRefGoogle Scholar
- 11.Carriero, M., Leaci, A., Tomarelli, F.: Second order variational problems with free discontinuity and free gradient discontinuity. Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat., Vol. 14, Dept. Math., Seconda Univ. Napoli, Caserta, pp. 135–186, 2004Google Scholar
- 14.De Giorgi E., Ambrosio L.: Un nuovo tipo di funzionale del calcolo delle variazioni. Atti. Accad. Naz. Lincei, 82, 199–210 (1988)Google Scholar
- 17.Del Piero, D. R., Owen, D. R.: Multiscale Modeling in Continuum Mechanics and Structured Deformations. CISM Courses and Lecture Notes, Vol. 447, Springer, Berlin, 2004Google Scholar
- 22.Evans, L. C., Gariepy, R. F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992Google Scholar
- 28.Owen, D. R.: Elasticity with gradient-disarrangements: a multiscale geometrical perspective for strain-gradient theories of elasticty and of plasticity. J. Elast. (submitted)Google Scholar
- 31.Paroni, R.: Second-order structured deformations: approximation theorems and energetics. Multiscale Modeling in Continuum Mechanics and Structured Deformations, Vol. 447 (Eds. Del Piero G. and Owen D.R.) Springer, Berlin, 2004Google Scholar