3D-1D reduction for a large deformation viscoelastic model. (Réduction 3D-1D d’un modèle viscoélastique en grandes déformations). (English) Zbl 1119.74011
Summary: We present a Cosserat-based 3D-1D dimensional reduction for the viscoelastic finite strain model introduced by P. Neff [Z. Angew. Math. Phys. 56, No. 1, 148–182 (2005; Zbl 1079.74039)]. The reduced 1D model preserves observer invariance. We prove the existence and uniqueness of the solution of the reduced coupled minimization/evolution problem.
MSC:
| 74D10 | Nonlinear constitutive equations for materials with memory |
| 74H20 | Existence of solutions of dynamical problems in solid mechanics |
| 74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |
Citations:
Zbl 1079.74039References:
| [1] | Antman, S., Nonlinear Problems of Elasticity, Applied Mathematical Sciences, vol. 107 (2005), Springer: Springer New York · Zbl 1098.74001 |
| [2] | Ciarlet, P. G., Mathematical Elasticity, Volume I: Three-Dimensional Elasticity (1988), North-Holland: North-Holland Amsterdam · Zbl 0648.73014 |
| [3] | Neff, P., Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation, Contin. Mech. Thermodyn., 15, 2, 161-195 (2003) · Zbl 1035.74015 |
| [4] | Neff, P., Local existence and uniqueness for a geometrically exact membrane-plate with viscoelastic transverse shear resistance, Math. Methods Appl. Sci., 28, 9, 1031-1060 (2005) · Zbl 1071.74034 |
| [5] | Neff, P., A geometrically exact viscoplastic membrane-shell with viscoelastic transverse shear resistance avoiding degeneracy in the thin-shell limit. Part I: The viscoelastic membrane-plate, Z. Angew. Math. Phys., 56, 1, 148-182 (2005) · Zbl 1079.74039 |
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