TY - JOUR
T1 - Large Time Existence for Thin Vibrating Plates
JF - Communication in Partial Differential Equations 36 (2011) 2062-2102
Y1 - 2011
A1 - Helmut Abels
A1 - Maria Giovanna Mora
A1 - Stefan Müller
AB - We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large\\r\\ntimes under appropriate scaling of the initial values such that the limit system as h --> 0 is either the nonlinear von Karman plate equation or the linear fourth order Germain-Lagrange equation. In the case of the\\r\\nlinear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.
PB - Taylor & Francis
UR - http://hdl.handle.net/1963/3755
U1 - 562
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity
JF - Calculus of Variations and Partial Differential Equations 41 (2011) 241-259
Y1 - 2011
A1 - Helmut Abels
A1 - Maria Giovanna Mora
A1 - Stefan Müller
AB - The asymptotic behaviour of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness $h$ of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of $h$, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von K\\\\\\\'arm\\\\\\\'an plate equation.
PB - Springer
UR - http://hdl.handle.net/1963/3835
U1 - 492
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Convergence of equilibria of three-dimensional thin elastic beams
JF - Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 873-896
Y1 - 2008
A1 - Maria Giovanna Mora
A1 - Stefan Müller
AB - A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\\\\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.
UR - http://hdl.handle.net/1963/1896
U1 - 2339
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Convergence of equilibria of planar thin elastic beams
JF - Indiana Univ. Math. J. 56 (2007) 2413-2438
Y1 - 2007
A1 - Maria Giovanna Mora
A1 - Stefan Müller
A1 - Maximilian G. Schultz
AB - We consider a thin elastic strip of thickness h and we show that stationary points of the nonlinear elastic energy (per unit height) whose energy is of order h^2 converge to stationary points of the Euler-Bernoulli functional. The proof uses the rigidity estimate for low-energy deformations by Friesecke, James, and Mueller (Comm. Pure Appl. Math. 2002), and a compensated compactness argument in a singular geometry. In addition, possible concentration effects are ruled out by a careful truncation argument.
UR - http://hdl.handle.net/1963/1830
U1 - 2386
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - RPRT
T1 - Derivation of a rod theory for phase-transforming materials
Y1 - 2007
A1 - Maria Giovanna Mora
A1 - Stefan Müller
AB - A rigorous derivation is given of a rod theory for a multiphase material,starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Gamma-convergence.
JF - Calc. Var. Partial Differential Equations 28 (2007) 161-178
UR - http://hdl.handle.net/1963/1751
U1 - 2793
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - CHAP
T1 - Recent analytical developments in micromagnetics
T2 - The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381.
Y1 - 2006
A1 - Antonio DeSimone
A1 - Robert V. Kohn
A1 - Stefan Müller
A1 - Felix Otto
JF - The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381.
SN - 978-0-12-480874-4
UR - http://hdl.handle.net/1963/2230
U1 - 2014
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Self-similar folding patterns and energy scaling in compressed elastic sheets
JF - Comput. Methods Appl. Mech. Engrg. 194 (2005) 2534-2549
Y1 - 2005
A1 - Sergio Conti
A1 - Antonio DeSimone
A1 - Stefan Müller
AB - Thin elastic sheets under isotropic compression, such as for example blisters formed by thin films which debonded from the substrate, can exhibit remarkably complex folding patterns. We discuss the scaling of the elastic energy with respect to the film thickness, and show that in certain regimes the optimal energy scaling can be reached\\nby self-similar folding patterns that refine towards the boundary, in agreement with experimental observations. We then extend the analysis\\nto anisotropic compression, and discuss a simplified scalar model which suggests the presence of a transition between a regime where\\nthe deformation is governed by global properties of the domain and another one where the direction of maximal compression dominates and the scale of the folds is mainly determined by the distance to the boundary in the direction of the folds themselves.
PB - Elsevier
UR - http://hdl.handle.net/1963/3000
U1 - 1333
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -