Connecting Atomistic and Continuous Models of Elastodynamics
- 107 Downloads
We prove the long-time existence of solutions for the equations of atomistic elastodynamics on a bounded domain with time-dependent boundary values as well as their convergence to a solution of continuum nonlinear elastodynamics as the interatomic distances tend to zero. Here, the continuum energy density is given by the Cauchy–Born rule. The models considered allow for general finite range interactions. To control the stability of large deformations we also prove a new atomistic Gårding inequality.
Unable to display preview. Download preview PDF.
- 3.Braun, J.: Connecting atomistic and continuum theories of nonlinear elasticity: rigorous existence and convergence results for the boundary value problems. Ph.D. thesis, Universität Augsburg 2016. To appear in Augsburger Schriften zur Mathematik, Physik und Informatik, Logos Verlag, BerlinGoogle Scholar
- 4.Braun, J., Schmidt, B.: Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory. Calc. Var. Partial Differ. Equ. 55(125) 2016. doi: 10.1007/s00526-016-1048-x
- 7.E, W., Ming P.: Cauchy–Born rule and the stability of crystalline solids: dynamic problems. Acta Math. Appl. Sin. (English Series) 23, 529–550 (2007). doi: 10.1007/s10255-007-0393
- 8.E, W., Ming P.: Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007). doi: 10.1007/s00205-006-0031-7
- 10.Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin 2001. doi: 10.1007/978-3-642-61798-0
- 14.Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton 1970Google Scholar
- 17.Valent, T.: Boundary Value Problems of Finite Elasticity. Springer, Berlin 1988. doi: 10.1007/978-1-4612-3736-5