Collisions and fractures: A predictive theory |
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| Authors: | Francesco Freddi Michel Frémond |
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| Institution: | 1. Laboratorio Lagrange, Dept. of Civil-Environmental Engineering, University of Parma, Parma, Italy;2. Laboratorio Lagrange, Dept. of Civil Engineering, University of Tor Vergata, Roma, Italy;1. Laboratório de Análise Numérica e Astrofísica, Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil;2. Ecole Nationale d''Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire, Menzel Abderrahman 7035, Tunisia;1. Faculty of Physical Chemistry, University of Belgrade, Serbia;2. Military Technical Institute in Belgrade, Serbia;1. Computational Chemistry Group, Department of Chemistry, Faculty of Science, University of Mauritius, Réduit, Mauritius;2. Department of Chemistry, Faculty of Science, Universiti Teknologi Malaysia, Johor, Malaysia;1. Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania;2. University of Bucharest, Faculty of Physics, Bucharest-Magurele, Romania;1. Department of Chemistry, Dr. Bhupendranath Dutta Smriti Mahavidyalaya, Hatgobindapur, Burdwan 713 407, India;2. Department of Chemistry, University of Burdwan, Golapbag, Burdwan 713 104, India;3. Department of Chemistry, University of Calcutta, 92 APC Road, Kolkata 700 009, India;4. Bio-Organic Division, Bhabha Atomic Research Centre, Mumbai 400 085, India;1. 231 W. Hancock St, Campus Box 17, Department of Mathematics, Georgia College & State University, Milledgeville, GA 31061, United States;2. 221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, Al 36849, United States |
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| Abstract: | We investigate collisions of solids which can fracture. Equations of motion and constitutive laws provide a predictive theory. Assuming the collision as instantaneous, the equations of motion are derived from the principle of virtual work introducing new interior forces which describe the very large stresses and the very large contact forces resulting from the kinematic incompatibilities. They are interior volume percussion stresses and interior surface percussions both on the unknown fractures and on the colliding surface. In order to approximate these equations, we assume solids are damageable. In this point of view, it results that velocity is continuous with respect to space but its strain rate is very large in a thin region where the material is completely damaged, so approximating a fracture. When the velocity before collision is very large, the damaged zone may be large accounting for parts of the solid completely transformed into powder. The constitutive laws result from dissipative functions satisfying the second law of thermodynamics and able to model the fracturation phenomenon at the macroscopic engineering level. Representative numerical examples confirm that the model accounts for the fracturation qualitative properties. |
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