A4_Deformations_in_a_Continuum_ans

# A4_Deformations_in_a_Continuum_ans - EATS 2470 Introduction...

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EATS 2470 Introduction to Continuum Mechanics Assignment #4-Solutions 1. Given: Lagrangen description of a flow: 4 1 1 2 3 y a a t a t = + + 2 2 2 1 y a a t = + 3 3 2 y a a t = + Find that: (a) the position of P (1,2,4) at time t = 2; Substitute 1 1 a = , 2 2 a = , 3 4 a = , t = 2 into the flow equations 4 1 1 (1) (2)(2) (4)(2) 41 y y = + + = 2 2 2 (2) (1)(2) 6 y y = + = 3 3 (4) (2)(2) 8 y y = + = Thus, the position is { } 41 6 8 y     =       (b) find the displacement of the point P at the same time, t = 2 The displacements are defined as i i i u y a = - or { } { } { } u y a = - Thus, 4 1 1 (2)(2) (4)(2) 40 u u = + = 2 2 2 (1)(2) 2 u u = = 3 2 (2)(2) 4 u u = = { } 40 4 4 u     =       (c) find the velocity of the same point at the same time The velocity is dt dy q i i = , thus

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3 1 2 3 1 4 68 q a t a q = + = 2 1 2 2 4 q a t q = = 3 2 3 2 q a t q = = or { } 68 4 2 q     =       (d) now find the acceleration of the same point at the same time, t = 2 The acceleration is i i dq d q dt dt = , then 2 1 1 2 12 96 dq dq a t dt dt = = 2 2 1 2 2 dq dq a dt dt = = 3 0 dq dt = or 96 2 0 dq dt       =           (e) find the Eulerian description of the displacement sector recall that we have given: { } [ ] { } y A a = or 4 1 1 2 2 2 3 3 1 1 0 0 1 y t t a y t a y t a         =             14243 It require to find the inverse of the matrix First find the matrix of cofactors 2 3 2 4 3 6 1 ( ) 1 (1 ) ij t t t t t t t t α - = - - - - Second, transpose this matrix of cofactor 2 4 2 3 3 6 1 ( ) 1 (1 ) T ij t t t t t t t t α - - = - - - Third, the value of the determinant of the original matrix [A] 4 2 2 4 2 4 6 1 1 0 1(1) ( ) 1 0 1 t t A t t t t t t t = = - - = + -
Finally, the inverse of the matrix [A] is [ ] 2 4 1 2 3 3 6 1 ( ) / 1 (1 ) T ij t t t A A t t t t t α - - - = = - - - 4 6 1 t t + - Thus, the displacement in Eulerian description is { } { } [ ] { } 1 u y A y - = - { } 4 6 2 4 1 2 3 2 4 6 3 1 2 3 3 4 1 2 3 ( ) ( ) ( ) t t y t t y ty u t y t t y t y

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