EN0175-08

EN0175-08 - EN0175 09 / 28 / 06 Continuing on concepts of...

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EN0175 09 / 28 / 06 Continuing on concepts of stress in a continuum: Traction on an arbitrary plane with normal vector n v i ij j n t σ = n t T v v = This equation indicates that the normal stress on a plane with normal vector n v is n n n t T n v v v v = = Accordingly, the shear stress on a plane with normal n v is ( ) n n n n n t t T T n v v v v v v v σσ τ = = ˆ n n v t v Force equilibrium (i.e. application of Newton’s law a m F v v = to a continuum; also called balance of linear momentum) X v V d v d x v u v 1
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EN0175 09 / 28 / 06 u v ––– displacement vector u t u & v v = ––– velocity vector u t u & & v v = 2 2 ––– acceleration vector V V u d & & v ρ ––– total inertia force on volume ( V V m d d = , u a & & v = ) Consider all the forces acting on the volume: S t S d v ––– total surface force on V V V f d v ––– total body force on V According to Newton’s law, = + V V S V u V f S t d d d & & v v v In index notation: = + V j V j S j V u V f S t d d d & & The first term is = = V i ij S i ij S j V S n S t d d d , σ (Divergence theorem) Thus ( ) 0 d , = + V j j i
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This note was uploaded on 01/10/2010 for the course EN 0175 taught by Professor Huajiangao during the Spring '06 term at Brown.

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EN0175-08 - EN0175 09 / 28 / 06 Continuing on concepts of...

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