An introduction to thermomechanics by Hans Ziegler

By Hans Ziegler

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A c c o r d i n g t o ( 1 . 7) or symbolically w = jcurlu. 33) yields v I n s e r t i n g t h i s in ( 2 . 11) w h e r e dr is t h e v e c t o r c o n n e c t i n g Ρ a n d P'. 2 t h a t t h e v e l o c i t y field ( 2 . 1 1 ) c o r r e s p o n d s t o a rotation of the 28 vicinity o f Ρ w i t h a n g u l a r v e l o c i t y w a b o u t P. This angular velocity, o b t a i n e d , a c c o r d i n g t o ( 2 . 8 ) , a s h a l f t h e c u r l o f v, is c a l l e d t h e vorticity t h e v e l o c i t y field.

T h e e x a c t t h e o r y o f t h e d i s p l a c e m e n t field will b e d e a l t w i t h in C h a p t e r 13. It s o m e t i m e s h a p p e n s , h o w e v e r , that the displacements are small c o m p a r e d to the dimensions of the b o d y . In this case, the theory m a y be s i m p l i f i e d . I n t h e first i n s t a n c e t h e d i s p l a c e m e n t s m a y b e t r e a t e d , b y w a y o f a p p r o x i m a t i o n , a s if t h e y w e r e i n f i n i t e s i m a l . y , y a s a r g u m e n t s o f a n y f u n c t i o n , m a y b e r e p l a c e d b y t h e Xj.

T h e s t r e s s e s give rise t o s e c o n d - o r d e r f o r c e s , w h e r e a s t h e b o d y a n d t h e inertia forces acting o n t h e element of Fig. 2 a r e of t h e third o r d e r a n d h e n c e m a y b e d i s r e g a r d e d ( a s well a s t h e s t r e s s d i f f e r e n c e b e t w e e n t h e o b l i q u e face a n d t h e s u r f a c e e l e m e n t p a r a l l e l t o it p a s s i n g t h r o u g h P). 5) w h i c h yields */ v ) =

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