An introduction to thermomechanics by Hans Ziegler

By Hans Ziegler

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7) S u b j e c t i n g t h e first e q u a l i t y ( 3 . 7 ) t o ( 2 . 7 6 ) , ( 2 . 5 0 ) a n d ( 1 . 9 3 ) , w e o b t a i n a n 46 Fig. 3. Stresses acting on an infinitesimal cuboid. alternate global f o r m of the t h e o r e m , \(QV*\o dV=\gf dV+fou-QVM)*, k dA. 8) T h i s is t h e l i n e a r m o m e n t u m t h e o r e m s t a t e d f r o m t h e p o i n t o f v i e w o f a n o b s e r v e r a t r e s t . I n f a c t , if A is i n t e r p r e t e d a s a s p a t i a l s u r f a c e , t h e i n t e g r a l o n t h e left is t h e i n c r e a s e o f l i n e a r m o m e n t u m w i t h i n A p e r u n i t t i m e , a n d the various terms o n the right represent the s u m of the external forces a n d t h e m o m e n t u m s u p p l y f r o m o u t s i d e b y flow t h r o u g h A.

37) kLm nJ T h e first t e r m o n t h e r i g h t is t h e l o c a l c h a n g e ; t h e s e c o n d o n e is d u e t o t h e d i s p l a c e m e n t o f t h e o b s e r v e r w i t h t h e p a r t i c l e a n d is r e f e r r e d t o a s t h e convective derivative change of D i v i d i n g ( 2 . 3 7 ) b y dt, w e o b t a i n t h e of the tensor as defined in Section 2 . n,0 + t i, . jVj. 38) m n a n d t h e s o - c a l l e d convective A p p l y i n g ( 2 . 3 8 ) t o t h e density v o l u m e , Q = dm/dV, material derivative.

G . , t h r o u g h o u t t h e classical t h e o r y o f elasticity. 2 . 3 . M a t e r i a l derivatives Let us return t o Section 2 . 1 , where the motion of a c o n t i n u u m was d e s c r i b e d b y a v e l o c i t y field v (Xj, k t), a n d let u s a s s u m e t h a t , w i t h i n t h i s field, a n a r b i t r a r y t e n s o r t „ (xj,t) kL is d e f i n e d a s a f u n c t i o n o f p o s i t i o n n and time. T h e local change of the tensor t _ kl i n t h e t i m e e l e m e n t at, i .

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