Tham khảo tài liệu 'elasticity part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sadd Elasticity Final Proof 3 00pm page 18 If f and c are scalar fields and u and v are vector fields several useful identities exist V fC rf C f W r2 fC r2f C f v2C 2rf rc r fu rf u f v u r X f u rf X u f r X u r u X v v r X u - u r X v r X rf 0 r-rf r2f r r X u 0 r X r X u r r u -r2u u X Vx u r u u u ru Each of these identities can be easily justified by using index notation from definition relations . Next consider some results from vector tensor integral calculus. We simply list some theorems that have later use in the development of elasticity theory. Divergence or Gauss Theorem Let S be a piecewise continuous surface bounding the region of space V. If a vector field u is continuous and has continuous first derivatives in V then B JJ u n dS V- u dV where n is the outer unit normal vector to surface S. This result is also true for tensors of any order that is B Us dS ---- k dV Stokes Theorem Let S be an open two-sided surface bounded by a piecewise continuous simple closed curve C. If u is continuous and has continuous first derivatives on S then j . u dr C JI r X u n dS where the positive sense for the line integral is for the region S to lie to the left as one traverses curve C and n is the unit normal vector to S. Again this result is also valid for tensors of arbitrary order and so j C JJs snr dS 18 FOUNDATIONS AND ELEMENTARY APPLICATIONS TLFeBOOK Sadd Elasticity Final Proof 3 00pm page 19 It can be shown that both divergence and Stokes theorems can be generalized so that the dot product in and or can be replaced with a cross product. Green s Theorem in the Plane Applying Stokes theorem to a planar domain S with the vector field selected as u f e1 ge2 gives the result K@x - @ dd Ld gdy Further special choices with either f 0 or g 0 imply IĨ fg - f fif .-i yr-dxdy gnxds .d x v fnyds JJs fx Jc JJs fy Jc Zero-Value Theorem .
