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1、Example : A Hole under Remote Shear,Solution in Polar Coordinates,Chapter 5.5,48,BCs :,Example : A Hole under Remote Shear,Solution in Polar Coordinates,Chapter 5.5,49,Uniform solution,Perturbation solution,Example : A Hole under Remote Shear,Solution in Polar Coordinates,Chapter 5.5,50,Stresses,Deg
2、eneracy (Solution),Solution in Polar Coordinates,Chapter 5.5,51,Degeneracy,Solution in Polar Coordinates,Chapter 5.5,52,Michells Solution,Solution in Polar Coordinates,Chapter 5.5,53,J. H. Michell, Proc. London. Math. Soc. Vol.31 (1899), 100-124,A Circular Hole under Tension,Solution in Polar Coordi
3、nates,Chapter 5.5,54,BCs :,Uniform solution,A Circular Hole under Tension(Details),Solution in Polar Coordinates,Chapter 5.5,55,From Michells solution, the terms that might match the uniform solution:,Terms that lead to decaying stresses :,Stress function,A Circular Hole under Tension,Solution in Po
4、lar Coordinates,Chapter 5.5,56,Stresses,Displacements for Circular Hole under Shear,Solution in Polar Coordinates,Chapter 5.5,57,Stresses,Displacements for Circular Hole under Shear,Solution in Polar Coordinates,Chapter 5.5,58,Strain,Displacements for Circular Hole under Shear,Solution in Polar Coor
5、dinates,Chapter 5.5,59,Displacements for Circular Hole under Shear,Solution in Polar Coordinates,Chapter 5.5,60,Displacement fields,Translation : A = B = 0,Rotation : C = 0,Displacements for Circular Hole under Shear,Solution in Polar Coordinates,Chapter 5.5,61,Radial displacement along the hole :,A
6、xi-symmetric Problems(Details),Solution in Polar Coordinates,Chapter 5.5,62,Displacement fields:,Displacements of Michells Solution,Solution in Polar Coordinates,Chapter 5.5,63,BCs :,Zero resultant forces and zero resultant torque:,Displacements of Michells Solution,Solution in Polar Coordinates,Cha
7、pter 5.5,64,Orthogonality of Fourier series dictates that the above conditions only bears influence on the terms of,QUESTION : Three independent equations WHILE four sets of coefficients?,ANSWER : Single-valued-ness requirement of the displacement fields, and that imposes restrictions on the multi-v
8、alued terms in ur uq that are linear in q.,Kolosov-Muskhelishvilli Method,Complex Variables,Chapter 5.6,65,Recall: in-plane elasticity formulation of an isotropic solid without body forces :,In-plane stress trace is harmonic The Airy stress function is biharmonic,Analogy to the anti-plane problems,C
9、omplex Variables,Kolosov-Muskhelishvilli Method,Chapter 5.6,66,Derivatives,Kolosov-Muskhelishivilli Potentials,Kolosov-Muskhelishvilli Method,Chapter 5.6,67,Fields by Complex Potentials,Kolosov-Muskhelishvilli Method,Chapter 5.6,68,Kolosov and Muskhelishivilli potentials :,Stress,Strain,Fields by Co
10、mplex Potentials,Kolosov-Muskhelishvilli Method,Chapter 5.6,69,Displacement,Fields by Complex Potentials,Kolosov-Muskhelishvilli Method,Chapter 5.6,70,Fields by Complex Potentials,Kolosov-Muskhelishvilli Method,Chapter 5.6,71,Rigid translation,Rigid body rotation,a,w,b,Complex displacement,Strain en
11、ergy density,Coordinate Transform,Kolosov-Muskhelishvilli Method,Chapter 5.6,72,Coordinate Transform,Kolosov-Muskhelishvilli Method,Chapter 5.6,73,Coordinate Transform,Kolosov-Muskhelishvilli Method,Chapter 5.6,74,Disks,Kolosov-Muskhelishvilli Method,Chapter 5.6,75,By Taylor series:,Complex displace
12、ment:,BCs:,Holes,Kolosov-Muskhelishvilli Method,Chapter 5.6,76,A is REAL.,Holes,Kolosov-Muskhelishvilli Method,Chapter 5.6,77,Single-valued analytical function can be expressed by Laurent series.,A is REAL, B, C and are COMPLEX.,LINKS TO the Michells solutions for holes,Increments of Complex Displac
13、ement by Circling the Hole,Kolosov-Muskhelishvilli Method,Chapter 5.6,78,Increment of the complex displacement,Inserted Wedge,Kolosov-Muskhelishvilli Method,Chapter 5.6,79,A,Be interpreted as inserting a wedge of an angle :,Kolosov-Muskhelishvilli Method,Chapter 5.6,80,Burgers vector,Dislocations,So
14、lutions of Single-valued Displacements,Kolosov-Muskhelishvilli Method,Chapter 5.6,81,For a problem free of inserted wedge or dislocations, The Kolosov-Muskhelishvilli solution is reduced to,Kolosov-Muskhelishvilli Method,Concentrated Forces,Chapter 5.6,82,Following the derivation in the textbook by
15、Lu and Luo, Section 8.1,For the single-valued-ness of , and the relation,Solutions without Wedges, Dislocations and Concentrated Forces,Kolosov-Muskhelishvilli Method,Chapter 5.6,83,Kolosov-Muskhelishvilli functions,Stresses at infinity :,Stresses along the Hole (Details),Kolosov-Muskhelishvilli Met
16、hod,Chapter 5.6,84,Traction distribution along a circle:,BCs can be expanded by Fourier series in complex form:,A Circular Hole under Tension(Details),Kolosov-Muskhelishvilli Method,Chapter 5.6,85,Remote tension,Kolosov-Muskhelishvilli stress functions,A Circular Hole under Tension(Details),Kolosov-Muskhelishvilli Method,Chapter 5.6,86,Stresses,Compare with Michells Solution,A Circular Hole under Tension(Details),Kolosov-Muskhelishvilli Method,Chapter 5.6,87,Gr
