东南大学：《固体力学基础》课程教学课件（英文讲稿）11 Simple Elastoplastic BVPs

Cylindrically Symmetric Elastoplastic Solids? Cylindrically symmetric geometryand loading (i.e. internal bodyAerforces, tractions or displacements7BCs, nonuniform temperaturedistribution)022. Cylindrical-polar bases: (e, e.,e.). Cylindrical-polar coordinates: (r, 0, z).Position vector: x=re,? Displacement vector: u = u, [r]e,. Body force vector: F = F, [r]e,? Acceleration vector: a =-o’re,6

Cylindrically Symmetric Elastoplastic Solids 6 • Cylindrically symmetric geometry and loading (i.e. internal body forces, tractions or displacements BCs, nonuniform temperature distribution). • Cylindrical-polar bases: • Cylindrical-polar coordinates: • Position vector: • Displacement vector: • Body force vector: • Acceleration vector: e e e r z , ,   r z , ,   r x e  r u e  u r r r   F e  F r r r   2 r a e   r

Cylindrically Symmetric Elastoplastic Solids. Cauchy stress: o =o, [rle,e, +oe[rlege,+o, [rle.e. Infinitesimal strain: =e, [rle,e, +ee[rlege。+8. [rle.eduru,? Strain-displacement relation: ,80dr: Stress-strain relation in elastic region (plane strain orgeneralized plane strain):E(1-v)e, +V8 +Ve:)a(1+v)(1-2v)Eve, +(1-v)8p +ve:)6(1+v)(1-2v)Eve, + Ve。 +(1-v)8.)a(1+v)(1-2v) von Mises yield criterion:0. = /(0, -0.) +(a。-0.) +(a -0,) ) =0y7

Cylindrically Symmetric Elastoplastic Solids • Cauchy stress: 7 • Strain-displacement relation: • Stress-strain relation in elastic region (plane strain or generalized plane strain): • von Mises yield criterion: • Infinitesimal strain: σ     r r r z z z r r r e e e e e e + +        ε     r r r z z z r r r e e e e e e + +        d , d r r r u u r r                          1 , 1 1 2 1 , 1 1 2 1 . 1 1 2 r r z r z z r z E E E                                                   1 2 2 2 2         e r z z r Y         

Cylindrically Symmetric Elastoplastic Solids: Stress-strain relation in plastic region? Strain partition: de, =de +dep,de = dee +ds,de, =dee +ds?do,.v(do。+do.)?Elastic strain: deeEE: Flow rule:de = d31S=2gv2 0ydo, +,-Oe+F =-po"r· Equations of motion:drr. Traction BCs: Cr[a]=a,Or[b]= Ob.BCs: u,[a]=ua, u,[b]=ub;or ,[a]=oa, o,[b]=oi. There is no clean, direct, and general method for integrating theseequations. Instead, solutions must be found using a combination ofphysical intuition and some algebraic tricks8

Cylindrically Symmetric Elastoplastic Solids • Stress-strain relation in plastic region 8 • Traction BCs: • Equations of motion:     R a R b a b    , .   • Strain partition: • Elastic strain: • Flow rule: • BCs: • There is no clean, direct, and general method for integrating these equations. Instead, solutions must be found using a combination of physical intuition and some algebraic tricks. , , e p e p e p r r r z z z d d d d d d d d d                     , , e r z r d d d d E E             3 3 1 1 1 + + + , , 2 2 3 2 p p p p r r r r z r z Y Y Y d d d d                                    r r 2 r d F r dr r          u a u u b u a b r a r b r a r b       , ; or ,          

Hollow Cylinder under Monotonic Pressure: We consider a long hollow cylinder. The sphere is stress free before it isverloaded.7? No body forces act on the cylinder.The cylinder has zero angular velocity02C. The cylinder has uniform temperature.: The cylinder does not stretch parallel toits axis.. The inner surface r = a is subjected to monotonicallyincreasing pressure pa: The outer surface r = b is traction free. Strains are infinitesimal.. We aim to find9

Hollow Cylinder under Monotonic Pressure • We consider a long hollow cylinder. • The sphere is stress free before it is loaded. • No body forces act on the cylinder. • The cylinder has zero angular velocity. • The cylinder has uniform temperature. • The cylinder does not stretch parallel to its axis. 9 • The inner surface r = a is subjected to monotonically increasing pressure pa . • The outer surface r = b is traction free. • Strains are infinitesimal. • We aim to find

Hollow Cylinder under Monotonic Pressure(1+v) pα26·Elastic solutionE(b?-α?2vp.ap.ap.a2/0。0b?-α?62-: von Mises effective stress:3p.a?b?3b4Paa?+1+4v2-4v, V~0.5=6deb? - α?(b2 -α°)r2. We see that the hollow cylinder first reaches yield at r = a,With the elastic limit: pa/oy ~(1-a2 /b2)/ /3 If the pressure is increased beyond yield, we anticipate thata region a < r < c will deform plastically, whereas a regionc < r < b remains elastic10

Hollow Cylinder under Monotonic Pressure • Elastic solution 10 • von Mises effective stress: • We see that the hollow cylinder first reaches yield at r = a, with the elastic limit: • If the pressure is increased beyond yield, we anticipate that a region a < r < c will deform plastically, whereas a region c < r < b remains elastic.       2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 ; 2 1 , 1 , . a r a a a r z p a r b u E b a r p a p a p a b b b a r b a r b a                                         2 2 2 4 2 2 2 4 2 2 2 3 3 1 4 4 , 0.5 a a e e p a p a b b b a r b a r                 2 2 1 3 a Y p a b   