Simple Elastoplastic BVPsmi@se.edu.cn
Simple Elastoplastic BVPs
Outline·Assumptions（假设）·Introduction（引言）·Summary of governing equations（弹塑性控制方程）· Cylindrically symmetric elastoplastic solids （轴对称）· Hollow cylinder under monotonic internal pressure (空心圆筒受单调增载）· Spherically symmetric elastoplastic solids （中心对称）· Hollow sphere under monotonic internal pressure （球壳受单调增载)·Hollow sphere under cyclic internal pressure（球壳循环受载)2
Outline • Assumptions（假设） • Introduction（引言） • Summary of governing equations（弹塑性控制方程） • Cylindrically symmetric elastoplastic solids（轴对称） • Hollow cylinder under monotonic internal pressure（空心 圆筒受单调增载） • Spherically symmetric elastoplastic solids（中心对称） • Hollow sphere under monotonic internal pressure（球壳受 单调增载） • Hollow sphere under cyclic internal pressure（球壳循环受 载） 2
Assumptions?Body force density is given? Prescribed boundary tractions and/or displacements All displacements are small. This means that we can usethe infinitesimal strain tensor to characterizedeformation; we do not need to distinguish betweenstress measures, and we do not need to distinguishbetween deformed and undeformed configurations of thesolid when writing equilibrium equations and boundaryconditions? The material is isotropic, elastic-perfectly plastic solid.? Neglect temperature changes3
Assumptions • Body force density is given. • Prescribed boundary tractions and/or displacements • All displacements are small. This means that we can use the infinitesimal strain tensor to characterize deformation; we do not need to distinguish between stress measures, and we do not need to distinguish between deformed and undeformed configurations of the solid when writing equilibrium equations and boundary conditions. • The material is isotropic, elastic-perfectly plastic solid. • Neglect temperature changes. 3
Lntroduction. The elastic limit: This is the load required to initiate plastic flow in the solid.Theplasticcollapseload:Atthisload,thedisplacements inthe solidbecomeinfinite..Residual stress:If a solid isloadedbeyond theelasticlimitandthenunloaded,asystem of self-equilibrated stress is established in the material.:Shakedown: If an elastic-plastic solid is subjected to cyclic loading and themaximum load during the cycle exceeds yield, then some plastic deformationmust occur in the material during the first load cycle. However, residual stressesare introduced in the solid, which may prevent plastic flow during subsequentcycles of load. This process is known as shakedown," and the maximum load forwhich it can occur isknown as the shakedownlimit. The shakedown limit is oftersubstantially higher than the elastic limit, so the concept of shakedown can oftenbe used to reduce the weight of a design.Cyclic plasticity: For cyclic loads exceeding the shakedown limit, aregion in thesolid will be repeatedly plastically deformed.4
Introduction • The elastic limit: This is the load required to initiate plastic flow in the solid. • The plastic collapse load: At this load, the displacements in the solid become infinite. • Residual stress: If a solid is loaded beyond the elastic limit and then unloaded, a system of self-equilibrated stress is established in the material. • Shakedown: If an elastic-plastic solid is subjected to cyclic loading and the maximum load during the cycle exceeds yield, then some plastic deformation must occur in the material during the first load cycle. However, residual stresses are introduced in the solid, which may prevent plastic flow during subsequent cycles of load. This process is known as “shakedown,” and the maximum load for which it can occur is known as the shakedown limit. The shakedown limit is often substantially higher than the elastic limit, so the concept of shakedown can often be used to reduce the weight of a design. • Cyclic plasticity: For cyclic loads exceeding the shakedown limit, a region in the solid will be repeatedly plastically deformed. 4
Summary of Governing EguationsDisplacement-strain relation:L? Strain partition: de, = de, +dep. Incremental stress-stain relation:0,S1 +vVEE3 02dep302 0yV2? Equations of static equilibrium: ji,j + F, = 0.. Traction BCs on S: O,n, =tj? Displacement BCs on Su: u, = ü5
Summary of Governing Equations • Displacement-strain relation: 5 , , 1 2 ij i j j i u u • Strain partition: e p ij ij ij d d d • Incremental stress-stain relation: 3 0, 1 2 ; 3 3 , 2 2 ij ij Y e p ij ij kk ij ij p ij ij ij Y Y d d d d E E d , 0. • Equations of static equilibrium: ji j i F • Traction BCs on St : • Displacement BCs on Su : ij i j n t i i u u