12.1 Plastic behavior in simple tension and compression （单轴拉伸下材料的塑性行为） Chapter Page Theory of Elasticity A B C D E P A0 l0 o s e sb s's se ee sp ss sp limit of proportionality(比例极限) se elastic limits (弹性极限) ss initial yield stress（初始屈服应力） ep sb strength limit（强度极限） subsequent yield stress（后继屈服应力） s's s''s Bauschinger Effect （ Bauschinger 效应） 12 3

12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为） Chapter Page Theory of Elasticity Loading, unloading and reloading(加载，卸载，再加载) Loading（加载）: Unloading（卸载）: 12 4

12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为） Chapter Page Theory of Elasticity 12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为） Definition of large plastic strains (对数、自然应变Ludwik, 1909) True strain Natural strain(对数，自然应变) For small deformations, Definition of true stress（真实应力） Assume the materials is incompressible（假设材料不可压缩） 12 5

12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为） Chapter Page Theory of Elasticity 12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为） Example 1 A bar o length l0 is stretched to a final length of 2 l0. Values of the engineering and true strain? If the bar is compressed again to its very initial length , compute the engineering and true strains. Physically impossible The natural strain yields the same magnitude while the engineering strain magnitude is different 真实应变关于原点对称。 12 6

12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为） Chapter Page Theory of Elasticity Example 2 A bar of 100 mm initial length is elongated to a length of 200 mm by drawing in three stages. The length after each stage are 120, 150 and 200mm, respectively: a) Calculate the engineering strain for each stage separately and compare the sum with the total overall value of ε b) Repeat (a) for the true strain Solution: But This illustrates the additive property of true strain.(对数应变可加) 12 7

12.2 Modeling of uniaxial behavior in plasticity Chapter Page Theory of Elasticity Idealized stress-strain curves (1)（理想应力－应变曲线） s e ss es E E1 Elastic linear strain-hardening (弹、线性强化) 12 8

12.2 Modeling of uniaxial behavior in plasticity Chapter Page Theory of Elasticity 12.2 Modeling of uniaxial behavior in plasticity Idealized stress-strain curves (2) （理想应力－应变曲线） s e ss es Elastic perfectly-plastic (理想弹塑性) 12 9

12.2 Modeling of uniaxial behavior in plasticity Chapter Page Theory of Elasticity Idealized stress-strain curves (3)（理想应力－应变曲线） Rigid-perfectly plastic (理想刚塑性) Rigid-linear strain-hardening (刚、线性强化) Exponential hardening (幂次强化模型) 12 10

12.2 Modeling of uniaxial behavior in plasticity Chapter Page Theory of Elasticity 12.2 Modeling of uniaxial behavior in plasticity Strain hardening / Working hardening （强化现象） The effect of the material being able to withstand a greater stress after plastic deformation（塑性变形后，材料的承载能力提高） Strain softening（软化） 12 11

12.2 Modeling of uniaxial behavior in plasticity Chapter Page Theory of Elasticity 12.2 Modeling of uniaxial behavior in plasticity Hardening rules(强化模型) Isotropic hardening(等向强化) The reversed compressive yield stress is assumed equal to the tensile yield stress.（反向压缩屈服应力等于拉伸屈服应力） 12 12

12.2 Modeling of uniaxial behavior in plasticity Chapter Page Theory of Elasticity Hardening rules(强化模型) Kinematic hardening(随动强化) The elastic range is assumed to be unchanged during hardening. The center of elastic region is moved along the straight line aa’ 12 13

12.2 Modeling of uniaxial behavior in plasticity Chapter Page Theory of Elasticity Hardening rules(强化模型) Mixed hardening(组合强化) A combination of kinematic and isotropic hardening (Hodge, 1957) Kinematic hardening(随动强化) Isotropic hardening等向强化 Mixed hardening组合强化 12 14

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity H-W stress space(H-W应力空间) Principal stress space （主应力空间） A possible stress state:P（1、2、3） Orientation of the stress state is ignored （忽略主应力方向） Hydrostatic axis（静水压力轴） Pass through the origin and making the same angle with each of the coordinate axes.（过原点和三个坐标轴夹角相等。） Deviatoric plane Perpendicular to ON π- plane 12 15

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） Division of stress state of a point 12 16

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） Yield criterion（屈服准则） Defines the elastic limits of a material under combined states of stress. （在复杂应力状态下，材料的弹性极限） Yield criterion of Simple states of stress（简单应力状态下的屈服准则） s Uniaxial tension or compression （单轴拉伸或压缩） t Pure shear（纯剪切） General Yield criterion（通常情况下的屈服准则） f (ij,k1,K2,K3,….) = 0 Kn are material constants 12 17

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） General Yield criterion （通常情况下的屈服准则） f (ij,k1,K2,K3,….) = 0 Kn are material constants f (1、2、3、1、2、3,k1,K2,K3,….) = 0 Isotropic, Orientation of principal stress is immaterial f (1，2，3, k1,K2,K3,…. )=0 1，2，3, can be expressed in terms of I1,J2,J3 f (I1，J2，J3 , k1,K2,K3,…. )=0 Experimental results: hydrostatic pressure not appreciable f(J2，J3 , k1,K2,K3,…. )= 0 12 18

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） Respresention of yield criteria(屈服准则的表达) f(J2，J3 , k1,K2,K3,…. )= 0 3D case Since hydrostastic pressure has no effect, the yield surface in stress space is a cylinder.静水压力不影响屈服，则屈服面在应力空间为柱形。 12 19

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） Tresca yield criterion（ Tresca 屈服准则） 1864,Tresca, first yield creterion f (ij) = π- plane: A straight line x= (s1  s3) = (1  3) = k1 12 = 2k1 13 = 2k1 23 = 2k1 12 20

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） Determination of the material constant（材料常数的确定） Simple tension, 1 =s，2 =3 =0 f (ij) = k1= s/2 Pure shear,  =s  1= s，2=0，3= s, f (ij) = k1= s s=2s 12 21

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） Von-mises yield criterion（ Von-mises 屈服准则） 2D case (π- plane) 1913, Von mises Yielding begins when the octahedral shearing stress reaches a critical value k f (ij) = J2, an invariant of the stress deviator tensor 3D case r= k2 =const， In terms of principal stresses 12 22

12.3 Basics of Yield Criteria （屈服准则概述） Chapter Page Theory of Elasticity 12.3 Basics of Yield Criteria （屈服准则概述） Determination of the material constant （材料常数的确定） Simple tension, 1 =s，2 =3 =0 Pure shear,  =s  1= s，2=0，3= s, J2= k2 k2 = s 12 23