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12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为) ChapterPage Theory of Elasticity P A0A0 l0l0 P o   A B C D E pp  p limit of.

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Presentation on theme: "12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为) ChapterPage Theory of Elasticity P A0A0 l0l0 P o   A B C D E pp  p limit of."— Presentation transcript:

1 12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为) ChapterPage Theory of Elasticity P A0A0 l0l0 P o   A B C D E pp  p limit of proportionality( 比例极限 ) ee  e  elastic limits ( 弹性极限 ) ss  s initial yield stress (初始屈服应力) bb  b strength limit (强度极限) 's's  '' s subsequent yield stress (后继屈服应力) 's's Bauschinger Effect ( Bauschinger 效应) 12 3 ee pp

2 12.1 Plastic behavior in simple tension and compression ( 单轴拉伸下材料的塑性行为) ChapterPage Theory of Elasticity Loading, unloading and reloading( 加载,卸载,再加载 ) Loading (加载) : Unloading (卸载) : 12 4

3 12.1 Plastic behavior in simple tension and compression ( 单轴拉伸下材料的塑性行为) ChapterPage Theory of Elasticity Definition of large plastic strains ( 对数、自然应变 Ludwik, 1909) True strain Natural strain( 对数,自然应变 ) Definition of true stress (真实应力) Assume the materials is incompressible (假设材料不可压缩) For small deformations, 5 12

4 12.1 Plastic behavior in simple tension and compression ( 单轴拉伸下材料的塑性行为) ChapterPage Theory of Elasticity Example 1 A bar o length l 0 is stretched to a final length of 2 l 0. Values of the engineering and true strain? If the bar is compressed again to its very initial length, compute the engineering and true strains. The natural strain yields the same magnitude while the engineering strain magnitude is different Physically impossible 真实应变关于原点对称。 6 12

5 12.1 Plastic behavior in simple tension and compression ( 单轴拉伸下材料的塑性行为) ChapterPage 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 ) 7 12

6 12.2 Modeling of uniaxial behavior in plasticity ChapterPage Theory of Elasticity Idealized stress-strain curves (1) (理想应力-应变曲线)   ss ss E E1E1 Elastic linear strain-hardening ( 弹、线性强化 ) 8 12

7 12.2 Modeling of uniaxial behavior in plasticity ChapterPage Theory of Elasticity Idealized stress-strain curves (2) (理想应力-应变曲线)   ss ss Elastic perfectly-plastic ( 理想弹塑性 ) 9 12

8 12.2 Modeling of uniaxial behavior in plasticity ChapterPage 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 ) 10 12

9 12.2 Modeling of uniaxial behavior in plasticity ChapterPage Theory of Elasticity 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) 11 12

10 12.2 Modeling of uniaxial behavior in plasticity Hardening rules ( 强化模型 ) ChapterPage Theory of Elasticity 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

11 12.2 Modeling of uniaxial behavior in plasticity ChapterPage 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’ 13 12

12 12.2 Modeling of uniaxial behavior in plasticity ChapterPage 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

13 12.3 Basics of Yield Criteria (屈服准则概述) ChapterPage 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

14 12.3 Basics of Yield Criteria (屈服准则概述) Division of stress state of a point ChapterPage Theory of Elasticity 12 16

15 12.3 Basics of Yield Criteria (屈服准则概述) ChapterPage Theory of Elasticity 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)  Uniaxial tension or compression (单轴拉伸或压缩) Pure shear (纯剪切) f (  ij,k 1,K 2,K 3,….) = 0 K n are material constants General Yield criterion (通常情况下的屈服准则)  12 17

16 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,k 1,K 2,K 3,….) = 0 K n are material constants Isotropic, Orientation of principal stress is immaterial f (  1 、  2 、  3 、  1 、  2 、  3,k 1,K 2,K 3,….) = 0 f (  1 ,  2 ,  3, k 1,K 2,K 3,…. )=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 ChapterPage Theory of Elasticity 12 18

17 12.3 Basics of Yield Criteria (屈服准则概述) Respresention of yield criteria( 屈服准则的表达 ) 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。 f(J2 , J3, k1,K2,K3,…. )= 0 3D case ChapterPage Theory of Elasticity 12 19

18 12.3 Basics of Yield Criteria (屈服准则概述) ChapterPage Theory of Elasticity Tresca yield criterion ( Tresca 屈服准则 ) f (  ij ) = x= (s 1  s 3 ) = (  1   3 ) = k 1 π- plane:A straight line  1  2 =  2k 1  1  3 =  2k 1  2  3 =  2k ,Tresca, first yield creterion 12 20

19 12.3 Basics of Yield Criteria (屈服准则概述) ChapterPage Theory of Elasticity Determination of the material constant (材料常数的确定) Simple tension,  1 =  s ,  2 =  3 =0 f (  ij ) = k 1 =  s /2 Pure shear,  =  s   1=  s ,  2=0 ,  3=  s, f (  ij ) = k 1 =  s  s =2  s 12 21

20 12.3 Basics of Yield Criteria (屈服准则概述) ChapterPage Theory of Elasticity Von-mises yield criterion ( Von-mises 屈服准则 ) 1913, Von mises f (  ij ) = r  = k 2 =const , 2D case (π- plane) 3D case Yielding begins when the octahedral shearing stress reaches a critical value k J 2, an invariant of the stress deviator tensor In terms of principal stresses 12 22

21 12.3 Basics of Yield Criteria (屈服准则概述) ChapterPage Theory of Elasticity 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, J 2 = k 2 k 2 =  s 12 23


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