12.1 plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为)
DESCRIPTION
D. s. B. s e. E. A. C. s p. Theory of Elasticity. s s. e. e p. e e. o. P. A 0. l 0. s ' s. subsequent yield stress (后继屈服应力). P. Chapter. Page. 12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为). s b. s ' s. s p limit of proportionality( 比例极限 ). - PowerPoint PPT PresentationTRANSCRIPT
12.1 Plastic behavior in simple tension and compression (单轴拉伸下材料的塑性行为)
Chapter Page
Th
eory
of
Ela
sti
cit
y
PA0
l0
P
0A
P
l
ll 0
o
A
B
C
D
Ep
p limit of proportionality( 比例极限 )
e
e elastic limits ( 弹性极限 )
s
s initial yield stress (初始屈服应力)
b
b strength limit (强度极限)
's
''ssubsequent yield stress (后继屈服应力)'s Bauschinger Effect
( Bauschinger 效应)
sss
12 3
ep
12.1 Plastic behavior in simple tension and compression( 单轴拉伸下材料的塑性行为)
Chapter Page
Th
eory
of
Ela
stic
ity
Loading, unloading and reloading( 加载,卸载,再加载 )
Loading (加载) :
0d
Unloading (卸载) :
0d
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
Th
eory
of
Ela
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ity
Definition of large plastic strains ( 对数、自然应变 Ludwik, 1909)
L
dld )(
00l
lIn
L
dld
l
l
True strainNatural strain( 对数,自然应变 )
Definition of true stress (真实应力)
A
P
_
Assume the materials is incompressible (假设材料不可压缩)
00lAAl el
l
A
P
00
)1()(0
Inl
lIn .....
!32
32
For small deformations,
512
12.1 Plastic behavior in simple tension and compression( 单轴拉伸下材料的塑性行为)
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Th
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Ela
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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.
0.12
0
00
l
ll
693.0)2
(0
0 l
lIn
5.02
2
0
00
l
ll
693.0)2
(0
0 l
lIn
The natural strain yields the same magnitude while the engineering strain magnitude is different
0
0
2
20.1
l
lL
0L
Physically impossible
真实应变关于原点对称。
612
12.1 Plastic behavior in simple tension and compression( 单轴拉伸下材料的塑性行为)
Chapter Page
Th
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Ela
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ity
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
33.0150
50,25.0
120
30,2.0
100
20321 Solution:
78.0321 sum But 0.1100
100200
overall
29.0)150
200(,22.0)
120
150(,18.0)
100
120( 321 InInIn
69.0321 sum 69.0)100/200( InoverallThis illustrates the additive property of true strain.( 对数应变可加 )
712
12.2 Modeling of uniaxial behavior in plasticity
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eory
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Ela
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Idealized stress-strain curves (1) (理想应力-应变曲线)
s
s
E
E1
Elastic linear strain-hardening( 弹、线性强化 )
sss
s
E
E
)(1
812
12.2 Modeling of uniaxial behavior in plasticity
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Th
eory
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Ela
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Idealized stress-strain curves (2) (理想应力-应变曲线)
s
s
Elastic perfectly-plastic( 理想弹塑性 )
ss
sE
912
12.2 Modeling of uniaxial behavior in plasticity
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eory
of
Ela
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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 )
1012
12.2 Modeling of uniaxial behavior in plasticity
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Th
eory
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Ela
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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)
1112
12.2 Modeling of uniaxial behavior in plasticity
Hardening rules(强化模型 )
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Isotropic hardening( 等向强化 )
*||
|||'| BCCB
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)
1212
12.2 Modeling of uniaxial behavior in plasticity
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Hardening rules(强化模型 )
Kinematic hardening( 随动强化 )
The elastic range is assumed to be unchanged during hardening.
|'||'| AABB The center of elastic region is moved along the straight line aa’
1312
12.2 Modeling of uniaxial behavior in plasticity
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Hardening rules(强化模型 )
Mixed hardening( 组合强化 )
1412
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
12.3 Basics of Yield Criteria (屈服准则概述)
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Th
eory
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O
N
P
Q
平面
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
ON3321
π- plane 0321
12 15
12.3 Basics of Yield Criteria (屈服准则概述)
O
N
P
Q
平面
OQONOP ),,(),,( 000 pppON
pI
nOPON
33
)3
1,
3
1,
3
1(),,(
1
321
),,(
)](),(),[(
),,(),,(
),,(
321
321
321
321
sss
ppp
ppp
ONOQ
822/12
32
22
1 32)( JsssOQ
Division of stress state of a point
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12 16
12.3 Basics of Yield Criteria (屈服准则概述)
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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
s
Uniaxial tension or compression(单轴拉伸或压缩)
Pure shear (纯剪切)
f (ij,k1,K2,K3,….) = 0 Kn are material constants
General Yield criterion (通常情况下的屈服准则)
12 17
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
Isotropic, Orientation of principal stress is immaterial
f (1 、 2 、 3 、 1 、 2 、 3,k1,K2,K3,….) = 0
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
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12 18
12.3 Basics of Yield Criteria (屈服准则概述)
Respresention of yield criteria( 屈服准则的表达 )
平面
屈服面
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
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12 19
12.3 Basics of Yield Criteria (屈服准则概述)
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Tresca yield criterion ( Tresca 屈服准则)
f (ij) = 02
131
k
x= (s1 s3) = (1 3) = k1 22
22
2
π- plane: A straight line
12 = 2k1
13 = 2k1
23 = 2k1
e' e'
e'
平面
屈服面
1864,Tresca, first yield creterion
12 20
12.3 Basics of Yield Criteria (屈服准则概述)
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Determination of the material constant (材料常数的确定)
Simple tension, 1 =s , 2 =3 =0
f (ij) = 02
131
k k1= s/2
Pure shear, =s 1= s , 2=0 , 3= s,
f (ij) = 02
131
k k1= s
s=2s
e' e'
e'
12 21
12.3 Basics of Yield Criteria (屈服准则概述)
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Von-mises yield criterion ( Von-mises 屈服准则)
1913, Von mises
0222 kJf (ij) =
r= k2 =const , 22 2 J
e' e'
e' 2D case (π- plane)
平面
屈服面
3D case
Yielding begins when the octahedral shearing stress reaches a critical value k
22 3
2
3
2kJoct
J2, an invariant of the stress deviator tensor
22
213
232
221 6k
In terms of principal stresses
12 22
12.3 Basics of Yield Criteria (屈服准则概述)
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Th
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ity
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,
22
2
23
kJ s
sk 3
12
J2= k2 k2 = s
ss 3
12 23