2 Plasticity If a body is deformed elastically,it returns to its original shape when the stress is removed.The stress and strain under elastic loading are related through Hooke's laws. Any stress will cause some strain.In contrast,no plastic deformation occurs until the stress reaches the yield strength of the material.When the stress is removed,the plastic strain remains.For ductile metals large amounts of plastic deformation can occur under continually increasing stress. In this text experimental observations are linked with mathematical expressions. Yield criteria are mathematical descriptions of the combination of stresses necessary to cause yielding. 2.1 YIELD CRITERIA A yield criterion is a postulated mathematical expression of the states of stress that will cause yielding.The most general form is: f(Ox,Oy,O:,tyz,t=x,Txy)=C. (2.1) For isotropic materials,this can be expressed in terms of principal stresses as f(o1,o2,o3)=C. (2.2) For most isotropic ductile metals the following assumptions are commonly made: 1.The yield strengths in tension and compression are the same.That is any Bauschinger*effect is small enough so it can be ignored. 2.The volume remains constant during plastic deformation. 3.The magnitude of the mean normal stress,does not affect yielding. 0m=1十2+a3 (2.3) 3 J.Bauschinger,Civilingenieur,27(1881),p.289. 17
2 Plasticity If a body is deformed elastically, it returns to its original shape when the stress is removed. The stress and strain under elastic loading are related through Hooke’s laws. Any stress will cause some strain. In contrast, no plastic deformation occurs until the stress reaches the yield strength of the material. When the stress is removed, the plastic strain remains. For ductile metals large amounts of plastic deformation can occur under continually increasing stress. In this text experimental observations are linked with mathematical expressions. Yield criteria are mathematical descriptions of the combination of stresses necessary to cause yielding. 2.1 YIELD CRITERIA A yield criterion is a postulated mathematical expression of the states of stress that will cause yielding. The most general form is: f (σx , σy , σz, τyz, τzx , τx y ) = C. (2.1) For isotropic materials, this can be expressed in terms of principal stresses as f (σ1, σ2, σ3) = C. (2.2) For most isotropic ductile metals the following assumptions are commonly made: 1. The yield strengths in tension and compression are the same. That is any Bauschinger∗ effect is small enough so it can be ignored. 2. The volume remains constant during plastic deformation. 3. The magnitude of the mean normal stress, does not affect yielding. σm = σ1 + σ2 + σ3 3 , (2.3) ∗ J. Bauschinger, Civilingenieur, 27 (1881), p. 289. 17
18 PLASTICITY 0 03 02 Figure 2.1.Mohr's circles for two stress states that differ only by a hydrostatic stress,am,and are therefore equivalent in terms of yielding. The yield criteria to be discussed involve these assumptions.Effects oftemperature, prior straining,and strain rate will be discussed in later chapters.The assumption that yielding is independent of om is reasonable because deformation usually occurs by slip or twining which are shear mechanisms.Therefore all yield criteria for isotropic materials have the form f[(o2-o3,(o3-1),(o1-o2]=C. (2.4) This is equivalent to stating that yielding depends only on the size of the Mohr's circles and not on their positions.Figure 2.1 shows this.If a stress state,1,2,o3,will cause yielding another stress state,of=o1-am,o2 =02-om,a3=o3-om that differs only by am will also cause yielding.The stresses,af,o,o,are called the deviatoric stresses. 2.2 TRESCA CRITERION The Tresca criterion postulates that yielding depends only on the largest shear stress in the body.With the convention,>2 >o3,this can be expressed as o1-o3= C.The constant C can be found by considering a tension test.In this case,o3 =0 and o1=Y,the yield strength at yielding,so C=Y.Therefore this criterion can be expressed as 1-03=Y. (2.5) Yielding in pure shear occurs when the largest shear stress,o =k and o3 -o1=k,where k is the yield strength in shear. 1-03=2k. (2.6)
18 PLASTICITY Figure 2.1. Mohr’s circles for two stress states that differ only by a hydrostatic stress, σm, and are therefore equivalent in terms of yielding. The yield criteria to be discussed involve these assumptions. Effects of temperature, prior straining, and strain rate will be discussed in later chapters. The assumption that yielding is independent of σm is reasonable because deformation usually occurs by slip or twining which are shear mechanisms. Therefore all yield criteria for isotropic materials have the form f [(σ2 − σ3), (σ3 − σ1), (σ1 − σ2)] = C. (2.4) This is equivalent to stating that yielding depends only on the size of the Mohr’s circles and not on their positions. Figure 2.1 shows this. If a stress state, σ1, σ2, σ3, will cause yielding another stress state, σ 1 = σ1 − σm, σ 2 = σ2 − σm, σ 3 = σ3 − σm that differs only by σm will also cause yielding. The stresses, σ 1, σ 2, σ 3, are called the deviatoric stresses. 2.2 TRESCA CRITERION The Tresca criterion postulates that yielding depends only on the largest shear stress in the body. With the convention, σ1 ≥ σ2 ≥ σ3, this can be expressed as σ1 − σ3 = C. The constant C can be found by considering a tension test. In this case, σ3 = 0 and σ1 = Y, the yield strength at yielding, so C = Y. Therefore this criterion can be expressed as σ1 − σ3 = Y . (2.5) Yielding in pure shear occurs when the largest shear stress, σ1 = k and σ3 = −σ1 = k, where k is the yield strength in shear. σ1 − σ3 = 2k. (2.6)
2.2 TRESCA CRITERION 19 0x=0 Figure 2.2.The Tresca criterion.In the six sectors the following conditions apply: 1y>0x>0:s0y=Y ∥ox>y>0:s0ox=Y Ill ax 0>dy so ax-ay =Y IV 0>ox>ay:so ay =-Y V 0>ay>ax so ax =-Y VI ay>0>ax:so ay-ax Y A yield locus is a plot of a yield criterion.Figure 2.2 is a plot of the Tresca yield locus,o vs.o,for o-=0,where ox,o,and a=are principal stresses. EXAMPLE 2.1:A thin-wall tube with closed ends is subjected to a maximum internal pressure of 35 MPa in service.The mean radius of the tube is 30 cm (a)If the tensile yield strength is 700 MPa,what minimum thickness must be specified to prevent yielding? (b)If the material has a yield strength in shear of k=280 MPa,what minimum thickness must be specified to prevent yielding? SOLUTION: (a)Hoop stress,a1 Pr/t =35(30 cm)/t omax,longitudinal stress =02 Pr/(2t)=(35 MPa)(30 cm)/(2t),omax,thickness stress,o30.Yielding occurs when o1=700,or t =(35 MPa)(30 cm)/700 MPa=1.5 cm
2.2 TRESCA CRITERION 19 I II III IV V VI σx σy σx σy σx σy σx σy σx σy σx σy σx σy σx = 0 Figure 2.2. The Tresca criterion. In the six sectors the following conditions apply: I σy > σx > 0 : so σy = Y II σx > σy > 0 : so σx = Y III σx > 0 > σy : so σx − σy = Y IV 0 > σx > σy : so σy = −Y V 0 > σy > σx : so σx = −Y VI σy > 0 > σx : so σy − σx = Y A yield locus is a plot of a yield criterion. Figure 2.2 is a plot of the Tresca yield locus, σx vs. σy for σz = 0, where σx, σy, and σz are principal stresses. EXAMPLE 2.1: A thin-wall tube with closed ends is subjected to a maximum internal pressure of 35 MPa in service. The mean radius of the tube is 30 cm. (a) If the tensile yield strength is 700 MPa, what minimum thickness must be specified to prevent yielding? (b) If the material has a yield strength in shear of k = 280 MPa, what minimum thickness must be specified to prevent yielding? SOLUTION: (a) Hoop stress, σ1 = Pr/t = 35(30 cm)/t = σmax, longitudinal stress = σ2 = Pr/(2t) = (35 MPa)(30 cm)/(2t), σmax, = thickness stress, σ3 ≈ 0. Yielding occurs when σ1 = 700, or t = (35 MPa)(30 cm)/700 MPa = 1.5 cm
20 PLASTICITY (b)o1-03=2k =560 MPa at yielding,so yielding occurs when t=(35 MPa) (30cm)/(560MPa)=1.875cm. 2.3 VON MISES CRITERION The von Mises criterion postulates that yielding will occur when the value of the root-mean-square shear stress reaches a critical value.Expressed mathematically, (a2-32+(a3-o12+(a1-02)2 =CI 3 or equivalently (o2-03)2+(o3-01)2+(a1-02)2=C2. Again,C2 may be found by considering a uniaxial tension test in the 1-direction. Substituting o1=Y,o2 =o3=0 at yielding,the von Mises criterion may be expressed as (o2-03)2+(o3-1)2+(o1-02)2=2Y2=6k2 (2.7) Figure 2.3 is the yield locus with o2 =0. In a more general form equation 2.7 may be written as a,-a+(a:-a}+o-+6(乐+孟+)=2Y2=6k2. (2.8) The Tresca and von Mises yield loci are plotted together in Figure 2.4 for the same values of y.Note that the greatest differences occur for a =-1,and 2. Three-dimensional plots of the Tresca and von Mises yield criteria are shown in Figure 2.5.The Tresca criterion is a regular hexagonal prism and the von Mises criterion 07+3-0103=Y2 03 03=201 02=0 01=203=1.155Y 03=0.577Y 01 +Y Pure shear 01=-03 01=0.577Y,03=-0.577Y -03 Figure 2.3.The von Mises yield locus
20 PLASTICITY (b) σ1 − σ3 = 2k = 560 MPa at yielding, so yielding occurs when t = (35 MPa) (30 cm)/(560 MPa) = 1.875 cm. 2.3 VON MISES CRITERION The von Mises criterion postulates that yielding will occur when the value of the root-mean-square shear stress reaches a critical value. Expressed mathematically, (σ2 − σ3) 2 + (σ3 − σ1) 2 + (σ1 − σ2) 2 3 = C1 or equivalently (σ2 − σ3) 2 + (σ3 − σ1) 2 + (σ1 − σ2) 2 = C2. Again, C2 may be found by considering a uniaxial tension test in the 1-direction. Substituting σ1 = Y, σ2 = σ3 = 0 at yielding, the von Mises criterion may be expressed as (σ2 − σ3) 2 + (σ3 − σ1) 2 + (σ1 − σ2) 2 = 2Y 2 = 6k2 . (2.7) Figure 2.3 is the yield locus with σ2 = 0. In a more general form equation 2.7 may be written as (σy − σz) 2 + (σz − σx ) 2 + (σx − σy ) 2 + 6 τ 2 yz + τ 2 zx + τ 2 x y = 2Y 2 = 6k2 . (2.8) The Tresca and von Mises yield loci are plotted together in Figure 2.4 for the same values of Y. Note that the greatest differences occur for α = −1, 1 / 2 and 2. Three-dimensional plots of the Tresca and von Mises yield criteria are shown in Figure 2.5. The Tresca criterion is a regular hexagonal prism and the von Mises criterion Figure 2.3. The von Mises yield locus
2.4 EFFECTIVE STRESS 21 a=+2 01 a=+1 02=0 ya=+ +a1(a=0) a=-1 Figure 2.4.Tresca and von Mises loci showing certain loading paths. 02 Radius of cylinder is Y√ 03 0 Yield surface Two dimensional yield locus 01 Figure 2.5.Three-dimensional plots of the Tresca and von Mises yield criteria. is a cylinder.Both are centered on a line,o1 =o2 =03.The projection of these on a plane o1+o2 +03 a constant is shown in Figure 2.6. EXAMPLE 2.2:Reconsider the capped tube in Example 2.1 except let t=1.5 cm. Use both the Tresca and von Mises criteria to determine the necessary yield strength to prevent yielding. SOLUTION: Tresca:1=Y=(700 MPa)(30 cm)/1.5 cm =1400 MPa. Von Mises:1=(2/3)Y.Y=(3/2)(700 MPa)(30 cm)/1.5cm 1212 MPa 2.4 EFFECTIVE STRESS It is useful to define an effective stress,G,for a yield criterion such that yielding occurs when the magnitude of reaches a critical value.For the von Mises criterion, 6=√(1/2[(o2-03)2+(a3-01)2+(o1-02)2]. (2.9)
2.4 EFFECTIVE STRESS 21 Figure 2.4. Tresca and von Mises loci showing certain loading paths. Figure 2.5. Three-dimensional plots of the Tresca and von Mises yield criteria. is a cylinder. Both are centered on a line, σ1 = σ2 = σ3. The projection of these on a plane σ1 + σ2 + σ3 = a constant is shown in Figure 2.6. EXAMPLE 2.2: Reconsider the capped tube in Example 2.1 except let t = 1.5 cm. Use both the Tresca and von Mises criteria to determine the necessary yield strength to prevent yielding. SOLUTION: Tresca: σ1 = Y = (700 MPa)(30 cm)/1.5 cm = 1400 MPa. Von Mises: σ1 = ( √ 2/3)Y. Y = (3/ √ 2)(700 MPa)(30 cm)/1.5cm = 1212 MPa. 2.4 EFFECTIVE STRESS It is useful to define an effective stress, ¯σ, for a yield criterion such that yielding occurs when the magnitude of ¯σ reaches a critical value. For the von Mises criterion, σ¯ = (1/2)[(σ2 − σ3)2 + (σ3 − σ1)2 + (σ1 − σ2)2]. (2.9)