260 C Plastic Flow Rate Inside the Neck of voids in Vi.With respect to Equations C.4 and 2.30,where Vo mroh, one obtains V兰4πa6 nekEp. By considering Equation 2.30(V=Vi)and denoting==1+ we get 4ra5ne1+)ep ()a( OeYerdEp. (C.5) When inserting Equation C.5 into Equation 2.33 one obtains dfm兰-Φjm+6ep, dep which is a linear differential equation.By solving this equation,a change of the fraction of mobile dislocations during the unstable deformation inside the neck can be expressed in the following form: 7-÷.a-j Here f"is the relative density of mobile dislocations at the onset of necking. After integration and some algebraic re-arrangement ( e+ 7+变e (C.6) Equation C.6 shows the dependence of the relative density of mobile disloca- tions on the plastic deformation during the necking process.By introducing Equation C.6 into the general relationship at Equation 2.29 one obtains =a+M四{} 21e-ep+22eYep）, (C.7) where=f-是，B=是and币=G品.Moreover,.P.and D.are the density of dislocations and the friction stress,respectively,at the onset of necking,(o)and are the mean effective stress and the mean value of during the necking process,respectively((is nearly constant).Finally, an appropriate algebraic re-arrangement of Equation C.7 leads to Equation 2.36
260 C Plastic Flow Rate Inside the Neck of voids in V1. With respect to Equations C.4 and 2.30, where V0 = πr2 0h, one obtains V2 ∼= 4πa2 0δneκεp . By considering Equation 2.30 (V = V1) and denoting Θ = 4πγa2 0δn V0 , γ = 1+κ we get d V2 V1 ∼= d 4πa2 0δn V0 e(1+κ)εp = Θeγεp dεp. (C.5) When inserting Equation C.5 into Equation 2.33 one obtains dfm dεp ∼= −Φfm + Θeγεp , which is a linear differential equation. By solving this equation, a change of the fraction of mobile dislocations during the unstable deformation inside the neck can be expressed in the following form: fm ∼= exp ⎧ ⎨ ⎩ εp 0 −Φdt ⎫ ⎬ ⎭ ⎡ ⎣fu m + εp 0 eγs exp ⎧ ⎨ ⎩ s 0 Φdt ⎫ ⎬ ⎭ ds ⎤ ⎦ . Here fu m is the relative density of mobile dislocations at the onset of necking. After integration and some algebraic re-arrangement fm ∼= fu m Θ γ + Φ e−Φεp + Θ γ + Φeγεp . (C.6) Equation C.6 shows the dependence of the relative density of mobile dislocations on the plastic deformation during the necking process. By introducing Equation C.6 into the general relationship at Equation 2.29 one obtains ε˙p = μbvc (ρu + Mεp) exp − Du σsk Ω1e−Φεp + Ω2eγεp , (C.7) where Ω1 = fu m − Θ γ+Φ¯ , Ω2 = Θ γ+Φ¯ and Φ¯ = H σsk . Moreover, ρu and Du are the density of dislocations and the friction stress, respectively, at the onset of necking, σsk and Φ¯ are the mean effective stress and the mean value of Φ during the necking process, respectively (σsk is nearly constant). Finally, an appropriate algebraic re-arrangement of Equation C.7 leads to Equation 2.36.
C.2 Real Model 261 C.2 Real Model The formation of cavities at secondary phase particles can be described by assuming either the energetic (matrix/particle interphase)or fracture- mechanics (breaking of the particle)criteria.In both cases,the analysis leads to the relation r a-1/2,where er is the critical applied stress for the cavity nucleation and a is the particle size (e.g.,[157,441,442].Therefore,the total initial surface Sp of nucleated cavities can be determined as Sp≈4πK1 a-2da. (C.8) (K2/er)2 Beyond the ultimate stress ou,the true stress o(Ep)is nearly a linear function ofep[236,237]: 4 a≈30u+Kap (C.9) where K3 is a constant.A combination of Equations C.9 and C.8 gives Sp≈4πK1 a-2da. (C.10) 12 K2 (4ou/3+K3ep可】 Nucleated voids grow according to Equation 2.24 and their total surface area increases in correspondence with Equation C.4.At the moment of reach- ing the ultimate stress,the total surface area can be estimated by means of Equation C.10 as Sg≈4rK1 a-2da. (C.11) (3K2/4au)2 Let us assume small increments of plastic strain Asp.Then,with respect to Equation C.4,the total surface area at pAsp can be approximated as follows: dsp dEp/o eep△ep十… e(e/2)△p+.+ dep AEp EB/2 =See+ e(ei-k△p)△ep k△p For△ep→0 the sum can be replaced by an integral:
C.2 Real Model 261 C.2 Real Model The formation of cavities at secondary phase particles can be described by assuming either the energetic (matrix/particle interphase) or fracturemechanics (breaking of the particle) criteria. In both cases, the analysis leads to the relation σcr ∝ a−1/2, where σcr is the critical applied stress for the cavity nucleation and a is the particle size (e.g., [157,441,442]. Therefore, the total initial surface SP of nucleated cavities can be determined as SP ≈ 4πK1 ∞ (K2/σcr)2 a−2da. (C.8) Beyond the ultimate stress σu, the true stress σ (εp) is nearly a linear function of εp [236, 237]: σcr ≈ 4 3 σu + K3εp, (C.9) where K3 is a constant. A combination of Equations C.9 and C.8 gives Sp ≈ 4πK1 ∞ K2 (4σu/3+K3εp) 2 a−2da. (C.10) Nucleated voids grow according to Equation 2.24 and their total surface area increases in correspondence with Equation C.4. At the moment of reaching the ultimate stress, the total surface area can be estimated by means of Equation C.10 as Su p ≈ 4πK1 ∞ (3K2/4σu)2 a−2da. (C.11) Let us assume small increments of plastic strain Δεp. Then, with respect to Equation C.4, the total surface area at ε∗ p Δεp can be approximated as follows: Sel ≈ Su p eκεp + dsp dεp 0 eκεpΔεp + ... + dSp dεp εp/2 eκ(ε∗ p/2)Δεp + ... + dSp dεp ε∗ p Δεp = Su p eκε∗ p + εp/ 2 Δεp k=0 dSp dεp kΔεp eκ(ε∗ p−kΔεp)Δεp. For Δεp → 0 the sum can be replaced by an integral:
262 C Plastic Flow Rate Inside the Neck Sa≈S0ee (i)dEp (C.12) By combining Equations C.11,C.10 and Equation C.12,the total surface area related to the plastic strain Ep reads Sel≈4πK1 eKEp a-2da+ er(ep-e)d a-2da de () K2 and,after integration,one obtains Se1≈4 K1 K e+ ke-+2Kr(层--r C.13 Since,for metallic materials,K3<,0.4<K<2.5 and =++号+… the last term in the brackets can be neglected (p<1).After a simple re-arrangement of Equation C.13(the constant+Ka is not too much different from ou)one finally obtains Sa=2xK1-1(3emEr -1), (C.14) aop where aop 3K2 is the size of voids that nucleate when reaching the ultimate stress. Let us further consider that the initial void size is determined by a dis- tribution function of particle sizes and,during the deformation,the number of nucleated voids increases.According to Bergh 443 and other authors (e.g.,444,445),the distribution function can be written as g(a)=ai 1 a≥a, where a*is a critical size of particles which start to nucleate voids.Conse- quently,the total number of particles which,at a given strain,have already nucleated voids reads
262 C Plastic Flow Rate Inside the Neck Sel ≈ Su p eκε∗ p + ε∗ p 0 dSp dεp eκ(ε∗ p−εp)dεp. (C.12) By combining Equations C.11, C.10 and Equation C.12, the total surface area related to the plastic strain εp reads Sel ≈ 4πK1 ⎡ ⎢ ⎢ ⎢ ⎣ eκεp ∞ ( 3K2 4σu ) 2 a−2da + εp 0 eκ(εp−e) d de ⎛ ⎜⎜⎜⎝ ∞ K2 (4/3)σu+K3e 2 a−2da ⎞ ⎟⎟⎟⎠ de ⎤ ⎥ ⎥ ⎥ ⎦ and, after integration, one obtains Sel ≈ 4π K1 K2 2 × × 4 3 σu 2 eκεp + 8 3κσuK3 (eκεp − 1) + 2 κ (K3) 2 1 κ eκεp − 1 κ − εp . (C.13) Since, for metallic materials, K3 < 4 3σu, 0.4 <κ< 2.5 and 1 κ eκεp = 1 κ + εp + 1 2 κε2 p + ..., the last term in the brackets can be neglected (εp < 1). After a simple re-arrangement of Equation C.13 (the constant 2 3σu + K3 κ is not too much different from σu) one finally obtains Sel = 2πK1 1 a0p (3eκεp − 1), (C.14) where a0p = 3K2 4σu 2 is the size of voids that nucleate when reaching the ultimate stress. Let us further consider that the initial void size is determined by a distribution function of particle sizes and, during the deformation, the number of nucleated voids increases. According to Bergh [443] and other authors (e.g., [444, 445]), the distribution function can be written as g(a) = 1 a4 , a ≥ a∗, where a∗ is a critical size of particles which start to nucleate voids. Consequently, the total number of particles which, at a given strain, have already nucleated voids reads
C.2 Real Model 263 h(a)=K1 atda, (C.15) where Ki is a constant,an is the smallest size of particles that nucleate voids in a particular deformation stage ep.With respect to Equations C.14 and C.15,the dependence of the volume V2 on the plastic strain is sp: 2≈2rK1i(3ep-1）. (C.16) aop When inserting Equation C.16 into Equation 2.33,and following the same reasoning as in the case of the ideal model,one finally obtains the strain rate formula Du p≈b.(pu+Mep)exp{-oj fom-0 1 e-ep十 +3(1+重 (C.17) 371+①) where曰*=6mK aopVo Since>1(H in units of GPa and ()k in hundreds of MPa)and 1.4 <7<3.5,the terms in both brackets can be neglected and Equation C.17 reduces to Equation C.7 (both the ideal and the real model lead to a similar result)
C.2 Real Model 263 h(a) = K1 ∞ an 1 a4 da, (C.15) where K1 is a constant, an is the smallest size of particles that nucleate voids in a particular deformation stage εp. With respect to Equations C.14 and C.15, the dependence of the volume V2 on the plastic strain is εp: V2 ≈ 2πK1 δ a0p (3eκεp − 1). (C.16) When inserting Equation C.16 into Equation 2.33, and following the same reasoning as in the case of the ideal model, one finally obtains the strain rate formula ε˙p ≈ μbvc (ρu + Mεp) exp − Du σsk fu om − Θ∗ 1 γ + Φ¯ − 1 3γ 1 + Φ¯ e−Φεp+ +Θ∗ 1 γ + Φ¯ eκεp − 1 3γ 1 + Φ¯ eεp , (C.17) where Θ∗ = 6πK1δγ a0pV0 . Since Φ > ¯ 1 (H in units of GPa and σsk in hundreds of MPa) and 1.4 <γ< 3.5 , the terms 1 3γ(1+Φ¯) in both brackets can be neglected and Equation C.17 reduces to Equation C.7 (both the ideal and the real model lead to a similar result).