《纺织复合材料》课程参考文献（Micromechanisms of Fracture and Fatigue）Appendix（Ab initio Computational Methods、Mixed-mode Criteria of Crack Stability、Plastic Flow Rate Inside the Neck）

254 A Ab initio Computational Methods wavefunction (and also the potentials)match above a certain cutoff radius Tcut· The pseudopotential approach has been implemented in the VASP (Vi- enna ab initio Simulation Package)code.This code was developed at the Institut fiir Materialphysik,Universitat Wien.The VASP currently supports three types of pseudopotentials:norm-conserving pseudopotentials,ultrasoft pseudopotentials and projector augmented wave pseudopotentials [436-438. In all three cases,the core electrons (at lower energy levels than valence elec- trons)are precalculated in an atomic environment and kept "frozen"during the remaining calculations

254 A Ab initio Computational Methods wavefunction (and also the potentials) match above a certain cutoff radius rcut. The pseudopotential approach has been implemented in the VASP (Vi￾enna ab initio Simulation Package) code. This code was developed at the Institut f¨ur Materialphysik, Universit¨at Wien. The VASP currently supports three types of pseudopotentials: norm-conserving pseudopotentials, ultrasoft pseudopotentials and projector augmented wave pseudopotentials [436–438]. In all three cases, the core electrons (at lower energy levels than valence elec￾trons) are precalculated in an atomic environment and kept “frozen” during the remaining calculations

Appendix B Mixed-mode Criteria of Crack Stability In order to describe the crack stability under mixed-mode loading,various concepts within the framework of LEFM were proposed (e.g.,[162-164).In this brief overview,the stress intensity factors are denoted only by capital letters KI,KI and KI,as is common for remote quantities.However,the criteria are relevant also to local stress intensity factors usually denoted as k1,k2 and k3. B.1 Energy Criterion The criterion postulates that the total energy Gr+Gr+Gr,released by the system to create a unit of a new surface,equals the crack growth resistance Ge: Ge=GI+GI+GI1I. (B.1) When the crack grows in its original plane(or propagates in a self-similar manner),the energy criterion can be expressed in terms of the effective K- factor as K=VK+K+年 (B.2) where Ke is the critical stress intensity factor (fracture toughness),K=3-4v for plane strain,K=(3-v)/(1+v)for plane stress and v is the Poisson's ratio. 255

Appendix B Mixed-mode Criteria of Crack Stability In order to describe the crack stability under mixed-mode loading, various concepts within the framework of LEFM were proposed (e.g., [162–164]). In this brief overview, the stress intensity factors are denoted only by capital letters KI , KII and KIII , as is common for remote quantities. However, the criteria are relevant also to local stress intensity factors usually denoted as k1, k2 and k3. B.1 Energy Criterion The criterion postulates that the total energy GI+GII+GIII , released by the system to create a unit of a new surface, equals the crack growth resistance Gc: Gc = GI + GII + GIII . (B.1) When the crack grows in its original plane (or propagates in a self-similar manner), the energy criterion can be expressed in terms of the effective K￾factor as Kc = K2 I + K2 II + 4 κ + 1 K2 III , (B.2) where Kc is the critical stress intensity factor (fracture toughness), κ = 3−4ν for plane strain, κ = (3 − ν)/(1 + ν) for plane stress and ν is the Poisson’s ratio. 255

256 B Mixed-mode Criteria of Crack Stability B.2 Criterion of Linear Damage Accumulation Each of Equations B.1 and B.2 can be understood as a condition of subse- quently reaching the critical level of material damage during loading.If the partial damages accumulate independently,the mixed-mode criterion +m+G皿≤1 GI GIe GIle GIlle can be accepted.Clearly,for the special case GIc GIIc GIIIe Ge, the criterion of linear damage accumulation transfers to the energy criterion. Under the dominant mode I loading,the problem of shear friction in modes II and III on the rough fracture surfaces is not crucial due to sufficiently large opening displacements of crack flanks.Consequently,the equality of critical crack driving forces in all modes might be assumed and the energy criterion can be accepted. Since the assumption of self-similar crack propagation does not hold un- der a general remote mixed-mode loading,one must be careful when applying both the above-mentioned criteria.They appear to be useful only for a predic- tion of the moment of unstable fracture or,generally,in the case of a dominant external mode I loading of microscopically tortuous cracks.The description of long fatigue crack propagation under external mixed-mode loading condi- tions by means of these criteria is rather limited in the sense that the growth direction cannot be predicted. B.3 Criterion of Minimal Deformation Energy The deformation energy density S is minimal in those elements near the crack front,where the ratio of the hydrostatic tensile stress and the octahedral stress is maximal [439.In such elements,therefore,the first local fracture is expected to occur and the crack propagates towards such damaged sites in the matrix.In the plane mixed-mode I+II,the condition of minimal S yields the stability condition as 16G Ka=2R-aea)K+2aaam)Kki+a2amKa, where ai(m)are angular functions and G is the shear modulus.The criterion generally enables us to predict the moment of crack nucleation as well as the crack growth direction in the 3D homogeneous continuum.Its disadvantage is a rather low transparency

256 B Mixed-mode Criteria of Crack Stability B.2 Criterion of Linear Damage Accumulation Each of Equations B.1 and B.2 can be understood as a condition of subse￾quently reaching the critical level of material damage during loading. If the partial damages accumulate independently, the mixed-mode criterion GI GIc + GII GIIc + GIII GIIIc ≤ 1 can be accepted. Clearly, for the special case GIc = GIIc = GIIIc = Gc, the criterion of linear damage accumulation transfers to the energy criterion. Under the dominant mode I loading, the problem of shear friction in modes II and III on the rough fracture surfaces is not crucial due to sufficiently large opening displacements of crack flanks. Consequently, the equality of critical crack driving forces in all modes might be assumed and the energy criterion can be accepted. Since the assumption of self-similar crack propagation does not hold un￾der a general remote mixed-mode loading, one must be careful when applying both the above-mentioned criteria. They appear to be useful only for a predic￾tion of the moment of unstable fracture or, generally, in the case of a dominant external mode I loading of microscopically tortuous cracks. The description of long fatigue crack propagation under external mixed-mode loading condi￾tions by means of these criteria is rather limited in the sense that the growth direction cannot be predicted. B.3 Criterion of Minimal Deformation Energy The deformation energy density S is minimal in those elements near the crack front, where the ratio of the hydrostatic tensile stress and the octahedral stress is maximal [439]. In such elements, therefore, the first local fracture is expected to occur and the crack propagates towards such damaged sites in the matrix. In the plane mixed-mode I+II, the condition of minimal S yields the stability condition as K2 eff = 16G 2 (κ − 1)  a11(θm)K2 I + 2a12(θm)KIKII + a22(θm)K2 II , where aij (θm) are angular functions and G is the shear modulus. The criterion generally enables us to predict the moment of crack nucleation as well as the crack growth direction in the 3D homogeneous continuum. Its disadvantage is a rather low transparency

B.4 Criterion of Maximal Principal Stress 257 B.4 Criterion of Maximal Principal Stress This criterion can be derived from the energy criterion by expressing the effective crack driving force Gef as a function of growth angle 0 and by searching its maximal value.It can be shown that this maximum corresponds to a crack perpendicular to the maximum principal stress.Considering the plane model,stresses in close vicinity of the crack tip can be expressed as follows: 1 00=- COS V2Tr Tre= cos(KIsin0+Kn(3cos0-1)). 2V2Tr If Tro=0,then the hoop stress oe becomes the principal stress.This condition determines the growth angle 0m as well as the related effective stress intensity value: +8 Kef =Ki cos3Kncos 0m 2 2 sin 2 The criterion can be used for predictions of crack stability and growth in homogeneous materials.The criterion does not involve the antiplane mode III and,consequently,it can be applied only in the framework of 2D models. Nevertheless,a very good applicability of the maximum principal stress crite- rion to propagation of long fatigue cracks was already proven(e.g.,[399,440). This success can be understood from the micromechanical point of view.In- deed,the fatigue crack tries to maximize the opening as well as the friction between the flanks.Consequently,it inclines to the maximum opening mode I perpendicularly to the maximal main stress direction,thereby also minimiz- ing the shear modes (friction)

B.4 Criterion of Maximal Principal Stress 257 B.4 Criterion of Maximal Principal Stress This criterion can be derived from the energy criterion by expressing the effective crack driving force Geff as a function of growth angle θ and by searching its maximal value. It can be shown that this maximum corresponds to a crack perpendicular to the maximum principal stress. Considering the plane model, stresses in close vicinity of the crack tip can be expressed as follows: σθ = 1 √ 2πr cos θ 2  KI cos2 θ 2 − 3 2 KII sin θ  , τrθ = 1 2 √ 2πr cos θ 2 (KI sin θ + KII (3 cos θ − 1)). If τrθ = 0, then the hoop stress σθ becomes the principal stress. This condition determines the growth angle θm as well as the related effective stress intensity value: tan θm 2 = KI 4KII ± 1 4  KI KII 2 + 8, Keff = KI cos3 θm 2 − 3KII cos2 θm 2 sin θm 2 . The criterion can be used for predictions of crack stability and growth in homogeneous materials. The criterion does not involve the antiplane mode III and, consequently, it can be applied only in the framework of 2D models. Nevertheless, a very good applicability of the maximum principal stress crite￾rion to propagation of long fatigue cracks was already proven (e.g., [399,440]). This success can be understood from the micromechanical point of view. In￾deed, the fatigue crack tries to maximize the opening as well as the friction between the flanks. Consequently, it inclines to the maximum opening mode I perpendicularly to the maximal main stress direction, thereby also minimiz￾ing the shear modes (friction)

Appendix C Plastic Flow Rate Inside the Neck C.1 Ideal Model When taking Equations2.25,2.26 and rangesλ1∈(0.8,2)，λ2∈(0,1)into account,the growth of the semi-axes of elliptical voids at the onset of necking can be described as a aoe0.8ep, b≈bo三ao (C.1) while a=a0e2sp,b≈a0ep (C.2) holds in the final stages of necking just before the final fracture.In order to calculate the active volume V2,the surface area of the elliptical void Se has to be considered as Se4兰2r(b2+ab). (C.3) By inserting Equations C.1 and C.2 into Equation C.3 and transferring the surface of the ellipsoid to that of the equivalent sphere of the same surface area one obtains aSel =4mage.,Se =4ndpe2.e. A corresponding error of this transfer is less than 10%for 0<p<1.Thus, the variation of the void surface during the necking process can be expressed as Se1≈4 rageRep, (C.4) where 0.4<K<2.5.Let us denote 6 the mean distance from the void surface to the outer boundary of the volume V2(see Figure 2.30)and n the number 259

Appendix C Plastic Flow Rate Inside the Neck C.1 Ideal Model When taking Equations 2.25, 2.26 and ranges λ1 ∈ 0.8, 2, λ2 ∈ 0, 1 into account, the growth of the semi-axes of elliptical voids at the onset of necking can be described as a = a0e0.8εp , b ≈ b0 ≡ a0 (C.1) while a = a0e2εp , b ≈ a0eεp (C.2) holds in the final stages of necking just before the final fracture. In order to calculate the active volume V2, the surface area of the elliptical void Sel has to be considered as Sel ∼= 2π  b2 + ab . (C.3) By inserting Equations C.1 and C.2 into Equation C.3 and transferring the surface of the ellipsoid to that of the equivalent sphere of the same surface area one obtains aSel = 4πa2 0e0,4εp , b Sel = 4πa2 0e2,5εp . A corresponding error of this transfer is less than 10% for 0 < εp < 1. Thus, the variation of the void surface during the necking process can be expressed as Sel ≈ 4πa2 0eκεp , (C.4) where 0.4 <κ< 2.5 . Let us denote δ the mean distance from the void surface to the outer boundary of the volume V2 (see Figure 2.30) and n the number 259