24 1 Deformation and Fracture of Perfect Crystals the crystal at each point of the deformation path.For the tetragonal path (68], the Jacobi matrix V1+2e 0 0 V1-2e0 01 and the corresponding deformation matrix e00 0-e0 .000 lead to the following change of the system energy(per volume unit)according to Equation 1.7: △Eu=(C1-C2)e2+. This deformation changes the tetragonal symmetry to the orthorhombic one but the calculated△Eu(e)curve is symmetric(△Eu(e)=△Eu(-e).With regard to the energy expansion,the tetragonal shear modulus C'=0.5(C11- C12)can be expressed as C=1OAE 40e2 The condition at Equation 1.14 corresponds to the shear instability related to the C44 modulus.Using the Jacobi matrix and strain as 0 0 2e 000 0 V1+2e+V1-2e JC44= 2 V1+2e+V1-2e C44 00e 2e W1+2e+W1-2e 0e0 01 1+2e+V1-2e 2 one obtains the dependence AEu(e)leading to △Em=2C44e2+.. and 18△Eu Ca4=40e2
24 1 Deformation and Fracture of Perfect Crystals the crystal at each point of the deformation path. For the tetragonal path [68], the Jacobi matrix JC = ⎛ ⎝ √1+2e 0 0 0 √1 − 2e 0 0 01 ⎞ ⎠ and the corresponding deformation matrix ηˆC = ⎛ ⎝ e 0 0 0 −e 0 000 ⎞ ⎠ lead to the following change of the system energy (per volume unit) according to Equation 1.7: ΔEu = (C11 − C12)e2 + ... This deformation changes the tetragonal symmetry to the orthorhombic one but the calculated ΔEu(e) curve is symmetric (ΔEu(e)=ΔEu(−e)). With regard to the energy expansion, the tetragonal shear modulus C = 0.5(C11− C12) can be expressed as C = 1 4 ∂2ΔEu ∂e2 . The condition at Equation 1.14 corresponds to the shear instability related to the C44 modulus. Using the Jacobi matrix and strain as JC44 = ⎛ ⎜⎜⎜⎜⎝ 10 0 0 √ 1+2e + √ 1 − 2e 2 2e √ 1+2e + √ 1 − 2e 0 2e √1+2e + √1 − 2e √1+2e + √1 − 2e 2 ⎞ ⎟⎟⎟⎟⎠ , ηˆC44 = ⎛ ⎝ 000 0 0 e 0 e 0 ⎞ ⎠ , one obtains the dependence ΔEu(e) leading to ΔEu = 2C44e2 + ... and C44 = 1 4 ∂2ΔEu ∂e2 .
1.1 Ideal Strength of Solids 25 The stability condition at Equation 1.15 can be tested by using V1+2e+V1-2e 2e 0 2 V1+2e+v1-2e 0e0 Jc66= 2e v1+2e+V1-2e e00 0 1+2e+vV1-2e 2 000 0 0 The corresponding energy change reads △Eu=2C66e2+. and the modulus 182△Eu C66=40e2 Let us also show the stability conditions applied for a cubic structure(with 48 symmetry operations)under isotropic loading(01=02=03=o).Here the form of the Wallace matrix(Equation 1.10)becomes symmetric: C11+0C12-0C12-00 0 0 C12-0C11+0C2-00 0 0 0 B= C2-0C12-0C1+00 0 0 0 0 0C44+0 0 0 00 0C44+0 0 0 0 00 0C44+0 and its determinant det|Bl=(C44+o)3(C1-C2+2o)2(C1+2C2-o) can be broken down into to a set of stability conditions: 1.C11+2C12-0>0 2.C11-C12+2o>0 (1.16) 3. C44+0>0. Stability criteria for a tetragonal system under biaxial (epitaxial)loading as well as those for the simplest case of an isotropic solid under hydrostatic loading can be found elsewhere [83]. Let us recall that,in addition to the violation of the mechanical stability conditions,some phonon (heterogeneous)instabilities may occur along the deformation path.A more detailed description of this problem lies beyond the scope of this book. The currently used methodology for the IS calculation can be,finally, briefly summarized in the following points:
1.1 Ideal Strength of Solids 25 The stability condition at Equation 1.15 can be tested by using JC66 = ⎛ ⎜⎜⎜⎜⎝ √1+2e + √1 − 2e 2 2e √1+2e + √1 − 2e 0 2e √ 1+2e + √1 − 2e √1+2e + √1 − 2e 2 0 0 01 ⎞ ⎟⎟⎟⎟⎠ , ηˆC66 = ⎛ ⎝ 0 e 0 e 0 0 000 ⎞ ⎠ . The corresponding energy change reads ΔEu = 2C66e2 + ... and the modulus C66 = 1 4 ∂2ΔEu ∂e2 . Let us also show the stability conditions applied for a cubic structure (with 48 symmetry operations) under isotropic loading (σ1 = σ2 = σ3 = σ). Here the form of the Wallace matrix (Equation 1.10) becomes symmetric: B = ⎛ ⎜⎜⎜⎜⎜⎜⎝ C11 + σ C12 − σ C12 − σ 000 C12 − σ C11 + σ C12 − σ 000 C12 − σ C12 − σ C11 + σ 000 000 C44 + σ 0 0 0000 C44 + σ 0 00000 C44 + σ ⎞ ⎟⎟⎟⎟⎟⎟⎠ and its determinant det|B| = (C44 + σ) 3(C11 − C12 + 2σ) 2(C11 + 2C12 − σ) can be broken down into to a set of stability conditions: 1. C11 + 2C12 − σ > 0 2. C11 − C12 + 2σ > 0 3. C44 + σ > 0. (1.16) Stability criteria for a tetragonal system under biaxial (epitaxial) loading as well as those for the simplest case of an isotropic solid under hydrostatic loading can be found elsewhere [83]. Let us recall that, in addition to the violation of the mechanical stability conditions, some phonon (heterogeneous) instabilities may occur along the deformation path. A more detailed description of this problem lies beyond the scope of this book. The currently used methodology for the IS calculation can be, finally, briefly summarized in the following points:
26 1 Deformation and Fracture of Perfect Crystals 1.construction of a suitable empirical interatomic potential or calculation of the electronic structure; 2.calculation of the energy-strain curve and the related stress vs strain de- pendence for a specific deformation path; 3.establishment of elastic and phonon instability ranges on the strain us energy and/or stress us strain curves; 4.determination of IS value as a stress related to the first instability point on the energy us strain curve. 1.1.3 Simple Loading Modes 1.1.3.1 Ideal Isotropic Strength A nearly isotropic,triaxial tensile stress state (1 021.603)builds up at the tip of cracks in solids stressed by uniaxial tension (e.g.,8).The value of oiht expresses a resistance to a brittle fracture (cleavage or tearing at a sharp crack tip),while the value of Tis reflects a defiance to a ductile response (blunting of the crack tip).Consequently,the ratio oiht/Tis can be used as a measure of the brittle/ductile behaviour of cracked perfect crystals 23-25(see also Section 1.2).This is why the search for isotropic IS values also becomes worthwhile from the engineering point of view.To our best knowledge,however,there are no available experimental data on oiht.The reason lies in difficulties in experimental realization of an isotropic tensile loading.Thus,a theoretical assessment remains to be the only applicable tool to gain such information.When the isotropic deformation is applied to a system,its volume changes but the symmetry remains unchanged (unless a phase transition takes place).Under such conditions,a rather simple LMTO- ASA method is particularly suitable for ab initio calculations(see Appendix A).Indeed,the error of ASA approximation is nearly independent of the volume. Let us consider a cubic crystal under applied isotropic stress o.A natural parameter for a description of deformation is the crystal volume V.The deviation from the equilibrium (unstressed)volume Vo can be expressed by the relative (normalized)volume vV/Vo.The isotropic stress can be simply derived from the dependence of the crystal energy U(see Figure 1.6) on the relative volume v as 1 dU Jiht= Vo dv The stress oiht reaches its maximum value when d'U du2 =0 (1.17)
26 1 Deformation and Fracture of Perfect Crystals 1. construction of a suitable empirical interatomic potential or calculation of the electronic structure; 2. calculation of the energy–strain curve and the related stress vs strain dependence for a specific deformation path; 3. establishment of elastic and phonon instability ranges on the strain vs energy and/or stress vs strain curves; 4. determination of IS value as a stress related to the first instability point on the energy vs strain curve. 1.1.3 Simple Loading Modes 1.1.3.1 Ideal Isotropic Strength A nearly isotropic, triaxial tensile stress state (σ1 = σ2 ≈ 1.6σ3) builds up at the tip of cracks in solids stressed by uniaxial tension (e.g., [8]). The value of σiht expresses a resistance to a brittle fracture (cleavage or tearing at a sharp crack tip), while the value of τis reflects a defiance to a ductile response (blunting of the crack tip). Consequently, the ratio σiht/τis can be used as a measure of the brittle/ductile behaviour of cracked perfect crystals [23–25] (see also Section 1.2). This is why the search for isotropic IS values also becomes worthwhile from the engineering point of view. To our best knowledge, however, there are no available experimental data on σiht. The reason lies in difficulties in experimental realization of an isotropic tensile loading. Thus, a theoretical assessment remains to be the only applicable tool to gain such information. When the isotropic deformation is applied to a system, its volume changes but the symmetry remains unchanged (unless a phase transition takes place). Under such conditions, a rather simple LMTOASA method is particularly suitable for ab initio calculations (see Appendix A). Indeed, the error of ASA approximation is nearly independent of the volume. Let us consider a cubic crystal under applied isotropic stress σ. A natural parameter for a description of deformation is the crystal volume V . The deviation from the equilibrium (unstressed) volume V0 can be expressed by the relative (normalized) volume v = V / V0. The isotropic stress can be simply derived from the dependence of the crystal energy U (see Figure 1.6) on the relative volume v as σiht = 1 V0 dU dv . The stress σiht reaches its maximum value when d2U dv2 = 0, (1.17)
1.1 Ideal Strength of Solids 27 U Figure 1.6 A schematic dependence of the crystal potential energy U on the relative volume v.The region with v<1 corresponds to compression,v>1 to tension and vip is assigned to the point of inflection.Ucoh represents the cohesive energy of a crystal when a zero energy is related to a system of isolated atoms lim U=0) Table 1.1 Theoretical isotropic strength oiht from ab initio and semi-empirical approximations Element Theoretical strength oiht (GPa) ab initio results Semi-empirical results LMTO VASP Polynomial Morse Sinus DVC [97] 97 [98] 99 [36 [36 36 1001 Li bcc 3.531 3.135 5.06 2.49 4.91 1.92 C dia 66.1a 53.2b 88.54 88.54 84.7 69.7 138 Na bcc 1.97a 1.55h 1.87 1.20 1.86 1.77 Al fcc 13.8a 12.0b 11.2b 22.2 11.9 23.0 Si dia 15.0 10.4b 15.54 15.4a 15.1 13.7 28.2 bcc 0.955a 0.701b 0.99 0.659 1.28 0.10 bcc 39.2a 33.2b 32.7b 32.6 23.5 45.4 38.3 Cr bcc 37.2a 21.0b 35.2 25.9 50.2 47.4 Fe bcc 37.7 26.7b 27.7b 28.5b 33.8 24.1 48.1 Ni fcc 39.5a 27.4b 28.9b 29.2b 44.7 26.9 51.2 Cu fcc 28.8a 20.9b 19.8b 20.4 32.7 19.9 38.4 Ge dia 11.02 6.46b 11.1a 11.3 10.1 21.4 Nb bcc 36.3a 31.6b 31.6b 35.5 25.5 49.4 34.1 Mo bcc 49.3a 42.7b 42.9b 43.2 48.2 35.0 68.9 42.2 Ag fcc 19.01 12.6b 17.6a 20.2 13.7 26.7 Ba bcc 2.69 1.93 2.36 1.64 3.17 Ta bcc 41.3 36.4b 39.2 28.5 55.1 41.3 W bcc 57.0a 50.6- 50.7b 50.2b 56.1 42.2 80.1 53.1 Ir fcc 40.1b 61.2 45.6 85.6 Pt fcc 42.7a 33.6b 39.6￥ 48.5 35.1 68.0 Au fcc 25.5a 17.6 23.2 23.54 28.4 20.9 39.9 Pb fcc 8.7a 6.98b 7.91 5.47 10.6 a LDA bGGA
1.1 Ideal Strength of Solids 27 1 vip v U 0 -Ucoh Figure 1.6 A schematic dependence of the crystal potential energy U on the relative volume v. The region with v < 1 corresponds to compression, v > 1 to tension and vip is assigned to the point of inflection. Ucoh represents the cohesive energy of a crystal when a zero energy is related to a system of isolated atoms ( lim v→∞ U = 0) Table 1.1 Theoretical isotropic strength σiht from ab initio and semi-empirical approximations Element Theoretical strength σiht (GPa) ab initio results Semi-empirical results LMTO VASP Polynomial Morse Sinus DVC [97] [97] [98] [99] [36] [36] [36] [100] Li bcc 3.53a 3.13b 5.06 2.49 4.91 1.92 C dia 66.1a 53.2b 88.5a 88.5a 84.7 69.7 138 Na bcc 1.97a 1.55b 1.87 1.20 1.86 1.77 Al fcc 13.8a 12.0b 11.2b 22.2 11.9 23.0 Si dia 15.0a 10.4b 15.5a 15.4a 15.1 13.7 28.2 K bcc 0.955a 0.701b 0.99 0.659 1.28 0.10 V bcc 39.2a 33.2b 32.7b 32.6 23.5 45.4 38.3 Cr bcc 37.2a 21.0b 35.2 25.9 50.2 47.4 Fe bcc 37.7a 26.7b 27.7b 28.5b 33.8 24.1 48.1 Ni fcc 39.5a 27.4b 28.9b 29.2b 44.7 26.9 51.2 Cu fcc 28.8a 20.9b 19.8b 20.4b 32.7 19.9 38.4 Ge dia 11.0a 6.46b 11.1a 11.3 10.1 21.4 Nb bcc 36.3a 31.6b 31.6b 35.5 25.5 49.4 34.1 Mo bcc 49.3a 42.7b 42.9b 43.2b 48.2 35.0 68.9 42.2 Ag fcc 19.0a 12.6b 17.6a 20.2 13.7 26.7 Ba bcc 2.69a 1.93b 2.36 1.64 3.17 Ta bcc 41.3a 36.4b 39.2 28.5 55.1 41.3 W bcc 57.0a 50.6b 50.7b 50.2b 56.1 42.2 80.1 53.1 Ir fcc 40.1b 61.2 45.6 85.6 Pt fcc 42.7a 33.6b 39.6a 48.5 35.1 68.0 Au fcc 25.5a 17.6b 23.2a 23.5a 28.4 20.9 39.9 Pb fcc 8.7a 6.98b 7.91 5.47 10.6 a LDA b GGA
28 1 Deformation and Fracture of Perfect Crystals i.e.,at the inflection point of the U(v)dependence (v-vip).Considering the bulk modulus as a parameter expressing the elastic response of the crystal to the volume change as B=doht、1dPU do Vo dv2' (1.18) the volume at which the equality at Equation 1.17 is valid corresponds to vanishing of the bulk modulus.In other words,the first (volumetric)condition of a stability set at Equation 1.16 related to the isotropic loading of cubic crystals is fulfilled.Indeed,the bulk modulus is defined by the combination C11+2C12 of elastic constants.In the further analysis,the crystal potential energy U will be substituted by a total energy Etot that is evaluated from the electronic structure of investigated crystals.Thus,the first stability condition is related to the inflection point of the dependence of the total energy on the volume.The violation of the second stability condition corresponds to a shear instability (vanishing of the tetragonal shear modulus)when we can expect a bifurcation from the primary deformation (isotropic)path to a secondary one,where the lattice acquires tetragonal or orthorhombic symmetry.The third condition corresponds to another shear instability related to the trigonal shear modulus. The values of oiht for various cubic crystals,as determined by ab initio approaches,are reported in our papers (60-62]and displayed in Table 1.1.The stability analysis,performed for crystals of Cu,Al,Ag,Fe,Ni and Cr,revealed that all these crystals fail under the volumetric instability [62].This means that the values of the stress related to the point of inflection really correspond to oiht.The associated relative change of volume is about 1.5,which means that the relative elongation of the lattice parameter is about 1.15.In the case of aluminium crystal,however,the ab initio approach indicated a break in the trigonal shear stability before reaching the inflection point.Because this is in disagreement with results achieved by several other authors [80,94],the problem of the aluminium crystal is open for further investigations.For a majority of crystals,nevertheless,the values of oiht are higher than Tis and oiut.Indeed,the values of Tis are generally much lower since,to reach this value,the atoms in the shear plane need not be separated by higher distances. During uniaxial tension,the shear instabilities appear well before reaching oiut at the inflection point (see next subsections)
28 1 Deformation and Fracture of Perfect Crystals i.e., at the inflection point of the U(v) dependence (v → vip). Considering the bulk modulus as a parameter expressing the elastic response of the crystal to the volume change as B = dσiht dv = 1 V0 d2U dv2 , (1.18) the volume at which the equality at Equation 1.17 is valid corresponds to vanishing of the bulk modulus. In other words, the first (volumetric) condition of a stability set at Equation 1.16 related to the isotropic loading of cubic crystals is fulfilled. Indeed, the bulk modulus is defined by the combination C11 + 2C12 of elastic constants. In the further analysis, the crystal potential energy U will be substituted by a total energy Etot that is evaluated from the electronic structure of investigated crystals. Thus, the first stability condition is related to the inflection point of the dependence of the total energy on the volume. The violation of the second stability condition corresponds to a shear instability (vanishing of the tetragonal shear modulus) when we can expect a bifurcation from the primary deformation (isotropic) path to a secondary one, where the lattice acquires tetragonal or orthorhombic symmetry. The third condition corresponds to another shear instability related to the trigonal shear modulus. The values of σiht for various cubic crystals, as determined by ab initio approaches, are reported in our papers [60–62] and displayed in Table 1.1. The stability analysis, performed for crystals of Cu, Al, Ag, Fe, Ni and Cr, revealed that all these crystals fail under the volumetric instability [62]. This means that the values of the stress related to the point of inflection really correspond to σiht. The associated relative change of volume is about 1.5, which means that the relative elongation of the lattice parameter is about 1.15. In the case of aluminium crystal, however, the ab initio approach indicated a break in the trigonal shear stability before reaching the inflection point. Because this is in disagreement with results achieved by several other authors [80, 94], the problem of the aluminium crystal is open for further investigations. For a majority of crystals, nevertheless, the values of σiht are higher than τis and σiut. Indeed, the values of τis are generally much lower since, to reach this value, the atoms in the shear plane need not be separated by higher distances. During uniaxial tension, the shear instabilities appear well before reaching σiut at the inflection point (see next subsections)