1.1 Ideal Strength of Solids 19 The standard Voigt notation for strain nj=n(1+)is used here.The corresponding energy expansion is then 1 U=Uo(V)+Voon+VCoanana +(n). (1.7) Note that the energy expansion at Equation 1.7 contains a double sum instead of a quadruple sum in Equation 1.6. 1.1.2.1 Elastic Moduli The 6x6 matrix of elastic moduli generally contains 21 independent elements that do not transform like the second-rank tensor components.According to the number of point group symmetry operations,the amount of independent elastic moduli can be lower.With respect to the energy expansion,the elastic moduli can be defined as 82U ongonB or 1/82U Ca8= VdEadEB From now on,the Cj will denote the elastic moduli defined on the basis of the finite strain and cij will stand for the elastic moduli based on the small strain. When,for example,the crystal is subjected to small isotropic deformation, the lattice parameter a is related to the reference parameter ar as a=ar(1+ e).Here e is a small stretch that represents diagonal components of the small strain tensor.Then,the finite strain is related to e according to m=n2= n3-n=e+,and the deformation gradient can be expressed as V1+2万 0 0 1+2万 0 0 √1+2 The corresponding energy expansion at Equation 1.7 gives U=U6()+3V,m+23C12+6C12)+ from which the following combination of elastic moduli: 1 d2U C11+2C12= 3V,d7
1.1 Ideal Strength of Solids 19 The standard Voigt notation for strain ηij = 1 2 ηα(1 + δij ) is used here. The corresponding energy expansion is then U = U0(V ) + V σαηα + 1 2 V Cαβηαηβ + O(η3). (1.7) Note that the energy expansion at Equation 1.7 contains a double sum instead of a quadruple sum in Equation 1.6. 1.1.2.1 Elastic Moduli The 6×6 matrix of elastic moduli generally contains 21 independent elements that do not transform like the second-rank tensor components. According to the number of point group symmetry operations, the amount of independent elastic moduli can be lower. With respect to the energy expansion, the elastic moduli can be defined as Cαβ = 1 V ∂2U ∂ηα∂ηβ or cαβ = 1 V ∂2U ∂εα∂εβ . From now on, the Cij will denote the elastic moduli defined on the basis of the finite strain and cij will stand for the elastic moduli based on the small strain. When, for example, the crystal is subjected to small isotropic deformation, the lattice parameter a is related to the reference parameter ar as a = ar(1 + e). Here e is a small stretch that represents diagonal components of the small strain tensor. Then, the finite strain is related to e according to η1 = η2 = η3 = η = e + e2 2 , and the deformation gradient can be expressed as Jiso = ⎛ ⎝ 1 + e 0 0 0 1+ e 0 0 0 1+ e ⎞ ⎠ or Jiso = ⎛ ⎝ √1+2η 0 0 0 √1+2η 0 0 0 √1+2η ⎞ ⎠ . The corresponding energy expansion at Equation 1.7 gives U = U0(Vr)+3Vrση + 1 2 Vr(3C11η2 + 6C12η2) + ... from which the following combination of elastic moduli: C11 + 2C12 = 1 3Vr d2U dη2
20 1 Deformation and Fracture of Perfect Crystals can be derived.Thus,this combination defines the bulk modulus B(Cu +2C) which expresses the elastic response of a crystal to isotropic (hydrostatic) loading. Several other elastic moduli can be related to simple types of loading. For example,the modulus Ci can be determined by a simple lattice stretch in the [100]direction (see Figure 1.5).The Young's modulus expresses the crystal response to uniaxial loading and,therefore,it depends on the crystal orientation with respect to the loading direction.For particular orientations one can derive E(100= (C1-C2)(C11+2C2) (1.8) C11+C12 E110)= C44 2C44 1+ (C11-C12)(C11+2C12). E11)= 3C44(C11+2C12) C44+C11+2C12 The shear modulus G can be expressed as 3C44(C11-C2) 4C44+C1-C12 for{111}(②11)，{111}(110)and{110}(11）slip systems, G=(Cu-Cu2) for {110)(110)slip system and G=C44 for {110}(001)slip system. [010] 00 [100j [001] Figure 1.5 Illustration of a lattice distortion for calculation of the elastic modulus C11
20 1 Deformation and Fracture of Perfect Crystals can be derived. Thus, this combination defines the bulk modulus B = 1 3 (C11 + 2C12) which expresses the elastic response of a crystal to isotropic (hydrostatic) loading. Several other elastic moduli can be related to simple types of loading. For example, the modulus C11 can be determined by a simple lattice stretch in the [100] direction (see Figure 1.5). The Young’s modulus expresses the crystal response to uniaxial loading and, therefore, it depends on the crystal orientation with respect to the loading direction. For particular orientations one can derive E100 = (C11 − C12)(C11 + 2C12) C11 + C12 , (1.8) E110 = 4C44 C11 1 + 2C44 (C11 − C12)(C11 + 2C12) −1 , E111 = 3C44(C11 + 2C12) C44 + C11 + 2C12 . The shear modulus G can be expressed as G = 3C44(C11 − C12) 4C44 + C11 − C12 for {111}¯211, {111}¯110 and {110}¯111 slip systems, G = 1 2 (C11 − C12) for {110}¯110 slip system and G = C44 for {110}001 slip system. a0 a0 a0 a [100] [010] [001] Figure 1.5 Illustration of a lattice distortion for calculation of the elastic modulus C11
1.1 Ideal Strength of Solids 21 1.1.2.2 Mechanical Stability Hereafter,the crystal potential energy U is determined from the electronic structure as Etot.Thus,by using the internal energy per unit volume E= Etot/V,one can also write 82Eu Ci)二0m0ni and 8Eu 0:=1 Oni (1.9) When no other instability(unstable phonon modes,phase transformations, elastic shear instabilities,etc.)occurs in the crystal,the relation at Equation 1.9 shows that IS corresponds to the first point of inflection on the energy vs strain curve (Cij=0,0i=dimar). Probably the first attempt to formulate general criteria of crystal stability based on its elastic moduli was made in 1940 in the work of Born 64 and Born and Fiirth [65].They showed that by expanding the internal energy of a crystal in a power series in the strain and requiring positivity of the energy,one obtains a set of conditions on the elastic constants appropriate to the crystal that must be satisfied to maintain structural stability.It can be briefly said that,in the Born criterion,the system is stable if the matrix of elastic moduli Cij is positive definite,i.e.,all its eigenvalues are positive. Their results are valid only when the lattice is not under external stress. In general,two basic cases of elastic (homogeneous)instability behaviour related to IS can be distinguished when analysing the crystal deformation: 1.instability occurs along the original deformation path, 2.instability changes the loading mode or the type of the deformation path. The instability of the first kind (so-called volumetric instability)means that the process of unstable crystal collapse starts at the above-mentioned point of inflection on the original deformation path.Assuming the constant stress ensembles (i.e.,the stress-controlled loading),the crystal starts to disinte- grate spontaneously after reaching this point.During this process,however, strain induced phase transformations (so-called displacive transformations) may appear along the deformation path [90,91].These transformations pro- ceed by means of cooperative displacements of atoms away from their lattice sites and alter the crystal symmetry without changing the atomic ordering or composition.They are of the first order and,therefore,accompanied by a symmetry-dictated extrema on the stress-strain curve.For example,the tetragonal Bain's path also induces typical displacive transformations(see Section 1.1.3).Moreover,additional extrema that are not dictated by the symmetry may occur,and reflect properties of the specific material.Conse- quently,more"IS values"can be found related to different points of inflection
1.1 Ideal Strength of Solids 21 1.1.2.2 Mechanical Stability Hereafter, the crystal potential energy U is determined from the electronic structure as Etot. Thus, by using the internal energy per unit volume Eu = Etot/V , one can also write Cij = ∂2Eu ∂ηi∂ηj and σi = ∂Eu ∂ηi . (1.9) When no other instability (unstable phonon modes, phase transformations, elastic shear instabilities, etc.) occurs in the crystal, the relation at Equation 1.9 shows that IS corresponds to the first point of inflection on the energy vs strain curve (Cij = 0, σi = σimax). Probably the first attempt to formulate general criteria of crystal stability based on its elastic moduli was made in 1940 in the work of Born [64] and Born and F¨urth [65]. They showed that by expanding the internal energy of a crystal in a power series in the strain and requiring positivity of the energy, one obtains a set of conditions on the elastic constants appropriate to the crystal that must be satisfied to maintain structural stability. It can be briefly said that, in the Born criterion, the system is stable if the matrix of elastic moduli Cij is positive definite, i.e., all its eigenvalues are positive. Their results are valid only when the lattice is not under external stress. In general, two basic cases of elastic (homogeneous) instability behaviour related to IS can be distinguished when analysing the crystal deformation: 1. instability occurs along the original deformation path, 2. instability changes the loading mode or the type of the deformation path. The instability of the first kind (so-called volumetric instability) means that the process of unstable crystal collapse starts at the above-mentioned point of inflection on the original deformation path. Assuming the constant stress ensembles (i.e., the stress-controlled loading), the crystal starts to disintegrate spontaneously after reaching this point. During this process, however, strain induced phase transformations (so-called displacive transformations) may appear along the deformation path [90, 91]. These transformations proceed by means of cooperative displacements of atoms away from their lattice sites and alter the crystal symmetry without changing the atomic ordering or composition. They are of the first order and, therefore, accompanied by a symmetry-dictated extrema on the stress–strain curve. For example, the tetragonal Bain’s path also induces typical displacive transformations (see Section 1.1.3). Moreover, additional extrema that are not dictated by the symmetry may occur, and reflect properties of the specific material. Consequently, more “IS values” can be found related to different points of inflection
22 1 Deformation and Fracture of Perfect Crystals on the energy-strain curve.However,the IS is determined by the stress as- sociated with the first point of inflection on the original energy-strain curve which corresponds to the maximal energy gradient.Note that atomic con- figurations related to energy minima behind the first point of inflection may mimic stable or metastable atomic arrangements that could be encountered when investigating thin films or extended defects such as interfaces or disloca- tions.Similar configurations can also be reached during the strain-controlled deformation path of a crystal (the constant strain ensembles),provided that they are not preceded by any instabilities of the second kind. The instabilities of the second kind (so-called shear instabilities)can be derived by considering a requirement that the free energy (and at T=0 also the total internal energy)be minimum in subsequent constant stress ensembles in accordance with the second law of thermodynamics 66,78-80, 87,92-94.The main point of such an analysis was the assumption that the crystal is subjected both to the applied load and to an infinite variety of small perturbing forces.Any of the forces can make the crystal fail in a different mode than that related to the main loading force.The proposed stability assessment requires information about an elastic response of the system to small deviations from the current state (let us call it the reference state). Therefore,in the case of a quasi-static loading,the stability assessment is in a sense independent of the deformation path which led the system to this state,because the same atomic arrangement can be obtained via many various transformations of an arbitrary original state.For that reason,the further deformations used to investigate the stability have nothing to do with the original deformation path.If the solid is strained infinitesimally from the reference state associated with the stress oij (in the standard notation)by a strain tensor sij,the related Cauchy(true)stress Tij can be written as T=0十Bk1ekl, where 1 BL=CL+2(6k01+dk01+6M0jk+d10ik-20k0) (1.10) is the elastic stiffness matrix (i,j,k,l=1,2,3)introduced by Wallace [95] that is generally asymmetric with respect to the interchange of indices.Con- struction of this matrix is crucial for the stability assessment.As was shown in [92,94,96],the system becomes unstable once its symmetrized counterpart A=专B+B) attains a zero determinant,i.e., Al=0 (1.11)
22 1 Deformation and Fracture of Perfect Crystals on the energy–strain curve. However, the IS is determined by the stress associated with the first point of inflection on the original energy–strain curve which corresponds to the maximal energy gradient. Note that atomic con- figurations related to energy minima behind the first point of inflection may mimic stable or metastable atomic arrangements that could be encountered when investigating thin films or extended defects such as interfaces or dislocations. Similar configurations can also be reached during the strain-controlled deformation path of a crystal (the constant strain ensembles), provided that they are not preceded by any instabilities of the second kind. The instabilities of the second kind (so-called shear instabilities) can be derived by considering a requirement that the free energy (and at T = 0 also the total internal energy) be minimum in subsequent constant stress ensembles in accordance with the second law of thermodynamics [66, 78–80, 87, 92–94]. The main point of such an analysis was the assumption that the crystal is subjected both to the applied load and to an infinite variety of small perturbing forces. Any of the forces can make the crystal fail in a different mode than that related to the main loading force. The proposed stability assessment requires information about an elastic response of the system to small deviations from the current state (let us call it the reference state). Therefore, in the case of a quasi-static loading, the stability assessment is in a sense independent of the deformation path which led the system to this state, because the same atomic arrangement can be obtained via many various transformations of an arbitrary original state. For that reason, the further deformations used to investigate the stability have nothing to do with the original deformation path. If the solid is strained infinitesimally from the reference state associated with the stress σij (in the standard notation) by a strain tensor εij , the related Cauchy (true) stress τij can be written as τij = σij + Bijklεkl, where Bijkl = Cijkl + 1 2 (δikσjl + δjkσil + δilσjk + δjlσik − 2δklσij ) (1.10) is the elastic stiffness matrix (i, j, k, l = 1, 2, 3) introduced by Wallace [95] that is generally asymmetric with respect to the interchange of indices. Construction of this matrix is crucial for the stability assessment. As was shown in [92,94,96], the system becomes unstable once its symmetrized counterpart A = 1 2 BT + B attains a zero determinant, i.e., |A| = 0 (1.11)
1.1 Ideal Strength of Solids 23 during the loading.It should be emphasized that the elastic moduli in Equa- tion 1.10 are the local ones,i.e.,corresponding to different points of the deformation path.Thus,in order to assess the stability,their values should be determined by introducing a sufficient number of independent small devi- ations (strain increments)away from the original deformation path at each point,in accordance with the symmetry of the particular crystal lattice.The solution of Equation 1.11 gives a different number of possible stability condi- tions for different crystal lattice symmetries as well as different loading modes. The higher the symmetry (the more point group symmetry operations),the fewer the stability conditions are to be tested.The smallest possible number of necessary stability conditions (only two)corresponds to the isotropic solid. The stability conditions for cubic crystals loaded in uniaxial tension or compression along the [001]direction,the so-called Bain's path,can serve as a suitable example.The tetragonal symmetry induced by the uniaxial loading means C11=C22≠0，C33≠0，C12≠0，C3=C23≠0，C44=C55≠ 0,C660 and Cj=0,other,and a simple relationship oij =00i36j3 stands for the stress tensor.By introducing these relations into Equation 1.11,one obtains the following stability conditions: (C33+o)(C12+C1)-2(C13-0/2)2>0, (1.12) C11-C2>0, (1.13) 2C44+0>0, (1.14) C66>0. (1.15) The left-hand side of Equation 1.12 differs from the tetragonal Eoo modu- lus for the stress-free state o =0 only by a multiplication constant.Therefore, the violation of that so-called volumetric condition is closely related to the first inflection point on the energy vs strain curve along the 001]deformation path(the instability of the first kind).In any case,the maximum value of the stress determining the IS along the [001]path is associated with the point of inflection.This is the main reason why the testing of the volumetric insta- bility can actually be omitted.All the other conditions prevent the crystal from shear (second-kind)instabilities.Breaking the condition at Equation 1.13 causes a shear bifurcation from the tetragonal deformation path to the orthorhombic one [46,85.In the case of the fcc crystal,this instability in- duces branching to the tetragonal face centred orthorhombic path-it is the so-called Born's instability.The instabilities at Equations 1.14 and 1.15 are related to C44 and C66 shear moduli,respectively. In order to test the shear-related criteria,special local Lagrangian defor- mations(determined by corresponding Jacobi matrices)are to be applied to
1.1 Ideal Strength of Solids 23 during the loading. It should be emphasized that the elastic moduli in Equation 1.10 are the local ones, i.e., corresponding to different points of the deformation path. Thus, in order to assess the stability, their values should be determined by introducing a sufficient number of independent small deviations (strain increments) away from the original deformation path at each point, in accordance with the symmetry of the particular crystal lattice. The solution of Equation 1.11 gives a different number of possible stability conditions for different crystal lattice symmetries as well as different loading modes. The higher the symmetry (the more point group symmetry operations), the fewer the stability conditions are to be tested. The smallest possible number of necessary stability conditions (only two) corresponds to the isotropic solid. The stability conditions for cubic crystals loaded in uniaxial tension or compression along the [001] direction, the so-called Bain’s path, can serve as a suitable example. The tetragonal symmetry induced by the uniaxial loading means C11 = C22 = 0, C33 = 0, C12 = 0, C13 = C23 = 0, C44 = C55 = 0, C66 = 0 and Cij = 0, other, and a simple relationship σij = σδi3δj3 stands for the stress tensor. By introducing these relations into Equation 1.11, one obtains the following stability conditions: (C33 + σ)(C12 + C11) − 2(C13 − σ/2)2 > 0, (1.12) C11 − C12 > 0, (1.13) 2C44 + σ > 0, (1.14) C66 > 0. (1.15) The left-hand side of Equation 1.12 differs from the tetragonal E[001] modulus for the stress-free state σ = 0 only by a multiplication constant. Therefore, the violation of that so-called volumetric condition is closely related to the first inflection point on the energy vs strain curve along the [001] deformation path (the instability of the first kind). In any case, the maximum value of the stress determining the IS along the [001] path is associated with the point of inflection. This is the main reason why the testing of the volumetric instability can actually be omitted. All the other conditions prevent the crystal from shear (second-kind) instabilities. Breaking the condition at Equation 1.13 causes a shear bifurcation from the tetragonal deformation path to the orthorhombic one [46, 85]. In the case of the fcc crystal, this instability induces branching to the tetragonal face centred orthorhombic path – it is the so-called Born’s instability. The instabilities at Equations 1.14 and 1.15 are related to C44 and C66 shear moduli, respectively. In order to test the shear-related criteria, special local Lagrangian deformations (determined by corresponding Jacobi matrices) are to be applied to