Chapter 1 Deformation and Fracture of Perfect Crystals Perfect crystals constitute an idealization of real single crystals that always contain some crystal defects.This holds even for almost perfect single crys- tals,so-called whiskers,where the presence of an equilibrium concentration of vacancies is inevitable.On the other hand,the mechanical behaviour of perfect crystals can be very successfully simulated and predicted by theoret- ical ab initio approaches based on electronic structure calculations.Because many of the crystallographic,elastic,electric,magnetic and thermodynamic characteristics of crystals do not depend on crystal defects,the ab initio re- sults can still be experimentally verified.The practical importance of such studies lies in several general aspects. First,knowledge of the behaviour of perfect crystals clearly identifies the role of crystal structure and chemical composition.This enables us to sep- arate the role of defects due to the difference in the mechanical behaviour of perfect and real crystals.Second,the ideal (theoretical)strength of per- fect crystals of a particular chemical composition represents an upper bound of the strength of solids.Consequently,engineers can see the gap remaining between the strength of contemporary high-strength materials and that of the theoretical limit.Third,the models of processes of dislocation creation near stress concentrators such as cracks in perfect crystals provide us with lower bounds of macroscopic characteristics used in fracture mechanics.Thus, the ab initio results yield both upper and lower benchmarks defining phys- ically possible ranges of mechanical characteristics of engineering materials. Fourth,the characteristics of ideal crystals can be utilized in multiscale mod- els of deformation and fracture processes of engineering materials.Finally, these results can be used as fitting data for the construction of sophisticated semi-empirical interatomic potentials that are utilized for prediction of the behaviour of extended defects in real crystals and polycrystals. All of the above-mentioned aspects are demonstrated in the three sec- tions of this chapter devoted to the mechanical behaviour of perfect crystals In the first section,the application of ab initio methods to calculations of elastic stress-strain response and,particularly,to the determination of ideal 9
Chapter 1 Deformation and Fracture of Perfect Crystals Perfect crystals constitute an idealization of real single crystals that always contain some crystal defects. This holds even for almost perfect single crystals, so-called whiskers, where the presence of an equilibrium concentration of vacancies is inevitable. On the other hand, the mechanical behaviour of perfect crystals can be very successfully simulated and predicted by theoretical ab initio approaches based on electronic structure calculations. Because many of the crystallographic, elastic, electric, magnetic and thermodynamic characteristics of crystals do not depend on crystal defects, the ab initio results can still be experimentally verified. The practical importance of such studies lies in several general aspects. First, knowledge of the behaviour of perfect crystals clearly identifies the role of crystal structure and chemical composition. This enables us to separate the role of defects due to the difference in the mechanical behaviour of perfect and real crystals. Second, the ideal (theoretical) strength of perfect crystals of a particular chemical composition represents an upper bound of the strength of solids. Consequently, engineers can see the gap remaining between the strength of contemporary high-strength materials and that of the theoretical limit. Third, the models of processes of dislocation creation near stress concentrators such as cracks in perfect crystals provide us with lower bounds of macroscopic characteristics used in fracture mechanics. Thus, the ab initio results yield both upper and lower benchmarks defining physically possible ranges of mechanical characteristics of engineering materials. Fourth, the characteristics of ideal crystals can be utilized in multiscale models of deformation and fracture processes of engineering materials. Finally, these results can be used as fitting data for the construction of sophisticated semi-empirical interatomic potentials that are utilized for prediction of the behaviour of extended defects in real crystals and polycrystals. All of the above-mentioned aspects are demonstrated in the three sections of this chapter devoted to the mechanical behaviour of perfect crystals. In the first section, the application of ab initio methods to calculations of elastic stress-strain response and, particularly, to the determination of ideal 9
10 1 Deformation and Fracture of Perfect Crystals strength under various loading conditions is demonstrated.One should note that these concepts are,in fact,two scale models based on both electronic and atomic structures.Section 1.2 deals with another important problem concerning the physics of brittle/ductile behaviour of perfect crystals.The models are three-scale approaches dealing with electronic,atomic and crys- tallographic structures of crystals.The third section illustrates the modelling of nanoindentation tests with regard to the physical interpretation of pop-ins observed at the end of the elastic part in the force-indentation depth diagram. These multiscale models couple electronic,atomic,crystallographic and con- tinuum approaches to provide a unique tool for experimental measurements of the ideal shear strength. 1.1 Ideal Strength of Solids The strength of any solid has an upper limit called the ideal (theoretical) strength (IS).This value corresponds to the failure of an infinite perfect single crystal loaded in a defined mode.The strength of engineering materials is usually controlled by nucleation and propagation of dislocations and/or microcracks.If such defects were not present,the material would only fail when the IS is reached.Until recently loads of this magnitude were only approached in studies of the mechanical behaviour of whiskers of very pure metals and semiconductors [1,13,14.Starting from the beginning of the last century,there has been a more or less continuous effort expended in order to obtain theoretical and experimental data concerning IS of various solids. The IS values set an upper limit to the envelope of attainable stresses and its knowledge enables us to assess the gap remaining to upper strength values of advanced engineering materials in each period of time.However,this is not the only reason for IS investigation.From the theoretical point of view,the IS plays a decisive role in the fundamental theory of fracture.For example, the stress necessary for nucleation of a dislocation loop can be identified with the shear IS value.This has been proved most eloquently by nanoindentation experiments(see e.g.,[15-18)which suggest that the onset of yielding at the nanoscale is controlled by a homogeneous nucleation of dislocations in a small, dislocation free,volume under the nanoindenter,where the stresses approach the shear IS.Similarly,the local stress needed for unstable propagation of a cleavage crack should overcome the value of the tensile IS [19-22].The ratio of shear and tensile IS expresses a tendency of the crystal matrix to become brittle or ductile 23-25(see Section 1.2).The values of IS may also be used in the construction or checking of semi-empirical interatomic potentials.From the practical point of view,the shear IS appears to control both the onset of fracture and the dislocation nucleation in defect-free thin films and,in particular,in nano-structured materials that are currently being developed. Perfect single crystal wires(whiskers)are used as reinforcements in advanced
10 1 Deformation and Fracture of Perfect Crystals strength under various loading conditions is demonstrated. One should note that these concepts are, in fact, two scale models based on both electronic and atomic structures. Section 1.2 deals with another important problem concerning the physics of brittle/ductile behaviour of perfect crystals. The models are three-scale approaches dealing with electronic, atomic and crystallographic structures of crystals. The third section illustrates the modelling of nanoindentation tests with regard to the physical interpretation of pop-ins observed at the end of the elastic part in the force-indentation depth diagram. These multiscale models couple electronic, atomic, crystallographic and continuum approaches to provide a unique tool for experimental measurements of the ideal shear strength. 1.1 Ideal Strength of Solids The strength of any solid has an upper limit called the ideal (theoretical) strength (IS). This value corresponds to the failure of an infinite perfect single crystal loaded in a defined mode. The strength of engineering materials is usually controlled by nucleation and propagation of dislocations and/or microcracks. If such defects were not present, the material would only fail when the IS is reached. Until recently loads of this magnitude were only approached in studies of the mechanical behaviour of whiskers of very pure metals and semiconductors [1, 13, 14]. Starting from the beginning of the last century, there has been a more or less continuous effort expended in order to obtain theoretical and experimental data concerning IS of various solids. The IS values set an upper limit to the envelope of attainable stresses and its knowledge enables us to assess the gap remaining to upper strength values of advanced engineering materials in each period of time. However, this is not the only reason for IS investigation. From the theoretical point of view, the IS plays a decisive role in the fundamental theory of fracture. For example, the stress necessary for nucleation of a dislocation loop can be identified with the shear IS value. This has been proved most eloquently by nanoindentation experiments (see e.g., [15–18]) which suggest that the onset of yielding at the nanoscale is controlled by a homogeneous nucleation of dislocations in a small, dislocation free, volume under the nanoindenter, where the stresses approach the shear IS. Similarly, the local stress needed for unstable propagation of a cleavage crack should overcome the value of the tensile IS [19–22]. The ratio of shear and tensile IS expresses a tendency of the crystal matrix to become brittle or ductile [23–25] (see Section 1.2). The values of IS may also be used in the construction or checking of semi-empirical interatomic potentials. From the practical point of view, the shear IS appears to control both the onset of fracture and the dislocation nucleation in defect-free thin films and, in particular, in nano-structured materials that are currently being developed. Perfect single crystal wires (whiskers) are used as reinforcements in advanced
1.1 Ideal Strength of Solids 11 composite materials and large metallic and ceramic single crystals start to be important in special engineering components,e.g.,in turbine blades 26. Section 1.1.1 provides a picture of the historical development including re- marks on sophisticated calculation methods.The principles of ab initio meth- ods and stability procedures,utilized in recent computations,are presented in Section 1.1.2.Some results of ab initio calculations of IS under simple loading modes reported in our works are outlined in Section 1.1.3.Section 1.1.4 is dedicated to IS of crystals under multiaxial loading.An analysis of mechanical properties of ideal nanocomposites is outlined in Section 1.1.5.A brief discussion concerning the influence of lattice defects and temperature on the crystal strength is presented in the last subsection. Section 1.1 is complemented by two appendices focused on ab initio meth- ods and mixed-mode criteria of crack stability. 1.1.1 From Classics to Recent Concepts For every particular loading,the stress state is characterized by six stress tensor components and,consequently,an infinite number of ideal strengths exists for a given crystal.For practical reasons,therefore,the IS was usually evaluated only for several special cases of loading,each defined by a single value of the stress tensor component,specifically,for the uniaxial tensions and compressions along various crystallographic directions,the isotropic (hy- drostatic)tension and compression and for the pure shear in certain planes and directions.The respective IS values denoted here as oiut,oiue,oiht,oihe and Tis cover,to a reasonable degree,the most important cases occurring in the engineering practice. 1.1.1.1 Classical Theories Historically the first calculations of Tis were performed in 1926 by Frenkel 27. The model of an ideal crystal subjected to block-like shear is,along with the related behaviour of the shear stress r under applied shear deformation,ex- pressed by the plane shift s in Figure 1.1.The values of ideal shear strength for the block-like model will be denoted Tis.6.The r(s)dependence was as- sumed to be of a sinusoidal shape. According to specification of variables in Figure 1.1,the stress behaviour could be described by the relation 2T T Tmaz sin -s. a For a small shift(sin2红s≈红s),the shear modulus should be G=dr/d, where =s/b.Since Tis,=Tmaz this leads to
1.1 Ideal Strength of Solids 11 composite materials and large metallic and ceramic single crystals start to be important in special engineering components, e.g., in turbine blades [26]. Section 1.1.1 provides a picture of the historical development including remarks on sophisticated calculation methods. The principles of ab initio methods and stability procedures, utilized in recent computations, are presented in Section 1.1.2. Some results of ab initio calculations of IS under simple loading modes reported in our works are outlined in Section 1.1.3. Section 1.1.4 is dedicated to IS of crystals under multiaxial loading. An analysis of mechanical properties of ideal nanocomposites is outlined in Section 1.1.5. A brief discussion concerning the influence of lattice defects and temperature on the crystal strength is presented in the last subsection. Section 1.1 is complemented by two appendices focused on ab initio methods and mixed-mode criteria of crack stability. 1.1.1 From Classics to Recent Concepts For every particular loading, the stress state is characterized by six stress tensor components and, consequently, an infinite number of ideal strengths exists for a given crystal. For practical reasons, therefore, the IS was usually evaluated only for several special cases of loading, each defined by a single value of the stress tensor component, specifically, for the uniaxial tensions and compressions along various crystallographic directions, the isotropic (hydrostatic) tension and compression and for the pure shear in certain planes and directions. The respective IS values denoted here as σiut, σiuc, σiht, σihc and τis cover, to a reasonable degree, the most important cases occurring in the engineering practice. 1.1.1.1 Classical Theories Historically the first calculations of τis were performed in 1926 by Frenkel [27]. The model of an ideal crystal subjected to block-like shear is, along with the related behaviour of the shear stress τ under applied shear deformation, expressed by the plane shift s in Figure 1.1. The values of ideal shear strength for the block-like model will be denoted τis,b. The τ(s) dependence was assumed to be of a sinusoidal shape. According to specification of variables in Figure 1.1, the stress behaviour could be described by the relation τ = τmax sin 2π a s. For a small shift (sin 2π a s ≈ 2π a s), the shear modulus should be G = dτ /dξ, where ξ = s/b. Since τis,b = τmax this leads to
12 1 Deformation and Fracture of Perfect Crystals (a) (b) Figure 1.1 (a)Model of a block-like shear deformation (dark spheres represent atomic positions within the upper block,that is,as a whole,shifted by s towards the lower block along the shear plane,light spheres show their original positions),and (b)shear stress T as a function of the shift s of two adjacent planes Ga Tis,b= 2xb (1.1) Equation 1.1 gave valuesis,G for the {111)(112)shear of face cen- tred cubic (fcc)metals,and Tis,G for {110)(111)shear of bcc metals as well as for the {111(110)shear of fcc metals.Because the yield stress of real crystals was found to be about three orders lower,the only plausible explana- tion of this discrepancy was the presence of line defects (dislocations).Thus, the Frenkel's result created a milestone for a development of the dislocation theory. First attempts to compute the ideal strength in uniaxial tension oiut were performed by Polanyi [28 and Orowan 29.They were based on an assump- tion of tearing fracture of a stretched crystal along a crystallographic plane. Forces between two adjacent atomic planes of a perfect solid vary with the interplanar distance as in Figure 1.2.This dependence was approximated by a sinusoidal function =dmax sinnto d and the expected deviation from this trend for high strain values was ne- glected.The function was parametrized according to the following assump- tions:1)the work of deformation per unit area corresponds to energy 2y of the two new surfaces ao+d odx 2y; ao
12 1 Deformation and Fracture of Perfect Crystals a s b a s 0 max (a) (b) Figure 1.1 (a) Model of a block-like shear deformation (dark spheres represent atomic positions within the upper block, that is, as a whole, shifted by s towards the lower block along the shear plane, light spheres show their original positions), and (b) shear stress τ as a function of the shift s of two adjacent planes τis,b = Ga 2πb . (1.1) Equation 1.1 gave values τis,b ≈ 1 9G for the {111}11¯2 shear of face centred cubic (fcc) metals, and τis,b ≈ 1 5G for {110}1¯11 shear of bcc metals as well as for the {111}1¯10 shear of fcc metals. Because the yield stress of real crystals was found to be about three orders lower, the only plausible explanation of this discrepancy was the presence of line defects (dislocations). Thus, the Frenkel’s result created a milestone for a development of the dislocation theory. First attempts to compute the ideal strength in uniaxial tension σiut were performed by Polanyi [28] and Orowan [29]. They were based on an assumption of tearing fracture of a stretched crystal along a crystallographic plane. Forces between two adjacent atomic planes of a perfect solid vary with the interplanar distance as in Figure 1.2. This dependence was approximated by a sinusoidal function σ = σmax sin π x − a0 d and the expected deviation from this trend for high strain values was neglected. The function was parametrized according to the following assumptions: 1) the work of deformation per unit area corresponds to energy 2γ of the two new surfaces a0+d a0 σdx = 2γ;
1.1 Ideal Strength of Solids 13 and 2)in the vicinity of the equilibrium state (x =ao),the stress is pro- do portional to Young's modulus and the relation E de must be valid for the strain e o.Then,the maximum value of the tensile stress can be d simply evaluated as E (1.2) ao The corresponding ideal tensile (tear)strengths of metals are mostly very high(several tens of GPa). 0 a6 &+d X Figure 1.2 Stress as a function of the distance between atomic planes Mackenzie presented in 1949 a more extended study of the shear IS based on a variation of potential energy U per unit area of a shear plane with the plane shift s [30].The shear stress can be calculated from the energy U as dU T= (1.3) ds From this point of view,the Frenkel's approach described above refers only to the first two terms in the Fourier series for U(s).Therefore,Macken- zie took further terms into consideration.This theory gave a very low value sG for {111)(112)shear in fcc lattice [30].As found by Sandera and Pokluda [31],however,some assumptions used in that theory were not phys- ically legitimate.Indeed,more recent calculations 31,32 based on more so- phisticated atomistic approaches confirmed a much better validity of Frenkel's estimation. Further IS calculations were performed by means of so-called empirical interatomic potentials [33].Most of them used an analogy to Equation 1.3. The potential energy U was calculated as a sum of pair-potentials of various kinds such as the Morse potential,the Lennard-Jones potential,etc.(e.g.,23, 24,34-361).As an example,some results of calculations of oiht are introduced in Section 1.1.3
1.1 Ideal Strength of Solids 13 and 2) in the vicinity of the equilibrium state (x = a0), the stress is proportional to Young’s modulus and the relation E = dσ dε must be valid for the strain ε = x − a0 d . Then, the maximum value of the tensile stress can be simply evaluated as σiut = Eγ a0 . (1.2) The corresponding ideal tensile (tear) strengths of metals are mostly very high (several tens of GPa). a0 x 0 max a0+d Figure 1.2 Stress as a function of the distance between atomic planes Mackenzie presented in 1949 a more extended study of the shear IS based on a variation of potential energy U per unit area of a shear plane with the plane shift s [30]. The shear stress can be calculated from the energy U as τ = dU ds . (1.3) From this point of view, the Frenkel’s approach described above refers only to the first two terms in the Fourier series for U(s). Therefore, Mackenzie took further terms into consideration. This theory gave a very low value τis,b ≈ 1 30G for {111}11¯2 shear in fcc lattice [30]. As found by Sandera and ˇ Pokluda [31], however, some assumptions used in that theory were not physically legitimate. Indeed, more recent calculations [31, 32] based on more sophisticated atomistic approaches confirmed a much better validity of Frenkel’s estimation. Further IS calculations were performed by means of so-called empirical interatomic potentials [33]. Most of them used an analogy to Equation 1.3. The potential energy U was calculated as a sum of pair-potentials of various kinds such as the Morse potential, the Lennard–Jones potential, etc. (e.g., [23, 24,34–36]). As an example, some results of calculations of σiht are introduced in Section 1.1.3.