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THEORETICAL STRENGTH OF SOLIDS. Approved by: Dr. Mo Li, Advisor. School of Materials Science and. Engineering. Georgia Institute of Technology.
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A DissertationPresented to The Academic Faculty
by
Hao Wang
of the Requirements for the DegreeIn Partial Fulfillment Doctor of Philosophy in theSchool of Physics
Georgia Institute of Technology December 2010
Approved by: Dr. Mo Li, Advisor School of Materials Science andEngineering Georgia Institute of Technology
Dr. Arash Yavari School of Civill Engineering Georgia Institute of Technology
Dr. Mei-Yin Chou School of Physics Georgia Institute of Technology
Dr. Dragomir Davidovic School of Physics Georgia Institute of Technology
Dr. Andrew Zangwill School of Physics Georgia Institute of Technology
Date Approved: [August 27, 2010]
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I am delighted to have this opportunity to say thank you to people who have helped me during my everyday life as a Ph.D. student. First I am grateful to Professor Mo Li, my advisor. When I entered his group, I wasted quite some time complaining that there was no direction, no help, and how unfair it was. I assume most Ph.D. students have a similar experience, being in darkness and experiencing growing pains; some quit, most survive. Now I believe one can not make the transition to become an independent scientist without that type of experience. I appreciate that. I thank Professor Mo Li for his critiques and his patience with me. Besides that, he is a role model by showing me how to be a professional scientist. It was through him that I understood real physics is not only the principles in textbooks or popular legends of Newton and Einstein, but also it is a complex social activity that, in addition to working diligently to resolve physics problems, includes giving talks, presenting posters, writing articles, proposals, reports, rebutting referee comments, setting up a network of collaborators and interacting with colleagues. I saw the much broader landscape of physics through him. I thank Professor Mo Li again for giving me the opportunity to work in such a wonderful group. Members with different backgrounds brought their various work styles and knowledge. They work in quite diversified research areas. I thank Dr. Mustafa Uludogan, Dr. Qikai Li, Dr. Xianming Bai, Dr. Huaming Li, Mr. Ken Beyerlein, Mr. Ming Zhao, Mr. Xueqiang Wang, Mr. Yuzheng Guo, Mr. Yongbo Guo, and Miss Jie Feng who inspired stimulating discussions and brought lots of fun to me.
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Special thanks are due to Professor Mei-Yin Chou, Professor Andrew Zangwill, Professor Dragomir Davidovic, and Professor Arash Yavari for their service as my thesis committee members. I thank Mrs. Judy Melton for proofreading my thesis. The love of my parents, my sister, as well as that of my good friends Dr. Donghua Xu, Dr. Xurong Chen, Dr. Xiya Liu, Dr. Haijing Tu, has been of great comfort to me. Finally I wish to say thanks to my wife, Ao Yang, who has been with me during the past six years. She taught me to swim, reminded me to improve my everyday behaviors, and trained me, a nerd, to be a nerdless, decent cook, and cat fancier.
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3.1 Introduction................................ 30 3.2 Theory of nonlinear elasticity........................ 33 3.3 Methods of homogeneous deformation and ab initio computation.... 36 3.4 Results................................... 48 3.5 Conclusions................................. 56 4 NONLINEAR STRESS-STRAIN RELATIONS OF CRYSTAL SOLIDS WITH ARBITRARY INITIAL CONFIGURATIONS................. 58 4.1 Introduction................................. 58 4.2 Linear stress-strain relations........................ 59 4.3 Derivation of nonlinear stress-strain relations............... 61 4.4 Test the nonlinear stress-strain relations to gold.............. 63 4.4.1 Nonlinear shear stress-strain equations of gold under hydrostatic stress
.................. 64 4.4.2 An ab initio calculation of the rhombohedral shear stress-strain curves .................. 64 4.5 Results and discussion............................ 67 4.6 Conclusions................................ 69 5 THEORETICAL AND COMPUTATIONAL STUDY OF ELASTIC STABILITY CRITERIA........................... 71 5.1 Unifying the criteria of elastic stability in solids............. 71 5.2 Elastic stability and ideal strength of gold under uniaxial stress..... 78 5.2.1 Introduction............................. 78
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Table 3.1: For each type of strain modes, η α ( )ξ , α = A , B ,..., K , the internal energy is expressed as a polynomial function of ξ. The coefficients P 2 (^) , P P 3 , 4 in Eq. (3.17) are shownconstants, respectively.................. as the linear combinations of the second-, third-,.............. and fourth-order elastic 37
Table 3.2: The calculated and experimentally determined lattice constants for Cu, Al, Au,and Ag. The unit is Å................................ 47
Table 3.3: The calculated (a) second-, (b) third-, and (c) fourth-order elastic constants of Cu, Ag, Au, and Al. Experimental results and other theoretical calculations are alsoshown. The unit is in GPa............................... 52
Table 4.1: For each crystal configuration of gold with various hydrostatic deformations, ξ = -0.04, -0.02, 0.0, and 0.02, respectively, we have the hydrostatic pressure P , and
those elastic constants by ab initio calculation and polynomial fitting.......... 66 Table 5.1: Zero-pressure elastic stiffness coefficients Bij (in GPa), their pressure derivatives and elastic modulus BT , G 'and G (in GPa)............... 95
Table 6.1: The ideal strength and stable region of face-centered cubic crystal Au, Al, and Cu under uniaxial stress along [100] axis. The results from our analytic scheme, fromprevious ab initio calculation work, and from embedded atom method are listed.. 139
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Figure 1.1: A model is shown for the critical resolved shear stress in a single-crystalspecimen. P is an applied force, A the cross-sectional area of the specimen, φ the angle
between the slip plane normal and the tension direction, and λ the angle between the slip direction and the tension axis............................. 3 Figure 2.1: Schematic illustration of all-electron (solid lines) and pseudoelectron (dashedlines) potentials and their corresponding wave functions. The radius at which all-electron and pseudoelectron values match is designated rc [44]................. 22
Figure 3.1: (a) The dependence of the first-principles results of internal energy of Cu onthe k -points mesh size. The energy converges well when the k -point mesh size goes beyondwith the (^14) k ×-point mesh size. It shows that the energy converges to meV level at the choice 14 × 14. (b) The inset is the zoom-in picture of the internal energy variation of the k -point mesh size................................ 42 Figure 3.2: (a) The calculated internal energy of Cu as a function of the cutoff energy. (b) The inset is the zoom-in picture of the energy that converges within meV level when the cutoff energy is beyond 340 eV. In our calculation, we chose Ecutoff Cu = 490 eV.. .... 43
Figure 3.3: In the calculated internal energy of Cu as a function of lattice constant, the equilibrium lattice parameter is found to be 3.64 Å, which is determined from the corresponding minimum value of the internal energy.................. 43 Figure 3.4: The dependence of four fourth-order elastic constants C 1111 (^) , C 1112 (^) , C 4444 (^) , C 1155 on the Monkhost-Pack k -point mesh size. With Ecutoff Cu = 490 eV applied to all points, the relative difference between two successive values of examined constants in our test after 24 × 24 × 24 is lower than 1%............................. 45 Figure 3.5: The dependence of four fourth-order elastic constants C 1111 (^) , C 1112 (^) , C 4444 (^) , C 1155 on the cutoff energy. With 30 × 30 × 30 k -point mesh size applied to all points, the relative difference between two successive values of examined constants in our test islower than 1%..................................... 46
Figure 3.6: (a) Cu, (b) Au, (c) Al, and (d) Ag, the fourth-order elastic constants C 1111 vs the strain range ξ (^) max. Only at large enough strain range do those elastic constants become convergent. For Cu, Au, Al and Ag, we selected a strain range of 0.15, 0.10, 0.12, and 0.12 respectively................................... 50
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Figure 5.7: The change of the internal energy per atom as a function of strain η 1. It is shown that under compressive stress there is a metastable bct structure at η 1 = −0.25 and an unstable bcc structure at η 1 = −0.19....................... 110
Figure 5.8: Variation of the normal stress (^) σ 1 with strain (^) η 1. Under compression, (^) σ 1 = 0 where the metastable bct and unstable bcc structures should form. Under tensile elongation, σ 1 reaches its maximum value (18.44 GPa ) at η 1 = 0.38where the Young’s modulus approaches zero as shown by the solid line in the inset. The dotted line in theinset is the Young's modulus calculated according to the definition from Ref. 19. It vanishes at the strain of 0.35 that corresponds to the normal stress of 18.36 GPa, lowerthan the maximum value of normal stress....................... 111
Figure 5.9: The stability conditions (Eqs. (5.63-5.66)) plotted using the elastic constants calculated from themodulus goes to be zero first; under elongation, the tetragonal shear modulus goes to zero ab initio method. It is shown that under compression, the Young’s first. Stable region corresponding to the above stability limits is in the strain range, η 1 ~ ( 0.07, 0.07)− , while the corresponding ideal compressive and tensile strength are at - 1.6 and 4.2 GPa respectively............................ 112 Figure 6.1: The hydrostatic stress varies with strain η 1. Two of the stress-strain curves use Eq. (6.20), with two sets of data for the elastic constants in the nonlinear theory, onefrom the experiments and the other from our recent ab initio calculations. The last line comes from our previousother in a finite strain range........................ ab initio simulation [20]. The three lines agree well with each...... 131
Figure 6.2: The three types of elastic moduli of Au under hydrostatic stress vary with strain η 1. Under compression, the crystal is stable. While expansion, the rhombohedral shear stiffness modulus first reaches zero at η 1 ~ 0.05................. 132
Figure 6.3: The normal stress varies with strain η 1 when fcc Au is under uniaxial stress along [100] direction. Two of the stress-strain curves come from Eq. (6.22), with two setsof data for the elastic constants in the nonlinear theory, one from the experiments and the other from our recent initio simulation............................ ab initio calculations. The last line comes from our previous........ 135 ab
Figure 6.4: The four types of elastic modulus of Au under uniaxial stress vary with strain η 1. Under compression, the Young’s modulus first reaches zero at η 1 ~ −0.045 ; while under tensile stress, the tetragonal shear stiffness modulus reach zero firstly at η 1 ~ 0.048....................................... 136
Figure 6.5: Use Eq. (6.20), but keep the hydrostatic stress accurate to the second-, third-, and fourth-order elastic constants, respectively, and plot the stress-strain curves comparedwith our previous ab initio simulation results...................... 143
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Figure 6.6: Use Eqs. (6.20-6.21), and then obtain the bulk stiffness modulus with Eq.(6.9). Control the accuracy to the second-, third-, and fourth-order elastic constants, respectively, and plot the modulus-strain curves compared with our previous simulation result.................................. ab initio 144
Figure 7.1: When a cubic crystal is deformed by a uniaxial compressive or tensile stressalong [100] direction, a crystal with tetragonal symmetry results. The new elastic stiffness tensor is presented, as well as its determinant................. 148 Figure 7.2: When a cubic crystal is compressed or expanded hydrostatically, the elasticstiffness matrix and its determinant are presented.................... 153
Figure 7.3: The stability conditions of Eqs. (7.10-7.12) are tested. Among the three elasticmoduli, Young’s modulus is the first one to reach zero, in both uniaxially compressive and tensile cases, at strain -0.12 and 0.17, respectively................. 155 Figure 7.4: The stability conditions of Eqs. (7.14-7.15) are tested. Under hydrostaticcompression, bulk modulus reach zero firstly at strain -0.16, while in case of hydrostatic tension, shear modulus reach zero first at strain 0.09.................. 157
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constants to test the nonlinear theoretical formulation of elastic stability criterion. In addition, we derive a nonlinear stress-strain equation for solids under arbitrary initial configurations. After that, we try to unify the elastic stability criteria of solids, and with ab initio method, we test the elastic stability conditions of crystal Au. The phenomenon of bifurcation is observed: under hydrostatic expansion, the rhombohedral modulus reaches zero first of all; while under uniaxial tensile stress, the tetragonal shear modulus reaches zero first. In the next part, we develop the nonlinear theoretical formulation of elastic stability criterion and test it with both experimental values and our calculated ones of high order elastic constants. The results are compared with the ones from our previous ab initio simulation work. At last, we extend this nonlinear theoretical formulation to isotropic materials. We use metallic glass Zr (^) 52.5 Ti 5 Cu (^) 17.9Ni14.6Al 10 as a sample to test it.
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A primary objective of materials research is to understand, design, and control the mechanical properties of materials. It is well known that the strength of a usual material is dominated by the behavior of dislocations or microcracks. If such defects were not present, the material under loading would fail only if the theoretical strength, also called ideal strength, were reached. As an intrinsic property of materials, the theoretical strength has drawn considerable attention theoretically and experimentally.
Scientists have developed quite a few theoretical models to explain the ideal strength of materials. Some of models displayed enormous differences with the actual strength of materials, like Frenkel’s^1 and Orowan’s^2 models (those divergences were explained with the postulation of dislocations). Some of the theoretical models are widely accepted nowadays. When a material breaks under an external stress and the fracture direction is perpendicular to the applied stress, it is said to cleave. This process involves the separation of the atoms along the applied stress. Orowan 2-3^ proposed a simple method to obtain the theoretical tensile strength of a crystal. The stress required to separate two planes can be regarded as a function of the distance between these planes. Given the
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shear modulus. For face-centered cubic (FCC) metals with b = a 0 / 2 , a = a 0 3 / 8,
then τ (^) max G / 5.4.
Figure 1.1 A model is shown for the critical resolved shear stress in a single-crystal specimen. P is an applied force, A 0 the cross-sectional area of the specimen, φ the angle between the slip plane normal and the tension direction, and λ the angle between the slip direction and the tension axis. The value of the shear stress required to initiate slip in a pure and perfect single crystal is also known as the critical resolved shear stress, σ (^) CR , which is a constant for a
given slip system. This rule, known as Schmid’s Law, has been confirmed experimentally for a large number of single crystals. As shown in figure 1.1, let φ be the angle between
the slip plane normal and the tension direction, and λ be the angle between the slip direction and the tension axis, then the critical resolved shear stress is σ (^) CR = τ cos φ cosλ,
where τ is the applied tensile stress, τ = P / A 0.
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The elastic stability limit is formulated traditionally following Born’s original idea5-6^ that a crystal should remain stable when the change of the elastic energy with respect to the spontaneous strain exhibits convexity; otherwise, instability would occur consequentially. The condition of convexity leads to the stability criteria in the form of a set of relations involving elastic constants appropriate to the crystal symmetry. However, Born’s theory is formulated for systems without external load. For systems under external load, it was shown 7-17^ that the elastic stiffness coefficients Bijkl rather than elastic
constants Cijkl should be used in formulating the stability criteria.
The stability criterion based on the elastic stiffness constants14-16^ certainly provides a convenient and powerful recipe to measure stability limit. As done in those previous works, one follows a standard recipe. First calculate the elastic constants; then construct the elastic stiffness coefficients as a function of the applied stress to a system that is usually under some specific loading modes such as hydrostatic, uniaxial tensile or compressive, or shear strain; and finally obtain the stability limit at the strain where a principal minor of the elastic stiffness constant matrix first becomes non-positive, or
det Bijkl ≤ 0. Along some simple loading paths such as uniaxial tension, the stability limit
is found to relate to some shear strain modes18-19^. Under hydrostatic loading, it is observed that the stability is dominated by the rhombohedral shear modulus 20. Born 5- and Hill 7-11^ argue that some perturbations, fluctuations, as well as sample loading conditions would make the deformation path astray from the primary loading path, causing a measured stability limit different from that intended originally. This
