Most metallic elements have a crystal structure that is either body-centered cubic (bcc), face-centered close packed, or hexagonal close packed. If the bcc lattice is the thermodynamically most stable structure, the close-packed structures usually are dynamically unstable, i.e., have elastic constants violating the Born stability conditions or, more generally, have phonons with imaginary frequencies. Conversely, the bcc lattice tends to be dynamically unstable if the equilibrium structure is close packed. This striking regularity essentially went unnoticed until ab initio total-energy calculations in the 1990s became accurate enough to model dynamical properties of solids in hypothetical lattice structures. After a review of stability criteria, thermodynamic functions in the vicinity of an instability, Bain paths, and how instabilities may arise or disappear when pressure, temperature, and/or chemical composition is varied are discussed. The role of dynamical instabilities in the ideal strength of solids and in metallurgical phase diagrams is then considered, and comments are made on amorphization, melting, and low-dimensional systems. The review concludes with extensive references to theoretical work on the stability properties of metallic elements.

https://perssongroup.lbl.gov/papers/rmp2012-instabilities.pdf

Lattice instabilities in metallic elements

Using pseudopotential density functional theory within the local-density approximation, we calculate the ideal shear strengths of W for slip on {110}, {112} and {123} planes allowing for complete structural relaxation orthogonal to the applied shear. The strengths in the weak directions on all planes are found to be very nearly equal (about 18GPa, or 11% of the shear modulus G). Moreover, the shear instability occurs at approximately the same applied shear strain (17–18%). This unusual isotropy is explained in terms of the atomic configurations of high-energy saddle points reached during shear. Analysis of these saddle points may also offer a simple explanation for the prevalence of the pencil glide of dislocations on planes containing a (111) direction in bcc metals. Finally, we calculate the ideal cleavage strengths of W on {100} and compare our calculated ideal shear and cleavage strengths with experimental nanoindentation and whisker measurements. All these results can be rather simply unders...

The ideal strength of tungsten

This paper investigates the conditions of elastic stability that set the upper limits of mechanical strength Following Gibbs, we determine the conditions that ensure stability against reconfigurations that leave the boundary of the system unchanged The results hold independent of the nature of properties of the loading mechanisms but are identical with those derived previously for a solid in contact with a reservoir that maintains the Cauchy stress Mechanisms that control the stress in some other way may add further conditions of stability; nonetheless, the conditions of internal stability must always be obeyed and can be consistently used to define the ultimate strength The conditions of stability are contained in the requirement that λ ijkl δe ij δe kl ≥ 0 for all infinitesimal strains, where λ ijkl = 1/2(B ijkl + B klij ), and B is the tensor that governs the change in the Cauchy stress t during incremental strain from a stressed state τ: t ij = T ij + B ijkl δe kl  Since λ has full Voigt symmetry, it can be written as the 6 x 6 matrix λ ij with eigenvalues λ α  Stability is lost when the least of these vanishes The conditions of stability are exhibited for cubic (hydrostatic), tetragonal (tensile) and monoclinic (shear) distortions of a cubic crystal and some of their implications are discussed Elastic stability and the limits of strength are now being explored through first-principles calculations that increment uniaxial stretch or shear to find the maximum stress We discuss the nature of this limiting stress and the steps that may be taken to identify orthogonal instabilities that might intrude before it is reached

The internal stability of an elastic solid

Using a first-principles computational tensile test, we show that the ideal tensile strength of an Al grain boundary (GB) is reduced with both Na and Ca GB segregation We demonstrate that the fracture occurs in the GB interface, dominated by the break of the interfacial bonds Experimentally, we further show that the presence of Na or Ca impurity, which causes intergranular fracture, reduces the ultimate tensile strength when embrittlement occurs These results suggest that the $\mathrm{Na}∕\mathrm{Ca}$-induced intergranular embrittlement of an Al alloy originates mainly from the GB weakening due to the $\mathrm{Na}∕\mathrm{Ca}$ segregation

http://www.eng.utah.edu/~fliu/pdfs/Phys.%20Rev.%20B%2073,%20224115.pdf

Origin of intergranular embrittlement of Al alloys induced by Na and Ca segregation: Grain boundary weakening

First-principles studies on the intrinsic mechanical properties of various materials and systems through ab initio tensile and shear testing simulations based on density-functional theory are reviewed. For various materials, ideal tensile and shear strength and features of the deformation of bulk crystals without any defects have been examined, and the relation with the bonding nature has been analyzed. The surfaces or low-dimensional nano-structures revealpeculiarstrengthanddeformationbehaviorduetolocaldifferentbonding nature. For grain boundaries and metal/ceramic interfaces, tensile and shear behaviors depend on the interface bonding, which impacts on the research of real engineering materials. Remaining problems and future directions in this research field are discussed. (Some figures in this article are in colour only in the electronic version)

https://iopscience.iop.org/article/10.1088/0965-0393/17/1/013001/pdf

First-principles approaches to intrinsic strength and deformation of materials: perfect crystals, nano-structures, surfaces and interfaces

On the basis of the pseudopotential plane-wave(PP-PW) method in combination with the local-density-functional theory(LDFT), complete stress-strain curves for the uniaxial loading and uniaxial deformation along the [001] and [111] directions, and the biaxial proportional extension along [010] and [001] of aluminium are obtained. During the uniaxial loading, certain general behaviors of energy versus stretch and the load versus the stretch are confirmed; in each acse, there exist three special unstressed structures: f.c.c., b.c.c. and f.c.t. for [001]; f.c.c., s.c. and b.c.c. for [111]. Using stability criteria, we find that all of these state are unstable, and always occur together with shear instability, except the natural f.c.c. structure. A Bain transformation from the stable f.c.c. structure to the stable b.c.c. configuration cannot be obtained by uniaxial compression along any equivalent [001] and [111] direction. The tensile strength are similar for the two directions. For the higher energy barrier of [111] direction, the compressive strength is greater than that for the [001] direction. With increase in the ratio of the biaxial proportional extension, the stress and tensile strength increase; however, the critical strain does not change significantly. Our results add to the existing ab initio database for use in fitting and testing interatomic potentials.

Weixue Li

Papers

Ab initio Investigation of Elasticity and Stability of Metal Aluminum