Principles of stability analysis of ideal crystals

Hill, R.; Milstein, Frederick

doi:10.1103/physrevb.15.3087

Cited by 177 publications

(135 citation statements)

References 10 publications

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“…Material elastic stability requires that the speed of propagation of all acceleration waves in a solid should be non-negative [32]. This condition is satisfied when the acoustic tensor is positivedefinite, which is equivalent to the condition that the continuum elasto-dynamic equations are hyperbolic.…”

Section: Predictive Capability: Elastic Stability Limits For Some Biamentioning

confidence: 99%

On the hyperelastic softening and elastic instabilities in graphene

Kumar

Parks

2015

Proc. R. Soc. A.

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Elastic material instabilities are precursors to failure in defect-free graphene single crystals. Elastic instabilities originate from softening in the material response (decay of tangent moduli) induced by dilatant mechanical deformation. Here, we characterize the softening in the constitutive response of graphene within the framework of hyperelasticity based on symmetry-invariants of the two-dimensional logarithmic strain tensor E (0) . The use of symmetry-invariants provides significant functional simplification in representation of the strain energy function of graphene; ab initio calculations of deformation energy are well-fit using half the number of elastic constants of previous formulations. For a set of large homogeneous deformations comprising uniaxial stretch/stress along the armchair and the zigzag directions, and equi-biaxial tension, stress values predicted by the model compare well with those directly calculated from ab initio solutions. Using acoustic tensor analysis, we show that the constitutive model accurately captures elastic stability limits for a number of biaxial deformation modes, providing results that compare well with independent phonon calculations carried out using linear response density functional perturbation theory. In the case of equi-biaxial deformation, an elastic shearing instability is identified which occurs prior to the configuration of maximum true stress. Potential implications of the present results on the interpretation of limiting deformations achieved in nanoindentation experiments are briefly noted.

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Section: Predictive Capability: Elastic Stability Limits For Some Biamentioning

confidence: 99%

On the hyperelastic softening and elastic instabilities in graphene

Kumar

Parks

2015

Proc. R. Soc. A.

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“…As Hill and Milstein 51,52 predicted, a material becomes unstable when its elastic stiffness tensor loses its positive definiteness. The general condition for stability is det 1 2…”

Section: Resultsmentioning

confidence: 99%

Negative Poisson’s ratios in metal nanoplates

et al. 2014

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The Poisson's ratio is a fundamental measure of the elastic-deformation behaviour of materials. Although negative Poisson's ratios are theoretically possible, they were believed to be rare in nature. In particular, while some studies have focused on finding or producing materials with a negative Poisson's ratio in bulk form, there has been no such study for nanoscale materials. Here we provide numerical and theoretical evidence that negative Poisson's ratios are found in several nanoscale metal plates under finite strains. Furthermore, under the same conditions of crystal orientation and loading direction, materials with a positive Poisson's ratio in bulk form can display a negative Poisson's ratio when the material's thickness approaches the nanometer scale. We show that this behaviour originates from a unique surface effect that induces a finite compressive stress inside the nanoplates, and from a phase transformation that causes the Poisson's ratio to depend strongly on the amount of stretch.

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“…Progressive softening of the elastic moduli with dilatant lattice deformation is often responsible for triggering an elastic instability. Acoustic tensor analysis [11,12], which constitutes a useful means of predicting such instabilities, asserts that if, for some pair of unit vectors m and n, Λ(m, n) ≡ (m ⊗ n) : A : (m ⊗ n) ≤ 0,…”

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confidence: 99%

Strain Shielding from Mechanically Activated Covalent Bond Formation during Nanoindentation of Graphene Delays the Onset of Failure

Kumar

Parks

2015

Nano Lett.

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Mechanical failure of an ideal crystal is dictated either by an elastic instability or a soft-mode instability. Previous interpretations of nano-indentation experiments on suspended graphene sheets [1,2], however, indicate an anomaly: the inferred strain in the graphene sheet directly beneath the diamond indenter at the measured failure load is anomalously large compared to the fracture strains predicted by both soft-mode and acoustic analyses. Through multi-scale modeling combining the results of continuum, atomistic, and quantum calculations; and analysis of experiments, we identify a strain-shielding effect initiated by mechanochemical interactions at the graphene-indenter interface as the operative mechanism responsible for this anomaly. Transmission electron micrographs (TEM) and a molecular model of the diamond indenter's tip suggest that the tip surface contains facets comprising crystallographic {111} and {100} planes. Ab initio and molecular dynamics (MD) simulations confirm that a covalent bond (weld) formation between graphene and the crystallographic {111} and {100} facets on the indenter's surface can be induced by compressive contact stresses of the order achieved in nano-indentation tests. Finite element analysis (FEA) and MD simulations of nano-indentation reveal that the shear stiction provided by the induced covalent bonding restricts relative slip of the graphene sheet at its contact with the indenter, thus initiating a local strain-shielding effect. As a result, subsequent to stress-induced bonding at the graphene-indenter interface, the spatial variation of continuing incremental strain is substantially redistributed, locally shielding the region directly beneath the indenter by limiting the buildup of strain while imparting deformation to the surrounding regions. The extent of strain shielding is governed by strength of the shear stiction, which depends upon the level of hydrogen saturation at the indenter's surface. We show that, at intermediate levels of hydrogen saturation, the strain-shielding effect can enable the graphene to support experimentally-determined fracture loads and displacements without prematurely reaching locally limiting states of stress and deformation.Keywords: Graphene, ideal strength, lattice stability, nano-indentation, mechanochemistry, strain-shielding. Lee, et al. [1] measured the fracture strength of graphene using nano-indentation experiments and reported an unprecedentedly high intrinsic strength, measuring orders of magnitude greater than those of conventional materials. The nano-indentation involves instrumented indentation of a suspended graphene sheet by a nanoscale diamond indenter up to the point of failure. However, the local stress and strain beneath the indenter are not directly measurable in such nano-indentation experiments; instead, the indentation load (F ) as a function of indentation depth (u z | r=0 ) during the course of indentation is recorded. Finite Elements Analysis (FEA) based on an appropriate constitutive description of graphe...

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Principles of stability analysis of ideal crystals

Cited by 177 publications

References 10 publications

On the hyperelastic softening and elastic instabilities in graphene

On the hyperelastic softening and elastic instabilities in graphene

Negative Poisson’s ratios in metal nanoplates

Strain Shielding from Mechanically Activated Covalent Bond Formation during Nanoindentation of Graphene Delays the Onset of Failure

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