The Elastic Behaviour of a Crystalline Aggregate

Hill, Robert S.

doi:10.1088/0370-1298/65/5/307

Cited by 9,347 publications

(3,899 citation statements)

References 5 publications

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“…Reuss found lower bounds, the Reuss bulk modulus (B R ) and the Reuss shear modulus (G R ), for all lattices [34,35], while Voigt discovered upper bounds, the Voigt bulk modulus (B V ) and the Voigt shear modulus (G V ) [34,36]. Hill has shown that the Voigt and Reuss averages are limits and suggested that the actual effective moduli could be approximated by the arithmetic mean of the two bounds [37]. We also calculated the Young's modulus, E, and Poisson's ratio, ν, which are frequently measured for polycrystalline materials when investigating their hardness.…”

Section: Elastic Propertiesmentioning

confidence: 99%

First-principles study of structural, electronic and elastic properties of Nb4AlC3

Bouhemadou¹

2010

Braz. J. Phys.

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Using First-principles calculations, we have studied the structural, electronic and elastic properties of Nb 4 AlC 3 , a new compound belonging to the MAX phases. Geometrical optimization of the unit cell is in good agreement with the experimental data. The effect of high pressures, up to 20 GPa, on the lattice constants shows that the contractions are higher along the c-axis than along the a-axis. We have observed a quadratic dependence of the lattice parameters versus the applied pressure. The band structure shows that this compound is electrical conductor. The analysis of the site and momentum projected densities shows that bonding is due to Nb d-C p and Nb d-Al p hybridizations. The Nb d-C p bond is lower in energy and stiffer than Nb d-Al p bond. The elastic constants are calculated using the static finite strain technique. We derived the bulk and shear moduli, Young's modulus and Poisson's ratio for ideal polycrystalline Nb 4 AlC 3 aggregate. We estimated the Debye temperature of Nb 4 AlC 3 from the average sound velocity. This is the first quantitative theoretical prediction of the elastic properties of Nb 4 AlC 3 compound, and it still awaits experimental confirmation.

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Section: Elastic Propertiesmentioning

confidence: 99%

First-principles study of structural, electronic and elastic properties of Nb4AlC3

Bouhemadou¹

2010

Braz. J. Phys.

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“…Since the elastic anisotropy of tantalum is weak (A Zener =1.56) [64], even relatively simple homogenization techniques work well, including the Voigt (constant strain) [65] and Reuss (constant stress) [66] averages, G V and G R , respectively. More accurate estimates of the polycrystalline shear modulus include the Voigt-Reuss-Hill average shear modulus, G VRH , [67], equal to the arithmetic mean of the Voigt and Reuss shear moduli, and the more sophisticated formula due to Hershey, G H [68]. The single-crystal elastic constants for the Finnis-Sinclair potential we have used are: B=196 GPa, C'=(C 11 -C 12 )/2=52.4 GPa, C 44 = 82.4 GPa at T=0K [39].…”

Section: =ε Yy (T)=(-1/2) ε Zz (T) + O(ε Zz 2 ) Using Eq (2) the Stmentioning

confidence: 99%

High-Rate Plastic Deformation of Nanocrystalline Tantalum to Large Strains: Molecular Dynamics Simulation

Rudd

2009

MSF

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Recent advances in the ability to generate extremes of pressure and temperature in dynamic experiments and to probe the response of materials has motivated the need for special materials optimized for those conditions as well as a need for a much deeper understanding of the behavior of materials subjected to high pressure and/or temperature. Of particular importance is the understanding of rate effects at the extremely high rates encountered in those experiments, especially with the next generation of laser drives such as at the National Ignition Facility. Here we use large-scale molecular dynamics (MD) simulations of the high-rate deformation of nanocrystalline tantalum to investigate the processes associated with plastic deformation for strains up to 100%. We use initial atomic configurations that were produced through simulations of solidification in the work of Streitz et al [Phys. Rev. Lett. 96, (2006) 225701]. These 3D polycrystalline systems have typical grain sizes of 10-20 nm. We also study a rapidly quenched liquid (amorphous solid) tantalum. We apply a constant volume (isochoric), constant temperature (isothermal) shear deformation over a range of strain rates, and compute the resulting stress-strain curves to large strains for both uniaxial and biaxial compression. We study the rate dependence and identify plastic deformation mechanisms. The identification of the mechanisms is facilitated through a novel technique that computes the local grain orientation, returning it as a quaternion for each atom. This analysis technique is robust and fast, and has been used to compute the orientations on the fly during our parallel MD simulations on supercomputers. We find both dislocation and twinning processes are important, and they interact in the weak strain hardening in these extremely fine-grained microstructures.

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“…One very famous approximation scheme is due to Hill [3]. The idea is to take the known Voigt and Reuss averages of the elastic system stiffnesses or compliances, and then make direct use of this information by computing either the arithmetic or geometric mean of these two limiting values.…”

Section: Estimation Schemes Based On Bounds For Elasticitymentioning

confidence: 99%

“…These results range from rigorous bounds such as the Voigt [1], Reuss [2], Hill [3], and Hashin-Shtrikman [4,5] bounds to the fairly popular and mostly well-justified (for sufficiently small concentrations of inclusions [6]) approximate methods such as the explicit approximations of Kuster and Toksöz [7] and Mori and Tanaka [8,9] and the implicit methods such as the differential effective medium (DEM) method [10,11] and the self-consistent [12,13] or the coherent potential approximation for elastic composites [14][15][16][17]. Older reviews [18] and both early [19,20] and more recent textbooks and research monographs [21][22][23][24] survey the state of the art.…”

Section: Introductionmentioning

confidence: 99%

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

Berryman

2006

Mechanics of Materials

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The most commonly discussed measures of microstructure in composite materials are the spatial correlation functions, which in a porous medium measure either the grain-to-grain correlations, or the pore-to-pore correlations in space. Improved bounds based on this information such as the Beran-Molyneux bounds for bulk modulus and the Beran bounds for conducticity are well-known.It is first shown here how to make direct use of this information to provide estimates that always lie between these upper and lower bounds for any microstructure whenever the microgeometry parameters are known. Then comparisons are made between these estimates, the bounds, and two new types of estimates. One new estimate for elastic constants makes use of the PeselnickMeister bounds (based on Hashin-Shtrikman methods) for random polycrystals of laminates to generate self-consistent values that always lie between the bounds. A second new type of estimate for conductivity assumes that measurements of formation factors (of which there are at least two distinct types in porous media, associated respectively with pores and grains) are available, and computes new bounds based on this information. The paper compares and contrasts these various methods in order to clarify just what microstructural information and how precisely that information needs to be known in order to be useful for estimating material constants in random and heterogeneous media. * berryman1@llnl.gov

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The Elastic Behaviour of a Crystalline Aggregate

Cited by 9,347 publications

References 5 publications

First-principles study of structural, electronic and elastic properties of Nb4AlC3

First-principles study of structural, electronic and elastic properties of Nb4AlC3

High-Rate Plastic Deformation of Nanocrystalline Tantalum to Large Strains: Molecular Dynamics Simulation

Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

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