Matlab Code associated with "Determination of Hashin-Shtrikman Bounds on the Isotropic Effective Elastic Moduli of Polycrystals of any Symmetry" Two files are included in this repository:

test_HSBounds.m HSBounds.m

The first contains example data, makes the call to the function, and includes a listing of expected results The second is the implementation determining the Hashin-Shtrikman Bounds:

function [hs,vrh,ko_go,ustart]=HSBounds(cij) % This function finds the Hashin-Shtrikman bounds for material with elastic % moduli cij in a 6x6 matrix. It is the implementation of the variational % equations first derived by Hashin and Shtrikman in 1963 and later % articulated in papers by Peselnick and Meister and Watt and Peselnick. % Usage: % [hs,vrh,ko_go,ustart]=HSBounds(cij) % where: % cij is a 6x6 matrix of elastic constants of any symmetry (GPa units) % hs is a 2x2 matrix with upper and lower "optimal" bounds for K and G % vrh gives, for reference, the Voigt-Reuss-Hill bounds % ko_go gives the properties of the reference material at the optimal point % ustart gives the starting point used for "go" in the calculations % % The following "nested" custom functions are included: % 1. [xmax,hs]=edgeu('pos' or 'neg'); finds the positive definite % boundary at small ko and the negative definite boundary at large ko % 2. [k,hsl]=edgek(y,'pos' or 'neg'); finds the positive/negative % boundary at fixed uo with increasing ko % 3. y=lowerbound(x) returns the hs value at the boundaries - used in % search for optimal values % 4. y=upperbound(x) returns the hs value at the boundaries - used in % search for optimal values % 5. [hs,value]=hscalc(ko,go,cij) does the Hashin-Shrtikman % calculations % 6. [K,G]=VRH(C) calculates Voigt-Reuss-Hill bounds % The built-in to MATLAB search function fminbnd is used to find the optimal bounds % JMB 2013