Tham khảo tài liệu 'kinetics of materials - r. balluff s. allen w. carter (wiley 2005) ww part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 16 CHAPTER 1 INTRODUCTION which is a requirement that the homogeneous equation Eq. has a nontrivial solution. After the eigenvalues have been determined the directions of the eigenvectors ế can be determined by solving Eq. . A rank-two property tensor is diagonal in the coordinate system defined by its eigenvectors. Rank-two tensors transform like 3x3 square matrices. The general rule for transformation of a square matrix into its diagonal form is diagonalized matrix eigenvector column matrix square matrix eigenvector column matrix where the zth member of the diagonal matrix is the eigenvalue corresponding to the eigenvector used for the ith column vector of the transformation matrix. Nearly all rank-two property tensors can be represented by 3 X 3 symmetric matrices and necessarily have real eigenvalues. Bibliography 1. . Allen and . Thomas. The Structure of Materials. John Wiley Sons New York 1999. 2. R. Clausius. The Mechanical Theory of Heat With Its Applications to the SteamEngine and to the Physical Properties of Bodies. Van Voorst London 1867. 3. . Gibbs. On the equilibrium of heterogeneous substances 1876 . In Collected Works volume 1. Longmans Green and Co. New York 1928. 4. L. Onsager. Reciprocal relations in irreversible processes. II. Phys. Rev. 38 12 2265-2279 1931. 5. . Carter . Taylor and . Cahn. Variational methods for microstructural evolution. JOM 49 12 30-36 1997. 6. . Nye. Physical Properties of Crystals. Oxford University Press Oxford 1985. EXERCISES The concentration at any point in space is given by c A xy yz zx 1-37 where A constant. a Find the cosines of the direction in which c changes most rapidly with distance from the point 1 1 1 . b Determine the maximum rate of change of concentration at that point. Solution. a The direction of maximum rate of change is along the gradient vector Vc given by Vc A y z i z x j x y z Therefore Vc l l l 2A i j z and the direction cosines are 1 y 3 l y 3 l 3
