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Some Properties of Copper – Gold and Silver – Gold Alloys ...

Proceedings of the International MultiConference of Engineers and Computer Scientists 2011 Vol II, IMECS 2011, March 16 - 18, 2011, Hong Kong some Properties of Copper gold and Silver . gold Alloys at different % of gold Fae`q A. A. Radwan*. Abstract: The norm of elastic constant tensor and the norms Where ij is the Kronecker delta. The Einstein of the irreducible parts of the elastic constants of Copper , Silver and gold metals and Copper - gold and Silver - gold summation convention over repeated indices is used Alloys at different percentages of gold are calculated. The and indices run from 1 to 3 unless otherwise stated. relation of the scalar parts norm and the other parts norms and the anisotropy of these metals and their Alloys are By applying the symmetry conditions (2) to the presented.

Some Properties of Copper – Gold and Silver – Gold Alloys at Different % of Gold Fae`q A. A. Radwan* Abstract: The norm of elastic constant tensor and the norms of the irreducible parts of the elastic constants of Copper,

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Transcription of Some Properties of Copper – Gold and Silver – Gold Alloys ...

1 Proceedings of the International MultiConference of Engineers and Computer Scientists 2011 Vol II, IMECS 2011, March 16 - 18, 2011, Hong Kong some Properties of Copper gold and Silver . gold Alloys at different % of gold Fae`q A. A. Radwan*. Abstract: The norm of elastic constant tensor and the norms Where ij is the Kronecker delta. The Einstein of the irreducible parts of the elastic constants of Copper , Silver and gold metals and Copper - gold and Silver - gold summation convention over repeated indices is used Alloys at different percentages of gold are calculated. The and indices run from 1 to 3 unless otherwise stated. relation of the scalar parts norm and the other parts norms and the anisotropy of these metals and their Alloys are By applying the symmetry conditions (2) to the presented.

2 The norm ratios are used to study anisotropy of decomposition results obtained for a general fourth- these metals and their Alloys . rank tensor, the following reduction spectrum for the elastic constant tensor is obtained. It contains two Index Terms - Copper , Silver , gold , Alloys , Anisotropy, scalars, two deviators, and one-nonor parts: Elastic Constants. 0;1 0; 2 2;1 . C ijkl C ijkl C ijkl C ijkl 2; 2 4;1 . I. ELASTIC CONSTANT TENSOR DECOMPOSITION C ijkl C ijkl (4). The constitutive relation characterizing linear Where: anisotropic solids is the generalized Hook's law [1]: 0;1 1. ij Cijkl kl , ij S ijkl kl (1) Cijkl ij kl C ppqq , (5). 9. Where ij and kl are the symmetric second rank 0; 2.

3 C ijkl . 1. 3 ik jl 3 il jk 2 ij kl . stress and strain tensors, respectively Cijkl is the 90. fourth-rank elastic stiffness tensor (here after we call 3C pqpq C ppqq (6). it elastic constant tensor) and S ijkl is the elastic compliance tensor. 2;1 . Cijkl . 1. ik C jplp jk Ciplp il C jpkp jl Cipkp . There are three index symmetry restrictions on these 5. tensors. These conditions are: . 2. ik jl il jk C pqpq (7). C ijkl C jikl , C ijkl C ijlk , C ijkl C klij (2) 15. Which the first equality comes from the symmetry 2; 2 . Cijkl 7 7.. ij 5 Cklpp 4 Ckplp kl 5 Cijpp 4 Cipjp 1 1.. of stress tensor, the second one from the symmetry of strain tensor, and the third one is due to the presence of a deformation potential.

4 In general, a ik 5C jlpp 4C jplp jl 5 Cikpp 4 Cipkp . 2 2. fourth-rank tensor has 81 elements. The index . symmetry conditions (2) reduce this number to 81. 35 35. Consequently, for most asymmetric materials il 5C jkpp 4 Ciplp jk 5 Cilpp 4 Ciplp . 2 2. (triclinic symmetry) the elastic constant tensor has . 21 independent components. 35 35. Elastic compliance tensor S ijkl possesses the same . 2. 2 jk il 2 ik jl 5 ij kl . symmetry Properties as the elastic constant tensor 105. C ijkl and their connection is given by [2,3,4,5]: 5C ppqq 4C pqpq (8). Cijkl S klmn =. 1. 2. im jn in jm (3) 4;1 . C ijkl 1. (C ijkl C ikjl C iljk ). 3.. ij C klpp 2C kplp ik C jlpp 2C jplp.

5 * Faculty of Engineering - Near East University 1. KKTC Lefkosa: Box: 670, Mersin 10 - TURKEY. (email: . 21. ISBN: 978-988-19251-2-1 IMECS 2011. ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online). Proceedings of the International MultiConference of Engineers and Computer Scientists 2011 Vol II, IMECS 2011, March 16 - 18, 2011, Hong Kong il C jkpp 2C jpkp jk C ilpp 2C iplp isotropic but the material symmetry group itself may have very few symmetry elements. We know that, jl C ikpp 2C ipkp kl C ijpp 2C ipjp ]. for isotropic materials, the elastic compliance tensor has two irreducible parts, , two scalar parts, so the norm of the elastic compliance tensor for.)

6 1. 105.. ij kl ik jl il jk C ppqq 2C pqpq (9) isotropic materials depends only on the norm of the Ns scalar parts, N N s , Hence, the ratio 1. These parts are orthonormal to each other. Using N. for isotropic materials. For anisotropic materials, the Voigt's notation [1] for C ijkl , can be expressed in 6. elastic constant tensor additionally contains two by 6 reduced matrix notation, where the matrix deviator parts and one nonor part, so we can define coefficients c are connected with the tensor Nd Nn for the deviator irreducible parts and for components C ijkl by the recalculation rules: N N. nonor parts. Generalizing this to irreducible tensors c C ijkl ; (ij 1.)

7 ,6, kl 1,..,6) up to rank four, we can define the following norm That is: Ns Nv ratios: for scalar parts, for vector parts, 11 1 , 22 2 , 33 3 , 23 32 4 N N. 31 13 5 , 12 21 6 . Nd N sc for deviator parts, for septor parts, and II. THE NORM CONCEPT. N N. Generalizing the concept of the modulus of a vector, Nn for nonor parts. Norm ratios of different norm of a Cartesian tensor (or the modulus of a N. tensor) is defined as the square root of the contracted irreducible parts represent the anisotropy of that product over all indices with itself: particular irreducible part they can also be used to asses the anisotropy degree of a material property as N T Tijkl.

8 Tijkl .. 1/ 2 a whole, we suggest the following two more rules: Rule 2. When Ns is dominating among norms of Denoting rank-n Cartesian Tijkl .. , by Tn , the Ns square of the norm is expressed as [5]: irreducible parts: the closer the norm ratio is N. T j ;q T n T n T j ;q . T n j , q . 2. N2 T. 2. n to one, the closer the material property is isotropic. j ,q n . n , j ,q Rule3. When N s is not dominating or not present, This definition is consistent with the reduction of the tensor in tensor in Cartesian formulation when all norms of the other irreducible parts can be used as a the irreducible parts are embedded in the original criterion. But in this case the situation is reverse; the rank-n tensor space.

9 Larger the norm ratio value we have, the more Since the norm of a Cartesian tensor is an invariant anisotropic the material property is. quantity, we suggest the following: The square of the norm of the elastic stiffness tensor Rule1. The norm of a Cartesian tensor may be used (elastic constant tensor) C mn is: as a criterion for representing and comparing the overall effect of a certain property of the same or N. 2. C mn . 0;1 . C . 2 0; 2. mn 2. different symmetry. The larger the norm value, the mn mn more effective the property is. It is known that the anisotropy of the materials, , 2 C mn ..C mn C mn 0;1 0; 2 2;1 . C . 2 2; 2. mn 2. mn mn the symmetry group of the material and the anisotropy of the measured property depicted in the 2 C mn.

10 C mn C mn 2;1 2; 2 4;1 .. 2. (10). same materials may be quite different . Obviously, mn mn the property, tensor must show, at least, the Let us consider the irreducible decompositions of symmetry of the material. For example, a property, the elastic stiffness tensor (elastic constant tensor) in which is measured in a material, can almost be the following elements and Alloys : ISBN: 978-988-19251-2-1 IMECS 2011. ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online). Proceedings of the International MultiConference of Engineers and Computer Scientists 2011 Vol II, IMECS 2011, March 16 - 18, 2011, Hong Kong By using table1, table 2, and table 3 and the Table 2, Elastic Constants (GPa) [2].


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