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Elastic Properties and Anisotropy of Elastic Behavior Figure 7-1 Anisotropic materials: (a) rolled material, (b) wood, (c) glass-fiber cloth in an epoxy matrix, and (d) a crystal with cubic unit cell. • Real materials are never perfectly isotropic. In some cases (e.g. composite materials) the differences in properties for different directions are so large that one can not assume isotropic behavior - Anisotropic. • There is need to discuss Hooke’s Law for anisotropic cases in general. This can then be reduced to isotropic cases - material property (e.g., elastic constant) is the same in all directions. • In the general 3-D case, there are six components of stress and a corresponding six components of strain. • In highly anisotropic materials, any one component of stress can cause strain in all six components. • For the generalized case, Hooke’s law may be expressed as: i Cij j (7-1) i Sij j (7-2) where, C Stiffness (or Elastic cons tan t ) S Compliance • Both Sijkl and Cijkl are fourth-rank tensor quantities. • Expansion of either Eqs. 7-1 or 7-2 will produce nine (9) equations, each with nine (9) terms, leading to 81 constants in all. • It is important to note that both ij and ij are symmetric tensors. • Symmetric tensor Means that the off-diagonal components are equal. For example, in case of stress: 13 31 , 12 21 23 32 We can therefore write: 11 12 13 11 12 13 21 22 23 12 22 23 31 32 33 13 23 33 (7-3) Similarly, the strain tensor can be written as: 11 12 13 11 12 13 21 22 23 12 22 23 31 32 33 13 23 33 (7-4) • Symmetry effect leads to a significant simplification of the stress-strain relationship of Eqs. 7-1 and 7-2. We can write: ij Sijkl kl or ij Sijlk lk and since Sijkl kl Sijlk lk ; kl lk ; and Sijkl Sijlk We can also write: ij Sijkl kl ji S jikl kl Sijkl S jikl • The direct consequence of the symmetry in the stress and strain tensors is that only 36 components of the compliance tensor are independent and distinct terms. • Similarly, only 36 components of the stiffness tensor are independent and distinct terms. Additional simplification of the stress-strain relationship can be realized through simplifying the matrix notation for stresses and strains. We can replace the indices as follows: 11 1 23 4 22 2 13 5 33 3 12 6 11 12 13 22 23 33 Notation I 1 = 5 2 4 3 6 Notation II • The foregoing transformation is easy to remember: In other to obtain notation II, one must proceed first along the diagonal (1 2 3 ) and then back ( 4 5 6 ). • Notation II method makes life very easy when correlating the stresses and strains for general case, in which the elastic properties of a material are dependent on its orientations. We now have the stress and strain, in general form, as 1 6 5 6 2 4 1 5 4 and 6 2 3 5 2 6 2 2 4 2 5 2 4 2 3 It should be noted that and 3 33, but 22 2 ,11 1 , 4 2 23 23 5 213 13 6 212 12 (7-5) In matrix format, the stress-strain relation showing the 36 (6 x 6) independent components of stiffness can be represented as: 1 C11 2 C21 C 3 31 4 C41 C 5 51 6 C61 C12 C13 C14 C15 C22 C23 C24 C25 C32 C33 C34 C35 C42 C43 C44 C45 C52 C53 C54 C55 C62 C63 C64 C65 C16 1 C26 2 C36 3 C46 4 C56 5 C66 6 Or in short notation, we can write: i Cij j and i Sij j (7-6) Further reductions in the number of independent constants are possible by employing other symmetry considerations to Eq. 7-6. • Symmetry in Stiffness and Compliance matrices requires that: Cij C ji and Sij S ji Of the 36 constants, there are six constants where i = j, leaving 30 constants where i j. But only one-half of these are independent constants since Cij = Cji Therefore, for the general anisotropic linear elastic solid there are: 30 2 6 21 independent elastic constant. • The 21 independent elastic constants can be reduced still further by considering the symmetry conditions found in different crystal structures. • In Isotropic case, the elastic constants are reduced from 21 to 2. • Different crystal systems can be characterized exclusively by their symmetries. Table 7-1 presents the different symmetry operations defining the seven crystal systems. • The seven crystalline systems can be perfectly described by their axes of rotation. For example, a threefold rotation is a rotation of 120o (3 x 120o = 360o); after 120o the crystal system comes to a position identical to the initial one. Table 7.1 Minimum Number of Symmetry Operations in Various Systems ______________________________________________ System Rotation ______________________________________________ Triclinic None (or center of symmetry) Monoclinic 1 twofold rotation Orthorhombic 2 perpendicular twofold rotation Tetragonal 1 fourfold rotation around [001] Rhombohedral 1 threefold rotation around [111] Hexagonal 1sixfold rotation around [0001] Cubic 4 threefold rotations around <111> abc 90 o abc 90 o abc 90 o The hexagonal system exhibits a sixfold rotation around the [0001] - c axis; after 60 degrees, the structure superimposes upon itself. In terms of a matrix, we have the following: Orthorhombic Tetragonal 11 12 13 0 0 0 11 12 13 0 0 16 . 22 23 0 0 0 . 11 13 0 0 16 . 33 0 0 0 . . 33 0 0 0 . , . . . 44 0 0 . . . 44 0 0 . . . 55 0 . . . . 44 0 . . . . . 66 . . . . . 66 . (7.7a) Hexagonal 11 12 13 0 0 . 11 13 0 0 . 33 0 0 . . . . 44 0 . . . 44 . 0 0 0 0 x (7.7b) where x 2( s11 s12 ), or 1 x (C11 C12 ) 2 • Laminated composites made by the consolidation of prepregged sheets, with individual piles having different fiber orientations, have orthotropic symmetry with nine independent elastic constant. • This is analogous to orthorhombic symmetry, and possess symmetry about three orthogonal (oriented 90o to each other) planes. The elastic constants along the axes of these three planes are different. Cubic 11 12 12 0 0 0 . 11 12 0 0 0 . 11 0 0 0 . . 44 0 0 . . . 44 0 . . . . 44 (7.7c) The number of independent elastic constants in a cubic system is three (3). For isotropic materials ( most polycrystalline aggregates can be treated as such) there are two (2) independent constants, b/c : C44 C11 C12 2 (7.8) The stiffness matrix of an isotropic system is: C11 C12 C12 . C C12 11 . C11 . . . . . . . . . . 0 0 0 0 0 C11 C12 2 0 0 . C11 C12 2 . . 0 0 0 0 C11 C12 2 0 (7.9) For cubic systems, Equation (7-8) does not apply, and we define an anisotropy ratio (also called the Zener anisotropy ratio, in honor of the scientist who introduced it): 2C44 A C11 C12 (7.10) • Several metals have high “A” anisotropy ratio. • Aluminum and tungsten, have values of A very close to 1. Single crystals of tungsten are almost isotropic. Elastic compliances - for the isotropic case: 0 0 0 S11 S12 S12 . S S 0 0 0 11 12 . S11 0 0 0 . . . . 2( S11 S12 ) 0 0 . . . 2( S11 S12 ) 0 . . . . . . 2 ( S S ) 11 12 (7.11) Similarly, the 81 components of elastic compliance for the cubic system have been reduced to three (3) independent ones while for the isotropic case, only two (2) independent elastic constants are needed. The elastic constants for an isotropic material are given by: Young’s modulus 1 E S11 (7.12) Rigidity or Shear modulus 1 1 G 2( S11 S12 ) S4 (7.13) Compressibility (B) and bulk modulus (K): 1 11 22 33 B K 1 ( ) 11 12 13 3 (7.14) Poisson’s ratio S12 v S11 (7.15) 1 1 u C44 (C11 C12 ) G 2 S44 (7.16) C12 (7.17) Lame’s constants: • The equation to determine the compliance of isotropic materials can be written as (by using Eqs. 7-2 and 7-11): 1 S11 2 S12 S 3 12 4 0 0 5 6 0 S12 S12 0 0 0 S11 S12 0 0 0 S12 S11 0 0 0 0 0 2S11 S12 0 0 0 0 0 2S11 S12 0 0 0 0 0 2S11 1 2 3 4 5 S12 6 (7.18) The relationship of Eq. 7-18 can be expanded and equated to Eq. 6-9 to give: 1 1 S11 1 S12 2 S12 3 1 2 3 E 1 2 S12 1 S11 2 S12 3 2 1 3 E 1 3 S12 1 S12 2 S11 3 3 1 2 E (7.19) Also, 4 2S11 5 2S11 6 2S11 1 S12 4 4 , G 1 S12 5 5 , G 1 S12 6 6 , G (7.19) • Expressing the strains as function of stresses, we have 1 C111 C12 2 C12 3 2 1 2 3 , 2 C121 C11 2 C12 3 1 2 2 3 , 3 C121 C12 2 C11 3 1 2 2 3 , 1 4 C11 C12 4 4 2 1 5 C11 C12 5 5 2 1 6 C11 C12 6 6 2 (7.20) Note : G • A great number of materials can be treated as isotropic, although they are not microscopically so. • Individual grains exhibit the crystalline anisotropy and symmetry, but when they form a poly-crystalline aggregate and are randomly oriented, the material is microscopically isotropic. • If the grains forming the poly-crystalline aggregate have preferred orientation, the material is microscopically anisotropic. • Often, material is not completely isotropic; if the elastic modulus E is different along three perpendicular directions, the material is Orthotropic; composites are a typical case. In a cubic material, the elastic moduli can be determined along any orientation, from the elastic constants, by the application of the following equations: 1 1 S11 2( S11 S12 S44 )(li21l 2j 2 l 2j 2lk23 li21lk23 Eijk 2 (7.21a) 1 1 S44 4( S11 S12 S44 )(li21l 2j 2 l 2j 2lk23 li21lk23 Gijk 2 (7.21b) Eijk and Gijk are the Young’s and shear modulus, respectively, in the [ijk] direction; li1 , l j 2 , lk 3 are the direction cosines of the direction [ijk] Table 7-2 Stiffness and compliance constants for cubic crystals ___________________________________________________ Metal C11 C12 S11 S44 C44 S12 ___________________________________________________ Aluminum 10.82 6.13 2.85 1.57 -0.57 3.15 Copper 16.84 12.14 7.54 1.49 -0.62 1.33 Iron 23.70 14.10 11.60 0.80 -0.28 0.86 Tungsten 50.10 19.80 15.14 0.26 -0.07 0.66 ___________________________________________________ Stiffness constants in units of 10-10 Pa. Compliance in units of 10-11 Pa Using the direction cosines l, m, n (as described in the text book) the equation for determining the Elastic Moduli along any direction is given by: 1 1 S11 2[(S11 S12 ) S44 ](l 2 m 2 m 2 n 2 l 2 n 2 ) E 2 (7.22) Typical values of elastic constants for cubic metals are given in Table 7.2. All the relations described in Eqs. 7-12 to 7-20 for obtaining Elastic constants are applicable. This include: 1 S11 E v S12 E 1 S44 G Example A hydrostatic compressive stress applied to a material with cubic symmetry results in a dilation of -10-5. The three independent elastic constants of the material are C11 = 50 GPa, C12 = 40 GPa and C44 = 32 GPa. Write an expression for the generalized Hooke’s law for the material, and compute the applied hydrostatic stress. SOLUTION Dilation is the sum of the principal strain components: = 1 + 2 + 3 = -10-5 Cubic symmetry implies that 1 = 2 = 3 = -3.33 x10-5 and 4 = 5 = 6 = 0 From Hooke’s law, i = Cijj 1 C111 C12 2 C12 3 and the applied hydrostatic stress is: p = 1 = (50 + 40 + 40)(-3.33) 103 Pa = -130 x 3.33 x 103 = -433 kPa Example: Determine the modulus of elasticity for tungsten and iron in the <111> and <100> directions. What conclusions can be drawn about their elastic anisotropy? From Table 7.1 ____________________________ Fe: W: S12 S44 S11 ________________ 0.80 -0.28 0.86 0.26 -0.07 0.66 SOLUTION The direction cosines for the chief directions in a cubic lattice are: _______________________________________ Directions li1 l j2 lk 3 _______________________________________ <100> 1 0 0 <110> 0 1/ 2 1/ 2 1/ 3 <111> 1/ 3 1/ 3 For iron: 1 1 1 1 0.80 2{(0.80 0.28) 0.86 / 2} E111 9 9 9 1 1 1 0.80 2(1.08 0.43) 0.80 1.30 E111 3 3 0.80 0.43 1 E111 2.70 x1011 Pa 0.37 1 0.80 1.30(0) 0.80 E100 E100 1.25x1011 Pa For tungsten: 1 0.66 1 0.26 2(0.26 0.07) E111 2 3 1 1 0.26 20.33 0.33 0.26 E111 3 1 E111 3.85x1011 Pa 385GPa 0.26 1 0.66 0.26 2(0.26 0.07) 0 0.26 E100 2 E100 1 3.85x1011 Pa 0.26 Therefore, we see that tungsten is elastically isotropic while iron is elastically anisotropic.