MATERIAL ELASTIC ANISOTROPIC command

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1 MATERIAL ELASTIC ANISOTROPIC command.. Synopsis The MATERIAL ELASTIC ANISOTROPIC command is used to specify the parameters associated with an anisotropic linear elastic material idealization. Syntax The following syntax is used to describe an anisotropic elastic material idealization: MATerial ELAstic ANIsotropic NUMber # (EScription string ) ( C11 #.# ) ( C12 #.# ) ( C13 #.#) ( C14 #.# ) ( C15 #.#) ( C16 #.#) ( C22 #.#) ( C23 #.#) ( C24 #.#) ( C25 #.#) ( C26 #.#) ( C33 #.#) ( C34 #.#) ( C35 #.#) ( C36 #.#) ( C44 #.#) ( C45 #.#) ( C46 #.#) ( C55 #.#) ( C56 #.#) ( C66 #.#).. 1 V. N. Kaliakin

2 Explanatory Notes The NUMBER keyword is used to specify the (global) number of the material associated with the incompressible isotropic elastic idealization. The default material number is one (1). The optional alphanumeric string associated with the ESCRIPTION keyword must be enclosed in double quotes ( ). It is used solely to describe the material being idealized to the analyst. The ESCRIPTION string is printed as part of the echo of the material. For a three-dimensional state of stress, this constitutive relation is written in the following form: σ 11 C 11 C 12 C 13 C 14 C 15 C 16 ε 11 σ 22 C 12 C 22 C 23 C 24 C 25 C 26 ε 22 σ 33 = C 13 C 23 C 33 C 34 C 35 C 36 ε 33 σ 12 C 14 C 24 C 34 C 44 C 45 C 46 γ 12 σ 13 C 15 C 25 C 35 C 45 C 55 C 56 γ 13 C 16 C 26 C 36 C 46 C 56 C 66 σ 23 For conditions of plane stress, plane strain and torsionless axisymmetry, the constitutive relation reduces to σ 11 σ 22 σ 33 σ 12 C 11 C 12 C 13 C 14 = C 12 C 22 C 23 C 24 C 13 C 23 C 33 C 34 C 14 C 24 C 34 C 44 where loading has been assumed to be applied only in the 1-2 plane. Note that in the above expressions, the engineering measure of shearing strains, which is twice the tensorial values, has been employed; i.e., γ 12 = 2ε 12, γ 13 = 2ε 13, and γ 23 = 2ε 23. Orthotropic Elastic Material Idealization: For a material through each point of which pass three mutually perpendicular planes of elastic symmetry. If similar planes are parallel at all points in the material, then taking (x 1, x 2, x 3 ) coordinate axes normal to these planes (i.e., along the principal directions) it follows that there should be no interaction between the various shear components or between the shear and normal components. The material is described by twelve elastic constants, only nine of which are independent. The entries in the three-dimensional constitutive matrix now become ε 11 ε 22 ε 33 γ 12 γ 23 (1) (2) C 11 = E 1(1 ν 23 ν 32 ) ; C 12 = C 21 = E 1(ν 21 + ν 31 ν 23 ) (3) C 13 = C 31 = E 1(ν 31 + ν 21 ν 32 ) ; C 22 = E 2(1 ν 13 ν 31 ) (4) 2 V. N. Kaliakin

3 C 23 = C 32 = E 2(ν 32 + ν 12 ν 31 ) ; C 33 = E 3(1 ν 12 ν 21 ) (5) C 44 = G 12 ; C 55 = G 13 ; C 66 = G 23 (6) where = 1 ν 12 ν 21 ν 13 ν 31 ν 23 ν 32 2ν 12 ν 31 ν 23. Examples of orthotropic materials include fiber reinforced composite beams, wood (as a first approximation), sheet metal and honeycomb. Transversely Isotropic Elastic Material Idealization: Such materials possess the following properties: through all points there pass parallel planes of elastic symmetry in which all directions are elastically equivalent (i.e., planes of isotropy). Thus at each point there exists one principal direction and an infinite number of principal directions in a plane normal to the first direction. Five parameter values characterize a transversely isotropic elastic material. Taking the x 3 -axis normal to the plane of isotropy, with the x 1 and x 2 axes directed arbitrarily in this plane, these are E 1 is the elastic modulus for compression or tension in the plane of isotropy. E 2 is the elastic modulus for compression or tension in a direction normal to the plane of isotropy. ν 11 is Poisson s ratio 1 characterizing transverse contraction in the plane of isotropy when tension is applied in the plane of isotropy. ν 21 is Poisson s ratio characterizing transverse contraction when tension is applied in the plane of isotropy. Similarly, ν 12 is Poisson s ratio characterizing contraction normal to the plane of isotropy when tension is applied in this plane. G 2 is the shear modulus associated with the plane of isotropy and any plane perpendicular to it. The entries in the three-dimensional constitutive matrix are thus C 11 = C 22 = C 12 = E 1 [E 2 E 1 (ν 21 ) 2 ] (1 + ν 11 ) [E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 ] E 1 [E 1 (ν 21 ) 2 + E 2 ν 11 ] (1 + ν 11 ) [E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 ] (7) (8) C 13 = C 23 = C 33 = 1 Named in honor of S.. Poisson ( ). E 1 E 2 ν 21 E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 (9) (E 2 ) 2 (1 ν 11 ) E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 (10) 3 V. N. Kaliakin

4 C 44 = E 1 2(1 + ν 11 ) ; C 55 = C 66 = G 2 (11) Isotropic Elastic Material Idealization: The simplest linearly elastic material, for which the elastic behavior is independent of the orientation of the coordinate axes, is called isotropic. For such a material the constitutive relations simplify in that only two material constants are required to completely describe the material behavior; e.g., the elastic (Young s) modulus E and Poisson s ratio ν, the Lamé constants λ and µ, or the bulk modulus K and the shear modulus G. The entries in the three-dimensional constitutive matrix now become C 11 = C 22 = C 33 = C 12 = C 13 = C 23 = C 44 = C 55 = C 66 = E(1 ν) (1 + ν)(1 2ν) = λ + 2µ = K G (12) Eν (1 + ν)(1 2ν) = λ = K 2 3 G (13) E 2(1 + ν) = µ = G (14) Further details pertaining to the finite element analysis of elastic bodies are given in Chapter 12 of [1]. 4 V. N. Kaliakin

5 Example of Command Usage Idealization of an Anisotropic Elastic Material The following commands are sufficient to describe the anisotropic elastic material idealization material elastic aniso num 2 desc hypothetical hybrid material c e+07 c e+07 c e+07 c & c e+07 c e+07 & c c e+07 c c e+07 & Idealization of Wood (an Orthotropic Elastic Material) In this example the linear anisotropic material idealization is used to describe an orthotropic material (ouglas Fir). Twelve parameters, nine of which are independent, characterize an orthotropic material. The three principal axes of material symmetry are oriented with respect to the grain directions and growth rings [2]. enoting the longitudinal, tangential and radial directions by the letters L, T, and R, respectively, the following values are used to characterize ouglas Fir at approximately 12% moisture content: [2] E T /E L = E R /E L = ν LR = ν RL = ν LT = ν T L = ν RT = ν T R = G LR /E L = G LT /E L = G RT /E L = where E L = 12, 600 MP a (an average value, obtained from bending tests). enoting the longitudinal, transverse and radial directions by 1, 2, and 3, respectively, it follows that and E 1 = 12, 600 MP a E 2 = 0.050(12, 600) = MP a E 3 = 0.068(12, 600) = MP a 5 V. N. Kaliakin

6 and G 12 = 0.078(12, 600) = MP a G 13 = 0.064(12, 600) = MP a G 23 = 0.007(12, 600) = 88.2 MP a Finally, since ν 12 = 0.449, ν 13 = 0.292, and ν 23 = 0.390, it follows that ν 21 = E 2ν 12 E 1 = (0.050)(0.449) = ν 31 = E 3ν 13 = 0.068(0.292) = E 1 ν 32 = E ( ) 3ν = (0.390) = E For an orthotropic material the entries in C are computed as follows: = 1 ν 12 ν 21 ν 13 ν 31 ν 23 ν 32 ν 12 ν 31 ν 23 ν 21 ν 13 ν 32 C 11 = E 1(1 ν 23 ν 32 ) C 21 = E 2(ν 12 + ν 13 ν 32 ) C 31 = E 3(ν 13 + ν 12 ν 23 ) Thus, for the present material idealization, C 12 = E 1(ν 21 + ν 31 ν 23 ), C 13 = E 1(ν 31 + ν 21 ν 32 ), C 22 = E 2(1 ν 13 ν 31 ), C 23 = E 2(ν 32 + ν 12 ν 31 ), C 32 = E 3(ν 23 + ν 21 ν 13 ), C 33 = E 3(1 ν 12 ν 21 ) C 44 = G 12 ; C 55 = G 13 ; C 66 = G 23 = 1 (0.449)( ) (0.292)( ) (0.390)(0.5304) 2(0.449)( )(0.390) = C 11 = MP a C 12 = MP a C 13 = MP a C 22 = MP a C 23 = MP a C 33 = MP a C 44 = G 12 = 0.078(12, 600) = MP a C 55 = G 13 = 0.064(12, 600) = MP a C 66 = G 23 = 0.007(12, 600) = 88.2 MP a The following commands are sufficient to describe the orthotropic elastic material idealization: 6 V. N. Kaliakin

7 material elastic aniso num 1 & desc orthotropic material entered using anisotropic command c c c & c c c & c c c & Idealization of an Isotropic Elastic Material In this example the linear anisotropic material idealization is used to describe an isotropic material. Two parameters characterize such a material; in this case an elastic modulus E = 20, 000 and a Poisson s ratio of ν = The entries in C for an isotropic material are computed as follows: E(1 ν) C 11 = C 22 = C 33 = = 24, 000 (1 + ν)(1 2ν) Eν C 12 = C 13 = C 23 = = 8, 000 (1 + ν)(1 2ν) E C 44 = C 55 = C 66 = = 8, 000 2(1 + ν) The following commands are sufficient to describe the isotropic elastic material idealization: material elastic aniso num 1 & desc isotropic material entered using anisotropic command c c c & c c c & c c c & 7 V. N. Kaliakin

8 Bibliography [1] Kaliakin, V. N., Approximate Solution Techniques, Numerical Modeling and Finite Element Methods. New York: Marcel ekker, Inc. (2001). [2] U. S. epartment of Agriculture, The Encyclopedia of Wood. New York, NY: Skyhorse Publishing (2007). 8

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