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F. Write down all equations (relating the material constants) that must be satisfied. g. If the above relations are proposed for an isotropic material, then how many independent relations among the above material constants must exist? Do not write down those equations. book Page 34 Monday, December 1, 2003 3:06 PM 34 Ultrasonic Nondestructive Evaluation h. If the material is transversely isotropic, then how many independent relations among the above material constants must exist? Do not write down those equations.
In other words, if the arguments of F are negative and G is positive, then the function values are A and -A, respectively, where A can be any positive or negative constant. 81 one can see that for t Ê x ˆ x1 , the argument Á t - 1 ˜ of function F is negative, and the argucP ¯ cP Ë Ê Ê x ˆ Ê x ˆ x ˆ ment Á t + 1 ˜ of function G is positive. Thus, u1 = FÁ t - 1 ˜ + GÁ t + 1 ˜ = 0 , cP ¯ Ë Ë cP ¯ Ë cP ¯ Ê Ê x x x ˆ x ˆ for t < 1 . For t > 1 , both arguments Á t - 1 ˜ and Á t + 1 ˜ are greater cP cP cP ¯ cP ¯ Ë Ë < than zero.
Book Page 30 Monday, December 1, 2003 3:06 PM 30 Ultrasonic Nondestructive Evaluation instead of traditional notation of two subscripts. The material constants have been written with two subscripts instead of four. 59 has 21 independent elastic constants in absence of any plane of symmetry. Such material is called general anisotropic material or triclinic material. However, if the material response is symmetric about a plane or an axis, then the number of independent material constants is reduced.