Negated Electric Field Strength

In the derived expression for the virtual work of internal contributions, Eq. (3.63), as well as in the definition of the electroelastic energy density, Eq. (3.64), the electric fields strength appears in a negated form represented by the vector E. With the adoption of such an arrangement in the associate constitutive relation, to be found on the left-hand side of Eq. (4.10a), it is possible to compensate for the uns3Tnmetry in the matrix of constitutive 

Correspondingly, the consequent modifications of the constitutive relation may be treated. For the case of planar stress and unidirectional electric in transformed coordinates, this yields  

In Section 3.5.1, Dirichlet s principle of minimum potential energy has been extended to electromechanically coupled problems. With the exception of referring to the required potential property, the electroelastic energy density Uq has not yet been further specified. With Eq. (3.64) a second relation linking the fields of the constitutive relation is available, such that it may be used 

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Typically, the fiber material is at most transversely isotropic, Eq. (4.17) with Eqs. (4.3), (4.8), and (4.18), while the matrix material is isotropic, Eq. (4.17) with Eqs. (4.4), (4.9), and vanishing piezoelectric moduli. The subsequent theories are not confined to such a behavior. However, it will be presumed that the distinguished axes are aligned. Deviating cases may be considered in conjunction with an appropriate transformation, see Section 3.2.5. In accordance with the considerations of Section 4.4.4, the notation with a negated electric field strength will be utilized throughout the entire chapter. 

These assumptions can be applied immediately in the directions ei and 2 of the cross-sectional plane of the fibers. As the constitutive relation of the considered piezoelectric materials exhibits only a partial electromechanical coupling, normal modes and each of the shear modes may be treated independently, as illustrated by Eqs. (4.19) to (4.21). In accordance with the considerations of Section 4.4.4, the notation with negated electric field strength and the associated constitutive submatrices E, Gi, and G2 will be utilized. 

The mechanical stiffness coefficients, which are unchanged in comparison to the classical lamination theory, have been arranged just as they appear in the latter. They may be determined in accordance with Eqs. (6.4b). As the remaining coefficients are involved with the negated electric field strengths E and appertaining resultants G3 of the electric flux density, they have to be established in consideration of the group association represented by the... 

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