Compliance matrix symmetric

As [fi] = 0 in this symmetric laminate, the compliance matrix is obtained by inverting [A]... 

For symmetric laminates it is possible to define effective in-plane moduli in terms of the in-plane stiffness or extensional compliance matrix, since there is no coupling between in-plane and bending response. The effective... 

Similarly, the flexural elastic moduli of symmetric laminates are readily obtained from the bending compliance matrix. 

The unit cell of the orthorhombic crystal is brick-shaped. The elastic properties are therefore symmetric with respect to three perpendicular planes. In a coordinate system that is parallel to the edges of the unit cell, the compliance matrix (equation (2.31)) takes the form... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

In a more complete, tensor form, one would write compliance matrix) to generalize Hooke s law (Equation (1.9b)). Indeed, reducing y and Voigt notation in mathematics, one... 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

Since the elastic strain energy is a unique function of state, which is independent of how that state was reached, it is possible to demonstrate that the elastic-compliance and elastic-constant matrixes, as defined above, must be symmetrical. This follows directly from the observation, for example, that... 

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