By Larry A Taber
Delicate organic tissues usually endure huge elastic (or approximately elastic) deformations that may be analyzed utilizing the nonlinear thought of elasticity. as a result of assorted techniques to nonlinear elasticity within the literature, a few facets of the topic will be tricky to understand. This e-book makes an attempt to explain and unify these remedies, illustrating the benefits and downsides of every via a variety of examples within the mechanics of soppy tissues. purposes comprise muscle, arteries, the guts, embryonic tissues, and organic membranes.
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Additional info for Nonlinear Theory of Elasticity: Applications in Biomechanics
6 illustrates a set of covariant and contravariant base vectors at a point in two-dimensional space. The vectors g 1 and g 2 are orthogonal to g2 and gi, respectively. Moreover, the magnitude and directions of g 1 and g 2 are determined by the requirement g*gi = g2-g2 = +1The g l can be computed in terms of the g; as follows. First, consider g 1 . Since it is orthogonal to both g2 and g3, we can write y/gg1 = g2Xg3 according to the properties of the cross product. 16) Vectors 15 --£. a = a'g; = ajg' aig 1 V Fig.
In general, the only components of A and B we need are those indicated in Eqs. 46). 48) with the understanding that dyadic operations are to be carried out with A and B in the forms of Eqs. 47). Then, dotting Eqs. 49) which can be considered defining relations for the mixed components of the transformation tensors. If the barred and unbarred systems coincide, Eqs. 49) show that A) = B) = 5), and Eqs. 47) show that A = B = I, as expected. The transformation tensors A and B are related. Since the base vectors in any coordinate system must satisfy Eq.
In addition, we use the following conventions: (1) For a second-order tensor, the first index (or base vector) corresponds to the row and the second to the column of the matrix. (2) If we leave the place-holding dot out of the mixed components of a second-order tensor and write Tlj = Tj or Tf = Tj, then the contravariant index gives the row and the covariant index the column. ] (3) If the basis is not specified, then components are associated with the natural base vectors gi and g l . 3 give some useful relations involving tensors.
Nonlinear Theory of Elasticity: Applications in Biomechanics by Larry A Taber