By Larry A Taber

Smooth organic tissues frequently endure huge elastic (or approximately elastic) deformations that may be analyzed utilizing the nonlinear conception of elasticity. a result of different techniques to nonlinear elasticity within the literature, a few elements of the topic should 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 number of examples within the mechanics of sentimental tissues. functions contain muscle, arteries, the guts, embryonic tissues, and organic membranes.

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For instance, using Eq. 44)i, we can transform g2 = gy to g2 = Se by gs = A'gy — A*e y ~ —rsin# r cos# 0 cos( smt 0 —r sin 9 0 r cos 9 0 0 1 = - r sin 6 ex + r cos 9 ey which agrees with gg in Eq. 10). Components of the Coordinate Transformation Tensors In practice, the components of the transformation tensors usually are computed from the equations that relate the unbarred and barred curvilinear coordinates. For a unique mapping of one system to the other, we can write xl = xl(xj), xl = xl(xj).

In fact, comparing Eqs. 115) shows that this rotation can be considered a coordinate transformation from the unbarred to the barred base vectors. In matrix form, inserting Eqs. 117) = e^e^ + eyey + ezez = I can be verified easily. Physical Components In general curvilinear coordinates, the components of vectors and tensors do not always have a direct physical meaning. If the base vectors have different dimensions, for example, then the components have different dimensions, since their combination must be consistent with the dimensions of the vector or tensor.

132) =^ij9ki- Similarly, dotting both sides of Eq. 133) Hence, although Tijk and Tk, are not tensor components, the components of the metric tensor can be used to raise or lower the third index. A useful formula for computing the Christoffel symbols in a given coordinate system can be found by differentiating the expression 9ij — Si'gj with respect to xk to obtain 9ijik giik ' g j ~r -L ikj i J- jki gi'gjik 46 Vectors, Dyadics, and Tensors where Eq. 128)i has been used. Permuting the subscripts and using Eqs.