In this paper, following the Backus (1962) approach, we examine expressions for elasticity parameters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic layers. These expressions reduce to the results of Backus (1962) for the case of isotropic and transversely isotropic layers.In the over half-a-century since the publications of Backus (1962) there have been numerous publications applying and extending that formulation. However, neither George Backus nor the authors of the present paper are aware of further examinations of the mathematical underpinnings of the original formulation; hence this paper.We prove that-within the long-wave approximation-if the thin layers obey stability conditions then so does the equivalent medium. We examine-within the Backus-average context-the approximation of the average of a product as the product of averages, which underlies the averaging process.In the presented examination we use the expression of Hooke's law as a tensor equation; in other words, we use Kelvin's-as opposed to Voigt's-notation. In general, the tensorial notation allows us to conveniently examine effects due to rotations of coordinate systems.
This version contains corrections of typographical errors in Bos, L., Danek, T., Slawinski, M.A., Stanoev, T. (2018) Statistical and numerical considerations of Backus-average product approximation. Journal of Elasticity 132(1), 141-159.Abstract. In this paper, we examine the applicability of the approximation, f g ≈ f g , within Backus [1] averaging. This approximation is a crucial step in the method proposed by Backus [1], which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions.Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al.[2], we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions.Even though, according to the upper bound, Backus [1] averaging might not appear as a viable approach, we show that-for cases representative of physical scenarios modelled by such an averaging-the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy.In particular, we examine a special case for which the approximation results in spurious values. However, such a case-though physically realizable-is not likely to appear in seismology, where Backus [1] averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus [1] averaging is also considered.
For a constant power output, the mean ascent speed (VAM) increases monotonically with the slope. Also, to maximize the ascent speed, the slope needs to be constant. These properties constitute a mathematical-physics background upon which various strategies for the VAM maximization can be examined in the context of the maximum sustainable power as a function of both the gear ratio and cadence.
For a moving bicycle, the power meters respond to the propulsion of the centre of mass of the bicycle-cyclist system. Hence, an accurate modelling of power measurements -on a velodrome -requires a distinction between the trajectory of the wheels and the trajectory of the centre of mass. We formulate and examine an individual-pursuit model that takes into account the aforementioned distinction. In doing so, we provide the details of the invoked physical principles and mathematical derivations.
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