By Robert Asaro, Vlado Lubarda
This ebook blends either leading edge (large pressure, pressure expense, temperature, time based deformation and localized plastic deformation in crystalline solids, deformation of organic networks) and conventional (elastic conception of torsion, elastic beam and plate theories, touch mechanics) subject matters in a coherent theoretical framework. wide use of rework tips on how to generate suggestions will make this ebook of curiosity to structural, mechanical, and aerospace engineers. Plasticity theories, micromechanics, crystal plasticity, and energetics of elastic platforms also are lined, in addition to an total evaluate of math and thermodynamics.
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Extra resources for Mechanics of Solids and Materials
W. H. Freeman and Company, New York. , and Toupin, R. (1960), The Classical Field Theories. In Handbuch der Physik (S. ), Band III/l, Springer-Verlag, Berlin. Wrede, R. C. (1972), Introduction to Vector and Tensor Analysis, Dover, New York. 1 Fourier Series Let f (x) be a continuous, integrable function deﬁned on the interval [−c, c]. , f (x) = (a0 /2) + ∞ [ak cos(kπ x/c) + bk sin(kπ x/c)]. 1) k=1 The coefﬁcients, ak and bk, indexed by the integers k, can be identiﬁed as follows. 1) by cos(nπ x/c), n being an integer, and integrate over [−c, c] to obtain c c f (x) cos(nπ x/c) dx = −c −c + (a0 /2) cos(nπ x/c) dx c ∞ −c k=1 [ak cos(kπ x/c) cos(nπ x/c) + bk sin(kπ x/c) cos(nπ x/c)] dx.
These are clearly orthogonal on the square deﬁned by −π ≤ x ≤ π and −π ≤ y ≤ π. For reference we note that the norms are 1 = 2π, cos(mx) = sin(mx) = √ 2π , cos(mx) cos(ny) = sin(mx) sin(ny) = cos(mx) sin(ny) = π. 19) f (x, y) cos(mx) sin(ny) dx dy, f (x, y) sin(mx) sin(ny) dx dy, for m, n = 1, 2, . . For the cases where either m = 0 or n = 0, we have Am0 = R = 1 2π 2 f (x, y) cos(mx) dx dy cos(mx) 2 f (x, y) cos(mx) dx dy, (m = 1, 2, . 3. Integral Transforms 39 A0n = R = 1 2π 2 Bm0 = R = 1 2π 2 B0n = R = 1 2π 2 f (x, y) cos(ny) dx dy cos(ny) 2 f (x, y) cos(ny) dx dy, (n = 1, 2, .
0 Now let η ≡ αx, and write ∞ ∞ K(αx)α s−1 dα = x −s 0 K(η)ηs−1 dη = x −s K(s). 33) 0 = K(s)F(1 − s). 35) ∞ I f (α) dα 0 x s−1 K(αx) dx. 3. 38) as desired. 40) Fc (α) cos(αx) dα. 41) 0 with its inverse f (x) = ∞ 2/π 0 As another example, let K(αx) be deﬁned as K(αx) = 2/π sin(αx). 43) Fs (α) sin(αx) dα. 47) where K(s) and H(s) are the Mellin transforms of K(x) and H(x), respectively. 48) f (x) = π 0 0 or, more generally, f (x) = 1 π ∞ −∞ ∞ dα −∞ f (η) cos(αη) cos(αx) dη. 49) Before exploring this, it is necessary to establish some integrability properties of f (x).