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Component Stresses on the Linear Element ∆s

Component Stresses on the Linear Element ∆s

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Research Proposal
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Recently translated father's 1968 Ph.D. Thesis, appears to be related to Hilbert's Problem #21, in the context of Theory of Elasticity using Complex Variables. The Hilbert kernel is stressed on various points and a complete solution for its elastic profile is given up to compromise. The crack is described using analytic functions.

Contexts in source publication

Context 1
... the linear elements are parallel to the axes x and y respectively, we say that the stress tensor T on the given point has components the linear magnitudes σ x , σ y , τ xy and τ yx . (Fig. ...
Context 2
... we "project" the above tensor against the linear element ∆s (passing through the initial point), which is signed with respect to direction as ⃗ n, we find by using the Cauchy relations the two components X n and Y n of T according to ⃗ n as follows 5 : (Fig. 2) ...
Context 3
... to be a unit vector, then the component of n in the x direction is precisely the cosine of the angle between them. So this also matches the interpretation which has συν(n, * ) = cos(n, * ) in the case that n is a unit, in which case the more austere notation cos ⟨ˆn⟨ˆn, ˆ * ⟩ could have been used. 6 Stresses are linear magnitudes and hence Fig. 2 depicts vectors having components equal to the stresses. 7 In the above as well as in what follows, we accept that there are no forces due to Mass. ...
Context 4
... and 21, respectively. Figure 20: σ x (θ) for upper disk side Figure 21: σ x (θ) for lower disk side ...

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