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Research Papers

A Theory in Finite Differences for Computation of Interchanging Principal Stresses and Principal Axes Rotation

[+] Author and Article Information
S. H. Stefanov

Department of Mechanical Engineering and Automation,
University of Forestry (LTU),
Faculty of Forest Industry,
10 Kliment Ohridski Blvd.,
Sofia 1756,Bulgaria
e-mail: stefanov_sh@yahoo.com;
stefanst@ltu.bg

Manuscript received February 27, 2012; final manuscript received September 19, 2012; accepted manuscript posted October 10, 2012; published online May 16, 2013. Assoc. Editor: Daining Fang.

J. Appl. Mech 80(4), 041009 (May 16, 2013) (8 pages) Paper No: JAM-12-1083; doi: 10.1115/1.4007796 History: Received February 27, 2012; Revised September 19, 2012; Accepted October 10, 2012

Suppose consecutive ordinates of three arbitrary and nonproportional stress-time functions of plane state of stress are entered into a computer by a little finite time step. The theory proposed solves the following problem: correct computation of the ordinates of the principal stress-time functions and the angle of principal axes rotation. This problem is not as simple as researchers approached it prior to the computer era. First of all, the correct solution for the principal stresses and the principal axes rotation require correct interchange of the principal stresses while computing them, i.e., correct interchange of the plus/minus signs in the well-known equations for them. For the interchange analysis, an ellipse of stress transformation in the three-dimensional stress-coordinate space is revealed. By changing a coordinate scale, the ellipse turns into a circumference that is an analog to, but different from, a Mohr circle. The correct solution also requires treating the principal axes rotation in little finite differences per little time differences during which little finite elements appear as building the stressing path in the three-dimensional stress-coordinate space. Based on the ellipse/circumference mentioned, three interchange conditions are revealed. The third one is the most important. And, a necessity is also revealed for dividing some stressing path's elements into two subelements. Based on all the findings, the main commands of an algorithm for computing the ordinates of the principal stress-time functions and the angle of principal axes rotation are presented. The correct solution of the problem has been achieved thanks to new notions taken from the so-called integration of damage differentials (IDD) theory. In fact, the paper presents a new contribution to the variable plane stress state analysis.

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Figures

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Fig. 1

(a) An initial immovable cuboid of plane stress state with variable stresses σx = σx(t), σy = σy(t), and τxy = τxy(t); (b) another immovable cuboid at an angle α; (c) rotating principal cuboid at the angle α′ = α′(t)

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Fig. 2

(a) Three arbitrary and nonproportional oscillograms σx(t), σy(t) and τxy(t); (b) σxyxy loading path called variant trajectory (Sv) and its finite element Δsv per Δt

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Fig. 3

Transforming ellipse

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Fig. 4

A view of the transforming ellipse from the arrowhead of the τxy axis

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Fig. 7

(a) Preceding (sp,0) stress state at tp = t − Δt on immovable cuboid that will be principal at t; (b) invariant (Δs) little finite stress state difference during Δt on the same cuboid; (c) invariant (s) stress state at t when the same cuboid is principal

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Fig. 6

Transformation of variant Δsv ≡ MpM finite element from σxyxy coordinate space into invariant Δs ≡ Mp,0M′ element in σ′-σ″-Δτ coordinate space, and other additional illustrations

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Fig. 5

Transforming circumference and principal axes

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Fig. 8

Circular transformation of (projection of) MpM variant element into invariant Mp,0M′, and other additional illustrations

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Fig. 9

Transformation of MpM variant element projection into invariant Mp,0M′, and other additional illustrations in Oξv*τxy coordinate plane

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Fig. 10

Variant (Sv) trajectory intersecting ηv axis (O point)

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