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By A. A. Ilyushin and V. S. Lensky (Auth.)

ISBN-10: 008011461X

ISBN-13: 9780080114613

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We note that the axis Ox is here a principal axis. Any direction perpendicular to the axis Ox is also a principal axis, since a2 = o3 = 0. 34) we have that: 3 SM 56 STRENGTH OF MATERIALS (we note that eyy and εζζ are not zero, but their increments do not appear in the expression for the work since ayy = σζζ = 0). If we put εχχ = ει, the work done by the internal forces per unit volume after a time /, during which the strain εχχ increases from zero to ε{, is W = J σ, del. If the longitudinal stress σ1 is a single-valued function of the strain εί, the work done per unit volume is given by the area under the σχ — εί curve (Fig.

26) has three real roots εί9ε2,ε3, which are the principal extensions. The directions of fibres which experience these principal extensions are called principal directions or principal axes of strain. The extensions of these fibres are the extremal extensions of all the many fibres radiating from the same point. The principal axes are orthogonal, and there is no shearing strain between the principal fibres. Let us consider a unit cube with edges directed along the principal axes 1, 2, 3. Since there is no shearing strain between the principal axes, the cube, after deformation, becomes a rectangular parallelepiped with edges 1 + εί9 1 + ε2, 1 + ε3.

18). e. to the maximum shearing stresses, and if σ1 > σ2 > σ3 (this can always be assumed since the order of numbering the principal axes is arbitrary), then τ3ί is the greatest of the maximum shearing stresses. 5. OCTAHEDRAL STRESSES The principal normal stresses are a natural physical characteristic of the state of stress at a point. However, the stress state can be described just as naturally in another way—a way which is more closely related to the physical state in the neighbourhood of the point considered.

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Strength of Materials by A. A. Ilyushin and V. S. Lensky (Auth.)

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