Fundamentals of the Mechanics of Solids by Paolo Maria Mariano & Luciano Galano

Author:Paolo Maria Mariano & Luciano Galano
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

As a consequence, we rewrite equation (5.30) as

(5.31)

Its trace is given by

i.e.,

Then equation (5.30) becomes

(5.32)

which reduces to

(5.33)

when b is constant. Both (5.32) and (5.33) are called the Beltrami–Donati–Michell equations after the work of Eugenio Beltrami (1892), L. Donati (1894), and John Henry Michell (1899, 1900).

5.9 Plane Problems

5.9.1 Plane-Strain States

Take a narrow right but not necessarily circular cylinder loaded in a way that the displacement along the cylinder’s axis vanishes, while its orthogonal components remain in the cross-section plane at each point. In this case, we say that the cylinder is in what we commonly call a plane-strain state. Here we discuss this state without accounting for inertial effects, i.e., focusing attention on the statics. We refer analysis to an orthonormal frame with the x 3-axis coinciding with the cylinder’s axis.

Formally, the plane-strain state in the plane is defined by the conditions

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