By C. N. Reid
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Additional info for Deformation Geometry for Materials Scientists
Such reversible strains are called linear elastic. All glassy or crystalline materials show linear elastic behaviour at small strains. 3(b) shows the response of another brittle elastic solid such as rubber. In this case, non-linear elastic behaviour occurs until the stress reaches σ>, when sudden fracture occurs. Again, strains induced by stressing are reduced to zero by releasing the stress. Note the large differences between the magnitudes of the elastic strains in Figs. 3(a) and (b). In Fig.
THE STRAIN ELLIPSOID Let us consider what happens to a unit sphere, centred on the origin, when it is deformed homogeneously. ) Calling this new site (x'l9 x'2, X3), we can say Xi = Xi (1 + */) and substituting this for xt in the equation of the sphere, we get M* x'i xi* · + ε2)2- + - + ε3)2 1 ' (1 (l + £ l ) 2 · '+ (1 This is obviously the equation of a central ellipsoid, called the strain ellipsoid, and its shape is that of the deformed sphere. The semi-axes are always positive, with magnitudes of (Ι + ε,), which may be either less than or greater than unity, depending on the sign of eh A section through the sphere and ellipsoid is drawn in Fig.
It was suggested (Irwin 1953) that the front lies along a direction of zero extension, in order that the yielded and unyielded regions may fit together. With this geometry the two regions are fully compatible with one another. Let us refer to the test-piece and coordinate axes illustrated in Fig. 10. 10 Schematic diagram of a Lüders band in a tensile testpiece. 14) ε22 = 0 = a2iCi2jSij The direction cosines, aij9 relating the two sets of axes are Oxi Ox2 Ox[ Ox2 cos 0 sin 0 -sin 0 cos0 OX3 0 0 Ox3 0 0 1 Using these values of ai} and eqn.