Lamé's stress ellipsoid
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Lamé's stress ellipsoid is an alternative to
Mohr's circle Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor. Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineer ...
for the graphical representation of the stress state at a point. The surface of the
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the radius-vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
. Once the equations of the ellipsoid is known, the magnitude of the stress vector can then be obtained for any plane passing through that point. To determine the equation of the stress ellipsoid we consider the coordinate axes x_1, x_2, x_3\,\! taken in the directions of the principal axes, i.e., in a principal stress space. Thus, the coordinates of the stress vector \mathbf T^\,\! on a plane with normal unit vector \mathbf n\,\! passing through a given point P\,\! is represented by :T_1^=\sigma_1n_1, \qquad T_2^=\sigma_2n_2, \qquad T_3^=\sigma_3n_3\, And knowing that \mathbf n\,\! is a unit vector we have :n_1^2+n_2^2+n_3^2=\frac+\frac+\frac=1\, which is the equation of an ellipsoid centered at the origin of the coordinate system, with the lengths of the semiaxes of the ellipsoid equal to the magnitudes of the principal stresses, i.e. the intercepts of the ellipsoid with the principal axes are \pm\sigma_1, \pm\sigma_2, \pm\sigma_3\,\!. * The first stress invariant I_1\,\! is directly proportional to the sum of the principal radii of the ellipsoid. * The second stress invariant I_2\,\! is directly proportional to the sum of the three principal areas of the ellipsoid. The three principal areas are the ellipses on each principal plane. * The third stress invariant I_3\,\! is directly proportional to the volume of the ellipsoid. * If two of the three principal stresses are numerically equal the stress ellipsoid becomes an
ellipsoid of revolution A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...
.Timoshenko Thus, two principal areas are ellipses and the third is a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
. * If all of the principal stresses are equal and of the same sign, the stress ellipsoid becomes a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
and any three perpendicular directions can be taken as principal axes. The stress ellipsoid by itself, however, does not indicate the plane on which the given traction vector acts. Only for the case where the stress vector lies along one of the principal directions it is possible to know the direction of the plane, as the principal stresses act perpendicular to their planes. To find the orientation of any other plane we used the ''stress-director surface'' or ''stress director quadric''Timoshenko represented by the equation :\frac+\frac+\frac=1 The stress represented by a radius-vector of the stress ellipsoid acts on a plane oriented parallel to the tangent plane to the stress-director surface at the point of its intersection with the radius-vector.


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Bibliography

* * {{DEFAULTSORT:Lame's Stress Ellipsoid Classical mechanics Materials science Elasticity (physics) Solid mechanics Mechanics