Lame's stress ellipsoid
Encyclopedia
Lame's stress ellipsoid is an alternative to Mohr's circle for the graphical representation of the stress state at a point. The surface of the ellipsoid 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
(Figure coming).
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
taken in the directions of the principal axes, i.e., in a principal stress space. Thus, the coordinates of the stress vector 
on a plane with normal unit vector 
passing through a given point 
is represented by
And knowing that
is a unit vector we have
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
.
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 represented by the equation
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.
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
(Figure coming).
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
taken in the directions of the principal axes, i.e., in a principal stress space. Thus, the coordinates of the stress vector on a plane with normal unit vector passing through a given point is represented byAnd knowing that
is a unit vector we havewhich 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
.- The first stress invariant
is directly proportional to the sum of the principal radii of the ellipsoid. - The second stress invariant
is directly proportional to the sum of the three principal areas of the ellipsoid. The tree principal areas are the ellipses on each principal plane. - The third stress invariant
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 revolutionSpheroidA spheroid, or ellipsoid of revolution 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....
. Thus, two principal areas are ellipses and the third is a circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. - If all of the principal stresses are equal and of the same sign, the stress ellipsoid becomes a sphereSphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
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 represented by the equation
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.