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To derive the equation of the Mohr circle for the two-dimensional cases of plane stress and plane strain, first consider a two-dimensional infinitesimal material element around a material point (Figure 4), with a unit area in the direction parallel to the -plane, i.e., perpendicular to the page or screen.
Figure 7.1 Plane stress state in a continuum. In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 ...
For two-dimensional, plane strain problems the strain-displacement relations are = ; = [+] ; = Repeated differentiation of these relations, in order to remove the displacements and , gives us the two-dimensional compatibility condition for strains
The first index i indicates that the stress acts on a plane normal to the X i-axis, and the second index j denotes the direction in which the stress acts (For example, σ 12 implies that the stress is acting on the plane that is normal to the 1 st axis i.e.;X 1 and acts along the 2 nd axis i.e.;X 2). A stress component is positive if it acts in ...
A three-dimensional problem can then be reduced to a one-dimensional problem (for beams) or a two-dimensional problem (for plates and shells). Stress resultants are defined as integrals of stress over the thickness of a structural element. The integrals are weighted by integer powers the thickness coordinate z (or x 3).
Lamé'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 ...
Mohr's circle, Lame's stress ellipsoid (together with the stress director surface), and Cauchy's stress quadric are two-dimensional graphical representations of the state of stress at a point. They allow for the graphical determination of the magnitude of the stress tensor at a given point for all planes passing through that point.
Schematic of the loading on a plane by force P at a point (0, 0) A starting point for solving contact problems is to understand the effect of a "point-load" applied to an isotropic, homogeneous, and linear elastic half-plane, shown in the figure to the right. The problem may be either plane stress or plane strain.