|
Mohr's circle, named after Christian Otto Mohr, is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor. After performing a stress analysis on a material body assumed as a continuum, the components of the Cauchy stress tensor at a particular material point are known with respect to a coordinate system. The Mohr circle is then used to determine graphically the stress components acting on a rotated coordinate system, i.e., acting on a differently oriented plane passing through that point. The abscissa, , and ordinate, , of each point on the circle, are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate system. In other words, the circle is the locus of points that represent the state of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress element. Karl Culmann was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. Mohr's contribution extended the use of this representation for both two- and three-dimensional stresses and developed a failure criterion based on the stress circle. Alternative graphical methods for the representation of the stress state at a point include the Lame's stress ellipsoid and Cauchy's stress quadric. The Mohr circle can be applied to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors. ==Motivation for the Mohr Circle== Internal forces are produced between the particles of a deformable object, assumed as a continuum, as a reaction to applied external forces, i.e., either surface forces or body forces. This reaction follows from Euler's laws of motion for a continuum, which are equivalent to Newton's laws of motion for a particle. A measure of the intensity of these internal forces is called stress.〔Chen〕 Because the object is assumed as a continuum, these internal forces are distributed continuously within the volume of the object. In engineering, e.g., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a rock mass around a tunnel, airplane wings, or building columns, is determined through a stress analysis. Calculating the stress distribution implies the determination of stresses at every point (material particle) in the object. According to Cauchy, the ''stress at any point'' in an object (Figure 2), assumed as a continuum, is completely defined by the nine stress components of a second order tensor of type (2,0) known as the Cauchy stress tensor, : : After the stress distribution within the object has been determined with respect to a coordinate system , it may be necessary to calculate the components of the stress tensor at a particular material point with respect to a rotated coordinate system , i.e., the stresses acting on a plane with a different orientation passing through that point of interest —forming an angle with the coordinate system (Figure 3). For example, it is of interest to find the maximum normal stress and maximum shear stress, as well as the orientation of the planes where they act upon. To achieve this, it is necessary to perform a tensor transformation under a rotation of the coordinate system. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation law. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circle for stress. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mohr's circle」の詳細全文を読む スポンサード リンク
|