Tham khảo tài liệu 'introduction to contact mechanics part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Elasticity 23 convention Ơ1 ơ2 ơ3 is not strictly adhered to. Note that two of the principal stresses Ơ1 and ơ3 lie in the rz plane with 0 a constant . The directions of the principal stresses with respect to the r axis are given by tan 20 p In Eq. a positive value of 0p is taken in an anticlockwise direction from the r axis to the line of action of the stress. However difficulties arise as this angle passes through 45 and a more consistent value for 0p is given by Eq. in Chapter 5. The planes of maximum shear stress bisect the principal planes and thus tan20s ơ ơz 2t rz Equations of equilibrium and compatibility Cartesian coordinate system Equations of stress equilibrium and strain compatibility describe the nature of the variation in stresses and strains throughout the specimen. These equations have particular relevance for the determination of stresses and strains in systems that cannot be analyzed by a consideration of stress alone . statically indeterminate systems . For a specimen whose applied loads are in equilibrium the state of internal stress must satisfy certain conditions which in the absence of any body forces . gravitational or inertial effects are given by Navier s equations of equilibrium1 2 dơx dTxy dzxI L UJxL 0 de dy dz dĩ dơ dĩ __ y 0 d d dz dx dTzy dơ z ZL ._ 0 dx dy dz Equations describe the variation of stress from one point to another throughout the solid. Displacements of points within the solid are required to satisfy compatibility conditions which prescribe the variation in displacements throughout the solid and are given by1 2 3 24 Mechanical Properties of Materials d2sx y d X2 d2 y d2s z 2 d e - d y d2sx 2 d xy dXdy Tyz dYz dzdx The compatibility relations imply that the displacements within the material vary smoothly throughout the specimen. Solutions to problems in elasticity generally require expressions for stress components which satisfy .
