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Analysis of Stress
Published in Abdel-Rahman Ragab, Salah Eldin Bayoumi, Engineering Solid Mechanics, 2018
Abdel-Rahman Ragab, Salah Eldin Bayoumi
which represents a circle in the σ–τ plane with its center at (σx + σy)/2 and a radius equal to the square root of [((σx+σy)/2)2+σxy2] as shown in Figure 1.19c. The circle represents the locus of all pairs of stress components (σ, τ) defining the state of plane stress at a point for any set of axes (x′ y′) and is known as Mohr’s circle, presented in books of strength of materials* as a graphical means for stress transformation.
Multiphase Toughening of Plastics
Published in Charles B. Arends, Polymer Toughening, 2020
By definition plane stress is a stress condition in which there is stress in only a single direction [13]. Conversely, plane strain refers to a stress condition in which stresses naturally develop perpendicular to an applied stress. This results in a triaxial stress state toward the center of the specimen that opposes shear flow and results in embrittlement of the specimen. Typically one observes plane stress in very thin specimens and plane strain in thick. Figure 11 is a pictorial representation of how plane strain can arise under uniaxial load [14]. From this picture one sees a shear response near the sample surface and a brittle response in the center. In a practical experiment the low energy state dominates. The shear response dimension is material dependent (intrinsic), not sample dependent (extrinsic); that is, it does not depend on the size of the piece being tested. Thus, if the sample is thin enough, it only experiences the plane stress response near the surface. As a result, we have relatively high energy absorption potential in thin sections and relatively low potential in thick. The absolute magnitude of the plane stress dimension varies greatly from polymer to polymer, from millimeters in polycarbonate to submicrons in polystyrene. Another technique for utilizing plane strain to plane stress conversion has to do with using a second phase to reduce the matrix dimensions internally. Wu [15] has demonstrated the effect of interparticle distance on the toughness of rubber-modified nylon. He observed a critical maximum spacing for toughness in the mixture and relates this observation to plane strain to plane stress conversion. In keeping with the concept of a critical dimension for plane stress we can readily see that the formation of thin ligaments between second-phase particles can provide a significant contribution to energy absorption potential. Yee et al. [16] have demonstrated the process in polyethylene-modified polycarbonate. This mechanism seems to be most useful for materials that are normally ductile but can be embrittled easily.
A new paradigm to explain the development of instability rutting in asphalt pavements
Published in Road Materials and Pavement Design, 2020
Reebie Simms, David Hernando, Reynaldo Roque
Mohr circle is commonly used to study the complete in-plane stress state of a material at a single point. However, when multiple locations and their corresponding stress states are to be investigated, Mohr circle becomes impractical. A convenient alternative to simultaneously analyse the stress state of multiple locations is the use of a p-q space diagram, where each point represents the top of the corresponding Mohr circle. The coordinates p and q are the average confinement and maximum shear stress at a point, respectively: where σ1 and σ3 represent the major and minor principal in-plane stresses, respectively.
Numerical investigation of the hull girder ultimate strength under realistic cyclic loading derived from long-term hydroelastic analysis
Published in Ships and Offshore Structures, 2023
Going back to the literature, most ultimate strength investigations are considering a perfect plastic material model, where the yield surface remains unchanged. However, in reality, the yield surface may change size, shape, and position. In the case of plane stress, the yield surface can be seen as the boundary of a region in the stress space in which both loading and unloading of a structure produces only elastic strains. A typical Von Mises yield surface for plane stress is illustrated in Figure 4 with a solid grey line.