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Một trong những vấn đề phổ biến nhất trong cơ học của vật liệu liên quan đến việc chuyển đổi của trục. Ví dụ, chúng tôi có thể biết sức ép hoạt động trên máy bay xy, nhưng thực sự quan tâm nhiều hơn trong những sức ép hoạt động trên máy bay theo định hướng, nói rằng, 30 ◦ với trục x như trong hình. | Transformation of Stresses and Strains David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA 02139 May 14 2001 Introduction One of the most common problems in mechanics of materials involves transformation of axes. For instance we may know the stresses acting on xy planes but are really more interested in the stresses acting on planes oriented at say 30 to the x axis as seen in Fig. 1 perhaps because these are close-packed atomic planes on which sliding is prone to occur or is the angle at which two pieces of lumber are glued together in a scarf joint. We seek a means to transform the stresses to these new x y planes. Figure 1 Rotation of axes in two dimensions. These transformations are vital in analyses of stress and strain both because they are needed to compute critical values of these entities and also because the tensorial nature of stress and strain is most clearly seen in their transformation properties. Other entities such as moment of inertia and curvature also transform in a manner similar to stress and strain. All of these are second-rank tensors an important concept that will be outlined later in this module. Direct approach The rules for stress transformations can be developed directly from considerations of static equilibrium. For illustration consider the case of uniaxial tension shown in Fig. 2 in which all stresses other than ơy are zero. A free body diagram is then constructed in which the specimen is cut along the inclined plane on which the stresses labeled ơy and Tx y are desired. The key here is to note that the area on which these transformed stresses act is different than the area normal to the y axis so that both the areas and the forces acting on them need to be transformed. Balancing forces in the y0 direction the direction normal to the inclined plane 1 Figure 2 An inclined plane in a tensile specimen. y A cos 0 7y A cos 0 ơy ơy cos2 0 1 Similarly a force balance in the .