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Unformatted text preview: Section 3.5 Solid Mechanics Part I Kelly 55 3.5 Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress . It is important for two reasons: (1) it has practical application in the analysis of thin components and (2) it is a two dimensional state of stress, and thus serves as an excellent introduction to more complicated three dimensional stress states. As discussed further below, plane stress arises in many situations, but particularly in thin materials. First, consider stress boundary conditions. 3.5.1 Stress Boundary Conditions When solving problems, information is usually available on what is happening at the boundaries of materials. This information is called the boundary conditions . Information is usually not available on what is happening in the interior of the material information there is obtained by solving the equations of mechanics. A number of different conditions can be known at a boundary, for example it might be known that a certain part of the boundary is fixed so that the displacements there are zero. This is known as a displacement boundary condition On the other hand the stresses over a certain part of the material boundary might be known. These are known as stress boundary conditions this case will be examined here. General Stress Boundary Conditions It has been seen already that, when one material contacts a second material, a force, or distribution of stress arises. This force F will have arbitrary direction, Fig. 3.5.1a, and can be decomposed into the sum of a normal stress distribution N and a shear distribution S , Fig. 3.5.1b. One can introduce a coordinate system to describe the applied stresses, for example the y x axes shown in Fig. 3.5.1c. Figure 3.5.1: Stress boundary conditions; (a) force acting on material due to contact with a second material, (b) the resulting normal and shear stress distributions, (c) applied stresses as stress components in a given coordinate system S F N contact region (a) (b) (c) xy yy y x Section 3.5 Solid Mechanics Part I Kelly 56 Figure 3.5.2 shows the same component as Fig. 3.5.1. Shown in detail is a small material element at the boundary. From equilibrium of the element, stresses yy xy , , equal to the applied stresses, must be acting inside the material, Fig. 3.5.2a. Note that the tangential stresses , which are the xx stresses in this example, can take on any value and the element will still be in equilibrium with the applied stresses, Fig. 3.5.2b. Figure 3.5.2: Stresses acting on a material element at the boundary, (a) normal and shear stresses, (b) tangential stresses Thus, if the applied stresses are known , then so also are the normal and shear stresses acting at the boundary of the material. This can be summarised as follows: Stress boundary conditions: Stress boundary conditions involve the normal and shear stresses acting on a surface Stress Boundary (Interface) Conditions between Two Materials...
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