3DElasticity_02_3D_StressStrain

3DElasticity_02_3D_StressStrain - Section 7.2 7.2 Analysis...

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Section 7.2 Solid Mechanics Part II Kelly 116 7.2 Analysis of Three Dimensional Stress and Strain The concept of traction and stress was introduced and discussed in Part I, §3.1-3.5. For the most part, the discussion was confined to two-dimensional states of stress. Here, the fully three dimensional stress state is examined. There will be some repetition of the earlier analyses. 7.2.1 The Traction Vector and Stress Components Consider a traction vector t acting on a surface element, Fig. 7.2.1. Introduce a Cartesian coordinate system with base vectors i e so that one of the base vectors is a normal to the surface and the origin of the coordinate system is positioned at the point at which the traction acts. For example, in Fig. 7.1.1, the 3 e direction is taken to be normal to the plane, and a superscript on t denotes this normal: 3 3 2 2 1 1 ) ( 3 e e e t e t t t + + = (7.2.1) Each of these components i t is represented by ij σ where the first subscript denotes the direction of the normal and the second denotes the direction of the component to the plane. Thus the three components of the traction vector shown in Fig. 7.2.1 are 33 32 31 , , : 3 33 2 32 1 31 ) ( 3 e e e t e + + = (7.2.2) The first two stresses, the components acting tangential to the surface, are shear stresses whereas 33 , acting normal to the plane, is a normal stress. Figure 7.2.1: components of the traction vector Consider the three traction vectors ) ( ) ( ) ( 3 2 1 , , e e e t t t acting on the surface elements whose outward normals are aligned with the three base vectors j e , Fig. 7.2.2a. The three (or six) surfaces can be amalgamated into one diagram as in Fig. 7.2.2b. In terms of stresses, the traction vectors are ) ( 3 e t 2 x 1 x 3 x ) ˆ ( n t 1 e 2 e 3 e 32 31 33
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Section 7.2 Solid Mechanics Part II Kelly 117 () 3 33 2 32 31 3 23 2 22 21 3 13 2 12 11 3 2 e e e t e e e t e e e t 1 e 1 e 1 e 1 σ + + = + + = + + = or ( ) j ij i e t e = (7.2.3) Figure 7.2.2: the three traction vectors acting at a point; (a) on mutually orthogonal planes, (b) the traction vectors illustrated on a box element The components of the three traction vectors, i.e. the stress components, can now be displayed on a box element as in Fig. 7.2.3. Note that the stress components will vary slightly over the surfaces of an elemental box of finite size. However, it is assumed that the element in Fig. 7.2.3 is small enough that the stresses can be treated as constant, so that they are the stresses acting at the origin. Figure 7.2.3: the nine stress components with respect to a Cartesian coordinate system The nine stresses can be conveniently displayed in 3 3 × matrix form: 21 11 31 12 22 32 23 33 13 3 x 2 x 1 x 1 e 1 e t 0 1 = x 0 2 = x 0 3 = x 2 e 3 e 3 e t ( ) 2 e t 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3 x 3 x 2 x 2 e 3 e 1 e ( ) 1 e t ( ) 2 e t ( ) 3 e t 1 x ) a ( ) b (
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Section 7.2 Solid Mechanics Part II Kelly 118 [] = 33 32 31 23 22 21 13 12 11 σ ij (7.2.4) It is important to realise that, if one were to take an element at some different orientation to the element in Fig. 7.2.3, but at the same material particle , for example aligned with the axes 3 2 1 , , x x x shown in Fig. 7.2.4, one would then have different tractions acting and
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3DElasticity_02_3D_StressStrain - Section 7.2 7.2 Analysis...

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