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ElasticityRectangular_02_StressFunction - Section 3.2 3.2...

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Section 3.2 Solid Mechanics Part II Kelly 46 3.2 The Stress Function Method An effective way of dealing with many two dimensional problems is to introduce a new “unknown”, the Airy stress function φ , an idea brought to us by George Airy in 1862. The stresses are written in terms of this new function and a new differential equation is obtained, one which can be solved more easily than Navier’s equations. 3.2.1 The Airy Stress Function The stress components are written in the form y x x y xy yy xx = = = σ 2 2 2 2 2 (3.2.1) Note that, unlike stress and displacement, the Airy stress function has no obvious physical meaning. The reason for writing the stresses in the form 3.2.1 is that, provided the body forces are zero , the equilibrium equations are automatically satisfied, which can be seen by substituting Eqns. 3.2.1 into Eqns. 2.2.3 { Problem 1}. On this point, the body forces, for example gravitational forces, are generally very small compared to the effect of typical surface forces in elastic materials and may be safely ignored (see Problem 2 of §2.1). When body forces are significant, Eqns. 3.2.1 can be amended and a solution obtained using the Airy stress function, but this approach will not be followed here. A number of examples including non-zero body forces are examined later on, using a different solution method. 3.2.2 The Biharmonic Equation The Compatability Condition and Stress-Strain Law In the previous section, it was shown how one needs to solve the equilibrium equations, the stress-strain constitutive law, and the strain-displacement relations, resulting in the differential equation for displacements, Eqn. 3.1.4. An alternative approach is to ignore the displacements and attempt to solve for the stresses and strains only . In other words, the strain-displacement equations 3.1.2 are ignored. However, if one is solving for the strains but not the displacements, one must ensure that the compatibility equation 1.3.1 is satisfied. Eqns. 3.2.1 already ensures that the equilibrium equations are satisfied, so combine now the two dimensional compatibility relation and the stress-strain relations 3.1.1 to get { Problem 2}
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Section 3.2 Solid Mechanics Part II Kelly 47 () 0 1 2 : strain plane 0 2 : stress plane 4 4 2 2 4 4 4 4 4 2 2 4 4 4 = + + = + + ν φ y y x x y y x x (3.2.2) Thus one has what is known as the biharmonic equation: 0 2 4 4 2 2 4 4 4 = + + y y x x The biharmonic equation (3.2.3) The biharmonic equation is often written using the short-hand notation 0 4 = .
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ElasticityRectangular_02_StressFunction - Section 3.2 3.2...

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