### Finite Element Formulation for Vector Field Problems - Linear Elasticity [chapter]

2007 A First Course in Finite Elements
The discipline underlying linear stress analysis is the theory of elasticity. Both linear and nonlinear elasticity have been studied extensively over the past three centuries, beginning with Hooke, a contemporary of Newton. Hooke formulated what has come to be known as Hooke's law, the stress-strain relation for linear materials. Linear elasticity is used for most industrial stress analyses, as under operating conditions most products are not expected to undergo material or geometric
more » ... eometric nonlinearities. Linear elasticity also deals with many important phenomena relevant to materials science, such as the stress and strain fields around cracks and dislocations. These are not considered in this course. We start by presenting the basic assumptions and governing equations for linear elasticity in Section 9.1, followed by the exposition of strong and weak forms in Section 9.2. Finite element formulation for linear elasticity is then given in Section 9.3. Finite element solutions for linear elasticity problems in 2D concludes this chapter. LINEAR ELASTICITY The theory of linear elasticity hinges on the following four assumptions: 1. deformations are small; 2. the behavior of the material is linear; 3. dynamic effects are neglected; 4. no gaps or overlaps occur during the deformation of the solid. In the following, we discuss each of these assumptions. The first assumption is also made in any strength of materials course that is taught at the undergraduate level. This assumption arises because in linear stress analysis, the second-order terms in the straindisplacement equations are neglected and the body is treated as if the shape did not change under the influence of the loads. The absence of change in shape is a more useful criterion for deciding as to when linear analysis is appropriate: when the application of the forces does not significantly change the configuration of the solid or structure, then linear stress analysis is applicable. For structures that are large enough so that their behavior can readily be observed by the naked eye, this assumption implies that A First Course in Finite Elements J. Fish and T. Belytschko # 2007 John Wiley & Sons, Ltd ISBNs: 0 470 85275 5 (cased) 0 470 85276 3 (Pbk) the deformations of the solid should not be visible. For example, when a car passes over a bridge, the deformations of the bridge are invisible (at least we hope so). Similarly, wind loads on a high-rise building, although often felt by the occupants, result in invisible deformations. The deformations of an engine block due to the detonations in the cylinders are also invisible. On the other hand, the deformation of a blank in a punch press is readily visible, so this problem is not amenable to linear analysis. Other examples that require nonlinear analysis are a. the deformations of a car in a crash; b. the failure of an earth embankment; c. the deformations of skin during a massage. As a rough rule, the deformations should be of the order of 10 À2 of the dimensions of a body to apply linear stress analysis. As we will see later, this implies that the terms that are quadratic in the deformations are of the order of 10 À2 of the strains, and consequently, the errors due to the assumption of linearity are of the order of 1%. Many situations are just barely linear, and the analyst must exercise significant judgment as to whether a linear analysis should be trusted. For example, the deformations of a diving board under a diver are quite visible, yet a linear analysis often suffices. Sometimes these decisions are driven by practicality. For example, you have probably seen the large motions of the wingtip of a Boeing 747 on takeoff. Would a linear analysis be adequate? It turns out that the design of the aircraft is still primarily analyzed by linear methods, because the errors due to the assumption of linearity are small and thousands of loadings need to be considered, and this becomes much more complex with nonlinear analysis. The linearity of material behavior is also a matter of judgment. Many metals exhibit a relationship between stress and strain that deviates from linearity by only a few percent until the onset of plastic yielding. Until the yield point, a linear stress-strain law very accurately reproduces the behavior of the material. Beyond the yield point, a linear analysis is useless. On the contrary, materials such as concrete and soils are often nonlinear even for small strains, but their behavior can be fit by an average linear stress-strain law.