4.1 Continuum Mechanics In computational sciences, deformable objects are often modeled as continuous (three di- mensional) objects. For the description of their behavior, the three quantities displacement , strain and stress play a major role. In one dimensional problems, these quantities are all one dimensional and have intuitive interpretations. Let us consider a beam with cross sectional area A as shown in Fig. 4.1. When a force f n is applied in the direction of the beam perpen- dicular to the cross section, the beam with original length l expands by Δ l . These quantities are related via Hooke’s law as f n A = E Δ l l . (4.1) The constant of proportionality E is Young’s modulus. For steel E is in the order of 10 11 N / m 2 while for rubber it lies between 10 7 and 10 8 N / m 2 . The equation states that the stronger the force per area, the larger the relative elongation Δ l / l as expected. Also, the magnitude of the force per area needed to get a certain relative elongation increases with increasing E so Young’s modulus describes the beam’s stiffness . Hooke’s law can be written in a more compact form as σ = E ε , (4.2) where σ = f n / A is the applied stress and ε = Δ l / l the resulting strain or the other way around, σ the resulting internal stress due to applied strain ε . It follows that the unit of stress is force per area and that strain has no unit. This is true in three dimensions as well. A three dimensional deformable object is typically defined by its undeformed shape (also called equilibrium configuration, original, rest or initial shape) and by a set of material parameters that define how it deforms under applied forces. If we think of the rest shape as a continuous connected subset Ω of R 3 , then the coordinates x ∈ Ω of a point in the object are called material coordinates of that point. When forces are applied, the object deforms and a point originally at location x (i.e. with material coordinates x ) moves to a new location p ( x ) , the spatial or world coordinates of that point. Since new locations are defined for all material coordinates x , p ( x ) is a vector
CHAPTER 4. THE FINITE ELEMENT METHOD 23 field defined on Ω . Alternatively, the deformation can also be specified by the displacement field, which, in three dimensions, is a vector field u ( x ) = p ( x ) - x defined on Ω . 4.1.1 Strain In order to formulate Hooke’s law in three dimensions, we have to find a way to measure strain, i.e. the relative elongation (or compression) of the material. In the general case, the strain is not constant inside a deformable body. In a bent beam, the material on the convex side is stretched while the one on the concave side is compressed. Therefore, strain is a function of the material coordinate ε = ε ( x ) . If the body is not deformed, the strain is zero so strain must depend on the displacement field u ( x ) . A spatially constant displacement field describes a pure translation of the object. In this situation, the strain should be zero as well. Thus, strain is derived from the spatial variation or spatial derivatives of the displacement field. In three dimensions the displacement field has three components
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