Well as more restricted situations, for example, such as that represented by Equation 11.7.
#Flex2d geodynamics full
To study full 3D flexure with in-plane forces, spatially variable loads, and variable flexural rigidity, as The single program listed in the following text can be used
#Flex2d geodynamics how to
In the chapter, we demonstrate how to solveĮquation 11.1 with the FEM using Matlab. In addition, the equationsĬan be used to constrain the elastic properties of the lithosphere (and in particular Te appearing inġ1.4) in cases when the loads and deflections are known. Simple manner how the lithosphere responds to different load scenarios. These various forms of the flexure equation are important because they enable one to predict in a Of isotropy and a constant flexural rigidity D, Equation 11.1 reduces to Produced as the plate deflects vertically and replaces material of different densities above and below,Īnd the fourth term accounts for vertical loads applied on the surface or base of the plate. The second term accounts for imposed in-plane forces, the third term accounts for restoring forces The first term of 11.1 represents the elastic resistance of the lithosphere to bending, Horizontal directions x or y), E is Young’s modulus, 𝜈 is Poisson’s ratio, and Te is the elastic thickness Matrix of imposed in-plane forces per unit length (Pa m, i.e., acting parallel to the lithosphere in the Is the acceleration due to gravity, q (Pa) is the vertical load imposed on the surface of the plate, P is the Where Δ𝜌 (kg m−3 ) is the density difference between the material below and above the plate, g (m s−2 ) Published 2017 by John Wiley & Sons Ltd.Īpplications of the Finite Element Method in Earth Science
Practical Finite Element Modeling in Earth Science Using Matlab, First Edition. (see Timoshenko and Woinowsky-Krieger (1959) and Turcotte and Schubert (1982)): With zero viscosity) is governed by the following fourth-order partial differential equation or PDE The vertical deflection (w) of an elastic lithospheric plate underlain by a dense inviscid fluid (i.e., one Involved in deriving the flexural equation are relaxed. While Chapter 12 deals with the mechanics of “classic” solids where the simplifying assumptions This chapter treats this simplified “flexural” model, This model is quite accurate forĬertain situations, and it results in a large reduction in computational cost compared to solving theįull displacement field in the entire 3D volume. Quantity, w, the vertical deflection of the mid-plane of the plate. Lithosphere itself (which is often the case), deformation of the full 3D solid can be reduced to a single In situations when the vertical deflection is small in comparison to the thickness of the Which enables various practical problems to be investigated.Įarth’s lithosphere becomes warped when subjected to vertical or horizontal loads, related, forĮxample, to faulting, volcanism, erosion, deposition, and glaciation (Turcotte and Schubert, 1982 A single stand-alone Matlab script is listed and discussed in detail, Thus, the discretized flexural model has several degrees of freedom per node, as in the coupled In addition, we show that even though flexure is governed by a single variable w, the FEMĪpproximation of this continuous variable involves not only its nodal values but also those of its gradients. The presence of fourth-order spatial derivatives in the equation governing flexure requires higher order shape functions than previouslyĬonsidered. flexural deformation of the lithosphere in three-dimensions (3Ds).
This chapter shows how the finite element method (FEM) can be applied to solve problems involving
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