# 统计代写|主成分分析代写Principal Component Analysis代考|”Formulating a 3D Finite Element for Stress Analysis: Strain-Displacement Relations, Potential Energy, and Equilibrium Equations”

The passage you’ve shared explains how to formulate a three-dimensional (3D) finite element for stress analysis when dealing with a general state of stress. Here’s a summary of the key steps:

Strain-Displacement Relations:

The six strain components {ε} are derived from the three displacement components u, v, w through differentiation as given by Equation 9.103. These can also be expressed compactly using a strain-displacement matrix [L] (Equation 9.104). Stress-Strain Relations:

The relationship between stresses {σ} and strains {ε} is given by Hooke’s Law (Equation 9.105), where the stress-strain matrix [D] depends on the elastic modulus E and Poisson’s ratio ν. Discretization of Displacements:

A 3D elastic element with M nodes has its displacement field represented by interpolation functions Ni(x,y,z) multiplied by the nodal displacements ui, vi, wi (Equation 9.106). The nodal displacements are grouped together into a single column vector {}, and the interpolation functions are arranged into the larger 3 × 3M matrix [N3]. Total Potential Energy:

The total potential energy Ue of the element includes the strain energy stored in the material and the work done by external forces -W. By substituting the strain-displacement relation and considering the nodal force vector { f}, the total potential energy can be written in terms of the nodal displacements and the element stiffness matrix [k] (Equation 9.117), which is derived from integrating the product of [B], [D], and [B]T over the element volume. Element Stiffness Matrix:

The element stiffness matrix [k] is calculated by integrating the product of the strain-displacement matrix [B] and the stress-strain matrix [D] with respect to the element volume. Global Equilibrium Equations:

For various element types (such as 4-node tetrahedrons and 8-node bricks), the local stiffness matrices are assembled into a global stiffness matrix [K]. The assembled global equilibrium equations are given by [K]{}={F}, where {} represents the global displacement vector and {F} represents the vector of externally applied nodal forces including body forces and pressure loads. In practice, depending on the complexity of the element (linear or higher-order), the integrals involved in computing the stiffness matrix may be evaluated exactly (for simple geometries and linear interpolation functions) or approximately using numerical integration techniques like Gaussian quadrature. The isoparametric mapping technique allows for transforming the parent element into elements of arbitrary shapes while maintaining linear behavior within the element.

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