# 统计代写|主成分分析代写Principal Component Analysis代考|”Mathematical Representation of Plane Strain in Finite Element Analysis: Derivation and Computation”

This passage discusses the concept of plane strain and its mathematical representation in the context of finite element analysis. Plane strain occurs when a solid body is very long in one direction (here, the z-direction) compared to its cross-sectional dimensions in the xy-plane, such that the normal strain in the z-direction and shearing strains involving the z-axis are negligible. The strain-energy density for a body in plane strain is derived, leading to an expression for the total strain energy Ue of a volume V in terms of the strain components {ε} and the constitutive matrix [D], which differs from the plane stress case due to the Poisson’s ratio effect.

The strain components are related to the nodal displacements through the matrix [B], where {ε} = [B]{δ}, where {δ} represents the nodal displacement vector. For a four-node rectangular element, the interpolation functions N1, N2, N3, and N4 are defined based on the natural coordinates r and s. The strain components are obtained by applying the chain rule to differentiate the interpolation functions with respect to the physical coordinates x and y.

The stiffness matrix [k] for the element is computed by evaluating the integral [k(e)] = ∫∫[B]^T[D][B] dV(e), which involves integrating the product of [B], [D], and the transpose of [B] over the element volume. Each term in the stiffness matrix requires integration of quadratic functions of the natural coordinates.

Gaussian quadrature is recommended for efficiently computing these integrals. With two integration points (±√3/3) in each dimension for a total of four points and associated weights (Wi=Wj=1), the stiffness matrix [k(e)] can be accurately approximated using Equation 9.67:

[k(e)] = tab * Σi=1^2 Σj=1^2 Wi*Wj*[B(ri,sj)]^T*[D]*[B(ri,sj)] This process can be generalized for higher-order elements by adapting the integration points and weights according to the increased polynomial order.

It’s emphasized that although the illustrations use triangular and rectangular elements for plane stress and plane strain respectively, any element type can be used to model either condition. The key distinction comes from the stress-strain relationship encoded in the [D] matrix, not the element shape or order itself.

### MATLAB代写

MATLAB 是一款高性能的技术计算语言，集成了计算、可视化和编程环境于一体，以熟悉的数学符号表达问题和解决方案。MATLAB 的基本数据元素是一个不需要维度的数组，使得能够快速解决带有矩阵和向量公式的多种技术计算问题，相比使用 C 或 Fortran 等标量非交互式语言编写的程序，效率大大提高。MATLAB 名称源自“矩阵实验室”（Matrix Laboratory）。最初开发 MATLAB 的目标是为了提供对 LINPACK 和 EISPACK 项目的矩阵软件的便捷访问，这两个项目代表了当时矩阵计算软件的先进技术。经过长期发展和众多用户的贡献，MATLAB 已成为数学、工程和科学入门及高级课程的标准教学工具，在工业界，MATLAB 是高效研究、开发和分析的理想选择。MATLAB 提供了一系列名为工具箱的特定应用解决方案集，这对广大 MATLAB 用户至关重要，因为它们极大地扩展了 MATLAB 环境，使其能够解决特定类别问题。工具箱包含了针对特定应用领域的 MATLAB 函数（M 文件），涵盖信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等诸多领域。