# 统计代写|主成分分析代写Principal Component Analysis代考|”Equilibrium Equations and Strain-Displacement Relations in Continuum Mechanics”

In continuum mechanics, the equations of equilibrium describe how internal and external forces balance out within a deformed solid body at any given point. Considering an infinitesimal differential element in the body, the forces acting on it include surface tractions (represented by stress components) and body forces (like gravity, magnetic forces, or centrifugal forces).

Equation B.16 shows the application of force equilibrium along the x-axis, where the sum of all forces acting on the differential element in the x-direction must equal zero. After simplification, this leads to the equilibrium equation in the x-direction (B.17):

∂σx∂x+∂τxy∂y+∂τxz∂z+Bx=0\frac{tial \sigma_x}{tial x} + \frac{tial \tau_{xy}}{tial y} + \frac{tial \tau_{xz}}{tial z} + B_x = 0 ∂x ∂σ x ​

​ + ∂y ∂τ xy ​

​ + ∂z ∂τ xz ​

​ +B x ​ =0

Similar equations apply for the y-axis (B.18):

∂τxy∂x+∂σy∂y+∂τyz∂z+By=0\frac{tial \tau_{xy}}{tial x} + \frac{tial \sigma_y}{tial y} + \frac{tial \tau_{yz}}{tial z} + B_y = 0 ∂x ∂τ xy ​

​ + ∂y ∂σ y ​

​ + ∂z ∂τ yz ​

​ +B y ​ =0

And for the z-axis (B.19):

∂τxz∂x+∂τyz∂y+∂σz∂z+Bz=0\frac{tial \tau_{xz}}{tial x} + \frac{tial \tau_{yz}}{tial y} + \frac{tial \sigma_z}{tial z} + B_z = 0 ∂x ∂τ xz ​

​ + ∂y ∂τ yz ​

​ + ∂z ∂σ z ​

​ +B z ​ =0

These are the Cauchy stress equilibrium equations, often referred to as the Navier-Cauchy equations in 3D space.

On the other hand, strain-displacement relations (such as Equations B.3-B.6) link the strain components to the displacement field (u, v, w) in a continuous body. Continuity and single-valuedness of the displacement field are critical assumptions in continuum mechanics.

However, solving for the displacement field directly from a given set of continuous strain components is not trivial. Six partial differential equations need to be solved, which can lead to potential issues with ensuring continuity and uniqueness of the solution. This is where the compatibility equations come into play.

The compatibility equation you’ve mentioned (B.20) is one example:

∂2ϵx∂y2+∂2ϵy∂x2=∂2γxy∂x∂y\frac{tial^2 \epsilon_x}{tial y^2} + \frac{tial^2 \epsilon_y}{tial x^2} = \frac{tial^2 \gamma_{xy}}{tial x tial y} ∂y 2

∂ 2 ϵ x ​

​ + ∂x 2

∂ 2 ϵ y ​

​ = ∂x∂y ∂ 2 γ xy ​

There are five more similar second-order partial differential equations that collectively guarantee that the calculated displacements are consistent with a continuous and single-valued strain field throughout the solid body. These compatibility conditions are especially relevant in higher-level analyses in elasticity theory and continuum mechanics.

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