In order to find the equilibrium equations in polar coordinates. Page 2 of: (soest.hawaii.edu/martel/Courses/GG703/GG711c_Lec_10.pdf)
Do you have to use the Taylor expansion to get from 10.4 to 10.5? Also, how is this equation altered if there are shear stresses?
To find the equilibrium equations in polar coordinates, you need to derive the equilibrium equations in Cartesian coordinates first. The document you mentioned (soest.hawaii.edu/martel/Courses/GG703/GG711c_Lec_10.pdf) is not accessible, so I will explain the general process for obtaining equilibrium equations in polar coordinates.
In the derivation process, the Taylor expansion is commonly used to approximate functions. However, without access to the specific page you mentioned, it is difficult to determine whether the Taylor expansion is used to transition from equation 10.4 to 10.5.
To alter the equilibrium equations in polar coordinates if shear stresses are present, you need to consider additional terms in the equations. In polar coordinates, the equilibrium equations describe the balance of forces and moments acting on an object. The equations are typically written as:
Radial Equilibrium: ∂σr/∂r + (σr - σθ/2) / r + ρgᵣ = 0
Tangential Equilibrium: (1/r) ∂σθ/∂θ + (∂σr/∂θ + σθ) / r = 0
Here, σr and σθ are the radial and tangential stresses, respectively, ρ is the mass density, gᵣ is the radial component of gravity, and r and θ are the radial and angular coordinate in the polar system.
In the presence of shear stresses, additional terms involving the shear stresses (τrθ, τθr) will appear in these equations. The specific form and derivation of these terms depend on the nature of the problem and the constitutive equations of the material. These additional terms consider the shear stresses and their effect on the equilibrium of the system.
It is important to consult reliable sources, textbooks, or lecture notes for the specific equations and derivations applicable to your problem, as the procedure may vary depending on the context.