Questions tagged [continuum-mechanics]
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.
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Stresses in shells of revolution with infinite radii
The stresses in pressurized shells of revolution are given by (see Roark's Formulas for Stress and Strain) $$\sigma_1\propto R_2\qquad\sigma_2\propto R_2\left(2-\frac{R_2}{R_1}\right),$$ where $R_1$ ...
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Trying to find nonlinear wave equation for a string by abusing notation (algebraically manipulating differentials)
I derived the wave equation for a string of mass density $\mu$ with an equal tension $T$ being applied at both ends at an angle $\theta_x$ relative to the $x$-axis by taking an infinitesimal length of ...
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Trying to find wave equation for a string by abusing notation (algebraically manipulating differentials)
I'm trying to derive the wave equation for a string of mass density $\mu$ with an equal tension $T$ being applied at both ends at an angle $\theta_x$ relative to the $x$-axis. I've taken an ...
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Why didn't Earth bounce upon collision during the formation of moon?
Simulations of moon formation show Earth splashed when a Mars-sized planet collided with it. Why doesn't Earth bounce like a rubber ball? What makes it behave like a liquid? I heard this depends on ...
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Why are SHM formulas applicable to waves on strings? Are they applicable for all transverse waves?
This is a very basic and conceptual doubt, I am in high school, they have taught us SHM formulas in school and told us to apply it on waves on strings, but I had a problem connecting the dots and in ...
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Upper bound for the lowest transverse oscillation frequency of a 2D string with non-uniform density
I'm working on Problem 4.5 from A First Course in String Theory by B. Zwiebach. The problem involves a nonrelativistic 2D string with fixed endpoints at $(x=0,y=0)$ and $(x=a,y=0)$, position-dependent ...
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Bernoulli equation for compressible inviscid flow [closed]
I am studying compressible flow and I want to derive the compressible inviscid flow Bernoulli equation. The main $3$ equations are: $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) =...
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Relationship between plane stress and Gaussian curvature
While studying perturbations of pressurized shells of revolution (in my case, given by a bulge forming at the pole of a sphere), I noticed that the hydrostatic stress (i.e., the trace of the plane ...
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Mass conservation relation for compressible flow from a continuity equation
I want to obtain the mass conservation relation: $$\dot m= \rho v A=const\tag1$$ where $A$ is the cross section area and $v$ is the velocity perpendicular to $A$, using a continuity equation of the ...
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Mechanism of Guitar String Tuning: Tension, Length, and Material Properties
I'm trying to understand the precise physical mechanism involved when tuning a guitar string, specifically from the perspective of the string's tension. When we turn the tuning pegs, we intuitively ...
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Why can internal stresses in solids be calculated from external forces, while in fluids they depend only on the velocity gradient? [duplicate]
In solid bodies, internal shear stresses can be calculated in two ways, and these are the simplest forms of the equations: Through external forces: $ \tau = \frac{F}{A} $. Through angular ...
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Euler-Lagrange equation for potential energy functional in continuum mechanics
I want to derive the Euler-Lagrange equation for the potential energy functional $\Pi$, by doing a variation $\delta\Pi=0$ on it, as one usually does. The potential energy functional is: $\Pi = \Pi_i +...
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Energy in Fourier modes for pseudo-quadratic fields
Consider the continuum analog of the Hooke's potential energy, $1/2\,\alpha\, s^2(x)$, where $s$ is the local displacement field and $\alpha$ is the spring constant. The potential energy in Fourier ...
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Why does shear modulus $\mu$ affect volumetric resistance in the Navier-Cauchy equation?
In the Navier-Cauchy equation, $$\mu \nabla^2 \mathbf{u} + (\lambda + \mu) \nabla (\nabla \cdot \mathbf{u}) = -\mathbf{f} $$ Why does the shear modulus $\mu$ influence the volumetric restitution force ...
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Are there any current theories of elastic materials in general relativity, *not* based on a notion of material projection?
[Probably a very niche question, but I hope someone here has useful information.] All treatments I've found so far of elastic materials in general relativity use a notion variously called "...
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