# 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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### Does a uniform loading of an elastic half space result in a uniaxial stress state or a uniaxial strain state?

Suppose for instance a soil is loaded by a building over an area of length $L$ (load is in the $z$ direction). In the neighborhood of a point at depth $h$, $h \ll,L$, in the soil under the loaded area,...
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### Are the stress and strain tensor covariant or contravariant?

My question is related to this question but I don't find the answer there to be completely satisfactory. The displacement of an elastic medium is a contravariant quantity, which I think is pretty ...
1 vote
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### What is the gradient of deformation gradient $F$?

Deformation gradient is defined as $$F_{iJ}=\frac{\partial x_i}{\partial X_J},\;\mathbf{F}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}},$$ where $\mathbf{x}$ is spatial coordinates; $\mathbf{X}$ is ...
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### Question about the elasticity matrix in metals

The most general anisotropic linear elastic material has 21 elastic constants. I am working with an HCP material and I found that it has 5 independent elastic constants. I am programming a subroutine ...
1 vote
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### In-plane stresses on the surface of a cylinder

The three principal stresses on the surface of a cylinder are the hoop, $\sigma_\theta=\frac{pR}{d}$, longitudinal, $\sigma_z=\frac{pR}{2d}$, and radial, $\sigma_r=-p$, stresses. However, what are the ...
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### Problem identifying type of equation (linear/nonlinear)

I've looked at the answer to this Math.SE question, but I still can't know the answer to my question here. The following is the equation of equilibrium: divergence of stress tensor that is the sum of ...
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### Conversion of fourth-order tensor multiplication from indices notation to matrix form

Considering a 2D fourth-order tensor $C_{IJKL}$ which can be represented in Voigt notation as:  C_{IJKL} = \begin{bmatrix}C_{1111}&C_{1122}&C_{1112}\\ C_{2211}&C_{2222}&C_{2212}\\ C_{...
1 vote
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### Understanding proof of stress tensor symmetry

Hi I'm trying to understand basic physics but with a more formal scheme. I'm reading P.K.Kundu book of mechanical fluids. In page 90 he proves that stress tensor is symmetric. But first applies ...
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### Why are the shear stresses corresponding to adjacent faces of the stress element equal?

I am studying the definition of the stress element from the book Theory of Elasticity from S. Timoshenko and J.N Goodier. In page 3 it is shown that from taking the moment of forces acting on the ...
1 vote
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### What do the different elements in strain tensors tell us?

I'm working with strain tensors of all sorts at the moment, and I think I've understood how they're derived. However, I'd like to get more intuition of what they're actually telling us. More ...
1 vote
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### Three dimensional classical continuum limit, wave equation

Many textbooks of classical mechanics or classical field theory mention that a three dimensional "string" (the continuum limit of a lattice) leads to/can be described by the 3 dimensional ...
1 vote
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### What are the references of this type of 2 dimensional Shallow Water Equation?

I'm reading the Wikipedia of Shallow Water Equations(SWEs) in 2 dimensional case. https://en.wikipedia.org/wiki/Shallow_water_equations I notice that the 2d case in the page was stated as: But there ...
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### Mass conservation in a deformed membrane in cylindrical coordinates

This is clearly an obvious question but here is my issue. Context : We assume an axisymmetric deformation of a membrane, and work with cylindrical coordinates $(r; \phi; y)$. At time $t = 0$ we let $r$...
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### Boundary Condition for conservation of mass

I'm interested in modelling the length of a rod as it's gradually heated up and shrinks in length and gets more dense, the rod is porous, and therefore compressible, and the density is variable ...
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### What really happends with the radial displacement at the origin of a disk and a cylinder under dynamic, uniform excitation?

I am trying to understand some properties of linearly elastic symmetric systems. Specifically, in the polar and cylindric coordinate systems. To be concrete, I am trying to understand the displacement ...
1 vote
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### Using Hertzian contact mechanics to explain fracture cones when an elastic sphere is pressed onto an elastic half space

I'm trying to understand the Hertzian cone fracture process from a continuum mechanics point of view. I'm considering a problem where an elastic sphere is pressed quasi-statically onto an elastic semi-...
1 vote
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### Critical force for a system comprised of a compressible and incompressible parts

What is the critical buckling force needed to be applied on a system of made out of two parts? The parts of the system are as depicted in the picture: incompressible elastic beam - on top ...
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What is the correct form of the momentum balance equation for a continuum body $\mathscr{B}$ whose particles are fixed, and occupies volume $V(t)$ at time $t$? \begin{align} &\frac{\mathrm{d}}{\...