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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What is a general definition of bulk modulus?
For a perfectly elastic body, Bulk modulus always remains constant and is defined as,
$$B=-V_i \frac{\Delta P}{\Delta V} \tag{1}$$
Which means,
$$B \left(\frac{V_f -V_i}{V_i}\right)= -(P_f-P_i)$$
But, ...
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Calculate bending moment for cantilever with uniformly distributed load
I am trying to understand how to calculate the bending moment for a cantilever with a uniformly distributed load so that I can build an equation of moments, as shown in this example:
I tried ...
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Surface integral Vs Volume Integral
Can I rewrite the continuity equation like this ? :
$$
\iiint\limits_{V}\dfrac{\partial\rho}{\partial t}\mathrm dV +\iiint\limits_{V}\rho\boldsymbol{\nabla\cdot}\mathbf v\,\mathrm dV + \iiint\limits_{...
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What shape does a rope make when compressed / coiled up?
Let's say we have a rope placed on the line $y=0$ on a flat table in the $(x,y)$ plane. We place one hand on the rope at $x=0$ and $x=l$ and then move our hands together along the $x$-axis so that ...
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Is natural frequency a local or global property?
Some objects have a natural frequency. This can be anything from a metal ball to a table, etc. When we hit such an object, it will start vibrating with a certain frequency $f$. Because of damping the ...
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What factors does a Spring depend on?
When we consider an ideal spring, the force applied by the spring is proportional to its extension $f_{sp} = kx$. Does the same apply in real life?
So I took it into my own hands. I got a spring ...
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Is it possible to label particles of a continuum body?
In basic continuum mechanics (e.g. fluid dynamics), we label particles of the continuum, i.e., each particle can be identified by a label, e.g., $p$. Then other quantities are defined accordingly, e.g....
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Given a stress tensor, how can you find the strain rate tensor?
What is the relation between stress and strain rate? If I had this as a stress rate:
$$\mathbf \epsilon= \left[ \begin {array}{ccc} 0& \tau_{{xy}}\ &0\\
\tau_{{yx}} & gh &0\\ 0&...
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How is it that on loading a spring, shearing stress is produced?
As far as I know, shearing stress is produced when layers shift parallel to each other, which means there is change in shape, and the layers are displaced in the direction of force applied. I imagine ...
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Does a locally volume-preserving deformation also preserves volume globally?
Is a locally volume-preserving deformation (in which the determinant of the deformation gradient (a.k.a Jacobian) is unity) is also globally volume preserving?
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Point load distribution inside elastic solid continuum medium in finite elements method
In finite elements method, when point load is applied to a particular node of elastic solid continuum medium (e.g. soil), does it affect nodal forces in the rest of the mesh (i.e. does each node ...
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Deformation Gradient in Fluid Mechanics and the Reynolds Transport Theorem
Fluids can be characterized by the fact, that initially neighboring material fluid particles do not remain neighboring during the course of deformation.
The deformation gradient in continuum mechanics ...
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Computing the strain in a cantilever beam under a known deflection
I intend to use a cantilever beam to calibrate a strain gauge, in a setup similar to the picture.
My idea is to introduce a small known vertical displacement $\delta$ on one side of the cantilever ...
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Cross section of a circle of elastic material under a point weight
If I've got a circle of elastic material that's fixed at the circumference, and I drop a point weight on it, what will the cross section look like?
For example, if I took some material from a balloon, ...
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Non-radial forces and the conservation of angular momentum
Previous discussions on this forum regarding the derivation of the law of conservation of angular momentum from Newton's Laws have pointed out that it supposes the strong form of Newton's Third Law. ...
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Why minus sign in definition of angular momentum field density tensor?
The following generalization of angular momentum is given on page 571 of the third edition of Goldstein's Classical Mechanics:$$\mathcal M^{ij}=-(x^iT^{j0}-x^jT^{i0}).\tag{13.44}$$The metric used in ...
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Why does the aggressively bowed string go sharp?
A vibrating string with fixed endpoints, such as on my fiddle, may be bowed (see Helmholtz motion, see stick and slip) with very little to a certain amount of pressure and proximity to the bridge (the ...
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How to calculate wave equation from a stretched string?
I am reading "Introduction to Electrodynamics" [Griffiths] and in section 9.1.1, there is an explanation for why a stretched string supports wave motion. It begins as follows:
It identifies ...
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Gradient Operator in Vector Spherical Harmonic Basis
The vector spherical harmonic basis (vector generalization to the scalar valued spherical harmonics) is a convenient spectral basis for problems involving vector fields with spherical symmetry (link ...
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What do we call a material property that has non-equal values when evaluated in opposite directions?
What do we call a material property that has non-equal values when evaluated in opposite directions?
That is, if the material property $k_{ij}$ has a value of $X$ along the direction defined by the ...
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Not all anisotropic porous media can be associated with a symmetric permeability tensor $K_{ij}$?
My intuition tells me that permeability $K_{ij}$ of Darcy's Law is symmetric, $K_{ij}=K_{ji}$, and I am looking for an answer to show me why this is not the case.
When measuring the permeability of a ...
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How to obtain spatial density from material density?
When you introduce the concept of the spatial density of an extensive property, a question of change of variables arises. Assume $\Omega$ is an extensive property, and $\omega$ and $\rho$ are its ...
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Buckling Question from Hibbler Textbook
Hello! I have a question from the Hibbler Textbook on Mechanics of Materials! The exact statement of the problem reads:
"A W150 * 24 steel column is 8 m long and is fixed at its ends as shown in ...
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Does theory impose a limitation on the size of the sample to be used for measuring permeability?
Theory imposes no limitation on the size of the sample to be used but
it is evident that in order to minimize effects due to local
inhomogeneities in the material such as concretions, small shale
...
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At what speed does information move through the atoms of a rigid object?
How fast does information travel on particles? I thought if you move a iron bar from one end it would take the speed of sound to move its other end. For example, theoretically if you hold an iron bar ...
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Uniquness of stress tensor
I am struggling to understand how to derive a stress tensor. In the picture above, I was asked to compute the stress tensor and the stress acting on the plane $\Sigma$. The x-axis and y-axis are in ...
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If the curl of the gradient is always zero why isn't it in vorticity definition? Kosterlitz - Thouless - Berezinsky topological transition
Is a well estabilished property that the curl of a gradient is always zero (i.e. $\nabla\times\nabla\Phi=0$) and it's possible to prove it in many ways. e.g.
If $(\nabla\times\nabla\Phi)_i = \...
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What's the difference between constitutive laws and equation of state?
While defining material properties in finite element modeling, when should we opt for constitutive models, such as Linear Elastic or Neohookean (that relate stresses and strains) over Equation of ...
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How is the work done by rotational dry friction distributed over the surface?
I stumbled upon the phenomena of pushing a door without closing it and it came back to me a little. I suppose it could be explained by pressure increasing in the other room for a moment or the doors ...
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Torque profile with fixed boundary condition
I am trying to find a simple treatment of elasticity theory of a scenario where I have a thin rod attached to a disc (radius $r_0$), and the disc is held fixed at its edge. Now I apply some torque to ...
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Why is the density in GR equal to $\rho_0\dfrac{dx^0}{ds}\sqrt{-g}$?
In general relativity, the continuity equation says
$$
\partial_{\mu}\left(\rho_0c\dfrac{dx^{\mu}}{ds}\sqrt{-g}\right) = 0
$$
with $\rho_0$ being the proper density, as seen by an observer who is at ...
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What is the the Implication of Navier-Stokes Millenium Problem to Practice?
As a student of meteorology, I wonder why Navier-Stokes equations (NSE) are still not understood in terms of whether or not there are unique solutions. In atmospheric dynamics, NSE is used as a basic ...
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Different Bernoulli equation from $F=dp/dt$ [closed]
If any part of the question needs clarifying, there might be explanation in the post-script and of course ask if needed.
I understand how the Bernoulli equation is derived for incompressible fluid in ...
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Compressible fluid equation
We know the continuity equation of a continuum (in this case I want to discuss fluids, equation reference):
$$\frac{\partial \rho}{\partial t} + \nabla . (\rho u) = 0$$
where $\rho$ is the mass ...
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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 ...
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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 ...
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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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In spatial description, why should Eulerian coordinates keep changing in material derivative?
I feel confused about spatial description, the text An introduction to continuum mechanics by J N Reddy says,
"For a fixed value of $\mathbf{x}\in\kappa$ (the current configuration), $\phi(\...
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The material time derivative of Jacobian of the deformation gradient
The key step in the derivation of Reynolds transport theorem is time derivative of $J$, the determinant of deformation gradient $F$. Its result says
$$\dot{J}=\frac{\partial J(\xi,t)}{\partial t}=\...
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Something wrong in Saint Venant–Kirchhoff model?
According to the Saint Venant–Kirchhoff model, the strain-energy density function is defined as
$$
W(\boldsymbol{E})=\frac{\lambda}{2}[\rm tr(\boldsymbol{E})]^2+\mu\rm tr(\boldsymbol{E}^2)
$$
$\...
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Is there specific form of Navier-Stokes equation for which mass can cross bounding surface?
In my textbook, we learned that Navier-Stokes (NS) equations can be derived from Reynolds transport theorem where the control volume is assumed to be fixed. But when the control volume is moving, can ...
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Question about the general Schmid’s law expression to calculate the critically resolved shear stress
I have a question about Schmid’s law for an arbitrary stress state. I found conflicting expressions and I would like to know which one is correct.
$\boldsymbol{s}$: slip direction
$\boldsymbol{n}$: ...
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Equations of motion for two masses connected by the Kelvin-Voigt Model
I have a system where two particles $x_1$ and $x_2$ in one dimension are connected by a spring and a dash in parallel. This is analogous to the Kelvin-Voigt model for viscoelastic materials. The two ...
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Regelation, and melting ice with pressure
It is an experimental fact (regelation) that if two weights are hung on the ends of a rigid bar, which passes over a block of ice, then the bar gradually passes through the block of ice. Moreover, the ...
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Volume Conservation in Shock Plastic Waves [closed]
SOLIDS: Plastic deformation is known to have constant volume. During shock, does the plastic wave not compress the volume? Should the only compression come from the elastic wave?
Edit-1: screenshot
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How does restoring shear forces arise (in elastic conditions)? Do they arise from central forces or not?
When you apply a shear force onto a solid piece of material (say a block on a surface or a cantilever
beam with a load) that creates shear stress in the elastic regime, there is a restoring force that ...
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How do I define the initial jerk and yank of a cylinder subjected to impact of a rigid body at it's base?
I am trying to solve an axisymmetric longitudinal wave propagation problem during the impact (collision) of a rigid body and the base of a cylinder. On one end the cylinder is rigidly bound to the ...
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Would anyone be able to provide a reference for the equations concerning plane strain and incompressibility?
I've been trying really hard to find a textbook or research paper that mentions the equations I mentioned in my question. Sadly, I haven't had any luck so far. Would it be possible for someone to ...