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 ...
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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 ...
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How to derive properly the stress-strain relation of an hyperelastic material?
I'm trying to implement a Yeoh hyperelastic model for a FEM simulation. The program requires a function that returns the second Piola-Kirchhoff stress tensor $\mathbf{S}$, with the Green-Lagrange ...
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Internal energy has excess translational energy excluded from it? [duplicate]
So I was going through this answer which states:
In a fluid, the internal energy is the sum of the internal energy (in
turn, the sum of the kinetic and potential energy) of each molecule
And I've ...
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Nonrelativistic limit of perfect fluid equations $\partial^a T_{ab} = 0$
On page 62 of his book “General Relativity”, Robert Wald writes the stress-energy tensor $T_{ab}$ for a perfect fluid in the form
$$T_{ab} = \rho u_au_b + P(\eta_{ab} + u_au_b), \tag{1}$$
and then ...
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The expression of elastic energy in this paper
Elastic properties of $\rm{Ni_{2}MnGa}$ from first-principles calculations
Hello,
I am reading a paper investigating the linear elasticity of a crystal. However, I am a little bit confused over the ...
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Is the expression of elastic energy in this paper correct?
Elastic properties of Ni2MnGa from first-principles calculations
I am reading a paper investigating the linear elasticity of a crystal. However, I am a little bit confused over the expression of ...
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Prove that an isotropic substance has only two elastic constants
I've been following the derivation from Ch. II-31 of Feynman's Lectures, and there is a bit that I don't understand. We start from the relation between the strain $T_{ij}$ and the stress $S_{ij}$, $$...
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What does each term of the deformation gradient tensor represent?
I am taking a course in introductory continuum mechanics, and we were being taught displacement and deformation gradients. I'm having a tough time understanding what each term of the deformation ...
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Forces of an elastic closed string
I want to calculate the forces acting on an element of a closed elastic string and derive a wave equation on a closed string. For this I have already figured out that if the closed shape of the string ...
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What does it mean for a material's elasticity to be non-linear?
Hooke's law only applies to materials with linear elasticity, usually for small displacements.
Now, if you imagine having a material that does not deform permanently when crossing a specific limit, ...
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Commuting material time derivative and material space derivative
Let's note $x$ the coordinates in the current configuration and $\nabla$ the associated gradient; similarly, let's note $X$ and $\nabla_0$ for the reference configuration. I will also note the ...
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How to solve dual boundary condition contradiction at the corner of an axially loaded cylinder?
Subject: Linear elasticity
Consider an isotropic elastic cylinder rigidly bound to the surface on one end. The other end is under uniformly distributed static load (pressure for example).
I am trying ...
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What exactly is a linear-elastic material?
Imagine a large, soft, 3D linear-elastic medium containing a small, hard object in the middle (e.g., a marble). If the marble is displaced from equilibrium and then released, it will oscillate back ...
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Why is a tetrahedron used in the derivation of Cauchy's stress formula
After looking at the derivation of Cauchy's stress formula, stated as:
$$ \vec{t} = \sigma \hat{n} $$
I've reflected upon the derivation and some questions have arised. I state them below:
First of ...
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Adjustment to finite difference scheme for continuity equation in axisymmetric coordinates
I am working on a problem in which I need to numerically solve the continuity equation in an axisymmetric coordinate system (i.e. cylindrical with no $\phi$ dependence). For concreteness, I will use ...
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Derivation of velocity profile between two parallel cylinders
Recently, I started learning about different flows such as Coutte, Poiseuille and Hagen-Poiseuille. When searching for the latter, I found this interesting image:
I thought that I might be able to ...
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Sign of the stress vector applied on the surface of a circle
In the following problem, concerning the stress vectors applied on the surface of the circle, why is there a minus sign in $t(- ê_r) = t(− \cos θ\ ê_1 − \sin θ\ ê_2)$?
In this problem, loads are ...
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Conceptual problem with incorporating constraints to a particular variational principle problem
Consider the following problem:
A vector field $\boldsymbol{F}(x)$ is defined over a finite region $V$. A functional of the form
\begin{equation}
U = \int_V u(\boldsymbol{F})\ d^3x
\end{equation}
is ...
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Does Reynolds transport theorem apply to all properties dependent on mass?
I'm looking into Reynolds transport theorem (RTT) stated as:
$$ \frac{dB}{dt} = \frac{d}{dt} \int_{CV} b\rho dV+\int_{CS}b\rho(\underline{v}\cdot\underline{n}) dA$$
where B is defined as an extensive ...
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How to prove that a drop of water in the weightlessness of space is round in shape?
How to prove that a drop of water in the weightlessness of space is round in shape theoretically? More specifically, how to prove that a drop of water in the weightlessness of space is round in ...
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Equilibrium of rods and stress
Reading through Theory of Elasticity, Landau & Lifshitz, I got stuck in the The equations of equilibrium of rods, on page 82. The part I do not understand is the following:
"We denote by $\...
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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_{...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 formulation of momentum balance for a body of continuum?
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}}{\...
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How does the stress vector change with respect to the radial coordinate at the radial surface of a cylinder?
Consider a linearly elastic vertical cylinder in the gravitational field with its base attached to a rigid wall while its other base is under uniformly distributed time-varying pressure. Its radial ...