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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Continuum fluid theory

It is written in this article: http://www.maths.ed.ac.uk/~yktsang/4520/basic_fluid.pdf "In the continuum model of fluids, physical quantities are considered to be varying continuously in space, for ...
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Why liquids and solids are mostly regarded as incompressible?

In many continuum-mechanical Problems it is assumed that liquid and solid substances cannot Change the total value of volume where it holds $\rho = const, \vec{\nabla}\cdot \vec{v} = 0$. In the 1-...
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What does it mean for shear modulus to be less than bulk modulus?

It is known that Shear Modulus is generally less than Young's modulus for most materials. What does this mean? Does this mean that it is easier to change shape of a fixed body by applying force than ...
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Point forces in linear elasticity and small strains

Consider a point force $\boldsymbol{F}=F\boldsymbol{e}_z$ in an infinite elastic material. In a linear approximation, the displacements can be calculated using Green's function for the Laplacian which ...
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893 views

In wave motion of a string both kinetic energy and potential energy are minimum at $y=y_\text{max}$ then why does the string comes down again?

In wave motion of a string both kinetic energy and potential energy are minimum at $y=y_\text{max}$ then why does the string comes down again? As everything in tries to attain lowest energy possible ...
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Elastic material with exponential behavior?

I know some constitutive models for elastic materials like Neo-Hooke or Mooney-Rivlin, which give a relation between elongation $\lambda=y/y_o$ (where $y$ and $y_o$ are the length of the elastic ...
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Metric Tensor and Strain Rate Tensor- Comparison of Units

Is there any way the metric tensor can have a dimension in general relativity? I ask because there is an equation where the strain rate tensor of continuum mechanics is expressed as a difference of ...
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Derivation of elastic energy per unit volume

So I basically asked this question a little while back and didn't get much help, but I really need help, so I'm coming back and asking again. Looking at the section on Continuum Systems on the ...
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How to do continuum approximation?

Assume you have $N$ matrix fields $T_{j}$ on a 1d lattice with lattice constant unity. Now consider a sum like the following (you can think of the traces as supertraces), and subject it to a continuum ...
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63 views

Why is temperature a function of $y$ and $t$ only?

Say you have an incompressible thermal conducting fluid contained between two infinite horizontal plates separated by a distance $H$. Initially both the plates and the fluid are at rest at ...
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Show that the boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$ [closed]

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by transformation techniques. Any help would be ...
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239 views

Derivative of deformation gradient with respect to Green-Lagrangian strain?

For hyperelastic material, the elastic energy $\Psi $ is related to the deformation gradient $F$ and other internal variables (e.g. temperature $ \theta$) In many literatures (including Malvern's and ...
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Jaumann deviatoric stress rate

Background about terms in this question: Hookes law and objective stress rates From my understading, the Jaumann rate of deviatoric stress is written as: $$dS/dt = \overset{\bigtriangleup}{{S}} = {\...
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Can a building get taller at night?

UberFacts recently tweeted "Office buildings are taller at night—a 1,300-foot-tall skyscraper shrinks about 1.5 millimeters under the weight of 50,000 occupants." Is what they are saying valid? It ...
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Terminology: Gauge stress?

When a material is loaded with a force, the stress at some location in the material is defined as the applied force per unit of cross-sectional area. If I have a material submerged in pressurized ...
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227 views

Understanding incompressibility in continuum plasticity

I am a beginner in continuum plasticity and wondering physical meaning of incompressibility in continuum plasticity. Referring to MIT OCW link the consequence of incompressibility condition is (eq ...
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854 views

How to determine plastic strain rate

Equivalent plastic strain rate is defined as $$ \dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p} } $$ Where, $ \dot{\bar{\epsilon}}$ is equivalent plastic strain ...
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Origin of a unique stress-strain relation

In a paper by Kees Wapenaar titled, "Retrieving the Elastodynamic Green's Function of an Arbitrary Inhomogeneous Medium by Cross Correlation" (2004), the following is stated: "In the space-frequency ...
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Seeking Reference on Transport of Momentum by Diffusion of Mass

I'm looking at continuum mechanics from the perspective of De Groot and Mazur's "Non-Equilibrium Thermodynamics" - the first reference that I've come across that seems to do a good job of bringing ...
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What is the meaning of symbols $\delta f$ and $\delta^2f$?

Professor was using these symbols to derive the continuity equation. He defined the infinitesimal mass as $\delta^2m=\rho \delta V$ and the mass that leaves some closed boundary $\partial V$ as $\...
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Does the definition of compressibility depend on the frame of reference?

According to many authors, a fluid is defined to be incompressible if the material derivative of the density $\frac{D\rho}{Dt}$ is zero, that is to say, that in an frame of reference following the ...
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219 views

Variation in spring constant with respect to the length and no. of coils

Do the spring constant depend upon the length of the spring? No. of coils? Like what happens to the spring constant if you cut it in the half?
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651 views

Why is elastic modulus greater than shear modulus?

I was looking at data for elastic modulus $E$ and shear modulus $G$, and found that $G$ is always lower than $E$. So I'm wondering what are the underlying principles that may be the cause of this. $$...
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In continuum mechanics, why is the stress vector $T=\sigma\cdot n$ not a covector?

In continuum mechanics, the stress vector (see Cauchy stress tensor) $T=\sigma\cdot n$ is the surface density of a force. Forces are covectors, since they map a displacement vector to a scalar energy. ...
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How to derive this equation? [closed]

How to derive this equation? It comes from a book named Seismic Wave Propagation in Stratified Media.
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What does the continuum hypothesis of fluid mechanics mean?

I'm a bit confused by the continuum hypothesis stating that fluid are continuous objects rather than made out of discrete objects. Say for $\rho (x,t)$ (density) is there more than one fluid ...
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665 views

Continuity equation in fluid mechanics

The continuity equation in fluid mechanics states that $$ \frac{\partial\rho}{\partial t} + \nabla\cdot(ρ\mathbf u)=0 $$ Can you explain to me what is the physical meaning of each term of the ...
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Why are stress forces considered as acting on a cross-sectional area through a solid?

I'm trying to understand the Cauchy-Stress tensor, in which the stress acting on a body at a point is analyzed by considering the cross-sectional area through which a force passes. And my question is -...
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353 views

Derive Equation For a Cantilever in SHM

I am currently investigating how a hacksaw blade's time period of oscillation changes when I add mass to the end of it or when I change the length it is clamped at. I found the following equation ...
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Can surface waves exist near a fixed surface

(I'll phrase this question in terms of waves in an elastic medium, but this is a more general question.) Surface waves are waves near the surface of a medium whose amplitude decreases as you go away ...
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Complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines. We need to take the value of m=1. I know how to find the stream function and velocity potential, ...
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Where does this formula for sagging of a beam come from?

In one of my physics textbooks there is a chapter on the elasticity of materials which contains pretty basic outline about Young's modulus, stress-strain, elastic potential energy and related stuff. ...
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Variational calculus, bending a stick and stationary states

We have a horizontal stick, one of its ends is on the wall, and we can apply a force to the other end. We assume that anything that we can do will leave this in the same plane. Our question is to ...
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Debye-Huekel Theory and the continuum approximation

This question stems from a problem I was doing on the Debye-Hueckel theory. It says that the continuum approximation which underlies the Debye-Hueckel theory is valid provided that $\lambda_D \gg r_{...
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origin of the major symmetry property of the elasticity tensor

In linear elasticity theory the stress tensor $\sigma$ is related to the strain tensor $\epsilon$ via the elastic tensor $C$. Specifically $$ \sigma_{ij} = C_{ijkl} \epsilon_{kl} $$ Because $\sigma$ ...
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Continuum Wave Function for the electron

I'm trying to understand certain processes like the photoelectric effect and Bremsstrahlung. In Bremsstrahlung I need to use the wave function of an electron coming from the continuum, and there is ...
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Understanding incompressibility (of rubber or viscoelastic material)

Literature gives a lot of explanation why rubber is incompressible. However, I still need some thinking to understand physical behavior of rubber or any such material. Often, incompressibility is ...
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Golf ball impact

A golf ball is said to be "compressed" when hit by a golf club and makes a characteristic "thwack-hiss" sound coming off of the club when impacted by professional golfers (whose impact conditions have ...
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Airfoils contradict the law of the lever?

The law of the lever says that "the less force you use, the more distance you have". It is often exemplified by referring to simple machines, but it should apply to all technical systems. But I do not ...
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Meaning of boundary conditions in solid mechanics

The Question is: A uniform horizontal beam OA, of length $a$ and weight $w$ per unit length is clamped horizontally at O and freely supported at A. The transverse displacement $y$ of the beam is ...
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Can $U_{ij}$ or $v_{ij}$ in continuum mechanics be negative?

In continuum mechanics, we have the deformation gradient $\mathbf F$ to be: $$d\mathbf x = \mathbf F d \mathbf X$$ And then, we do a polar decomposition (A good reference here would be http://www....
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Estimate the persistence length of a rubber band [closed]

Not much more to say here, it's all in the question. The best, most convincing estimate will be chosen as the correct answer. EDIT: Assume the rubber band is at room temperature, with thickness $t$ ...
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Relation between area elements in finite deformation theory (continuum mechanics)

There are relations for the line and volume elements in continuum mechanics. For example: \begin{align} \ \ \ \ \ \ \ \ \ \ \ \ \frac{V}{V_0}&={\rm det}(F)\tag{1}\\ \lambda^2&=(F^TFe_1\cdot ...
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Deriving general boundary conditions from first principles for elastodynamic scattering

It seems that most of the relevant books only give the linear case and the rest say something along the lines of "here are common examples of boundary conditions." What are the most general boundary ...
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Problem on bending plates in Newtonian Mechanics?

I am reading a book on interesting physics problems and demonstrations. One of the problems in the section on buildings, structures and equilibrium talks about a plate with one side attached to the ...
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On the isotropy of materials

I am working on honeycomb structures and first of all I would like to understand whether it is isotropic or not, and, if the latter holds, which kind of anisotropy does it have? How to do it? I don't ...
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385 views

Find the maximum allowable bending moment

A rolled steel universal I-section beam with a serial size of $406\times178$ has a mass of $60$kg/m. What is the maximum safe allowable bending moment this beam can sustain,given that the maximum ...
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Is there any viscoelastic model directly specified from strain energy function?

For most models I've seen, including the classic quasi-linear viscoelastic model, the parallel network viscoelastic model ABAQUS uses, and the model described in Holzapfel & Gasser 2000, they all ...
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Is there a way to calculate strain energy based on stress and deformation gradient?

We know that we can obtain stress from strain energy density and deformation gradient, for example: $$\mathbf P=\frac{\partial W}{\partial \mathbf F}$$ However, is there a way to calculate $W$ from $...
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Superior attachment of Möbius strip

In this DIY project, the Möbius strip is used to make a spill-proof coffee cup carrier. The author uses a Möbius strip as the handle of this carrier and says If you attach a Möbius strip to an ...