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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Shear flow in J section type beam

What does the distribution of shear flow look like in a J section type beam. I'm only interested in a qualitative picture of it. I'm not interested in the calculations themselves. It is the top of the ...
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1answer
37 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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43 views

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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76 views

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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84 views

buckling of tube - shell thickness vs. momentum of inertia optimum

is there any simple formula (perhabs semi emperical, or aproximatively derived model) for buckling of tube under axial compression load given its crossection and wall thickness? ( and naturraly ...
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1answer
121 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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1answer
241 views

Equations of motion of displacement field

We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\} $$ Where $u_{ij} = ...
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1answer
232 views

Volumetric and Deviatoric Strain Equation in 2D

Strain is defined as $$\epsilon=\frac{1}{2}\left( \nabla u + \nabla u^T\right).$$ I found a formula for the strain tensor in 3D decomposed into volumetric and deviatoric components: $$\epsilon= v + ...
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1answer
24 views

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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1answer
216 views

Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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173 views

Degree of anisotropy of crystal tensors

Does there exist a scalar that can describe how anisotropic the elasticity of a crystal is? What about other tensors such as the permittivity or susceptibility? I found a Wikipedia article that was ...
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71 views

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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35 views

Rate of deformation tensor vs. Rate of strain tensor

I have this conceptual problem(I'll use the notation of Malvern's book): when D=0 where $D_{ik}=\frac{1}{2}(\frac{\partial v_i}{\partial x_k}-\frac{\partial v_k}{\partial x_i})$ or ...
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2answers
214 views

Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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164 views

Strain and stress tensor

I have problem by definition of strain and stress. From Gockenbach's book that our reference for FEM, we have $$\epsilon=\frac{\nabla u+ \nabla u^T}{2},$$ that $u$ is vector displacement, and ...
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76 views

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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1answer
87 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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1answer
32 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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2answers
7k views

what is difference between homogeneous vs isotropic material?

When we say a material is isotropic? When properties such as density, Young's modulus etc. are same in all directions. If these properties are direction dependent, then we can say that the material is ...
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83 views

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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84 views

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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1answer
189 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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1answer
36 views

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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1answer
32 views

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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44 views

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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2answers
94 views

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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40 views

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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32 views

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 ...
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3answers
92 views

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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39 views

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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2answers
76 views

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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32 views

Dynamic mass or static mass?

I am testing a cantilever beam assuming it as a single degree of freedom system, therefore it can be described by the equation $$m_d \ddot{y}(d) + c_d \dot{y}(d) + k_d y(d) = p(t) $$ Where $d$ is ...
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46 views

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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1answer
34 views

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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130 views

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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1answer
377 views

Explain $\rho_{0}\dot{e} - \bf{P}^{T} : \bf{\dot{F}}+\nabla_{0} \cdot \bf{q} -\rho_{0}S = 0$

I am trying to understand the balance of energy -law from continuum mechanics, fourth law here. Could someone break this a bit to help me understand it? From chemistry, I can recall $$dU = \partial Q ...
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2answers
50 views

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 ...
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56 views

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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32 views

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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1answer
28 views

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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43 views

On the isotropy of materials

Good morning. 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 it has. How to do it? ...
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903 views

Symmetry of the $3\times 3$ Cauchy Stress Tensor

When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a ...
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164 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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4answers
191 views

Particles scattering on fluids: breakdown of the effective continuum description

When does the macroscopic continuum description of a medium like a fluid break down? Say I'm interested in a scattering process of some particles with momentum $p$ and energy $E$ off a fluid of ...
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1answer
110 views

Physics of Wrinkling: Understanding inextensibility condition

I'm reading this very cool paper on the formation of wrinkles in elastic materials. The key result of the paper is a set of scaling laws for the amplitude and wavelength of wrinkles based on the ...
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2answers
106 views

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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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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Like viscoelastic polymers, why there are not storage and loss moduli for cast iron?

Viscoelastic polymers have different paths upon loading and unloading, so there is energy dissipation, so they have storage and loss moduli. Plastic behavior is also shown by cast iron: loading and ...
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1answer
59 views

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 ...