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 the intuition behind this acceleration formula?

What is the intuition behind this acceleration formula? In another word, how to demonstrate this by using common sense without using chain rule?
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Why is maximum shear stress at 45 degrees instead of near 0?

I think I might havr an immensely stupid question but it's really bothering me so please be patient. I am not physicist smart. Look at this guy's question: ...
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How is a unidirectional lamina transversely isotropic?

What I don't understand specifically is that if there happen to be more fibers in the $x_2$ direction than the $x_3$ direction, wouldn't that make the material properties in those directions ...
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Equation Governing Small Lateral Deflections z of a Uniform Membrane

The equation governing small lateral deflections z of a uniform membrane subjected to a lateral (dimensionless) pressure p is given by $$ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 ...
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Infinite elastic half-space with point load (Mindlin's problem)

What is the equilibrium deformation of an infinite half-space (that is, an isotropic and homogeneous linearly-elastic three-dimensional medium, with a single planar surface) produced by a force which ...
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Is deviatoric strain associated with thermal effects?

Does temperature have any effects on deviatoric strain for a linearly elastic isotropic material?
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From in-plane strain to Poisson ratio

I have recently been trying to simulate a graphene cantilever and thus I need to know the Poisson ratio, Young's modulus, and density. From literature it is easy to find the Young's modulus and ...
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Traveling wave solutions for an irregular “waveguide”

I'm looking at solutions for the wave equation $$\frac{\partial^2 z}{\partial t^2}=c^2 \nabla^2z,$$ in a finite 2D domain. Say that I have periodic boundary conditions on the left and right edges ...
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How to derive an expression for entropy generation in a diffusive, reacting continuum

I'm trying to understand a derivation from "The Thermodynamics of Linear Fluids and Fluid Mixtures," by Miloslav Pekař and Ivan Samohýl (2014). The derivation produces an expression for the entropy ...
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28 views

The force of a spring

I am new in continuum mechanics and I want to prove the formula which gives the force given by a spring : $$F_{max}= \frac{Ed^4(L-nd)}{16(1+\nu)(D-d)^3 n}$$ where : $E$ – Young's modulus $d$ – ...
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28 views

Unicity of a stress (only) boundary value problem

A static problem in linear elasticity is typically written as the following boundary value problem: find $\boldsymbol u$ and $\boldsymbol \sigma$ such that: $\text{div} \boldsymbol \sigma + ...
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physical meaning of major symmetry of the stiffness tensor

What happens if a stiffness tensor does not have the "major symmetry" $C_{ijkl}=C_{klij}$? Background: In linear elasticity (generalising Hooke's law from a spring to a continuous medium), the ...
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What is “Accumulated plastic strain rate” in Current yield Norton law?

I'm doing FEA of steel under high strain rates and using Elasto-ViscoPlastic material model, with Von-mises yield criterion along with Isotropic hardening. The strain rate sensitivity is addressed by ...
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38 views

Difference between mechanical modes and phonons

As stated in this review article: Mechanical modes are long compared to the interatomic spacing. It is natural to make the distinction between nanomechanical modes and phonons: The former are ...
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assumptions about sound waves

When deriving the sound wave equation: $${1 \over c^2} {\partial^2 p' \over \partial t^2 }= \Delta^2 p' $$ by linearizing the Euler equation: $$\rho {d v \over dt }= - \nabla p $$ and the continuity ...
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What is meant by the Laminar boundary layer equations?

I have a question and it is to briefly explain (do not derive) the laminar boundary layer equations. I need to understand what the underlying ideas and how the equations are employed. Any help would ...
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Reference values for viscosity and density in incompressible NSE

I come from a pure mathematics background, so I have very limited physics knowledge. I'm currently working out the non-dimensional form for the Navier-Stokes equations and have some questions. Where ...
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78 views

Why did we make equations dimensionless? [closed]

I study a paper on propagation of plane wave, in which equations are made dimensionless. Equation of motion is \begin{equation*} c_{ijmn}u_{m,nj} = \ddot{u_i} \end{equation*} where $c_{ijmn}$ are ...
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Determine the pressure difference required to drive a prescribed constant volume flux $Q$ through a gap

Determine the pressure difference $P_{1}-P_{2}$ required to drive a prescribed constant volume flux $Q$ per unit width through a gap of thickness $\delta$ of length $L$. To do this introduce ...
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44 views

Internal energy and particle fluid

We know that the property of a fluid at a point is the mean of this quantity over a small volume centered around this point. For internal energy is it also the mean or it is the sum of the internal ...
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88 views

Derive the Boltzmann factor in classical statistical mechanics

In both quantum and classical statistical mechanics, the probability of an NVT system having an energy $E$ is proportional to $$ p(E)\propto e^{-E/T} $$ However, all of the derivations (that I can ...
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52 views

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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101 views

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 ...
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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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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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136 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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176 views

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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116 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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26 views

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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473 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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55 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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131 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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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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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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124 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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377 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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103 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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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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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 ...