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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Derivation of continuum expression of the first law of thermodynamics

Continuum expression of first law of thermodynamics: $$\frac{D E_t}{D t}=\nabla\cdot({\bf \sigma\cdot v}) - \nabla\cdot{\bf q}$$ (I've seen it in my physics book) How this equation is derived? ...
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Equivalent beam rigidity for a 1D lattice structure

To model the behavior of continua, often discrete lattice models with nodes joined by 2-point spring elements (which resist tensile forces) and 3-point beam elements (which resist bending moments) are ...
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Objective time derivative that is no Lie derivative

Summary Led by an interest into the concept of "Material Objectivity", I am asking myself: Are there objective time rates that are not Lie derivatives? The long read I am trying to understand the ...
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Discrete form of deformation gradient from vectors with finite length

I am writing some code for a deformable mesh and need to calculate a local deformation gradient within the material by using the vectors connecting material points. I think the method of solving ...
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401 views

Momentum of transverse waves on a string

In general, if a wave carries energy density $u$ with velocity $v$, it also carries momentum density $u/v$. I've seen this explicitly shown for electromagnetic waves and (longitudinal) sound waves. ...
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What is Lamè mode?

I am trying to read some of the articles that says Lamè mode. But I can't find in google that describes Lamè mode. Can anyone quote good reference for this term?
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Standing wave on a rope fixed at both sides: minus sign in the reflected wave

I'm studying stationary waves on a rope fixed at both sides. In some books I find that the wave function studied is the sum of incident wave $\xi_1(x,t)$ and of the reflected wave $\xi_2(x,t)$. $$\xi(...
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1answer
67 views

Wave speed of a hanging rope

Let us consider a homogeneous rope hanging from the ceiling. I will call the vertical direction $x$ and the horizontal displacement $y$. When we apply the second Newton's Law to a portion of mass $\...
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35 views

Confused about shear elasticity and complementary shear stress

I am a self learner of continuum mechanic. I am confused about simple shear stress in situation similar to figure 1, in case $F_\textrm{ext}$ is caused by external perturbation by i.e., human, what ...
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60 views

Momentum equation in a Lagrangian configuration

When writing the momentum equation in a lagrangian configuration is the the stress tensor used the first Piola-Kirchhoff stress tensor or the nominal stress tensor (which is the transpose of the 1st P-...
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61 views

Why an impact exerts so much force? [closed]

If an object of velocity $v$ and mass $m$ moves towards a resting object of mass $M$, then if the object which is hit might break. Why? What is the reason that a collision has more power than a ...
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Warping function for torsion of non-circular prism

I have a few questions regarding the case of torsion of a prism, as encountered in continuum mechanics. Specifically, a prism (which can be a cylinder, a rectangular prism, elliptical prism, etc.) has ...
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20 views

Gradient effects in continuum mechanics

What I have learned is that inhomogenous materials (materials with different material properties over space and time) can be treated by the homogenization technique (https://en.wikipedia.org/wiki/...
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26 views

Liquid Flow In A Vertical Tube

A cylindrical vertical tube has uniform cross section $A_1$, and length $l$. It is open at both ends. Water enters from the top with a constant velocity $v_1$, and allowed to flow out from the bottom. ...
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1answer
30 views

An expression for stress power

I have seen it written that for a continuum undergoing deformation, if we ignore body forces and heat transfer, the work done is equal to stress power: $\cfrac{dW}{dt}=\sigma_{ij}D_{ij}$, where $D_{...
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Linear viscoelastic differential operators

I am starting with differential operators: $P = \sum_{i=0}^{N}p_i \cfrac{d^i}{dt^i}$ $Q = \sum_{i=0}^{N}q_i \cfrac{d^i}{dt^i}$ $p_i$ and $q_i$ are functions of time only. $K$ is a constant that ...
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157 views

Why is it said that standing waves do not transfer energy?

The author of my physics textbook writes that standing waves, unlike travelling waves, do not transfer energy. He says that this is because a standing wave is composed of two travelling waves carrying ...
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41 views

How to calculate Lagrangian density function in classical field theory

In Lagrangian mechanics observing the possible degrees of freedom we first write down our Lagrangian. Then we use E-L equation to determine equation of motion and using sufficient boundary condition ...
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35 views

Identity in continuum mechanics [closed]

For a problem in the textbook I am reading, I need to prove that $$\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS,$$ where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a ...
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Pressure vessel analysis of transversely isotropic multilayer material

Suppose I have a transversely isotropic, hyperelastic material with known strain energy that is a fibrous composite. I am interested in an explicit formula for the displacements (so I can get the ...
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51 views

Question regarding the use of the Green-Naghdi Objective Stress Rate

The equation for the Green-Naghdi Stress Rate reads: $\boldsymbol{\sigma}^{GN} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol\Omega - \boldsymbol\Omega\cdot\boldsymbol{\sigma}$ ...
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1answer
43 views

How to prove this material derivative formula rigoriously with experiment or prove it without the chain rule? [closed]

How to prove this rigoriously by experiment or prove it with other mechanics law but without the chain rule? This is coming from reddy's introduction to continuum mechanics. Also please explain ...
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40 views

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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84 views

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: https://www.physicsforums.com/threads/mohrs-...
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26 views

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 different?...
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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 z}{\...
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67 views

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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32 views

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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1answer
29 views

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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38 views

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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32 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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1answer
37 views

Uniqueness 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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42 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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1answer
33 views

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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1answer
92 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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1answer
62 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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1answer
111 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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61 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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417 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 1-...
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1answer
150 views

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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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 ...