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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Interpretation of Evanescent Elastic Waves for Material Damage

In elastodynamic theory, when the slowness vector is imaginary, the resulting elastic waves are called evanescent. I have read that this corresponds to exponential decay. I have also read that complex ...
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Why aren't the lengths of the bars on a toy glockenspiel proportional to the wavelengths?

As you might already know, frequency of musical notes is arranged in a such a way that if, for example, an A note has frequency of $x$, another A note which is placed one octave higher would produce ...
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Parabola or Catenary in this case?

Exhibit A: the flexible film sinks into the box due to lower internal pressure inside the box. question is, does the film form a paraboloid or a 3D catenary or neither? this is the usual method used ...
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Which theoretical models are there between quantum mechanics and cosmology? [closed]

I'm an enthusiast/hobbyist right now and I'm quite curious about the subject of understanding which scales come between the quantum scale (ab initio/first principles) and the macroscopic scale. After ...
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necessary and sufficient conditions for linear elasticity

If it's experimentally observed that a particular elastic isotropic material has a linear relationship between stress and strain for a certain range of stresses and strains, does it follow that a ...
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64 views

the poisson ratio and conservation of volume

In order to measure the Poisson ratio of a rectangular sample of elastic material I subject it to a vertical load along its major axis using weights of gradually increasing importance and measure the ...
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70 views

Estimating Young's Modulus

In order to measure the Young's modulus of a rectangular sample of elastic material I subject it to a vertical load along its major axis using weights of gradually increasing importance and measure ...
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Standing waves on compound string

Please help with this question - No data is given as such. The 2 strings have different thickness. Initially, minimum frequency of the thick string is 120 Hz. Then if we push the cart such that only ...
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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 not a 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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35 views

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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3answers
416 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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80 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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1answer
36 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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1answer
61 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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2answers
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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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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27 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
37 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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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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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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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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99 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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27 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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72 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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1answer
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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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35 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
31 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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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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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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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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1answer
43 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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246 views

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
37 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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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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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 ...