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Questions tagged [continuum-mechanics]

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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2D Linear Elastostatics with Displacement Boundary Conditions

I am new to this type of problem and feel as if I am going in circle with regards to boundary conditions. I am interested in finding an analytic solution for: $\mu\nabla^{2}\underline{u}+(\lambda+\mu)...
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How are fluid membranes stretchable?

The Poisson's ratio of fluid lipid membranes is exactly $\nu=0.5$, because fluids can flow. As answered in this question, under the assumption of a finite bulk modulus $K$, this is the value of $\nu$ ...
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Aggregation phenomena : How to get from a discrete to a continuous point of view

I'm studying a diffusion limited aggregation phenomenon. The $N$ particles diffuse in a box and when there is a contact they stick with a probability $p$, and let's say to simplify $p=1$. Meaning that ...
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Divergence of a displacement vector field multiplied by delta function

I'm trying to work out why $$ \boldsymbol{\nabla\cdot[u}\,\delta^3(\mathbf{r})]=0, $$ where $\boldsymbol{u}$ is the displacement field of a source of stress, $\boldsymbol{\nabla\cdot u}\ne 0$, and $\...
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Are acoustic phonons always the lowest energy vibrational modes in solids?

In solids with unit cells containing more than one atom, the normal modes show acoustic and optical branches. The number of optical branches is proportional to the number of atoms in the unit cell, ...
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What type of stresses occur during bending of steel rod? [closed]

When an external weight (and/or sometime by weight of its own) is tied to a horizontal metal rod, whose ends are supported by two opposite walls, the rod sags, bends due to the weight. My question is ...
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SHM time period with spring mass [duplicate]

Prove that time period of SHM for spring block system for small displacement is $T=2\pi\sqrt{m_{eq}/k}$ where $$m_{eq}=m_{block}+m_{spring}/3.$$ I am able to prove for massless spring (by both ...
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Overdamped Approximation

I am reading the paper Information transfer and behavioural inertia in starling flocks (Attanasi et al, 2014) in the field of active matter for an undergraduate course I am taking. The paper describes ...
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What causes the wave velocity of longitudinal waves in a spring to changes as its length changes?

This video shows the velocity of longitudinal waves changing in a slinky spring as its length changes: https://youtu.be/y7qS6SyyrFU?t=17 Something else responsible for the change in velocity ...
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General question about making differential equations dimensionless

Suppose you have a set of differential equations that you wish to normalize/make dimensionless. From what I've seen, you can usually use dimensional analysis to figure out a good choice of constants ...
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Lorentz Reciprocal Theorem: Why No Pressure

So in my reading on the Lorentz reciprocal identity, I got lost in a step. Why can he remove the pressure term with the continuity equation? Any advice or guidance would be great! Pozrikidis
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Guitar strings struck out of phase

Do guitar strings struck out of phase with one another force each other to begin vibrating in phase with one another? I ask because wouldn’t chords sound more dissonant from time to time if this did ...
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Force analysis of falling spring [duplicate]

As picture below, I want to explain why the bottom of spring will be motionless a little while after release. But in my view, the bottom of spring will be falling in a low speed after release, since ...
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How to calculate the max load a metal bar can hold?

How would one calculate the amount of weight a steel bar could hold before breaking? Apologies for the terrible diagram. So if I had a steel bar of the length 18in, and the cross-section with a ...
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Relation between strain and velocity

The strain tensor writes $\epsilon_{ij}=\frac{1}{2}\Big(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_i}{\partial x_j}\Big)$ with $u_i$ the displacement in the $i$ direction. Then $\frac{\...
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Integral formula for inertia tensor

Writing down the balance of angular momentum, we introduce the inertia tensor by the formula \begin{equation} J(t)a \cdot b = \int_{S(t)} \rho (t,x)\left( a \times \left( x - X(t) \right)\right)\cdot ...
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Continuity equation in the Lagrangian flow picture approach

In deriving continuity equations using Lagrangian. We consider the element of fluid which occupied a rectangular parallelopiped having its centre at the point $(a,b,c)$ and its edges $\delta a$ , $\...
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determine the shear stress and bending moment of the next axis

I have problems to raise the equations. They could recommend me a text to be able to solve it.
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Young's modulus and shear modulus relation - temperature effect

Young's modulus and shear modulus are related by $E=2G(1+\nu)$ (for isotropic and homogeneous materials), $E$ is Young's modulus, $G$ is shear modulus and $\nu$ is Poisson's ratio. I can do experiment ...
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Terminology for how bendable an object is and what affects the bendable-ness of an object

I was wondering what the term is for how bendable an object is. Also, does this feature vary depending on the thickness of the object? Say, for example, I want to know how bendable a ruler is. Does ...
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Fixing the Poisson equation to match the deformation of elastic sheet with experimental observation

I am working on the calculation of the deformation of a circular elastic sheet with radius $R=1.2~m$ when a plate with mass $M$ and radius $r_0 = 4~cm$ is sitting in the center of the sheet. I used ...
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67 views

Helfrich energy derivation

The Helfrich elastic energy of membranes is given by $F = \int dS (\kappa H^2 + \kappa_G K)$ where $H$ is the mean curvature and $\kappa_G$ is the Gaussian curvature. The derivation in the original ...
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Is $\sin\left[2\alpha\right]\cos\left[2\alpha\right]\ge0$ a valid restriction on the angles of the principal stresses in 2D elasticity?

This question pertains to Elasticity: Tensor, Dyadic, and Engineering Approaches By: Pei Chi Chou, Nicholas J. Pagano, Section 1.4. The objective under discussion is to find the directions of ...
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Boundary condition: displacement

I have a controversial case. I have a rod, which is fixed from one end (constraint). From another end, I apply a compressive force, by pressing the rod down. So in a way I have a constraint, but at ...
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Why stress always flows through the shorter/stiffer path?

Why load has a preference at which way to act? What is the reason behind this preference? Any link or comment would be welcome
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Is continuum mechanics a generalization or an approximation to point particle mechanics?

Newtonian Mechanics is usually presented as a theory of point particles (and forces). My impression of the status of continuum mechanics is that it is mostly taken as an approximate description for ...
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Spring Constant of Two-Sided Fixed Beam [closed]

Let's suppose I have a two-sided fixed beam: ...and I want to find the equivalent spring constant... can I do the following: I know that the maximum deflection (at the center is): $$\delta =\frac{...
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eigenvectors of two deformation measures described in the same basis system

I have two deformation gradient tensors, corresponding to the simple shear of a cube; one at time t (say $\textbf{F0}$) and another at time $t+\Delta t$ (say $\textbf{F1}$). The basis of $\textbf{F1}$ ...
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What is wrong with my deformation gradient calculation?

I created two ellipses,$\hspace{150px}$,where the red ellipsis is as the blue one, except translated to the right and rotated by ${30}^{\circ} .$ Using rotation matrix, $$ \left[ \begin{array}{cc} ...
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Stress Energy Tensor in language of differential forms

The motivation for this is that quantities like the electric current $J$ in maxwell's equations of motion can be expressed as a differential 3-form, so that the continuity equation can be written just ...
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1answer
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Strain compatibility

From this link, strain compatibility for 2D problem with strains as $$ \varepsilon_{11} = \cfrac{\partial u_1}{\partial x_1} ~;~~ \varepsilon_{12} = \cfrac{1}{2}\left[\cfrac{\partial u_{1}}{\...
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Thermo-Elasticity for Irreversible Non-Linear Anisotropic Material (Continuum Mechanic)

I am interested by the thermodynamic behaviour for Irreversible non-linear and anisotropic material and by reading equations from this link, I see that following equation: Now, I'd like to know if ...
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Landau and Lifshitz argument for symmetry of stress tensor

In Landau and Lifshitz's book on the theory of elasticity (vol 7, theoretical physics series), specifically section 2 of the first chapter, the authors present an argument for justifying the symmetry ...
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Continuum Physics

How distinct is fluid dynamics from continuum physics? I've heard it is a subset and by the definition of the subject's name, it seems likely to be the case. Can anyone please clarify?
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Are there any proved consistent quantum continuous cellular automata/game of life model?

There are quantum cellular automata (continuous game of life, which is a type of cellular automata): https://hal.archives-ouvertes.fr/hal-00542373/document There are continuous cellular automata (...
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How to map a (virtual) truss deformation to a surrounding object

I am trying to craft a way of simulating deformable objects without using finite element method (despite it is the goldstar method for such purpose). The goal is to rapidly get an estimate of how a ...
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Why are diagonal elements of stress tensor equal to pressure for fluids?

In fluid mechanics it is assumed, that the normal components of the stress tensor are all the same and identical to the pressure p: $\sigma_{xx}= \sigma_{yy}=\sigma_{zz} = p$ Where does this come ...
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Statistical mechanics arguments for the Newtonian stress-strain model in continuum mechanics

In traditional continuum mechanics, it can be shown using frame-invariance and the Cayley-Hamilton theorem that an isotropic, inelastic* fluid embedded in Galilean space-time must possess the ...
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Making sense of the stress tensor for elastic deformations

I've seen this kind of formula a number of times, in the context of elastic deformations. $$ -\nabla \sigma = f $$ whete $\sigma$ is "the stress tensor" and $f$ is force. I never understood it even ...
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Simple Harmonic Motion - spring with mass [duplicate]

I'm reading about the following set-up: We have a body of mass $m$ attached to a spring. The restoring force is proportional to the displacement of the body from equilibrium. Using Newton's 2nd law ...
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Existence of solid mechanics problems that cannot be solved through Lax-Milgram approaches

Very often, solid mechanicians employ finite-element analyses to solve problems in linear solid mechanics. This approach is guaranteed to work because the Lax-Milgram theorem, along with some ...
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Why is the traction vector dependant on just normal vector instead of the gemoetry of the entire cuting surface?

Given a continuum G with surface forces on its surface S being cut arbitrarily (not necessarily by a plane) it is natural that the traction vector at any point along the cut depends on the cut itself....
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Deformation gradient of a 2D manifold

I have a 2D manifold (triangle) defined by points $\vec{X}=(X_1, X_2, X_3)$ in the reference configuration and have $X_3=C$ (moreover, the reference configuration is not the unitary simplex ); I ...
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Calculate the stress tensor on the surface of a tube

I am studying engineering and I was doing some preparation for the exam but stumbled accross the following problem: I would know how to solve task 2-4 but I am stuck at task 1. I do have multiple ...
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98 views

Stress tensor and traction vector

Hey all. I have this problem with the above figure. The lengths of the three sides are $AB=a, AG=b, GB=c$. Also, P,Q,R are forces perpendicular to each side as you can see and P,Q,R are their ...
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Potential energy in continuum physics: Why is this $0$ in the surface?

At the start of the continuum mechanics, we were calculating the potential energy. Here is the part that's not clear to me: Let $V$ be the potential energy and $U$ be its density function. Then: $$V=\...
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The approximate values of density for a flame with respect to heights

I'm aware that the difference in density with respect to height is low, but I'm in need of the values for a project: in which I explain that the continuity equation of fluid dynamics will work well ...
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Material tangent stiffness matrix for linear elasticity

I am trying to find material stiffness matrix for linear elasticity for finite element code, $$\mathbf{\sigma} = \lambda \hspace{1pt} \operatorname{tr}{\left(\mathbf{\epsilon}\right)}+ 2\mu\mathbf{\...
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1answer
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Trouble connecting stress and force in continuum mechanics with my concept of force from point mechanics

I'm not very familiar with continuum mechanics and have a hard time combining my knowledge of forces from simple mechanics with what I read about continuum mechanics. Let's suppose we have a metal ...
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Waves in $2$ different mediums - Why do we do it like this?

When we were solving the wave equations in the case of $2$ different mediums, we were assuming that, when a longitudinal wave reaches the border, there will be 1 reflected longitudinal and 1 reflected ...