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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Can Smoothed-Particle Hydrodynamics (SPH) be used to simulate porous media flow and deformation?

I am trying to use Smoothed-Particle Hydrodynamics (SPH) to study fluid flow in and around porous media. The aim is to observe how it causes erosion and failure. For this, from my understanding, there ...
4
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202 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
7k views

Neutral axis of T shaped beam? [closed]

I am not a mechanics or physics student but a computer science student. I came across a question related to find neutral axis of figure but I do not have slightest idea of what it is and how to find ...
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2answers
4k views

Conservation Vs Non-conservation Forms of conservation Equations

I understand mathematically how one can obtain the conservation equations in both the conservative $${\partial\rho\over\partial t}+\nabla\cdot(\rho \textbf{u})=0$$ ...
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1answer
191 views

How practical is fracture mechanics?

I have been reading fracture mechanics recently and have encountered many beautifully elegant theories. However, one thing keeps bothering me: How practical is fracture mechanics in the real world? ...
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171 views

Einstein Notation for Strain Energy Function

I encounter the following formula (for strain energy function) a lot in physics literature: $$ W(\epsilon_{kl}) = \int_0^{\epsilon_{kl}} \sigma_{ij} \textrm{d}\epsilon_{ij} $$ where all indices ...
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1answer
228 views

Forms of the first law of thermodynamics

The first law of thermodynamics states that $$\frac{D}{Dt}(K+U)=W+H,$$ where K is the kinetic energy, U is the internal energy, W is the power of the external forces and H is the heat flux. I have ...
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2answers
229 views

Extension to continuous in proofs of rigid body mechanics

I'm studying rigid body mechanics and I've seen several proofs of properties related to total angular momentum, kinetic energy, etc. that all regard discrete set of points. For example, to show that ...
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136 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 ...
3
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1answer
96 views

Can convection cells evolve in stably stratified fluid?

Assume stably stratified fluid but not in equilibrium, e.g. with non-constant temperature gradient for example. Can convection cells be present? Typical example of convection cells is Rayleigh–Bénard ...
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2answers
799 views

Symmetry of the 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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215 views

Explain the Föppl–von Kármán equations

I am a newbe to elasticity. Could someone please explain to me briefly how the Föppl–von Kármán equations work? What are we trying to solve for? Is there some kind of intuition to the way they look? ...
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250 views

Why is this thought experiment flawed: A vast lever rotating faster than the speed of light [duplicate]

If there were a vast lever floating in free space, a rigid body with length greater than the width of a galaxy, made of a hypothetical material that could endure unlimited internal stress, and this ...
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3answers
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Why is the (nonrelativistic) stress tensor linear and symmetric?

From wikipedia: "...the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. ...The linear ...
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1answer
675 views

What is Relativistic Navier-Stokes Equation Through Einstein Notation?

Navier-Stokes equation is non-relativistic, what is relativistic Navier-Stokes equation through Einstein notation?
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1answer
169 views

2-D Turbulence - how does it look like?

Consider parallel flow in the X direction over a 2D semi infinite flat plate. If turbulence is 2-D, in which axes should we expect the vortices to form. Also, are there any experimental/visualization ...
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4answers
674 views

Hooke's law limitation question

Let's consider a spring. I am a strong man(well, lets assume) and I am pulling the spring. the work I do is being stored in the spring in the form of its elastic potential energy. Then suddenly, ...
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1answer
136 views

Equivalence of turbulence in solid materials

The governing equations for a fluid and a solid are effectively the same and many times analysis can be done for a solid using the Navier-Stokes equations with the equation of state and/or the stress ...
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1answer
97 views

References on wave solutions in continuum mechanics [closed]

I am interested in literature on known wave solutions in continnum mechanics, precisely the following mechanical equation: $$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$ My interest is spread ...
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1answer
179 views

Dispersion relation in continuum mechanics

I'm looking at the vibration of a solid having a lattice structure, they obey the following equation: $$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$ with $u(\vec{x},t)$ the displacement to ...
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1answer
218 views

Normal modes of a flexible rod clamped at only one point

I am interested in the vibrations of a thin, flexible rod that would only be clamped at one point, properly I'd like to calculate its eigenvalue. But the way I learned it in wave mechanics doesn't ...
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1answer
643 views

Calculation of a bending moment

I'd like to calculate the bending moment of a cantilever, fixed at its base, and submitted to a certain stress on a specific spot, but I can't find the proper definition of this bending moment (first ...
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2answers
173 views

How local is the stress tensor?

I am confused by the definition of the stress tensor in a crystal (let's say a semi-conductor), I don't see how it could be "more local" than over an unit cell. I know that in field theory the stress ...
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346 views

How to solve fixed-fixed beam with finite difference method?

What equations to use on this system to form a matrix $A$ with dimensions $[n,n]$ and load vector $q$ with dimension $[n]$ ? I am trying to get vertical displacement $w$. $$w = A^{-1}\times q$$ ...
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114 views

Continuum mechanics and effects of stress

Going to word this question a bit more straightforward than I may have before. Also, I'm trying to use baby formulas so I can grasp exactly what's going on. Object A has an elasticity of ...
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259 views

Shape of wall's deformation wave caused by baseball's impact

Clicking through this year's top sports pictures, I stumbled upon this one. I was wondering about the shape the baseball is leaving on the wall. What phenomenon causes this peculiar shape? Why is ...
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1answer
208 views

(Botanical) branch bending under gravity

I'm a PhD student in maths, and attended my last physics class some 15 years ago, so you can imagine my competences in the field. My supervisor (also not a mechanist) cant tell me how to proceed ...
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3answers
760 views

Why are Navier-Stokes equations needed?

Can't we picture air or water molecules individually? Then, why are Navier-Stokes equations needed, after all? Can't we just aggregate individual ones? Or is it computationally difficult, or ...
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3answers
242 views

How wide does a wall of ice need to be to stay in place?

Let us say that we have unlimited manpower to construct a huge wall of water ice e.g. 200 m tall (700 feet). -and that the wall is placed in a climate, where the temperature never (for your purpose) ...
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4answers
179 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
383 views

A differential equation of Buckling Rod

I tried to solve a differential equation, but unfortunately got stuck at some point. The problem is to solve the diff. eq. of hard clamped on both ends rod. And the force compresses the rod at both ...
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1answer
338 views

What is the two dimensional equivalent of a spring?

I'm trying to model isotropic linear elastic deformation in two dimensions. In one dimension, I know that a linear elastic material can be thought of as a spring which obeys Hooke's law $F=-k\Delta ...
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298 views

Tensors: relations between physics and linear algebra

In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as $$ ...
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107 views

Difference between using displacement and current configuration as unknown?

We could use either the current configuration $x$ or the displacement $u$ as unknown while solving for the deformation, for example, of a solid object. I want to know what's the difference between ...
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204 views

Decomposition of deformation into bend, stretch and twist?

I'm wondering is there any way to decompose the deformation of an object into different components? For example, into stretching, bending and twisting part respectively? The decomposition could be ...
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2answers
118 views

A problem of approximation [duplicate]

Possible Duplicate: Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size? When we apply differentiation on charge being conducted with respect to ...
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1answer
1k views

Continuity equation for compressible fluid

A question is given as Consider a fluid of density $ \rho(x, y, z, t) $ which moves with velocity $v(x, y, z, t) $ without sources or sink. Show that $ \nabla \cdot \vec J + \frac{\partial \rho ...
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680 views

A conceptual problem with Euler-Bernoulli beam theory and Euler buckling

Euler-Bernoulli beam theory states that in static conditions the deflection $w(x)$ of a beam relative to its axis $x$ satisfies $$EI\frac{\partial^4}{\partial x^4}w(x)=q(x)\ \ \ \ (1)$$ where $E$ is ...
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1answer
421 views

Boundary conditions of Navier-Cauchy equation

I'm having difficulties with Neumann boundary conditions in Navier-Cauchy equations (a.k.a. the elastostatic equations). The trouble is that if I rotate a body then Neumann boundary condition should ...
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552 views

Interpretation of Stiffness Matrix and Mass Matrix in Finite Element Method

I would like to have a general interpretation of the coefficients of the stiffness matrix that appears in FEM. For instance if we are solving a linear elasticity problem and we modelize the relation ...
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85 views

Can a wave propagate in an elastic fluid in the absence of volume forces?

A motion (wave) $\mathbf{x}: \mathcal{B}_0 \times [t_0,t_1] \to \mathcal{E}:$ such that $q-o = \mathbf{x}(p,t)=(p-o)+\mathbf{a}_0 cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$ can propagate in an elastic ...
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105 views

Physics for taffy pulling?

I am creating a simulation and am interested in pulling stretchy things and when they break, like taffy. I imagine this is a bit tougher then a simple equation like gravity, but I have no idea. Is ...
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665 views

Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size?

Suppose we view fluids classically, i.e., as a collection of molecules (with some finite size) interacting via e&m and gravitational forces. Presumably we model fluids as continuous objects that ...
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Calculation of the maximum load to the bar

Looking for a way of calculating the maximum weight (W) to the rod with the given length (L) where the rod did not break and that only bend for (b) mm. Need only approximative solution (read: ...
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what is a difference in the width of the spinning bar?

The bar with length l, density r, diametr d, Young's modulus E, Poisson's ratio mu, is spinning around the cross-section, what is the change in the width of this bar?
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248 views

Does a thermally expanding torus experience internal stress?

I'm trying to learn continuum mechanics and thermo-mechanics. As we know, heating an object increases the mean atomic distance $a_0$ of the atoms in a rigid body. Let's assume it is a linear elastic ...
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1answer
356 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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In continuum mechanics, what is work potential in the context of total potential energy?

I'm reading a book on the finite element method. Specifically I'm looking at the background material where they are discussing potential energy, equilibrium, and the Rayleigh–Ritz method. The book ...
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2d soft body physics mathematics [duplicate]

Possible Duplicates: Modern references for continuum mechanics Good books on elasticity The definition of rigid body in Box2d is A chunk of matter that is so strong that the distance ...
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1answer
68 views

stress work of uniformly deforming continuum

I have a volume which is deforming (using explicit time-integration scheme) uniformly with velocity gradient $L$ and stress tensor $\sigma$. I would like to determine work done by the volume ...