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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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 ...
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Problem on bending plates in Newtonian Mechanics?

I am reading a book on interesting physics problems and demonstrations. One of the problems in the section on buildings, structures and equilibrium talks about a plate with one side attached to the ...
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On the isotropy of materials

Good morning. I am working on Honeycomb structures and first of all I would like to understand whether it is Isotropic or not, and , if the latter holds which kind of anisotropy it has. How to do it? ...
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185 views

Why plane stress condition is taken for thin plates

Why plane stress is taken for thin plates? It says in the books that the stress variation is small for thin components and is close to zero. Why is that so? Also why stress at free surface is zero? ...
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135 views

Is it possible that Cauchy stress be asymmetric?

According to conservation of linear momentum and angular momentum, one can derive that Cauchy stress tensor is symmetric and hence has only 6 independent components. Is it possible that, when breaking ...
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94 views

Why is general relativity only formulated in continuum terms?

So, when we are discussing Newtonian mechanics, we treat particles as point particles. In continuum mechanics, which I understand to be a version in which mass is continuously distributed, we have ...
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201 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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434 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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143 views

Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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66 views

buckling of tube - shell thickness vs. momentum of inertia optimum

is there any simple formula (perhabs semi emperical, or aproximatively derived model) for buckling of tube under axial compression load given its crossection and wall thickness? ( and naturraly ...
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59 views

If I roll an elastic plate into a cylinder, does it shrink?

Suppose I start with a rectangular elastic (to keep things simple, zero Poisson's ratio) sheet of length $2\pi R$, thickness $h$, and (immaterial) width $W$. I roll it up into a cylinder of radius ...
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1answer
146 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 ...
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180 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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184 views

Classical point particles to classical fields

I often hear that in the continuum limit we can study large numbers of particles as fields. I always imagined that by removing all bounds on the number of particles (while keeping total energy, ...
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251 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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1answer
174 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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43 views

Can $U_{ij}$ or $v_{ij}$ in continuum mechanics be negative?

In continuum mechanics, we have the deformation gradient $\mathbf F$ to be: $$d\mathbf x = \mathbf F d \mathbf X$$ And then, we do a polar decomposition (A good reference here would be ...
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54 views

Is there a way to calculate strain energy based on stress and deformation gradient?

We know that we can obtain stress from strain energy density and deformation gradient, for example: $$\mathbf P=\frac{\partial W}{\partial \mathbf F}$$ However, is there a way to calculate $W$ from ...
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1answer
77 views

Physics of Wrinkling: Understanding inextensibility condition

I'm reading this very cool paper on the formation of wrinkles in elastic materials. The key result of the paper is a set of scaling laws for the amplitude and wavelength of wrinkles based on the ...
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1answer
111 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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2answers
25 views

Understanding incompressibility (of rubber or viscoelastic material)

Literature gives a lot of explanation why rubber is incompressible. However, I still need some thinking to understand physical behavior of rubber or any such material. Often, incompressibility is ...
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1answer
108 views

Confusion in Euler-Bernoulli beam theory

Euler-Bernoulli beam equation is given by $$ EI \frac{\mathrm d^2 u}{\mathrm d x^2} = M'(x) \\ EI \frac{\mathrm d u}{\mathrm d x} = xM'(x) + C_1 $$ Where, $E$ is modulus, $I$ is second moment of ...
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1answer
52 views

Stress in horizontal bars

Imagine we have an horizontal bar. My teacher expresses the tensions along the longitudinal axis by this way $\sigma_{xx}=A(x)y+B(x)$ He doesn't give any motivation behind this. So, is this general? ...
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61 views

Resource(s) for developing a good understanding of surface tension?

I have read through several junior undergraduate level explanations of surface tension. Here is a typical presentation at that level: Molecules at the surface of a fluid experience approximately ...
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165 views

Derivation of normal shear stress

I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about. Specifically, what I don't understand is ...
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86 views

Show that detF(X,t) is positive in continuum mechanics?

I want to show that the determinant of the field $detF$ at the point $X \in B$ is positive, when the following motion. I think that time derivative of Jacobin is positive for $t > 0$. However, I ...
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317 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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2k 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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433 views

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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21 views

Golf ball impact

A golf ball is said to be "compressed" when hit by a golf club and makes a characteristic "thwack-hiss" sound coming off of the club when impacted by professional golfers (whose impact conditions have ...
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34 views

Deriving general boundary conditions from first principles for elastodynamic scattering

It seems that most of the relevant books only give the linear case and the rest say something along the lines of "here are common examples of boundary conditions." What are the most general boundary ...
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1answer
159 views

Divergence of Cauchy Stress Tensor

On the wikipedia page for the Cauchy Momementum Equation, it's stated that the equation can be written as $$\rho \frac{D\,\textbf{v}}{D\,t} = \nabla \cdot \sigma + \textbf{f}$$ Where $\sigma$ is ...
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106 views

Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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34 views

What is the criterion of stability of thick-walled spherical shell?

Is there the formula (if someone already has discovered it) or what is the algorithm (if a particular formula was not deduced), to calculate the critical pressure of thick-walled spherical shell $−$ ...
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70 views

Is there quantitative theory of cutting with edge or blade

I wonder if there is some simple theory of what determine efficiency ( speed, energy end force required ) of cutting by edge ( blade , knife, sword ) At least something phenomenological like in ...
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153 views

Solid-body rotation of fluid in polar coordinates: How to compute the stress tensor

In a course on continuum mechanics, we are given an exercise concerning solid-body rotation of a fluid in polar coordinates. In the first parts (feel free to correct any errors here) we are tasked ...
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438 views

Conservation of energy and continuity equation

When physicists say energy is conserved, do they mean that energy satisfies the continuity equation: $$\triangledown \cdot j+\dot{\rho}=0$$ On the internet there is plenty of talk of how the ...
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243 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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112 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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906 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
25 views

Airfoils contradict the law of the lever?

The law of the lever says that "the less force you use, the more distance you have". It is often exemplified by referring to simple machines, but it should apply to all technical systems. But I do not ...
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81 views

Why only vertical component of the stress tensor on vertically suspended bar?

EDIT: I am gonna rephrase the question entirely. Imagine we have a bar which we will analyze in the linear elastic regime. The shape of the cross section is irrelevant. The bar is suspended from a ...
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2k views

Calculate the weight a simple plank can support [closed]

I'd like to build a simple desk; just a single plank of wood (or a few side-by-side) with solid supports on each end of the desk. What I'm trying to figure out is how thick a plank I want to use for ...
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1answer
67 views

Curve of a rod bent by force on both sides

Suppose we have a flexible rod (i.e. it can be bent without breaking apart) and we excert a force on both sides, like this: If the force $F$ is not exactly horizontal, the rod will be bent and form ...
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1answer
1k views

Problem with Velocity of efflux [closed]

I am stuck in this problem- I need to find the velocity of efflux at the hole of the container. [We can assume that the area of the hole is negligible in comparison with the base area of the ...
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181 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
680 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
4k views

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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Continuum Wave Function for the electron

I'm trying to understand certain processes like the photoelectric effect and Bremsstrahlung. In Bremsstrahlung I need to use the wave function of an electron coming from the continuum, and there is ...
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20 views

Dynamic mass or static mass?

I am testing a cantilever beam assuming it as a single degree of freedom system, therefore it can be described by the equation $$m_d \ddot{y}(d) + c_d \dot{y}(d) + k_d y(d) = p(t) $$ Where $d$ is ...