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Questions tagged [elasticity]

Questions related to the Continuum Mechanics Division of Elasticity. The bending of beams, deflection of rods, or in general, applications of Hooke's Law generalized to three dimensions.

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My Elasticity Doubt

I had a question on elasticity which looked something like this. {: .center} We have small beads of mass say $m$ and Mass of rod say $M$. Now we need to find the extension in the rod after the beads ...
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1answer
37 views

What are inelastic elastic-bands?

In the third episode of Julius Sumner Miller's physics series he sets up an experiment with two masses connected by an array of elastic-bands. He comments that the elastic-bands are inelastic. ...
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Relation between elasticity, elastic limit and breaking point [on hold]

Is there any relationship between elasticity, elastic limit and breaking point?(I have very basic knowledge on the subject) Reference or proof is appreciated.
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Continuum limit for a bulk of discrete masses coupled by springs

I have a question that has been bothering me for a long time. I know very well how to take the continuum limit for a chain of masses connected by springs. But in my recent project I would like to ...
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3answers
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Hooke's full unapproximated law

It is known that the Hooke's law relating the restoring force of a spring to the distance of retraction from the equilibrium position, is only an approximation. That is, the equation $F=-kx$ is only ...
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1answer
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Why is the partial derivative of strain energy function with respect to strain equal to stress

In Elasticity, we have a strain energy function , $W$, that is a function of strain tensor, $E$. Then the cauchy stress tensor, $T$ can be determined by: $$T_{ij}=\frac{\partial W}{\partial E_{ij}} \...
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Confusion Regarding Hookes law

I am used to seeing Hookes law in the form: $F = kx$ but in another one of my books it gives the equation: $\\ F = \frac{\lambda x}{l}$ where $l$ is the unstretched length, $x$ is the extension and $\...
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Stress in a rod clamped between two rigid walls when the temperature is increased

The usual approach to calculate stress is to equate thermal expansion in the unclamped condition to the magnitude of contraction caused by strain produced due to the walls. I have some questions about ...
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Elongation of rod in two cases

Case 1: Suppose there is a rod hung vertically from the ceiling. It experiences gravitational force W (its weight).It's Young's Modulus is $Y$. Now I know that the end connected to the ceiling ...
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22 views

Deformation of a circle ring under tensile force

How will it deform when I apply a tensile force to a circle ring? Can I estimate the force applied on the ring by knowing the major and minor axis of the deformation elliptic ring? For simplicity we ...
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15 views

How is elastic hysteresis effected by time?

Here's how I understand the phenomenon: let's take the example of weights and rubber band, when a weight is put it does work on the band to stretch it, and heats it up, some of the heat gets ...
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1answer
47 views

When can viscosity be ignored?

I am working on understanding the motion of a solid ball in an infinite elastic (or possibly viscoelastic) medium when subjected to a sinusoidal driving force. The frequency-dependent complex wave ...
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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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Derive Porosity from Stiffness Value

I have the Young's modulus (stiffness) values of a hydro-gel scaffolds, and I need to know the porosity of those scaffolds. I read about Nielsen Equation, which is this: $$ E = \frac{E_0 (1 - P) ^ 2}...
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Surface formed by a heavy elastic material hung from a circular rim

The solution to the problem of a string hung from two points in a uniform gravitational field is well known, its solution is the catenary. A solution can also be found for the problem of a spring of ...
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2answers
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Work done in circular expansion of a rubberband or an elastic wire

What will be the work done in radially stretching a rubberband it can't be zero as there is potential energy being stored in it All I came up with it that there would be increase in overall length so ...
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2answers
188 views

How does temperature affect the elasticity and spring constant of a rubber band

Rubber bands expand when chilled and contract when heated (no stretching, just when heated/chilled and at rest) to my knowledge. Why are hot rubber bands able to be stretched longer than cold rubber ...
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1answer
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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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Elastic potential energy and springs

The formula for elastic potential energy(for a spring) has been derived by assuming the following things: 1.Work done by a deforming force on a spring from a relaxed state (state where spring is not ...
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Kelvin solid and deformation in transverse direction with Poisson ratio

A Kelvin-Voigt material is a material with such a behavior : I'm wondering if there is a way to model in a similar way the deformation in the transverse directions $y$. Meaning we have for the ...
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1answer
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Statics and dynamics in elasticity : how to add time in elasticity

In elasticity, there is are static relations of the form $R(\sigma,\epsilon)=0$. In fluid dynamics, there is a dynamics relation with the conservation of momentum leading to Navier-Stokes equation. ...
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impossible Poisson's ratio of Lennard-Jones potential type solid (cubic cristal for example), or any potential depending on $r$

We know (for instance : https://en.wikipedia.org/wiki/Poisson%27s_ratio ) that the Poisson ratio is $\nu=1/2-Y/(6B)$, with $Y$ the Young modulus and $B$ the bulk modulus. Let's assume we have a ...
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How can Young modulus depend on the geometry of the item & Lennard Jones (question related to an older one)

In the question Young's modulus and geometry of test material, the answer was that it cannot play any role. Nonetheless, for the Lennard-Jones solid, in many places the Young modulus is given as : ...
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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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Elongation of rod [closed]

If a force is applied on a unconstrained body lying on a smooth floor ,will it elongate? My thoughts-as force is applied on body ,it moves with some acceleration,but as it is not constrained it will ...
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Pressure on a compressible solid, dynamics of the relaxation

I have a sphere made of a solid of bulk compressibility B, and of volumic mass $\rho_0$. The pressure around that sphere is $P_0$. At time $t=0$, we apply a pressure on the sphere $P=P_0+\Delta P$. ...
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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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What decides ductility of a substance- youngs modulus or the plastic limit of the substance? [closed]

Given two materials A and B such that a has higher young modulus as well as plastic limit than B , which of the two is more ductile?
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143 views

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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1answer
67 views

Hooke's Law and Overstretched Springs

Recently been working with Hooke's Law at university. My experience of real springs tells me that stretching to a certain point causes the spring to lose "springiness". (Not sure if "elasticity" is ...
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2answers
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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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1answer
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(I-beam vs rectangular beam) Which sword blade cross section is less likely to break?

Original question Assumption: i made several swords with different cross sections (lenticular, single broad fuller as in viking swords, diamond, hollow ground diamond) the blades are made using the ...
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Relationship between strain energy function and strain or stress

How one can get the strain or stress from the strain energy function ? And if one cannot do it, what is the use of that function ?
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Is the Young's modulus the same for compressive and tensile test?

I want to know the Young's modulus for a compressive test, but I have just a tensile test. Are they of the same value?
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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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1answer
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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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2answers
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Collision between two non-deformable objects

Let say we have a non-deformable object and we release it into free fall and it hits the non-deformable floor. What would happen? Here is the way I think. Since the floor and the object are both ...
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Stress state in an elastic half space with uniform constant body force

Is there a closed form expression for the stress state in an elastic halfspace subject to a uniform constant body force? I know that the Green's function for this problem is given by Mindlin's ...
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2answers
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Can I use $v_{final}=0$ when I calculate impulsive force?

If a 1kg ball strikes a wall with 10m/s and it rebounds with 10m/s, the time of impact being 0.2s, what is the impulsive force? I know that the initial velocity is 10m/s but can I state that the final ...
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2answers
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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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Spring strain vs fractional extension

How do I show that the strain tensor $$\epsilon_{ij}=\partial_i u_j + \partial_j u_i$$ in the case of a one-dimensional spring is reduced to $$\epsilon= \frac{x-L}{L},$$ where L is the initial ...
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Does the resonance frequency of a cantilever beam depend on whether it is bending in-plane?

What is the in-plane resonance frequency equation for a single-beam cantilever of rectangular cross-section that is fixed on one end and has a mass M attached to the other end? Below are the only ...
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72 views

Why the resultant spring constant different in the following two cases?

In these two cases in the first case my book The Physics Of Waves And Oscillations by NK Bajaj says: That the restoring force exerted ...
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Why does different part of a spring having mass expand proportional to their distance while the spring has some mass hanged in the bottom?

Why do different parts of a massive spring expand proportional to their distance while the spring has some mass hung on the bottom which is comparably very less than the mass of the spring? If we take ...
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Motion along length of a spring [closed]

What's relation between velocity of each part of a massive spring undergoing Simple Harmonic Motion.
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2D plane stress problem: a bent rod with one curved side squeezed

Consider a bent rod (i.e., a sector of an annulus) of angle $\alpha$ and inner and outer radii $R_i$ and $R_o$. Now, we squeeze in azimuthally the lower face by an amount $\Delta$ (see the image below)...
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Stress/strain within an object experiencing a single applied force

If I push an object in space, will any strain/stress happen within the object, though there is only 1 force rather than a pair of opposite forces on the object? Can be any (even if trivial) stress/...
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1answer
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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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Oscillation of elastic rope hanging from the ceiling

I have to explore the problem of oscillations of an elastic rope hanging from the ceiling, which can move both in vertical and horizontal direction. I'm planing to solve this using a finite ...