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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Pressure Applied by Sphere Embedded in Elastic Medium

Consider a sphere embedded inside of an infinite elastic and isotropic medium. If a force $F$ is applied to the ball from a distance (e.g. via a magnetic field), then the ball will in turn apply a ...
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Bulk's Modulus in adiabatic process? [closed]

The bulk modulus in an isothermal process is given as $$\beta=-\frac{VdP}{dV},$$ where $$PV=\text{constant}.$$ So how is the bulk modulus defined in an adiabatic process where $PV^\gamma=\text{...
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Tensile strain produced on two thin rods of different lengths

Suppose there are two thin rods $Y$ and $Z$ with length $L_1$ and $L_2$ respectively. $L_2$ has larger magnitude than $L_1$. Both rods have same density $p$, cross sectional area $A$, Young's Modulus $...
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Proof that $det \, {\bf F} = 1 + tr \bf \, H$ in a linearized setting, where $\bf F$ and $\bf H$ are deformation and displacement gradients

I am looking for a proof of that the determinant of deformation gradient $\bf F$ in a linearized setting is $det \, {\bf F} = 1 + tr \bf \, H$ Where $\bf H$ is the displacement gradient.
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How to derive the equation of a spline inside a pipe

Let's suppose a pipe with some curves. We put an elastic bar inside. The diameter of the elastic bar is much smaller than the diameter of the pipe. We know that the elastic bar bends to minimize the ...
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Elastic moduli interrelations for homogeneous isotropic materials

The classical equation for the Young modulus in elasticity theory for a homogeneous isotropic material in one-dimension is commonly given in the formulation $$ E = \frac{\sigma}{\epsilon} \quad,$$ ...
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Young Modulus in Finn's Thermal Physics

In Finn's Thermal Physics (equation 2.4), the Young modulus $Y$ of a stretched wire with tension $F$ is given to be $$Y = \frac{L}{A} \left( \frac{\partial F}{\partial L}\right)_T$$ However, usually ...
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Area under stress–strain graph

What quantity does the area under the stress–strain graph (under the elastic limit) represent? As far as I know, the potential energy density due to strain is 1/2×(stress)×(strain). So does the area ...
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Elongation of a rod hinged at the end

Suppose a rod is rotating in a horizontal frictionless plane, hinged at one of its ends. If the body is non rigid, it would change its length, but I am not sure whether it would elongate or get ...
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Cycles to failure

$$ \text{Ramberg-Osgood equation:} \hspace{41mm} \varepsilon_{tot} = \underbrace{\frac{\sigma}{E}}_{\text{elastic}} + \underbrace{\left(\frac{\sigma}{K}\right)^{\frac{1}{n}}}_{\text{plastic}}$$ $$ \...
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How do you determine energy transferred to a trampoline when a ball is dropped on it? [closed]

Suppose a ball of mass $m$ is dropped over a trampoline(consider the young's modulus of the material to be $Y$) from some height $h$. The ball impacts the trampoline, gets slowed down due to the ...
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Dynamics of a football

Why can toe kick cause greater change in the ball's momentum than kicking with the whole instep in contact with the ball? Perhaps it has something to do with how the foot's momentum is transferred in ...
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Football Dynamics

Suppose a football is racing towards you at some speed $v$. What you are gonna do is try to stop it. There are two scenarios: 1.Suppose you hit it with your closed fist in the direction opposite to ...
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Name of the product between Young's modulus and coefficient of thermal expansion (CTE)

In solid mechanics, and especially thermo-elasticity, there are relationships with both the Young's modulus, $E$, and coefficient of thermal expansion, $\alpha$. Let's take the partitioning of strain ...
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Mechanism of heating during deformation [duplicate]

When a metal or rubber is bent and deformed, it heats up. What is the reason for this? I know that when deforming it, work is done on the object. If the object is perfectly elastic, all of the energy ...
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Why does a flat sheet roll back into a cylinder when having rolled it once?

For example, if you were to roll a piece of paper into the shape of a hollow cylinder and then try to return the paper to its original, flat shape, it would naturally roll to the cylinder when you let ...
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How mechanical Strain developed in metal bar at molecular level?

If I have metal bar fixed to a support at one end while I apply a tensile force at the other end, the bar elongates while its cross sectional area decreases. I want to know How strain is developed at ...
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1answer
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Oil spreading over water

The book says that oil spreads over water due to the greater surface tension of water as compared to oil, so the comparatively stronger water film stretches the oil surface and makes it spread... But ...
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Locating point of impact “during” elastic collision (1D)!

We have two objects moving opposite to each other on the x-axis. They started to collide on the location $x = 0$ on x-axis. We know that if object one $m_1 = 10 kg$, and $v_1 = 10 m/s$, and object ...
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Can a sound wave have such high frequency that it causes permanent displacement of medium?

Is it possible to have a sound wave with frequency so high that it causes permanent displacement of matter (the frequency is greater than the elasticity of the medium, so the medium is unable to ...
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2answers
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How does stress “propagate” in beams?

I'm studying structural mechanics and I have been stuck on a thought. I drew a very simple cantilever beam situation. If I make a fictional cut like in my fig.1, I will be in the situation of the ...
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The Electric Pressure Bomb: Can a conductor rupture due to its own electric pressure?

So I conducted a though experiment where I take a hollow spherical conductor and beef it up with a lot of electric charge. Here, I have ignored the ionization of air due to that huge amount of charge. ...
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Why does the string of a simple pendulum need to be perfectly flexible?

The book says: "The string should be perfectly flexible, if we like to neglect the effects of different velocities of the different parts of the string during the oscillation." Can anyone ...
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Time period for small vertical oscillations of bob

We have a small bob attached to an elastic rubber wire and we are given values of the wire’s Young’s modulus, length, and area. My doubt is not to know the answer specifically, just to review my ...
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Why is an incompressible isotropic elastic medium always a liquid [closed]

This was a claim made by my professor in my mechanical metallurgy class. I don't see how it is true. I could very well have a solid (which is incompressible) and have isotropic properties. We deal ...
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Strain on heating

A steel rod of some length rests on a SMOOTH horizontal base. If it is heated for a considerable temperature difference, say $0^\circ C$ to $100^\circ C$, what is the strain produced? Strain is ...
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Elastic limit tempered steel

What is the elastic limit of tempered steel? I have to calculate the maximum charge that can be applied to a tempered steel wire of $0.08\ m$ of diameter without it exceeding its elastic limit I am ...
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Do solids heat up on compression or extension? If so how?

Well, let's take the example of a compressible solid. We know that in order to compress a solid we need to apply a force "greater(and not really equal)" than the interatomic/intermolecular ...
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4answers
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Why is Poisson's ratio necessary, when we know that volume stays constant?

Why is the Poisson ratio necessary, when volume is conserved? I read that volume is conserved when a body is subjected to longitudinal (compressive or tensile) stress or shear stress, so given that ...
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Why is the continuity equation hardly used in solid mechanics when it is essential in fluid mechanics?

For any continuum, fluid or solid, we can express mass conservation through the continuity equation $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 ,$$ where $\rho$ is density ...
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Elasticity of collisions in Relativity

Is it possible that collisions which are elastic in the rest frame of reference are inelastic or partially elastic in some other constant velocity frame? If you need to invoke general relativity here, ...
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Meaning of elastic energy formulation

In Chaikin's Principles of Condensed Matter Physics, in chapter 6 ("Generalized Elasticity"), on pg. 290, there is a formulation of what he refers to as an elastic energy associated with gradients of ...
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Is shear strain an additive quantity?

Consider a box with rigid walls containing an elasic medium, subject possibly to some body forces or tractions. The volume is an additive quantity, in the sense that the total volume change of the ...
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1answer
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Modeling elastic tree branch (Double torsional pendulum?)

I’m trying to model “bending tree branch like motion” and it seems, that it can be described with some kind of «upward facing torsional pendulum” I guess. The construction is facing upward and start ...
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Willmore energy

Recently I saw an equation, which is called as Willmore energy, which gives an interpretation that nature tends to minimise this Willmore energy by changing its shape (elasticity). But, how does this ...
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Is Young's modulus of elasticity a measure of ductility?

I've learnt that the Young's modulus of elasticity is defined as the ratio of stress and strain when the material obeys Hooke's law. So it has no significance beyond the proportional limit in the ...
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Physical limits or empirical data on ratios of bulk modulus, shear modulus and density

The elastic dynamics of an isotropic continuum (solid or fluid) can be described in terms of the bulk modulus ($\kappa$), shear modulus ($\mu$, zero in a fluid) and density. (The two elastic moduli ...
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Why do we get a different result when we calculate it by parts?

Assume a wire is 5 m long and we want to find how much force is required for increasing the length of it 5 m and make it 10 m long. $$F = \frac{YAl}{L}\, .$$ If Y=5 and A=1 we will find the answer 5....
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When compressing liquid, how much energy is converted to increased temperature vs increased pressure?

If you compress liquid in an infinitely stiff and infinity insulated cylinder such that the cylinder does not expand and no heat can transpire, how much of the energy will converted to increase in ...
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Design an Experiment based on Castigliano’s Theorem to Determine Young’s modulus of the Unknown Member

The question states: "You need to design an experiment based on Castigliano’s theorem to determine Young’s modulus of the vertical member (unknown material) without destroying the structure." Is ...
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Hyperelasticity: relating two PK2 stress tensors in terms of two nonlinear displacements

I am working on numerical analysis for a nonlinear hyperelasticity problem. Given that the second Piola Kirchhoff stress tensor $S$ depends on the Green strain tensor $E$, which in turn depends on the ...
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How are body deformations modeled in Lagrangian mechanics?

With rigid-body systems, we choose a finite number of generalized coordinates to model a system, i.e. a pendulum. However, I've read that deformable bodies like elastomers have "infinite" degrees of ...
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Why does a ball bounce bounce higher when the pressure is higher? Why does it fly faster when kicked?

A student of mine is doing a project on this topic, and I have realized that I cannot answer the question. If we are talking about dropping the ball, and assuming a completely elastic ball, it ...
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What causes increment in volume when compressive forces are applied on an object

Doubt When compressive forces are applied on a body, what causes increment in volume. According to me the volume should remain constant since the mass and density of body are constant.
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Work done by elastic force

I have a doubt about the work done by elastic force. The general formula of the work is: $W_{Fe} = \frac{1}{2}k(x_0^2-x^2)$; if we take that the state in which the spring is at rest, we have: $W_{Fe} ...
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How to measure the displacement of a force over a sphere [closed]

I know it seems like a broad answer but since it is really straight forward to measure the displacement of a beam, how do you measure the displacement of a "steel" sphere when a force is placed over ...
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Independent Elements of Elastic Stiffness and Compliance Tensor for ALL Space Groups

In short: Does anybody know if there exists a compendium, a document, a book or a stone tablet listing the independent elements of the elastic stiffness and compliance tensors ( that is, naming the ...
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Why do towels go crispy when air-dried but not in the tumble-dryer; when other materials don't have this effect so noticeably?

Our tumble dryer recently broke, so we've resorted to hanging our clothes on banisters etc. I've noticed that, when air-dried, towels (as well as cotton "muslin" cloths) go crispy, kind of hard, ...
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What is the difference between elastic and acoustic waves?

On Wikipedia, it is written that acoustic waves are elastic waves. If they are the same then why do we have two different names? Can someone please explain the difference between these two types of ...
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Force acting on a cube on all sides [closed]

A metallic cube of length L is stretched out uniformly by force F normal to each of its faces. Given Young's modulus and Poisson ratio, what is the deformation in the cube?

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