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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Is it correct to say that it is theoretically impossible for perfect rigid bodies to exist?

If perfect rigid bodies were to exist, then consider a scenario in which two rigid bodies of equal masses moving with velocities of equal magnitude but opposite in direction colliding against one ...
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How to get local density field of a compressible solid Neo-Hookean material undergoing large deformation?

My understanding of a compressible material in the context of non-linear elasticity in continuum mechanics is that the volume of the body can change due to the applied forces. Due to the law of ...
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Hooke's law and modulus of elasticity [closed]

I'm having trouble in understanding the following question which state modulus of elasticity in Newton. Is it possible that we can define modulus of elasticity in Newton somehow?? I am getting a wrong ...
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1 answer
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$F=m*a$ accounting for pressure wave propagation

Imagine a long deformable rod which has just been hammered on the top end (the bottom end is clamped to Earth). Consider a time interval $dt$ = $t_{2}$ - $t_{1}$ in which the pressure wave is ...
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2 answers
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How do we justify that potential energy in a spring is Galilean invariant (to the extent that Newtonian mechanics holds)?

A spring can store elastic potential energy by elastically deforming and moving its atoms out of their minima potentials. The atoms themselves can be modeled as balls connected by Hooke-like springs ...
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Diffraction Elastic Constants

I'm currently trying to calculate the diffraction elastic constants for hcp material by the method of Voigt and Reuss, the Voigt method is quite straight forward, but with the Reuss method i have ...
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Requesting good reference to understand the Gurtin-Murdoch model of surface elasticity

I am searching for a good and understandable reference about the Gurtin-Mordoch model for surface elasticity. I was wondering if anyone can help me with the preliminaries and good references to fully ...
1 vote
1 answer
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Equilibrium and constitutive partial and algebraic equations describing stresses and deformation of an axisymmetric elastic thin shell over a hole

I want to design a vacuum table to clamp down a very thin plate and I want to know the stresses and deformations due to the atmospheric pressure. Consider the simplified model below: ...
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Stiffness for helical spring under lateral bending force

The stiffness for a helical spring under axial loads is $$k_\text{axial}=\frac{F_\text{axial}}{\delta_{axial}}=\frac{Gd^4}{8n D^3}\, ,$$ where $G$ is the shear modulus, $d$ the wire diameter, $D$ the ...
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About Volterra's displacement equation on dislocation: cancellation of a surface integral on stress

In the theory of dislocations, the displacement induced by a dislocation in an anisotropic solid media can be expressed by Volterra's displacement equation as follows: $$ u_j(\mathbf{x}) = \int\int\...
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Do inertia affect stiffness matrix of beam?

Refering to a pdf I uploaded at here. Why does these stiffness matrix displacement do not need to consider the inertia (determines the shape of cross-sectional area of each element) ? I have always ...
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1 answer
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Which part of a massless wire will break at the breaking stress?

Consider an ideal massless wire of length $L$, uniform cross-sectional area $A$, Young's modulus $Y$ suspended from the ceiling, with a load of weight $W$ suspended at the end. There is no variation ...
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Heat generated in a rod due to forces on its ends

Given is a uniform rod of length $l$,cross-sectional area $A$ , Young's modulus $\gamma$. Forces F and 2F are applied at it's ends. To find the heat generated in it, I first went in the frame of the ...
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Relation between elastic constant tensor $C_{\alpha\nu,\beta \lambda}$ and force constant matrix $\Phi_{\alpha \beta}(ij)$

In three-dimensional solid, there is a more general notion of pressure called the stress tensor $\sigma_{\alpha \gamma}$ which is symmetric. Linear elasticity theory relates the stress via the elastic ...
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How much should I dig a hole to set a post?

I need your help to solve this problem We suppose that we have a post pulled by $N=4$ barbed wires separated a distance $dh = 30 \text cm$, and that each wire pulls with a force of $F=300\text N$ . ...
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How does the shape of an object affect the coefficient of restitution?

I can understand how elasticity of an object plays a role in determining the coefficient of restitution, but various sources say that the shape of the object is important as well (and in particular I ...
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1 answer
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Which occurs first: stress or strain? [duplicate]

Of stress and strain, which is the cause and which is the effect? Is stress a cause of strain, or is stress an effect of strain, or do they occur together (by applying Newton's third law)?
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1 vote
2 answers
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How to find a bend curve in 2D?

I am looking for a solution on how to find the (approxmiate) shape when bending a rigid-flex circuit board. Please see the abstract sketch below. I have two solid objects ($A$, $B$) which are ...
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Form of a general solution to an equilibrium equation of elasticity in terms of displacement vectors

In the theory of elasticity, when the deformation of body is not induced by body forces, but by forces applied to its surface. The equation of equilibrium in terms of the displacement vector, $\mathbf{...
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1 answer
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Free massless spring pulled from one end

When two masses are attached to the ends of a spring and a force is applied on one of the masses, each point on the spring will move the same distance from its equilibrium position. This is also the ...
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Euler-Bernoulli equation for a periodically supported static beam

The Euler-Bernoulli equation for a homogeneous beam is $$ EI w^{(4)}(x) = q(x),$$ where $w$ is beam height and $q$ is load density. Inspired by the deflection in a multi-support cantilever bridge ...
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1 answer
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Stress tensor between spatially separated layers

If I have an interface between two bodies (1 and 2). I would expect that the stress tensor is such that $\boldsymbol \sigma_1\cdot \hat n=\boldsymbol\sigma_2\cdot \hat n$ at the interface, where $\hat ...
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2 votes
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Why doesn't pressure affect the speed of sound in air? [duplicate]

I keep getting the answer "because density increases when pressure increases" but that doesn't really make sense to me since in denser materials - like water - sound travels faster. And if ...
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1 answer
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Question about the modulus of elasticity of a half-string

Question The ends of a light elastic string of natural length 0.8m and modulus of elasticity $\lambda$ N are attached to fixed points A and B which are 1.2m apart at the same horizontal level. A ...
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How to transform from a discrete to integral equation?

In this picture, the red curve is an elastic rod that has resistance to bending and extension. I am trying to model the adhesions (contact) between the rod and the substrate (glass): the green dashed ...
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2 votes
2 answers
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Is the Young's Modulus different for Compressive and Tensile Stress?

My textbook says that the magnitude of strain produced is the same whether the stress is tensile or compressive. So the Young's modulus, which is the ratio of (tensile or compressive) stress to the ...
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Stress-strain relationship for linear viscoelastic solid

I am a bit confused about the definition of a linear viscoelastic (isotropic) solid. Following Landau and Lifshitz (Theory of Elasticity, section "viscosity of solids"), I would say that in ...
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1 vote
1 answer
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Relation between Youngs modulus $Y$ and Bulks modulus $B$ for a cube

lets suppose a cube of side $l$, Young's modulus $Y$, bulk's modulus $K$ under a force F across all sides. so $$Y=\frac{F*l}{\Delta l*l^2}$$ now $$\Delta v=l^3-(l-\Delta l)^3$$ now ignoring powers of $...
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Free response of plucked string

Based upon points $x=l/4$ and $x=l/2$, how would one determine the relative error between the free response at $t=0$ and the exact initial deflection as a function of mode numbers contributing to the ...
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Why are the components of elasticity tensor 21?

I know there's a duplicate but I didn't understand the whole answer which was: A 6×6 matrix has 36 different components. When you reduce it to a symmetric case it has 1+2+3+4+5+6=21, where we are ...
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1 answer
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How to calculate force required to push down on a supported wooden cylinder (arrow shaft), a distance of exactly 12.7mm (0.5″)?

trying to find out how many newtons do I need to push down an arrow shaft a distance of 0.5″ in order to measure the spine factor. the diameter of the wooden arrow is 7.9375mm (5/16") stiff ...
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1 answer
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Question regarding elasticity and deformation [duplicate]

Say I apply a deforming force (elongation to be specific) of 10N there is a restoring force of 10N. Now I apply additional 1N and there is elongation in direction of force because $fnet$ is in that ...
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6 votes
2 answers
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Why is stress invariant under a change in coordinate system?

Since stress is represented by a tensor, it should be invariant under a change in coordinate system. For something like a position vector $\mathbf{r}_{P/O}$ which is also invariant under a change in ...
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1 vote
0 answers
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Magnitude of Strain Tensor

What is the physical interpretation of "magnitude" of the Strain tensor? For the strain tensor $e_{ik}$ its magnitude is defined by: $$ e_{ik}e_{ik}=e_{11}^2 + e_{22}^2+e_{33}^2 + 2e_{12}^2+ ...
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1 answer
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Energy loss mechanisms for a bouncing ball

How does a bouncing ball loses energy? Are there different energy loss regimes, depending on the ball material/structure (e.g., tennis ball vs. a solid caoutchouc) and the surface from which it ...
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1 vote
1 answer
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Elastic Potential Energy Of Elastic Wire: Two contradictory methods [closed]

Suppose a wire of length $L$, cross sectional area $A$, density $\rho$ and Young's Modulus $Y$ is suspended at one of its ends from a ceiling, then find its total elastic potential energy due to its ...
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1 answer
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Should the air in a pneumatic tire be included in factors contributing to hysteresis, or should only the tire/tube material be considered?

From my understanding the elastomeric properties of a tire with its casing and rubber, and a butyl tube are considered as contributors to hysteresis. The thicker the tire and tube,the greater the ...
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1 vote
1 answer
28 views

Compression or elongation for Young's modulus

Suppose that a mass $m$ is suspended from a wire whose length now is $L$. When we remove the mass, the length of the qire becomes $L'$. In this condition how will be the young's modulus of the wire be ...
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How the bulk modulus and Lamé coeffiencts are connected in the book of Theory of Elasticity by Landau?

In the book of Theory of Elasticity written by Landau and Lifshitz (published in 1970), they make a connection of bulk modulus $K$ and Lamé coeffiencts, which are $\lambda$ and $\mu$, at the section 1....
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2 votes
1 answer
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Difficulty in proof of relationship between $Y$, $B$, $\sigma$

On internet, I was looking for the proof regarding the following relationship between $Y$ young's modulus,$\sigma$ poisson's ratio, and $B$ bulk modulus: $Y=3B(1-2\sigma)$, and I came across the ...
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1 vote
0 answers
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Why are the after effects of elasticity maximum for glass?

After effects of elasticity are maximum for A) Glass. B) Quartz C) Rubber. D) Metal The correct answer given is glass. The vague explanation provided is that in the case of metal, the metallic bonds' ...
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4 votes
2 answers
132 views

How can I calculate the compression of a ball?

I was thinking about elastic collisions, and then I thought, What causes more compression in a ball when it's hit? Is it the velocity of the thing hitting the ball? Does the mass of the thing hitting ...
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0 votes
1 answer
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Twisting Solid Steel Bar - Annealed, Hardened or Doesn't mater? [closed]

Workshop project underway... Been decades since I was in a physics class. Building something out of steel - options are mild steel 1018 and alloy steel 4140 that can be hardened by heat treatment. ...
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1 vote
0 answers
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Questions on dynamical matrix and reduction in number of elastic constants

I had two questions involving chapter 22 of Ashcroft and Mermin: On page 438, A&M discuss some symmetries of the real space dynamical matrix. Specifically, I can't see to figure out the argument ...
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4 votes
0 answers
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Resonant states in elastic scattering

My question is simple, and I'm hoping the answer is also simple! Lets say I have a neutron and I scatter it off of an iron nucleus. Lets say that this is non-relativistic elastic scattering and I ...
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1 answer
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Bending stiffness and flexural rigidity of idealized slender structures under stretch

In the literature of applied maths/computational physics interested in slender structures, I find that there are two choices for the basic behaviour of the bending stiffness of slender structures when ...
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0 votes
1 answer
33 views

Moment of Inertia in non-rotational context

I have read that Civil Engineers use moment of inertia to characterize the elastic properties (rigidity) of such structures as loaded beams. My doubt is how is Moment of Inertia helpful in explaining ...
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1 answer
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If Potential Energy Is only defined For Conservative forces, can Non-conservative forces Increase spring potential?

Consider Two Cases Of collision When Energy is conserved before And after collision,we can say That The forces that acted between them were Conservative. During deformation their kinetic converted to ...
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I have to figure out when someone comes to the first stop after bungee jumping. How would I do it?

This is the information I'm given. John is attached to an elastic rope which is unstretched before John steps off the beam, and another end is fixed to a high beam. John then steps off the beam from ...
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Quantifying Elastic "Resistance Band" Tension

I am undertaking a small investigation into the effects of resistance bands applied to dynamic activity (in this case a squat). As part of this I'd like to quantify band tension throughout the ...
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