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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Maxwell's relations and adiabats
I was trying to understand problem regarding finding the adiabatic modulus given the isothermal young's modulus. I'm still an amateur in thermodynamics.
I just didn't understand the final step where ...
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Are the stress and strain tensor covariant or contravariant?
My question is related to this question but I don't find the answer there to be completely satisfactory.
The displacement of an elastic medium is a contravariant quantity, which I think is pretty ...
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What is the result of this Newton's cradle experiment where the initial ball doesn't bounce (exactly as in the regular case) but it has glue on it?
I have a question about a Newton's cradle type collision, but it has a twist to it. First, I will describe two well-known results, and then I'll add my twist.
First, consider a cradle with five balls (...
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Question about the elasticity matrix in metals
The most general anisotropic linear elastic material has 21 elastic constants. I am working with an HCP material and I found that it has 5 independent elastic constants. I am programming a subroutine ...
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Longitudinal wave in a falling elastic body
Consider an elastic rod hung from a high point with density $\rho$ and Young's modulus $Y$, subject to gravitational acceleration $g$. The coordinate from the hanging point is $x$, while the ...
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Why the total work to move a spring from point A to B equals the integral of all forces needed to hold it balance at every point between A and B?
I'm reading a Calculus book that mentions the Hooke’s Law for Springs that says the force needed to hold a spring at $x$ cm from it normal position is: $F = kx$, where $k$ is a constant.
I can ...
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In-plane stresses on the surface of a cylinder
The three principal stresses on the surface of a cylinder are the hoop, $\sigma_\theta=\frac{pR}{d}$, longitudinal, $\sigma_z=\frac{pR}{2d}$, and radial, $\sigma_r=-p$, stresses. However, what are the ...
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Identifying the elastic limit? [closed]
Consider this problem
I understand that the elastic limit is the point at which the material no longer elastically deforms, that is it doesn't return to its original shape. However, I am struggling ...
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Spring constant of typical flute style champagne glass?
I am a high school student doing a lab report on the relationship between height and resonance frequency of champagne glasses, using the "singing glasses" method where you rub your finger ...
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How would I understand Hysteresis from scratch? [closed]
I wanted to learn about Hysteresis and I know the basic undergraduate level of 'Mechanical properties of matter".
Can anyone please help?
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Equations of motion for two masses connected by the Kelvin-Voigt Model
I have a system where two particles $x_1$ and $x_2$ in one dimension are connected by a spring and a dash in parallel. This is analogous to the Kelvin-Voigt model for viscoelastic materials. The two ...
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How does restoring shear forces arise (in elastic conditions)? Do they arise from central forces or not?
When you apply a shear force onto a solid piece of material (say a block on a surface or a cantilever
beam with a load) that creates shear stress in the elastic regime, there is a restoring force that ...
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How do I define the initial jerk and yank of a cylinder subjected to impact of a rigid body at it's base?
I am trying to solve an axisymmetric longitudinal wave propagation problem during the impact (collision) of a rigid body and the base of a cylinder. On one end the cylinder is rigidly bound to the ...
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Would anyone be able to provide a reference for the equations concerning plane strain and incompressibility?
I've been trying really hard to find a textbook or research paper that mentions the equations I mentioned in my question. Sadly, I haven't had any luck so far. Would it be possible for someone to ...
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How to derive properly the stress-strain relation of an hyperelastic material?
I'm trying to implement a Yeoh hyperelastic model for a FEM simulation. The program requires a function that returns the second Piola-Kirchhoff stress tensor $\mathbf{S}$, with the Green-Lagrange ...
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Contradiction of Bulk's Modulus and Pressure exerted by a static liquid
My question is about a scenario where a certain amount of air is passed through water. Now, this air will form a bubble and due to the buoyant force acting on it, it will rise.
We know that the ...
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Since when has the formula $I = \frac{b h^3}{12}$ (second moment of area) has been discovered and applied in building construction?
A question about history of physics: since when and by who has the formula for the second moment of area
$$I = \frac{b h^3}{12}$$
been discovered? When has it started being applied in building ...
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Potential energy term in lagrangian of waves on a string
I'm going through Quantum Field for the Gifted Amateur, in example 1.4, the wave equation is derived through the Lagrangian. A key fact used is that, the potential energy equation is given as:
$$ V = ...
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The expression of elastic energy in this paper
Elastic properties of $\rm{Ni_{2}MnGa}$ from first-principles calculations
Hello,
I am reading a paper investigating the linear elasticity of a crystal. However, I am a little bit confused over the ...
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Is the expression of elastic energy in this paper correct?
Elastic properties of Ni2MnGa from first-principles calculations
I am reading a paper investigating the linear elasticity of a crystal. However, I am a little bit confused over the expression of ...
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Non-linear spring systems
I've recently been re-learning some physics, and a question came to me when looking over Hooke's law:
In the following I am always assuming that the force required for permanent deformation is ...
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Relation between Youngs Modulus and Co-efficient Of resistitution [duplicate]
I was thinking that $Y$ (youngs modulus) should be dependent on $e$ (Co-efficient of Restitution) as COR explains the separation/approaching ratios of velocities of 2 objects colliding, The velocity ...
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Prove that an isotropic substance has only two elastic constants
I've been following the derivation from Ch. II-31 of Feynman's Lectures, and there is a bit that I don't understand. We start from the relation between the strain $T_{ij}$ and the stress $S_{ij}$, $$...
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Elastic potential energy formula
From the Wikipedia page on elastic energy, we can find a bunch of formulas to describe it. For example, in the continuum section it talks about energy per unit of volume (density?):
$U=\dfrac{1}{2}C_{...
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Elasticity tensor for non-isotropic materials
How are the components of the elasticity tensor $C_{ijkl}$ defined for ANY material, especially those that are not isotropic? Does it always depend on the metric tensor (which in orthonormal Cartesian ...
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What does it mean for a material's elasticity to be non-linear?
Hooke's law only applies to materials with linear elasticity, usually for small displacements.
Now, if you imagine having a material that does not deform permanently when crossing a specific limit, ...
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What are the exact conditions for 1D elastic collisions? Is there a list for characterizing when elastic collisions occur?
We know there are elastic, inelastic, and partially elastic collisions. In a fully elastic collision, both momentum and kinetic energy are fully conserved. However, that tells us nothing for which ...
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How to solve dual boundary condition contradiction at the corner of an axially loaded cylinder?
Subject: Linear elasticity
Consider an isotropic elastic cylinder rigidly bound to the surface on one end. The other end is under uniformly distributed static load (pressure for example).
I am trying ...
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Calculating energy of a spring? [closed]
A toy of mass 20g attached to a spring is compressed and when the toy is released it jumps 0.7 m into the air. Calculate the energy of the spring. Assume the limit of proportionality is not exceeded.
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What is the microstructural explanation for the high Poisson's ratio of rubber?
Natural rubber has the highest quoted Poisson's ratio I've seen for a real material, usually given as 0.4999. This obviously makes intuitive sense when you think about the behaviour of rubber, and I'...
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Assumption in Maxwell Spring-Dashpot model
In the Maxwell Spring-Dashpot model : Maxwell Material - Wikipedia,
It is assumed that,
$$\sigma_{\text{total}} = \sigma_{\text{dashpot}} = \sigma_{\text{spring}} \text{ ...(1)}$$
$$\epsilon_{\text{...
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How to calculate Young's modulus of wood?
I'm attempting to calculate the modulus of elasticity, or Young's modulus, of various types of wood.
However, I'm confused by the strain component. It's calculated ...
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Equilibrium of rods and stress
Reading through Theory of Elasticity, Landau & Lifshitz, I got stuck in the The equations of equilibrium of rods, on page 82. The part I do not understand is the following:
"We denote by $\...
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Can contact stresses become independent of the rolling velocity in a viscoelastic rolling-contact?
In nonlinear viscoelastic rolling contact problems (assuming a viscoelastic wheel rolling/slipping with zero slip angle on a rigid body with friction), is it possible that the contact stresses (...
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Conversion of fourth-order tensor multiplication from indices notation to matrix form
Considering a 2D fourth-order tensor $C_{IJKL}$ which can be represented in Voigt notation as:
$$
C_{IJKL} = \begin{bmatrix}C_{1111}&C_{1122}&C_{1112}\\ C_{2211}&C_{2222}&C_{2212}\\ C_{...
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Pressure distribution inside solid sphere
Imagine a spherical solid body with initially uniform density, to which we apply a uniform external pressure on its surface. How would the stress distribution look inside that sphere? Would the center ...
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Does Hooke's Law apply to all springs?
I understand that Hooke's Law is $F=-kx$, and that this law only applies when a spring is not "overstreched." However, does Hooke's Law apply to all springs, or only simple harmonic ...
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Newton's Third law for two objects in an isolated system with only one internal force acting between them
I have been thinking about Newton's Third law from the past three days and am not sure if i completely understand it.
I need some help answering/describing this situation i had been thinking about. ...
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What really happends with the radial displacement at the origin of a disk and a cylinder under dynamic, uniform excitation?
I am trying to understand some properties of linearly elastic symmetric systems. Specifically, in the polar and cylindric coordinate systems. To be concrete, I am trying to understand the displacement ...
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Why should bulk modulus always be positive?
The minus sign that appears in Equation 12.39 is for consistency, to ensure that $B$ is a positive quantity. Note that the minus sign ($–$) is necessary because an increase $\Delta p$ in pressure (a ...
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Critical force for a system comprised of a compressible and incompressible parts
What is the critical buckling force needed to be applied on a system of made out of two parts?
The parts of the system are as depicted in the picture:
incompressible elastic beam - on top
...
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How does the stress vector change with respect to the radial coordinate at the radial surface of a cylinder?
Consider a linearly elastic vertical cylinder in the gravitational field with its base attached to a rigid wall while its other base is under uniformly distributed time-varying pressure. Its radial ...
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What is a Poisson Solid?
I've found on a few places on the internet references to a solid where both Lamé parameters are equal ($\mu=\lambda$), making its Poisson ratio $\nu=0.25$. It seems to be some sort of an 'perfect ...
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Why is the extension in the rod not same in both case?
I know that when you pull a rod from one end as shown in the figure
such that it is accelerating on a smooth horizontal floor with no friction, the average tension would be $F\over{2}$. And So stress ...
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Find the tensions in the supporting ropes of an inclined heavy beam supported by two diferent kinds of ropes [closed]
Recently I was making some late-revisit (for self reference) to undergraduate physics topics from the book Ohanian's Physics, 2E expanded-1989, which I loved to use during my freshman undergraduate ...
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Why extension is not dependent on friction here? [closed]
A ideal rod has mass $m$ and length $l$) is pulled by a constant force $F$ on a rough horizontal floor (of coefficient of friction $\mu$). Let its Young's Modulus of elasticity of the rod to be $Y$
I ...
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Is the restoring force equal to the deforming force in the case of plastic deformation?
I've been told by my professor that the restoring force (a reaction force) is always equal to the deforming force due to Newton's third law, and this is why stress is calculated using deforming force.
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What is the difference between elasticity and stiffness? [closed]
Modulus of elasticity indicates Elasticity(E) but stiffness(specifically in case of axial deformations) is given as EA/L where A is area and L is length of body. These expressions led me to believe ...
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Is it easier to expand or compress a rubber
I know that rubber follow's Hook's law to a certain limit. So during this period is it easier to push or pull?
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Why stretching body do not have acceleration? [closed]
I was studying elasticity and my teacher said as we are streching it, we can take F(elastic) = F(ext) as acceleration is zero. My efforts are that acceleration is only zero when the object is ...
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