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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19 views

Is the force $F$ in formula of longitudinal stress applied on both sides of a rod?

Suppose a force $F$ is applied on one face of a rod (with cross-sectional area $A$) and one end is fixed. And now suppose another situation with the same rod, force $F$ is applied on both faces of the ...
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Strain rate and deformation in non-Cartesian coordinates

I have a question regarding the connection between deformation, strain, and strain rate: The deformation tensor $\textbf{F}$ is defined by $F_{ij} = \partial X_i / \partial x_j$ and its determinant ...
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Bow and Arrow problem '2'

Several days ago I posted the next question....Bow and Arrow problem. At the beginning the question got 3 downvotes but after a certain period it stabilized at equal number of downvotes and upvotes... ...
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Could you help me understand the notation of forces on the plate?

I am so confused about the notation in this book with the one I learned before. So could anybody help me comprehend the notation in this figure? I am not clear about is the notation of the moments. ...
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Physics of bursting balloons?

This is a reference request for a theory of bursting balloons: Let's say I have a balloon and this balloon has an impurity along a small strip. Let's assume for example the balloon had a hole there at ...
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1answer
28 views

Stress to make concrete stronger

In reinforced concrete steel rods are embedded in wet cement and held in tension while the mixture dries. Once fully dry the tension is removed, and the concrete should be stronger as a result of this ...
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44 views

Why is the number of independent elements of a stiffness tensor 21?

I think that the dilatation of an elastic material is proportional to the hydrostatic pressure. That is, ${\rm{tr}}(\boldsymbol\sigma)=3k\ {\rm{tr}}(\boldsymbol\epsilon)$ for small strains. If so, ...
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Elastic Modulus of Elastomers

For an elastomer, what could be the modulus of elasticity. My thought process is that the modulus of elasticity is the tangent of a point of the stress-strain graph. And therefore looking at the graph ...
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Chladni plates theory

I have come upon a very old post (Theory behind patterns formed on Chladni plates?). In this post, the computation of the eigenmodes of a rectangular plate with free edges is addressed, but one point ...
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49 views

Why do books deform?

I take a thick book, open it to the middle, put a pencil in it, and then close the book. If I then wait five minutes and take the pencil out, the book will go back to normal. If I instead wait five ...
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Regarding the Hooke's law and how it can be described using stress and strain [closed]

Can someone kindly explain to me the how the relationship marked as (2) below is obtained from the relationship marked as (1). In our Physics class, our lecture wrote down the following relationship, ...
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5answers
497 views

In a collision shouldn't objects of different mass have same acceleration?

Suppose two objects of different mass, A and B, collide with each other. Now, during the time of collision, they both apply forces on each other according to Newton's 3rd law. Therefore, their ...
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17 views

Impulse Time Elapsed in relation to Hardness and Momentum

I am trying to calculate the time elapsed of an impulse during a collision of two objects. I will give three examples below in hopes that they make clear what I am stuck on. Two object of the same ...
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What is the unit for modulus of elasticity?

I've previously asked a question regarding the modulus of elasticity which I am still unsure about. I've been told various formulae to calculate the modulus of elasticity, some of which give answers ...
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How do I calculate the modulus of elasticity of a spring when I don't know the cross-sectional area? [closed]

'An elastic spring of natural length 1.5 m has one end attached to a fixed point. A horizontal force of magnitude 4 N is applied to the other end and compresses the spring to a length of 1 m. Find the ...
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Energy conservation of longitudinal waves

Suppose I have an elastic body described by a vector field $\pmb{\Delta} \left( \pmb{x}, t \right) $ which gives the displacement of a point from equilibrium. This field can be separeted in a ...
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Does increasing the number of stretched elastic bands increase the total elastic potential energy?

Suppose I stretch a single elastic band to $x$ cm attached to an object like a paperclip. The elastic has potential energy modelled by $$U = \frac{1}{2}kx^2$$ where $U$ is potential energy, $k$ is the ...
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Can tablets/capsules bounce?

I'm doing a physics lab on rotational motion tomorrow and I wanted to know whether a capsule-shaped object (other than a tic-tac, i.e. a capsule/pill, since I don't have tic tacs and don't have time ...
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Rotational invariance of the the elastic deformation in spherical symmetry

I write the elastic deformation $E$ for an incompressible material in spherical symmetry (Have a look at here Eq.(5.32) ). Since we are incompressible, $\mathbf{det} \,E=1$, so in spherical ...
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1answer
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Chameleon's tongue as a harmonic oscillator

I was watching a documentary about reptiles, at some point, it showed how chameleons catch their prey. Approximately, their tongue has an acceleration of $2500 \frac{m}{s^2}$, length of $0.7m$. They ...
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Rule of the Work done by a force

I have been always taught that Work done = force x distance through which the force was applied. but recently in the elasticity I have been taught that W= (1/2) (F X D). why did we multiply it by a ...
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Completely elastic and inelastic colission

I know what is preserved in completely inelastic collision: $m_1v_1+m_2v_2=(m_1+m_2)v$. Now I would like to know what is preserved in completely elastic one.
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How much of the momentum is being transfer in case of collision relative to elastically and momentum difference?

if we take a look at two objects that has been collision ,we may use the the third law of newton to calculate the velocity of each one if we know the force they use on each other. for example if we ...
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Doubt on Cauchy Stress tensor: a partial derivative of metric tensor?

In the reference $[1]$ the author presented a definition of Stress tensor: $$ \sigma = 2 \rho \frac{\partial e}{\partial g} \tag{1}$$ In a local chart we have: $$ \sigma_{ab} = 2 \rho \frac{\...
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Question from Mechanics of Deformed Bodies

Question I tried cutting a section of width $dx$ at a distance $x$ from the left end, So, at steady state, the acceleration would be $ \frac{4F}{m} $ So, $ F-T= \frac{M dx }{L} (\frac{4F}{m}) ...
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How can ideal gases be “elastic” and “ideal” at the same time?

The elastic properties of any substance is because of the restoring intermolecular forces operating in it. And I read that the isothermal elasticity of an ideal gas is given by $E_{isothermal} = P$ ...
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Reference request for piezoelectricity

I'm looking for a mathematically rigorous treatment of piezoelectricity (preferably both the linear theory as well as nonlinear electroelasticity). I've already seen Yang's (2018) An Introduction to ...
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Sources for studying non linear elasticity and relevant tensor mathematics

What are some helpful sources for studying nonlinear elasticity, plasticity, and the relevant mathematics involved, primarily tensors?
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99 views

What exactly is Hooke's law defined for?

I used to think that Hooke's law was a relationship between how much a bar under uniaxial loading deformed and the internal force (per unit area) that developed within that bar. But this clearly isn't ...
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1answer
31 views

Ambiguity in taking the cross sectional area in Young's modulus

Let's say we stretch a wire and it's cross-sectional area decreased. When using the Young's modulus formula$$\frac{FL}{A\Delta L}$$ Where F is the force subjected on the body. Should we take the cross ...
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21 views

Elastic constants, transformation/basis change $P21/a\to P21/n$

I just calculated the stiffness tensor for a given material (a monoclinic molecular crystal) using molecular simulation. When trying to compare my results to published experimental results I found ...
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2answers
106 views

Is there a relationship between the complex modulus and the dynamic viscosity?

I have a material for which I know both the tensile storage modulus $E'$, and loss modulus $E''$ at a frequency of 1 Hz. As I understand it, $E'$ conceptually describes the elastic properties of the ...
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1answer
84 views

How do I account for damping on structural vibrations?

I want to know what terms to add to the differential equation for structural vibrations to account for structural damping. Not external damping, but internal damping. I consider the transfer of energy ...
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3answers
90 views

Meaning of the notation $\sigma_{ji,j}$

In page 28 of the book Introduction to Linear Elasticity, 4ed by Phillip L. Gould · Yuan Feng, it says $$ \int_V{\left( f_i+\sigma _{ji,j} \right) \text{d}V=0} $$ What does it mean by writing $\sigma ...
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Why is stress defined in the way as it is?

Stress is like pressure and it doesn't matter in which direction the force acts (given it is perpendicular to the surface). I read in my book that if we have a rope which is being pulled on both ...
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Fundamentals of body deformations (recommend some books? soft robots?)

I'm researching about soft robots but most of the papers I've found have a very complex theory, which is making my progress quite slow. I was wondering if any of you who have some background on ...
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How derive a spring constant of a helically turned helix based on elasticity of the material and the geometry? [closed]

The structure has helicities on two levels and looks like the tungsten wire in an incandescent light bulb: How, if at all, can the spring constant be derived from elastic properties of the material, ...
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20 views

Can a metal barrier mounted on a system of springs with a right elastic constant save a weak object from a bullet?

My question is motivated by the fact that a lot of objects or vehicles that are important can be hit by bullets.Will a barrier mounted on a system of springs efficiently attenuate the kinetic energy ...
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1answer
32 views

Rubber's resistance to compression along one axis

I'm concerned with application of rubber sealings. I need to find out what Force is needed to compress this piece of rubber (the rubber cannot move in the z direction, consider the width of rubber ...
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4answers
142 views

Why can a piece of paper be folded along a straight line but not along a curved line?

It seems that if something can be stretched then it can be folded along a curved line. Since paper can't be stretched I can't fold it along a curve. But it's just an observation not an answer to the ...
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1answer
25 views

Friction in continuum mechanics

How does one include friction forces in the context of continuum mechanics? I imagine that one could rewrite the relation $$ F\leq N $$ from the classical mechanics in terms the strain on one surface ...
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45 views

Counterforce of a stretched gummi

According to Newton's 3rd law, $ F_{A\to B} = -F_{B\to A} $, when you apply a force to an object, it applies the same force (against you?). When I stretch a strap with my hands, which force exactly is ...
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76 views

What is the maximum height from which an object can be dropped without breaking it?

Suppose we drop from rest, a body of mass $m$ and breaking stress $\sigma$ on the ground from height $h$ and it collides with the perfectly rigid ground and comes to rest within a short time $t$. The ...
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The Derivation of Thin Plate Energy function?

I am trying to derive the Thin Plate Energy Functional. Given a Thin Plate $z = z(x,y)$, how does one derive the energy function: $$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\partial ^2z}{\partial x^2}\...
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1answer
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Continuum Mechanics: Why would dashpot kinematics require 5 equations to specify deformation for a 2D axisymmetric problem?

I work with PEG hydrogels and use the material to recapitulate cartilage biology and I am really interested in modeling the soft tissue mechanics in COMSOL. I have been studying the work of Caccavo et ...
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Can anyone explain how this rod (experiencing shear stress and deflection) mysteriously resists the downward force from the weight of the mass?

This fascinating and, in my opinion, very unique question came out of a competition lab I did in a general engineering course. I will leave out or change info about the lab to make this question more ...
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Mathematical definition of elastic materials

Physically, elastic materials are materials which return to their original state upon complete removal of applied mechanical loads under isothermal conditions. In the book "Mechanics of ...
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21 views

mechanical energy for a membrane with natural curvature of both radii

I'd like to study the following mechanical energy : $$F_{my\,energy}=\int \Big(\frac{1}{R_1}-c_1\Big)^2+\Big(\frac{1}{R_2}-c_2\Big)^2 dA $$ I noticed that this energy cannot be written as a Canham-...
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1answer
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Energy of local rotation in elasticity theory

In the theory of elasticity there is an important object, the displacement increment vector $u_i$. The derivative of such an object can be decomposed into symmetric and antisymmetric parts: $$ \...
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Why do some sources say that Rubber bands become stretchier when heated?

For a physics project I'm changing the temperature of a rubber band and measuring elasticity. The rubber band will be tied to a mass and will be heated in a water bath. Some sources show data of the ...

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