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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Error of deriving the variable cross section rod wave equation

On page $519$ of the book Engineering Vibration (which can be downloaded from here), the following wave equation of a variable cross-section rod is derived: $${ {\partial}\over{\partial}x} \Big( EA(x){...
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Continuum mechanics from a simple spring model?

I was trying to see if a simple spring model would reproduce continuum mechanics. My reasoning was that (at least in metals), the atoms form a lattice held together by forces that can be well ...
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Is there any problem in having a stress-strain constitutive relation that relates time-derivative of stress with strain?

We usually use two empirical laws to model viscoelastic behaviour: Hooke's law of elasticity that relates stress with strain Newton's law of viscosity that relates stress with time-derivative of ...
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Dependence of speed of sound [duplicate]

Why does speed of sound only depend upon elasticity of medium and not on frequency of vibrating particles of the medium?
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Center hole in a helical coil spring - formula for tightening during extension?

A helical coil spring can be thought of a wire wrapped multiple times around a virtual cylinder. While extending the spring, the virtual cylinder becomes smaller, until at last the spring is as long ...
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Deriving the wave equation of a variable cross-section rod subjected to non-uniform pressure

On page $519$ of the book Engineering Vibration (which can be downloaded from here), the following wave equation of a variable cross-section rod is derived: $${ {\partial}\over{\partial}x} \Big( EA(x){...
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Why does an extension cause my impact gun to have less torque?

I have an impact gun that I can mount an extension onto. It appears the longer the extension, the less torque the impact gun delivers ? Does this make any sense , or am I just imagining things
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If water is nearly as incompressible as ground, why don't divers get injured when they plunge into it?

I have read that water (or any other liquid) cannot be compressed like gases and it is nearly as elastic as solid. So why isn’t the impact of diving into water equivalent to that of diving on hard ...
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Can elastic mediums snap at a faster speed than the wave which travels in it?

The speed of wave in an elastic medium is given by Velocity of Longitudinal Wave 1) Velocity of Sound in any Elastic Medium: It is given by $v=\sqrt{\frac{E}{\rho}}=\sqrt{\frac{Elasticity\,of\,the\,...
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Generalization of Airy points for a circular plate

Given a 1D structural beam, the Airy points are the points at which vertical supporsts minimize the vertical displacement due to bending. For a 2D structural circular plate, we need three points at ...
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Classical Lagrangian for an isotropic elastic solid

Professor Fradkin’s course (CHAPTER 1. SECOND QUANTIZATION- equation 1.65) presents the Lagrangian density as: $$L=\int d^3r\frac{\rho}{2}\left(\frac{\partial u}{\partial t}\right)^2-\frac{1}{2}\int d^...
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Why is bulk modulus positive in equilibrium?

$B=-PV/\Delta V$ The negative sign indicates that when pressure increases, the volume decreases. That is, if $P$ is positive, $\Delta V$ is negative. Thus for a system in equilibrium, the value of ...
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What formula is needed to calculate the shape of a rod with a fixed point?

I want to find a formula to calculate the change in shape of a rod with a fixed point. The data would be: Amount of force displacement from the fixed point where force is applied (d) Young's modulus ...
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Why does a spring work the way it does?

I am wondering if anyone can explain what exactly makes the shape of a spring so good at creating something so elastic and good at converting between kinetic and potential energy. The metal itself isn'...
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Transverse waves in a rope: Why does tension not increase?

So I've gone down the "waves" rabbit hole... I found out that the propagation speed of transverse waves in ropes depends on the tension in the rope. At a first glance, this intuitively ...
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Limitations of spring and dashpot models in terms of strain meaningfulness

It is common to explain viscoelastic materials with spring and dashpot 1D constructions, e.g: which represents a Maxwell rheology, usually explained by saying that $ \sigma = \eta \dot{\epsilon}_1 = ...
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Do Elastic implies Quadratic Energy

We know, that are materials that holds the Hooke Law under a specific range, this is, materials for which the deformation $dx$ is proportional to the force $dF$, $dF=-kdx$, which implies $E={1\over2}k{...
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What happens if you push and pull 5 balls simultaneously connected by springs at opposite ends?

Say you have 5 balls connected by Steel springs in space . Each ball is 1 KG Distance between each ball is 1 Meter. Total Object is 5 Meters 1--2--3--4--5 say you push 1 and pull 5 simultaneously with ...
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What shape does an elastic rod take when both ends are dragged to the same point?

Suppose we have an ideal elastic rod of some kind, where the energy at a point along the rod is proportional to the square of the curvature, and we drag the ends of this rod so that they touch, and ...
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Can we prove the Hooke's law? [duplicate]

I was learning about stress and strain and my text book suddenly mentions about this law so called hooke's law. It states that $$\text{stress}\propto \text{strain}$$ Or, $$ \text{stress}=k \times \...
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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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$F=m*a$ 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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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