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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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Hooke's Law and Overstretched Springs

Recently been working with Hooke's Law at university. My experience of real springs tells me that stretching to a certain point causes the spring to lose "springiness". (Not sure if "elasticity" is ...
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2answers
61 views

Terminology for how bendable an object is and what affects the bendable-ness of an object

I was wondering what the term is for how bendable an object is. Also, does this feature vary depending on the thickness of the object? Say, for example, I want to know how bendable a ruler is. Does ...
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1answer
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(I-beam vs rectangular beam) Which sword blade cross section is less likely to break?

Original question Assumption: i made several swords with different cross sections (lenticular, single broad fuller as in viking swords, diamond, hollow ground diamond) the blades are made using the ...
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How to write a set and a series of equation using indicial notations

The above is the physical problem wherein I have a beam connected with two springs. each spring can take two states either zero or infinity. Accordingly, I am getting an equation which comprises the ...
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1answer
26 views

Relationship between strain energy function and strain or stress

How one can get the strain or stress from the strain energy function ? And if one cannot do it, what is the use of that function ?
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2answers
34 views

Is the Young's modulus the same for compressive and tensile test?

I want to know the Young's modulus for a compressive test, but I have just a tensile test. Are they of the same value?
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0answers
39 views

Fixing the Poisson equation to match the deformation of elastic sheet with experimental observation

I am working on the calculation of the deformation of a circular elastic sheet with radius $R=1.2~m$ when a plate with mass $M$ and radius $r_0 = 4~cm$ is sitting in the center of the sheet. I used ...
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1answer
31 views

Helfrich energy derivation

The Helfrich elastic energy of membranes is given by $F = \int dS (\kappa H^2 + \kappa_G K)$ where $H$ is the mean curvature and $\kappa_G$ is the Gaussian curvature. The derivation in the original ...
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Hertzian contact problem with two cylinders - question about derivation in book by Landau and Lifshitz

In the book "Theory of Elasticity" by Landau and Lifshitz (1986, Volume 7 of Course of Theoretical Physics, Third English Edition, Revised) a derivation of the Hertzian contact problem for spheres is ...
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2answers
48 views

Collision between two non-deformable objects

Let say we have a non-deformable object and we release it into free fall and it hits the non-deformable floor. What would happen? Here is the way I think. Since the floor and the object are both ...
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0answers
14 views

Stress state in an elastic half space with uniform constant body force

Is there a closed form expression for the stress state in an elastic halfspace subject to a uniform constant body force? I know that the Green's function for this problem is given by Mindlin's ...
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38 views

Physical limits of multiplying the elasticity tensor of a material by an arbitrary constant

Can all the elements in the elasticity tensor $(C_{ijkl})$ be multiplied by the same, positive constant (say $\lambda$)? If so, what are the physical limits? I've done some research myself and ...
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Can I use $v_{final}=0$ when I calculate impulsive force?

If a 1kg ball strikes a wall with 10m/s and it rebounds with 10m/s, the time of impact being 0.2s, what is the impulsive force? I know that the initial velocity is 10m/s but can I state that the final ...
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2answers
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Is $\sin\left[2\alpha\right]\cos\left[2\alpha\right]\ge0$ a valid restriction on the angles of the principal stresses in 2D elasticity?

This question pertains to Elasticity: Tensor, Dyadic, and Engineering Approaches By: Pei Chi Chou, Nicholas J. Pagano, Section 1.4. The objective under discussion is to find the directions of ...
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2answers
42 views

Spring strain vs fractional extension

How do I show that the strain tensor $$\epsilon_{ij}=\partial_i u_j + \partial_j u_i$$ in the case of a one-dimensional spring is reduced to $$\epsilon= \frac{x-L}{L},$$ where L is the initial ...
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2answers
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Does the resonance frequency of a cantilever beam depend on whether it is bending in-plane?

What is the in-plane resonance frequency equation for a single-beam cantilever of rectangular cross-section that is fixed on one end and has a mass M attached to the other end? Below are the only ...
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2answers
53 views

Why the resultant spring constant different in the following two cases?

In these two cases in the first case my book The Physics Of Waves And Oscillations by NK Bajaj says: That the restoring force exerted ...
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3answers
82 views

Why does different part of a spring having mass expand proportional to their distance while the spring has some mass hanged in the bottom?

Why do different parts of a massive spring expand proportional to their distance while the spring has some mass hung on the bottom which is comparably very less than the mass of the spring? If we take ...
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1answer
35 views

Motion along length of a spring [closed]

What's relation between velocity of each part of a massive spring undergoing Simple Harmonic Motion.
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0answers
23 views

2D plane stress problem: a bent rod with one curved side squeezed

Consider a bent rod (i.e., a sector of an annulus) of angle $\alpha$ and inner and outer radii $R_i$ and $R_o$. Now, we squeeze in azimuthally the lower face by an amount $\Delta$ (see the image below)...
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2answers
74 views

Stress/strain within an object experiencing a single applied force

If I push an object in space, will any strain/stress happen within the object, though there is only 1 force rather than a pair of opposite forces on the object? Can be any (even if trivial) stress/...
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1answer
41 views

Spring Constant of Two-Sided Fixed Beam [closed]

Let's suppose I have a two-sided fixed beam: ...and I want to find the equivalent spring constant... can I do the following: I know that the maximum deflection (at the center is): $$\delta =\frac{...
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2answers
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Oscillation of elastic rope hanging from the ceiling

I have to explore the problem of oscillations of an elastic rope hanging from the ceiling, which can move both in vertical and horizontal direction. I'm planing to solve this using a finite ...
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0answers
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Euler Elastica with a bent rod

Here is a thought exercise. What would happen if one were to do Euler's Elastica with a naturally bent rod, i.e. one already with a spontaneous curvature? Consider two cases. First, put in on the ...
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2answers
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Shape of a compressed wristband

What is the curve of the top part of this wristband after I squish the two ends closer? It's a curve of fixed length with given start $(x_1, 0)$ and end $(x_2, 0)$, and zero slope at these points, ...
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1answer
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Strain compatibility

From this link, strain compatibility for 2D problem with strains as $$ \varepsilon_{11} = \cfrac{\partial u_1}{\partial x_1} ~;~~ \varepsilon_{12} = \cfrac{1}{2}\left[\cfrac{\partial u_{1}}{\...
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1answer
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Are the curvature Frank elastic constants of the liquid crystal the no-zero elements of the 6x6 elastic (stiffness) matrix?

I was reading Landau's book on elasticity theory, and it raised this doubt. Can anyone help me?
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2answers
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Calculating Elastic Potential Energy of a Stretched Sheet

Essentially, I'm trying to determine the amount of elastic potential energy stored in a thin, elastic sheet that has gone under some type of stretching (ex. A flag of stretchy fabric waving in the ...
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2answers
152 views

Conservation of energy when dropping a ball [duplicate]

If we drop a bowling ball from one meter above the earth's surface, we are converting its gravitational potential energy to kinetic energy. When it hits the floor, it transfers its kinetic energy to ...
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0answers
60 views

Shearing strain - difference between gamma-xy and epsilon-xy

What is the difference between $\epsilon$$_{xy}$ and $\gamma$$_{xy}$? How does the relation $\epsilon$$_{xy}$ = $\gamma$$_{xy}$/2 comes?
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46 views

Calculating shear stresses with multiple forces

I have a rigid body where multiple forces (green) of different magnitude are applied. If I sum up the parallel component of the forces and divide it by the surface area, I can get the average shear ...
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1answer
55 views

Landau and Lifshitz argument for symmetry of stress tensor

In Landau and Lifshitz's book on the theory of elasticity (vol 7, theoretical physics series), specifically section 2 of the first chapter, the authors present an argument for justifying the symmetry ...
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1answer
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Why do materials obey Hooke's law? [duplicate]

Why do materials extend proportionally to the force exerted on them (Hooke's law)? I thought that when materials are compressed or extended under force, their atoms become closer or further apart; ...
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4answers
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Where is the Elastic Strain Energy per Actual Volume Gone?

I have a probably real stupid question, but I still can not wrap my head around it. The change in inner energy per unit volume $v$ of the actual state of a stressed crystal is given by the well ...
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1answer
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How to stack padding materials to pad an edge [closed]

This is about how to combine materials of various elastic and deformation properties as a stack to show different properties as a stack. A cardboard box has two holes punched in the side of the box ...
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Calculation of Young's modulus for a three-layer stacked material

I am currently working on three-layer stacked material and want to calculate the effective Young's modulus. The spring constant of the three layer material would need the effective Young's modulus ...
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1answer
61 views

Change of system of coordinates for the stress matrix

I have a stress matrix in cartesian coordinates : $\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}$. How can I convert it to spherical coordinates ?
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1answer
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Which Formula for the Internal Elastic Energy is Correct?

I am confused by the internal energy of a crystal. Lets say I have a isotropically stressed crystal. The total differential of the internal energy per unit volume V is defined as $$du = \sigma d\...
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2answers
141 views

Why a short rope takes more load than a longer rope?

The captain of a ship told me that it should be avoided to put a long rope next to a short one because the short rope takes most of the load and the long becomes relatively inactive, although both ...
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1answer
80 views

Can a piece of glass get higher than the height the original cup of glass fell from?

If yes, what physics concepts make this possible? This isn't for any class or anything, is just a honest doubt of mine.
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How to map a (virtual) truss deformation to a surrounding object

I am trying to craft a way of simulating deformable objects without using finite element method (despite it is the goldstar method for such purpose). The goal is to rapidly get an estimate of how a ...
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2answers
47 views

After elastic collision, where position of center of stationary ball? Does ball stop instantly?

Suppose we have two balls, $A$ and $B$, of radius $1$ with equal mass. Ball $B$ is initially at (center is at) two on the $x$-axis, i.e. $(2,0,0)$, and has velocity $0$. Ball $A$ is initially at (...
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0answers
54 views

Compute stress tensor in finite element simulation

I have implemented a finite element simulation using a tetrahedral mesh that minimizes a linear elasticity potential energy. Among the main quantities I compute for the simulation is the deformation ...
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2answers
47 views

Making sense of the stress tensor for elastic deformations

I've seen this kind of formula a number of times, in the context of elastic deformations. $$ -\nabla \sigma = f $$ whete $\sigma$ is "the stress tensor" and $f$ is force. I never understood it even ...
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Existence of solid mechanics problems that cannot be solved through Lax-Milgram approaches

Very often, solid mechanicians employ finite-element analyses to solve problems in linear solid mechanics. This approach is guaranteed to work because the Lax-Milgram theorem, along with some ...
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129 views

Hertzian contact between cylinder: calculating the displacement

I'm trying to calculate the total displacement (also known as depth of indentation, total elastic compression and total surface separation in different literature) in a Hertzian model of a contact ...
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1answer
37 views

Material tangent stiffness matrix for linear elasticity

I am trying to find material stiffness matrix for linear elasticity for finite element code, $$\mathbf{\sigma} = \lambda \hspace{1pt} \operatorname{tr}{\left(\mathbf{\epsilon}\right)}+ 2\mu\mathbf{\...
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1answer
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Difference between bending elasticity and Stretching elasticity in complex solids

In a solid, elasticity is described by the Young modulus E. Let us consider a thin sheet of material of thickness h. The bending rigidity is usually Eh³. I wanted to know if they are "abnormal" ...
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1answer
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Can Hooke's law be derived?

Can we derive Hooke's law from the theory of elasticity? I know it is not a fundamental law and therefore can be derived from more basic considerations.
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1answer
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Trouble connecting stress and force in continuum mechanics with my concept of force from point mechanics

I'm not very familiar with continuum mechanics and have a hard time combining my knowledge of forces from simple mechanics with what I read about continuum mechanics. Let's suppose we have a metal ...