Questions tagged [stress-strain]

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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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Strength Calculation of a Simply Supported 3D rectangular beam Under Loading As Shown In Image Attached

For a 2D and 3D circular beam we can easily calculate the strength of beam and reactions using beam theory. The same can be used for 2D rectangular plate. Also for a 3D rectangular plate with uniform ...
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Symmetric stress tensor in fluid Mechanics for a cube

In textbooks and online I read that conservation of angular momentum of a cubic element of fluid guarantees the symmetry of the stress tensor. I am not sure how to make this work. See the figure below,...
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How does shear strain defined as $\tan θ$ capture the ratio of change in dimension to original dimension? [closed]

Strain is generally defined as the ratio of change in dimension to original dimension. However, when it comes to shear strain, it is defined as $\tan θ$. How does $\tan θ$ give us an idea of strain?
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Torque equilibrium of a cylinder

In the process of deriving the stress tensor of a cube which is shown in the picture below, it is concluded from the torque equilibrium equation that $\sigma_{ij}$ is equal to $\sigma_{ji}$. I am ...
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How to figure out if stress is distributed uniformly?

What causes us to think that the stress is distributed uniformly over the cross sections A and B? Figure: Force F applied to a stationary bar
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Why would the stress-strain diagram differ for tension and compression in inelastic region?

The question came to me when I read the following statement: "The stress-strain diagrams may differ in the inelastic region for tension and compression. But these differences may be reasonably ...
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Two Different Forearm Movements

I came up with this thought 2 3 days back that when my move my forearm in two different sense I feel different strain/pain. First start assuming that your hand is like this.(i.e. the forearm and arm ...
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2 answers
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Young's modulus for a string that is being applied by two unequal forces at the both ends

In space for example if we apply two forces 10,6 respectively at the end of an elastic string (10 at the right end and 6 at the other) then obviously that body will expand and accelerate. But how ...
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Computing the maximum force a rod can bear [closed]

Suppose I had a rod of diameter $d$ composed of some material with tensile strength $T$. If I then exterted a pulling force $F$ on the ends of the bar, how do I compute the force $F$ for which the ...
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What is the divergence $S^{k}{}_{i,k}$ of a force component of a stress tensor called?

The following is from Hermann Weyl's Space-Time-Matter. Notice that $$\mathfrak{p}=-\left\{ p_{i}=S^{k}{}_{i,k}\right\}$$ is equivalent to a volume force. But, from the infinitesimal view, it is the ...
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Electromagnetic stress-energy-momentum tensor and stress tensor

The purely spatial components of the energy-momentum-stress tensor for a perfect fluid are clearly related the components of the ordinary stress tensor inside the fluid. My question if this case is ...
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Stress and equilibrium

By newton's third law, all internal forces in a portion of volume must be zero. Thus the total force is due to other volume portions exerting force onto the surface of said portion. However, if one ...
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Continuum Mechanics: Momentum Balance Equation of infinitesimal square [duplicate]

Introduction: According to the theory, the stress tensor should be symmetric. On This page, and on many others, the explanation is: Taking the rotational equilibrium around the center of this 2D ...
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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){...
1 vote
0 answers
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What is the intuition behind the ${\nabla u^T}$ term of the stress/strain tensor for a Newtonian fluid?

$$\epsilon=\frac{\nabla u+ \nabla u^T}{2},$$ $u$ is vector displacement, and $\nabla u$ is the gradient matrix of $u$. Now for a Newtonian, incompressible fluid, this describes the shear stress forces ...
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How to interpret XRD results?

I am given several datapoints of XRD strain information. It comes from XRD measurement of a thin layer (100 nm) of semiconductor deposited on top of another semiconductor. The result I'm given is ...
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Navier-Stokes Equation Derivation in "A Mathematical Introduction to Fluid Mechanics" (Stress Tensor)

My question relates to the attached pictures, which contain text from the book noted in the question title. In particular, my question is related to the first assumption made in the text (See bottom ...
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Diagonalization of a particular operator

I was exploring a little bit on quantum mechanics and I asked myself why wavefunctions do not get deformed, and I constructed a Hamiltonian based on the strees tensor $\tau_{ij}$ that it is not ...
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Untwisting Strands of a Rope

Statement:: Topoisomerases help in relieving strain in the DNA ahead of the replication fork caused by the untwisting of the double helix (Topoisomerases are enzymes that participate in the over ...
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Viscous stress tensor for an incompressible Newtonian fluid

I am reading the cengel book and found that the viscous stress tensor for an incompressible Newtonian fluid with constant properties is given by: $$\tau_{ij} = 2 \mu \epsilon_{ij}$$ where $\epsilon_{...
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Potential energy of a bent spring

What is the potential energy for a large spring that is bent into an arbitrary shape? The solution should be of the form $V=\int (...) ds $ EDIT: I have reduced this problem to the problem of the 2D ...
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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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1 answer
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Axial Force and Normal Stress in Mechanics

A couple of questions about the behavior of normal stress. For both of these, assume that the axial force applied is ideal, meaning distributed along the whole face of the member on both sides, and ...
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Is a stress tensor still symmetric when the object is rotating?

I am trying to simulate a spinning & flying deformable football with FEM method. It is always accelerating, instead of keeping static. Let an undeformed nodal position on this football $X \in \...
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Does the response of a fiber-optic sensor to strain increase as temperature increases?

Fiber-optic sensors etched with Bragg diffraction gratings are used as temperature sensors in many applications. They are also used to measure strain. As the strain or temperature increase, the peak ...
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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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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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What curve does a rod form when bent to intersect 3 or more points?

Suppose that we have a sufficiently thin, flexible cylindrical rod of length $L$ made from a homogeneous, isotropic material, and that initially [at rest?] the central axis of the rod is a straight ...
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Nickel gauge factor and why is it so quirky?

Hi so I've been researching the piezo-resistance effect and experimenting with finding different metals' gauge factors by applying stress on a wire and measuring the change in resistance however I can'...
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How to obtain nodal forces from stress tensor?

I am working on a FEM code in which I need to obtain the force vector on each node of a triangular linear element from the stress tensor in 2D. I have the shape functions, and the stress is constant ...
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Maximum tensile strain at a point on a standing wave

Objective To find the maximum tensile stress on a wire (of length $l$) at a point $x$ which has a standing wave on it having equation $y=f(x,t); 0 \le x \le l$. My work When the wire is at rest, the ...
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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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What would be the Stokes hypothesis in a 1D flow?

In this paper, the author derives the Navier-Stokes equation for a Newtonian fluid starting from the Cauchy equation: $$\rho \frac{D\mathbf V}{Dt} = \rho \mathbf{f} + \nabla\cdot\mathbf{T}$$ where $\...
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In stress-strain diagrams, why is the dependent axis ($y$) used to represent stress and the independent axis ($x$) to represent strain?

why this convention is used despite the fact that strain is influenced by stress (cause -> stress, effect -> strain)
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Volumetric Strain In a Thin Spherical Pressure Vessel

Consider a thin spherical pressure vessel with a fluid inside at a gauge pressure of P. The normal stress developed in the pressure vessel is given by $$\sigma = \frac{Pd}{4t}$$ where t = thickness , ...
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Interpreting stress at the ends of a bar

Consider a bar loaded in tension by distributed loads applied on its ends as shown in the figure. The stress at any cross section of this bar will be $$\sigma = \frac{P}{A}$$ From what I know about ...
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Stress without strain [duplicate]

Suppose there are two rigid walls with some gap between them and a steel rod is placed within that gap firmly attached to both the walls(no strain can occur). If the temperature is increased by $T$,...
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Opposing uniform pressures/forces

If I have a vertical plate, with a uniform pressure of $10\,\text N/\text m^2$ acting on the left hand side, and a uniform pressure of $6\,\text N/\text m^2$ acting on the right hand side, can I treat ...
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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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Does this work to reduce eyes strain? Enlarge the PC monitor with a lens

I have an idea to reduce eyes strain, but don't know if it works. I work by looking at PC monitor all day long and I believe my myopia condition is getting worse and eyes strain occurs are due to ...
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Why is viscous stress a tensorial quantity?

In an incompressible fluid, the viscous stress (in Cartesian) is defined by \begin{align} \tau_{ij} = \eta(\partial_i v_j + \partial_j v_i) \end{align} for dynamic viscosity $\eta$ and velocity field $...
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Rate of deformation tensor in cylindrical coordinates

A question regarding the rate of deformation in cylindrical coordinates. The rate of deformation tensor, $D_{ij}$, is the symmetric component of the velocity gradient tensor, i.e. $D_{ij} = (\frac{\...
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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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Homogeneous strain-Atomic position in deformed state

I encountered today an equation in a quite old paper, which I am not able to derive. For the notation they define $\vec{\mathring{x}}$ as the reference position for any material point and $\vec{x}(\...
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Stress in a rigid body

Consider two bars one rigid and the other deformable, acted upon by two equal and opposite point loads P as shown. In either of the cases, if we cut the beam from an imaginary section, then, to bring (...
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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 is this basic assumption leads incorrect answer?

A thin cylindrical uniform metallic rod of Length 'L' and radius 'R' rotates with an angular velocity "omega" in a horizontal plane about a vertical axis passing through one of its end . The ...
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Shear stress vs normal stress in incompressible solid

According to my current understanding, if a non compressible solid cube is subjected to a uniaxial compressing force, it will essentially be flattened - it will get shorter along the axis that is ...
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What is the interpretation of the symmetric stress tensor in laminar flow?

The strain rate tensor $D_{ij}$ is defined as $$ D_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right) $$ For a Newtonian fluid the stress tensor $\...

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