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Strain energy density in index notation

The strain energy density is defined as $$dU = \int_0^{\epsilon_{ij}} \sigma_{ij} d \epsilon_{ij}$$ (see Reddy "Energy Principles and Variational Methods in Applied Mechanics", 2nd Ed, 4.11). Assuming ...
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1answer
66 views

Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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1answer
68 views

How far do atoms move in a wire when whirling it horizontally?

so the question is: I whirl a mass m attached to a wire with length L and diameter d around my head in the horizontal plane. The mass takes t seconds to move around a circle. How far do the ...
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1answer
29 views

Stress in horizontal bars

Imagine we have an horizontal bar. My teacher expresses the tensions along the longitudinal axis by this way $\sigma_{xx}=A(x)y+B(x)$ He doesn't give any motivation behind this. So, is this general? ...
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90 views

Derivation of normal shear stress

I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about. Specifically, what I don't understand is ...
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585 views

Stress due to the mismatch of thermal expansion coefficients of two different attached materials

I'm simulating the thermo-electro-mechanical behavior of a copper wire which is surrounded by silicon dioxide. In other words, the wire segments is under mechanical and thermal loads and at the same ...
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0answers
226 views

Does a thermally expanding torus experience internal stress?

I'm trying to learn continuum mechanics and thermo-mechanics. As we know, heating an object increases the mean atomic distance $a_0$ of the atoms in a rigid body. Let's assume it is a linear elastic ...
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0answers
32 views

buckling of tube - shell thickness vs. momentum of inertia optimum

is there any simple formula (perhabs semi emperical, or aproximatively derived model) for buckling of tube under axial compression load given its crossection and wall thickness? ( and naturraly ...
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0answers
70 views

Stress calculations in a perforated paper

You have a sheet of paper (torn out of a good quality foolscap notebook) as shown above, and you start pulling it apart with both your hands (forces indicating by the blue arrows). Its difficult to ...
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0answers
244 views

Stresses on an inclined plane

Background This issue has been bothering me for a few days now. It's actually part of my homework, but I've already finished them and in a sense they're not part of the question here. What I would ...
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0answers
57 views

If I roll an elastic plate into a cylinder, does it shrink?

Suppose I start with a rectangular elastic (to keep things simple, zero Poisson's ratio) sheet of length $2\pi R$, thickness $h$, and (immaterial) width $W$. I roll it up into a cylinder of radius ...
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0answers
244 views

Stress analysis of a cantilever beam using FDM

I am a CSE researcher with a not so in depth background of physics. As a part of my research in object modelling, I am trying to computationally figure out the stress for various objects by using ...
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0answers
47 views

What diameter is used to calculate stress in an elastomer whose cross sectional area changes continuously when streched?

Consider a cylindrical rope of rubbery elastomer with a radius (in inches) equal to the square root of the inverse of $\pi \rightarrow \sqrt \frac{1}{\pi} $ so that the cross sectional area is $1 \, ...
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0answers
257 views

Free body diagram of stresses governing fracture propagation in a solid material

Reading this, I can intuitively understand that fractures propagate along the path of least resistance, creating "width in a direction that requires the least force". However, it is less intuitive ...
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0answers
41 views

Double dot product in Cylindrical polar coordinates - Strain Energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$2W=σ_{ij}ε_{ij}$$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
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0answers
22 views

Solid state diffusion as a function of shear stress

I am hoping to get some clarification on the impact of shear stress or shear strain on atomic diffusion in a metallic solid. More specifically, will the kinetics for the dissolution of secondary phase ...
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0answers
110 views

Is there analysis library for stress-strain data?

I have three column data that has time-displacement-force from 1D tensile/compression test. Now I would like to get the standard mechanical properties of the material, like Young modulus, yield ...
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31 views

Young elasticity modulus anisotropic media

Im studying anisotropic system composed by a elastic matrix (Young modulus $E_m$) filled with oriented rods. Given this filler orientation, the material is elastic-anisotropic, with Young elastic ...
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33 views

Volumetric and Deviatoric Strain Equation in 2D

Strain is defined as $$\epsilon=\frac{1}{2}\left( \nabla u + \nabla u^T\right).$$ I found a formula for the strain tensor in 3D decomposed into volumetric and deviatoric components: $$\epsilon= v + ...
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45 views

Can't work out how the following free body diagram was created

I've been asked to draw a free body diagram of the following stepped bar setup: $ P = 10kN \;and\; Q = 30kN$ The one shown in the answer is the following: Could someone explain to me how this ...
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40 views

Types of semiconductor relaxation

Ok, so you've basically got tensile and compression strain in a lattice mismatched material right? So as you're epitaxially layering the material, the strained material is deforming more and more ...
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0answers
129 views

Caculating thermal stresses in a composite rod

The Question Between two rigid walls separated by a distance $l_1+l_2$, two rods of equal cross-section area $A$, and length, coefficient of linear expansion and Young's modulus $l_1, ...
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36 views

How do I determine $\phi$ in a hollow circular cross section using torsion equivalence?

I know that the maximum shear stress $$\tau = \frac{T}{J}\rho$$ where $\rho$ is the radial distance from the center of the cross section. I have also determined the torsional constant $J$, which is ...
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74 views

A distributed axial load acting on a bar

Normally, when a distributed force is acting on a bar, if it's 'rectengular' we multiply the length with q, but I have no idea what I'm supposed to do at this situation. I first need to calculate ...
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63 views

Stress and Mass Density Distribution

I'm simulating electromigration in a copper wire using COMSOL and trying to see the back-stress caused by material transport. However, I do not see any stress growth. In other words, the atomic ...
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34 views

Photoelasticity and existence of stress

Shall we consider photoelasticity a non-mathematical, or purely visual, proof of the existence of stresses in mechanical structures undergoing external loading?
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49 views

What is the relationship between strain and electric current?

Strain or stress can be caused by different sources. I categorized theses sources as mechanical, thermal and electrical loads and formulated the total stress as follows: $$ \epsilon_{total} = ...
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41 views

How do I minimize the the stress created by fore-aft forces?

My friends and I are designing a bench press spotter. Essentially, when the user needs help lifting the barbell due to muscle fatigue, an arm on each side of the user raises to provide assistance. ...
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371 views

Whats the contact area for elastic contact between a cylinder and a plane surface?

I am trying to determine the contact area between a cylinder and a plane surface of two different materials so that the plane lies tangent to the cylinder. There is elastic contact between the two ...
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0answers
102 views

Continuum mechanics and effects of stress

Going to word this question a bit more straightforward than I may have before. Also, I'm trying to use baby formulas so I can grasp exactly what's going on. Object A has an elasticity of ...