What is Stress tensor: Definition and 119 Discussions

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

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  1. Salmone

    A Calculate a tensor as the sum of gradients and compute a surface integral

    I am trying to compute the stress tensor defined as ##\vec{\Pi}=\eta(\nabla{\vec{u}}+\nabla{\vec{u}}^T)## where ##T## indicates the transpose. The vector field ##\vec{u}## is defined as follows: ##\vec{u}(\vec{r})=(\frac{a}{r})^3(\vec{\omega} \times \vec{r})## with ##a## being a constant...
  2. L

    A Going from Cauchy Stress Tensor to GR's Energy Momentum Tensor

    Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units? Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR? If so, What...
  3. D

    I Cauchy Stress Tensor in Applied Strength of Materials

    I am in a course in applied strength of materials and we often use the 3D stress tensor for stress analysis of materials i.e. Mohr's circles, bending, torsion, etc. Is the stress-energy tensor in relativity basically a 4-d extension to the Cauchy stress tensor commonly used in mechanical...
  4. A

    Dependence of the stress vector on surface orientation

    According to Cauchy's stress theorem, the stress vector ##\mathbf{T}^{(\mathbf{n})}## at any point P in a continuum medium associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e., in terms of...
  5. T

    Cauchy's Formula for Stress tensor

    Hello I am reviewing the proof of Cauchy's formula for the stress tensor and surface traction. Without exception, every book I look at gets to the critical point of USING the projection of a triangle onto one of the three orthogonal planes. However, I have never seen this proven. I have...
  6. B

    Frame indifference and stress tensor in Newtonian fluids

    During lecture today, we were given the constitutive equation for the Newtonian fluids, i.e. ##T= - \pi I + 2 \mu D## where ##D=\frac{L + L^T}{2}## is the symmetric part of the velocity gradient ##L##. Dimensionally speaking, this makes sense to me: indeed the units are the one of a pressure...
  7. B

    I Divergence of first Piola-Kirchoff stress tensor

    Hi everyone, studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93) where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$ How is the...
  8. wrobel

    A Stress Tensor: Definition, Ideas & Discussion

    Inspired by the closed thread about pressure:) Here is some of my fantasies about a definition of the stress tensor. Nothing here claims to be a correct theory but just as a matter for discussion.
  9. B

    Invariants of the stress tensor (von Mises yield criterion)

    Hello all, I am trying to understand the von Mises yield criterion and stumbled across two equations for the second stress invariant. Although the only difference is a difference in signs (negative and positive), it has been bothering me. Attached are the two versions. Which one is correct and...
  10. E

    What is the relationship between force lines and the stress tensor field?

    Force lines method is used in Solid Mechanics for visualization of internal forces in a deformed body. A force line represents graphically the internal force acting within a body across imaginary internal surfaces. The force lines show the maximal internal forces and their directions. But...
  11. Glenn Rowe

    Electromagnetic stress tensor from pressure and tension

    I'm puzzling over Exercise 1.14 in Thorne & Blandford's Modern Classical Physics. We are given that an electric field ##\boldsymbol{E}## exerts a pressure ## \epsilon_{0}\boldsymbol{E}^{2}/2## orthogonal to itself and a tension of the same magnitude along itself. (The magnetic field does the...
  12. C

    A How are the thermal expansion of a solid and the stress tensor related?

    My idea is this: tensor stress is directly related to the internal pressure of a solid. That is to the force that the neighboring atoms exert each other in relation to a unit of surface. When I heat a solid we can have the phenomenon of thermal expansion: this is connected to the fact that a...
  13. patrykh18

    Stress tensor for a parallel plate capacitor

    The question is partially taken from Griffith's book. I am confused about the physical meaning of momentum in fields. I have determined the solution and found that in part d the momentum crossing the x-y plane is some value in the positive z direction. I don't however understand the physical...
  14. lachgar

    How to form the stress tensor component from the equilibrium equation?

    Good evening everybody. This is my suggestion for answer. The tensor is diagonal and the compression is a plane stress equilibre equation div(σ)=0 so: So, does that means that = f(y.z) = Ay+Bz and =f(x.z)= Cx+Dz A,B,C and D are constants. Is that what the question meant? Thank you in...
  15. SamRoss

    I How can the stress tensor be non-zero where there is no matter?

    You're on Earth. You throw a ball and watch its trajectory. It's curved. That's because the Earth is curving space-time at every point along the trajectory. But the Earth itself is not present along the trajectory - there is no matter along the trajectory (let's ignore the air and any radiation...
  16. A

    Shear and the stress tensor of a Newtonian fluid

    Similarly the paper by @buchert and @ehlers https://arxiv.org/abs/astro-ph/9510056 Here the author has defined ##v_{ij}=\frac{\partial v_i}{\partial x_j}=\frac{1}{2}(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i})+\frac{1}{2}((\frac{\partial v_i}{\partial...
  17. JD_PM

    Understanding the Maxwell Stress Tensor

    The elecromagnetic force can be expressed using the Maxwell Stress Tensor as: $$\vec F = \oint_{s} \vec T \cdot d \vec a - \epsilon \mu \frac{\partial }{\partial t} \oint_{V} \vec S d\tau $$ (How can I make the double arrow for the stress tensor ##T##?) In the static case, the second term...
  18. N

    I What is the relationship between the Hamiltonian and Lorentz invariance?

    Hi, I hope this is in the right section. It's for EM which I guess is a relativistic theory but the question itself is not to do with any Lorentz transformations or anything similar. I'm reading through Jackson with my course for EM and I'm on the section where he is generalising the Hamiltonian...
  19. U

    Theories without a stress tensor

    Can someone tell me a theory in which the lowest twist operators are not the stress tensor and its derivatives? My aim is to work out the lightcone OPE for the theory and derive bounds like the averaged null energy condition. (as worked out in https://arxiv.org/pdf/1610.05308.pdf)
  20. M

    Maxwell Stress Tensor: Engineering Question Answered

    Hello! I was talking with a friend today about electrical motors and we started talking about theoretical designs. One question came up which was could the Maxwell Stress Tensor be used to calculate the torque on a rotor of a motor where the airgap is held constant and the magnetic circuit...
  21. A

    How Can We Derive the General Stress Tensor Without Assuming Linear Media?

    Homework Statement Hi everyone! My name is Alexandra, and I'm new in this forum. I am trying to determine the mentionated tensor without the assumption of linear media or vacuum ( ## \textbf{D} = \epsilon \textbf{E} ## and ## \textbf{B} = \mu \textbf{H} ##). What I want to obtain is the...
  22. S

    A Stressing Over Stress Tensor Symmetry in Navier-Stokes

    How do we know that the stress tensor must be symmetric in the Navier-Stokes equation? Here are some papers that discuss this issue beyond the usual derivations: Behavior of a Vorticity Influenced Asymmetric Stress Tensor In Fluid Flow http://www.dtic.mil/dtic/tr/fulltext/u2/a181244.pdf...
  23. Another

    Help with Maxwell stress tensor

    << Mentor Note -- OP has been reminded to use the Homework Help Template when posting schoolwork questions >> my think if ## \hat{r} = \sin(θ) \cos( φ) \hat{x} +\sin(θ) \sin( φ) \hat{y} +\cos(θ) \hat{z} ## ## da = R^2 \sin(θ) dθdφ \hat{r} = da_{x} \hat{x} + da_{x} \hat{y} + da_{z} \hat{z}##...
  24. Polyamorph

    A Viscosity from DFT (VASP) using the Green-Kubo relation

    Hello! In this paper https://pdfs.semanticscholar.org/e8a2/02f25555cd8c4f947bbbdff5a61a0ea0efd2.pdf the authors use VASP to determine MgSiO3 viscosity using the Green-Kubo relation ## \eta = \frac{V}{3k_{\rm{B}}T}\int_{0} \left<\sum_\limits{i<j}\sigma_{ij}(t+t_{0}).\sigma_{ij}(t_{0})\right>dt##...
  25. CantorsLuck

    A Momentum Energy tensor and Wilson Loop in Yang-Mills Theory

    Hello Everyone. I Was Wondering how excatly the Gauge invariance of the trace of the Energy-momentum tensor in Yang-Mills theory connects with the trace of an Holonomy. To be precise in what I'm asking: The Yang-Mills Tensor is defined as: $$F_{\mu \nu} (x) = \partial_{\mu} B_{\nu}(x)-...
  26. dRic2

    I Significance of the viscous stress tensor

    Hi, when working with NS equations the stress tensor can be written as ##\nabla \tau = - \nabla P + \nabla \tau_{v}##, where ##\tau_{v} ## is \begin{pmatrix} \tau_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{xy} & \tau_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \tau_{zz} \end{pmatrix} This...
  27. S

    Stiffness Stress Tensor Question

    Homework Statement I am given c11, c12, and c44. What is poissons ratio ν and the E modulus E [100] for a single crystal for uniaxial strain in [100] (if Fe is isotropic)? ii) What is the anisotropy factor A? (iii) There is: sigma=[100 0 0; 0 100 0; 0 0 0]Mpa What is the transverse strain in...
  28. Kushwoho44

    Understanding the Cauchy Stress Tensor: Clearing Up My Confusion

    I have been trying to fully grasp the concept of the Cauchy stress tensor and so I thought I'd make a post where I clear up my confusion. There may be subsequent replies as I pose more questions. I am specifically confused at how the stress tensor relates to the control volume in the image...
  29. Ron19932017

    I Pressure term in the energy stress tensor

    Hi all, I am reading Bernard Schutz's a first course in general relativity. In Chapter 4 it introduced the energy stress tensor in two ways: 1.) Dust grain 2.) Perfect fluid. The book defined the energy stress tensor for dust grain to be ## p⊗N ##, where ##p## is the 4 momentum for a single...
  30. C

    Continuum mechanical analogous of Maxwell stress tensor

    Maxwell stress tensor ##\bar{\bar{\mathbf{T}}}## in the static case can be used to determine the total force ##\mathbf{f}## acting on a system of charges contanined in the volume bounded by ##S## $$ \int_{S} \bar{\bar{\mathbf{T}}} \cdot \mathbf n \,\,d S=\mathbf{f}= \frac{d}{dt} \mathbf...
  31. Vectronix

    I Stress tensor and partial derivatives of a force field

    If F = Fxi + Fyj +Fzk is a force field, do the following derivatives have physical significance and are they related to the components of the stress tensor? I notice they have the same dimensions as stress. ∂2Fx / ∂x2 ∂2Fx / ∂y2 ∂2Fx / ∂z2 ∂2Fx / ∂z ∂y ∂2Fx / ∂y ∂z ∂2Fx / ∂z ∂x ∂2Fx / ∂x...
  32. V

    Why Isn't My EM Stress Tensor Calculation Giving the Expected Result?

    Homework Statement An electric field E exerts (in Gaussian cgs units) a pressure E2/8π orthogonal to itself and a tension of this same magnitude along itself. Similarly, a magnetic field B exerts a pressure B2/8π orthogonal to itself and a tension of this same magnitude along itself. Verify...
  33. P

    A Stress tensor in 3D Anti-De Sitter Space

    I am doing some mathematical exercises with 3D anti-de sitter face using the metric ds2=-(1+r2)dt2+(1+r2)-1+r2dφ2 I found the three geodesics from the Christoffel symbols, and they seem to look correct to me. d2t/dλ2+2(r+1/r)*(dt/dλ)(dr/dλ)=0...
  34. S

    I Understanding the Concept of Momentum Flux in Stress: Explained and Demystified

    I'm trtying to get a better understanding of the spatial part of the energy-momentum tensor, and although similar questions have been asked here, I think the point I do not fully grasp has not been covered so far. The stress tensor can be considered as "momentum flux density" tensor. If I...
  35. V

    I Infinitesimal cube and the stress tensor

    The Cauchy stress tensor at a material point is usually visualized using an infinitesimal cube. The stress vectors (traction vectors) on opposite sides of the cube are equal in magnitude and opposite in direction. As a result, the infinitesimal cube is in equilibrium. However, when we derive...
  36. H

    How to find the Piola-Kirchhoff stress tensor

    Homework Statement Hello, I am supposed to show that the quantity TR=JTF-t satisfies TR=∂W/∂F for some scalar function W(X, F, θ) in my continuum mechanics homework. The task is to identify this scalar function W(X, F, θ).Homework Equations This is part b) of a question. In part a), we get...
  37. E

    I Is the Energy Stress Tensor of Dust Always Zero Inside a Moving Cloud?

    If a large cloud of dust of constant ρ is moving with a given ##\vec v ## in some frame, then at any given time and position inside the cloud there should not be no net energy or i-momentum flow on any surface of constant ##x^i ## (i=1,2,3) because the particles coming in cancels those going out...
  38. C

    Exploring the Differences between τxy and τyx in the Cauchy Stress Tensor

    Homework Statement https://en.wikipedia.org/wiki/Cauchy_stress_tensor[/B] I don't understand the difference between τxy . τyx , τxz , τzx , τyz , τzy ..What did they mean ? Homework EquationsThe Attempt at a Solution taking τxy and τyx as example , what are the difference between them ? They...
  39. J

    Single shear element in stress tensor: Finding Von Mises

    When finding the Von Mises of given a stress tensor who's only element is a single shear component (τ): \begin{bmatrix} 0 & τ & 0\\ τ & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} the result is simply √3×τ. Is the Von Mises criterion not valid when considering a single component as in this example? I...
  40. C

    Stress tensor for non-Newtonian fluid

    How does one setup the stress tensor for a non-Newtonian fluid? I know that for any fluid the normals should be the pressure and for a power law fluid the shear stress in the direction of flow is related by K(du/dy)^n. Does this mean that all other components are 0 for a symmetric pipe or...
  41. M

    Decomposing Uniaxial Stresses

    Homework Statement I am having trouble decomposing a uniaxial compressive stress into hydrostatic and pure shear components. Homework EquationsThe Attempt at a Solution I am starting with ## \begin{pmatrix} -\sigma & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix} ## I then do ##...
  42. W

    How to get stress tensor from force

    Homework Statement I'm analyzing the landing gear of a plane for an extracurricular project. I know that the landing gear will undergo impact loading during touch down. Using what we've been taught, I converted the impact load to a static load, P that will act on the landing gear. Now I need to...
  43. D

    Finding principal axes of electromagnetic stress tensor

    Homework Statement In a certain system of units the electromagnetic stress tensor is given by M_{ij} = E_iE_j + B_i B_j - \frac12 \delta_{ij} ( E_kE_k + B_kB_k) where E_i and B_i are components of the 1-st order tensors representing the electric and magnetic fields \bar{E} and \bar{B}...
  44. E

    What stress tensor components mean?

    Hey! I'm reading a book Intermediate Physics for Medicine and Biology In it, there is a section that is describing shear forces and it says this as a side note: In general, the force F across any surface is a vector. It can be resolved into a component perpendicular to the sur- face and two...
  45. genxium

    Why is stress tensor (in this derivation) symmetric?

    First by "this derivation" I'm referring to an online tutorial: http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node9.html It's said in the above tutorial that the ##i-th## component of the total torque acting on a fluid element is ##\tau_i = \int_V \epsilon_{ijk} \cdot x_{j} \cdot F_{k}...
  46. sweet springs

    Sign of Maxwell's stress tensor

    Why Maxwell's stress tensor has minus sign to the corresponding components of electromagnetic momentum energy tensor ? From WP --- , where , is the Poynting vector, is the Maxwell stress tensor, and c is the speed of light. ----
  47. M

    MHB Unraveling the Stress Tensor: A Beginner's Guide

    Hey! :o Could you explain to me the stress tensor?? (Wondering) I haven't really understood what it is...
  48. fluidistic

    Maxwell stress tensor to calculate force (EM)

    Homework Statement A sphere with dielectric constant ##\varepsilon## and radius R is placed inside a homogenous external electric field ##\vec E_0##. The sphere is divided in 2 hemispheres such that their common interface is orthogonal to the external field. Using the energy-momentum tensor...
  49. R

    Calculating Force using the Maxwell Stress Tensor

    Homework Statement Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω, and surface charge density σ. Use the Maxwell Stress TensorHomework Equations F=\oint \limits_S \...
  50. C

    Stress tensor from action in Landau-Ginzburg field theory

    I would appreciate any help with the following question: I know that for relativistic field theories, the stress tensor can be obtained from the classical action by differentiating with respect to the metric, as is explained on the wikipedia page...
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