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. M

    Normal Stress/Shear Stress from stress tensor

    Homework Statement If \sigma_{ij} = \begin{pmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \end{pmatrix} represents a stress tensor, on what plane(s) will the normal stress be a minimum? On what plane(s) will the shear stress be a maximum? Homework EquationsThe Attempt at a Solution The first...
  2. R

    Understanding Composite Laminate Stresses

    Hi Guys I am trying to figure out what stresses in different planes mean physically for composite laminates. Say S stands for stress then SXX SYY SZZ would be longitudinal/axial stresses in respective Axes. (If we take the laminates to be in XY plane, failure stress SZZ would be equal to...
  3. A

    Is stress tensor symmetric in Navier-Stokes Equation?

    Hello, In CFD computation of the Navier-Stokes Equation, is stress tensor assumed to be symmetric? We know that in NS equation only linear momentum is considered, and the general form of NS equation does not assume that stress tensor is symmetric. Physically, if the tensor is asymmetric then...
  4. bcrowell

    Parity of stress tensor versus stress-energy tensor

    The stress-energy tensor is an actual tensor, i.e., under a spacetime parity transformation it stays the same, which is what a tensor with two indices is supposed to do according to the tensor transformation law. This also makes sense because in the Einstein field equations, the stress-energy...
  5. S

    Maxwell Stress Tensor -> Force between magnets and perfect iron

    (this is not a hw) Assume you have a magnet of dimensions x_m, h_m, remanent flux density Br, and coercive field density Hc. The magnet is placed in a magnetic "C" structure (perfect iron) such that it is connected on one side but there is an airgap on the other side. xxxxxxxx xx... xx...
  6. T

    Understanding Stress Tensor in MTW Ex. 5.4

    I've been working on Ex 5.4 in MTW. The maths is fairly straight forward, but I don't really understand what is going on! In part (b) what are the 'forces' pushing the volume through a distance? If they are forces, they must produce an acceleration but we have a constant velocity. Are these...
  7. M

    Understanding Viscous Stress Tensor in Incompressible Flow

    hey pf! in reading a book on viscous stresses i found the following: \tau_{ij}=2\mu\Big(s_{ij}-\frac{1}{3}s_{kk}\delta_{ij}\Big) where einstein summation is used. now we have s_{ij}=\frac{1}{2}\Big(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\Big) and then the claim is...
  8. T

    Stress tensor transformation and coordinate system rotation

    Homework Statement Hi, I am not sure if this is the right place for my question but here goes! The stress tensor in the Si coordinate system is given below: σ’ij = {{-500, 0, 30}, {0, -400, 0}, {30, 0, 200}} MPa Calculate the stress tensor in the L coordinate system if: cos-1a33=45°, and...
  9. TrickyDicky

    Is the Maxwell stress tensor a true stress?

    Should it be added to the Cauchy stress to calculate a "total stress", or it doesn't have such a physical interpretation as a surface force(EM field force is usually considered more of a "body force")? Certainly when the MST was first derived before aether theories were made superfluous by...
  10. M

    Understanding the Cauchy Stress Tensor for Beginners

    Hello, I am not sure what the first indice in the cauchy stress tensor indicates For example, σ_xy means that the stress in the y direction, but does x mean the cross sectional area is normal to the x direction?
  11. D

    Maxwell stress tensor in different coordinate system

    Hi guys, I would like to know if the answer given to this thread is correct https://www.physicsforums.com/showthread.php?t=457405 I got the same doubt, is the expression for the tensor given in cartesian coordinates or is it general to any orthogonal coordinate system? Thanks in advance
  12. R

    Solve the Stress Tensor Problem Now

    http://im32.gulfup.com/u6n3f.jpg
  13. S

    Stress tensor rotation/shear stress

    Hi. I have a huge problem and without solving it I can't move forward. I will appreciate any help. Having the stress tensor S: 163.666557052527 -63.0272557558942 0.000000000000000E+000 -63.0272557558942 70.3802282767392 0.000000000000000E+000...
  14. P

    Maxwell Stress Tensor in the absence of a magnetic field

    I'm having some trouble calculating the stress tensor in the case of a static electric field without a magnetic field. Following the derivation on Wikipedia, 1. Start with Lorentz force: \mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B}) 2. Get force density \mathbf{f} =...
  15. J

    Stress tensor vanishes on cylinder edge

    Homework Statement Given a cylinder in the Ox1x2x3 coordinate system, such that x1 is in the Length direction and x2 and x3 are in the radial directions. The stress components are given by the tensor $$ [T_{ij}] = \begin{bmatrix}Ax_2 + Bx_3 & Cx_3 & -Cx_2 \\ Cx_3 & 0 & 0 \\ -C_2 & 0 &...
  16. T

    Stress tensor transformation

    Hi all, I have been reading up about continuum mechanics recently, and have a question regarding the reduction in stiffness coefficients in the stiffness matrix. I am aware of how the stiffness matrix is reduced to 21 coefficients. However, in order to reduce it from 21 to 13, one has to...
  17. M

    Stress Tensor in Spherical Coordinates

    Homework Statement Calculate the deformation of a sphere of radius R and density \rho under the influence of its own gravity. Assume Hooke's law holds for the material. Homework Equations Not applicable; my question is simply one of understanding. The Attempt at a Solution I want...
  18. R

    Stress Tensor Homework: Finding Traction Vector

    Homework Statement http://img842.imageshack.us/img842/9577/stresstensor.png Homework Equations 'traction vectors' are just the rows of the stress tensor. that is, the first row of the stress tensor(the i-component of the tensor) is the first traction vector, second row is the second,etc...
  19. bcrowell

    Asymmetric stress tensor gives asymmetric stress-energy tensor?

    I'm sure there's a trivial explanation for this, but it's escaping me. The space-space components of the stress-energy tensor are interpreted as the 3x3 stress tensor. But WP claims that the symmetry of the stress tensor need only hold in the case of equilibrium: "However, in the presence...
  20. S

    Get the stress tensor using virtual displacement principle

    Homework Statement Using method of virtual displacement get the stress tensor for this problem. http://img32.imageshack.us/img32/7272/problemba.png The attempt at a solution The given system is 4-times tatically indeterminate. Since it is symetrical we can split it in half vertically...
  21. H

    Maxwell stress tensor for a nonlinear media

    Hi all, It seems to me that the derivation of Maxwell stress tensor is independent of the permeability of the media or the nonliterary of its B-H relation. By this I mean that we use μ0 in the equations rather than μ. Would you please confirm that?
  22. H

    Is stress tensor a 3x3 or 3x3x3 tensor?

    Hi all, I have a fundamental question about ( mechanical) stress tensor. Stress tensor a 3x3 tensor whose 9 entries looks "scalars" but in figures, the stress is illustrated by nine "vectors". Does it mean the stress tensors is in fact a 3x3x3 tensor of scalars whose nonzero entries are ignored?
  23. Peeter

    Fluid stress tensor in cylindrical coordinates?

    For fluid with viscosity \mu our stress strain relationship takes the form \sigma_{ij} = -p \delta_{ij} + 2 \mu u_{ij}. I was wondering how to express this in cylindrical coordinates. The strain tensor I can calculate in cylindrical coordinates (what I get matches eq 1.8 in [1]). But how...
  24. F

    Cauchy stress principle & eigenvalues of stress tensor

    First of all, thanks for all the helpful comments to my previous posts. I'm trying to get a grasp of stress tensors and have been doing some studying. In the literature I've been looking at, it says something about the eigenvalues of stress tensors and the principle stresses. This is...
  25. H

    Maxwell Stress Tensor: Application & Force Calculation

    Hello everyone, I have a confusion about the application of Maxwell stress tensor: I have read some materials about Maxwell stress tensor and its application in calculating electromagnetic force on a body. To this end, a closed surface is assumed around the body and a surface integral on...
  26. W

    The way textbooks talk about stress tensor confused me

    In fluid dynamics, always when some textbook talks about stress tensor, there is a figure like this: http://www.fea-optimization.com/ETBX/hooke_help_files/stress.gif it shows how stress tensor is defined based on a small cubic volume. I kind of understand why the shear stress τxz should be...
  27. A

    Direction of the Maxwell stress tensor

    Homework Statement Show that in vacuum the pressure tensor of a (complex) plane electromagnetic wave only has a contribution for both directions in this bivector being along the direction of motion, and that contribution is equal in magnitude to the energy density. HINT: Choose 3 orthogonal...
  28. Matterwave

    E&M Maxwell equations and Stress tensor

    Hello, so I was asked a question in two parts (Peskin & Schroeder problem 2.1). The first part asked me to derive the source-free Maxwell's equations from the action: S=\int{d^4 x \frac{-1}{4}F_{\mu\nu}F^{\mu\nu}} Given that the vector potential itself is the dynamical variable. I derived...
  29. A

    Perfect Fluid Energy Stress Tensor

    in a perfect fluid the stress energy tensor is: T_{AB} = (P + \rho) u_A u_B + P g_{AB} queation1 : always u_A =1, \vec{0}? question2: if the space time have a line element h_{AB}dx^A dx^B...for the calculus of T_{AB}, ¿ g_{AB} = h_{AB}? ¿can i to use g_{AB}=\eta_{AB} if h_{AB}...
  30. Z

    Stress tensor : tension VS Pressure

    My question is regarding exercise 5.1 on page 141 of MTW. How come the tension and the pressure have the same value ? The field lines here are they the field lines of the electromagnetic force ? Thanks,
  31. L

    Maxwell Stress Tensor: Find Elements for Plane Wave in Z Direction

    Homework Statement find all elements of maxwell stress tensor for a monochromatic plane wave traveling in z direction and linearly polarized in x. Homework Equations Tij=\epsilono(EiEj-(1/2)\deltaij E2+1/\muo(BiBj-(1/2)\deltaB2 The Attempt at a Solution So i found what E and B is well not...
  32. M

    Stress Tensor in Classical Field Theory

    Hi, I have a problem in classical field theory. I have a Lagrangian density \mathcal{L}=\frac{1}{2}\partial_\lambda \phi \partial^\lambda \phi + \frac{1}{3}\sigma\phi^3 . Upon solving the Euler-Lagrange equation for this density, I get an equation of motion for my scalar field \phi (x), where...
  33. C

    Stress tensor and viscosity relation

    Can anyone tell me why this is true? I can't find an explanation anywhere, and it doesn't make sense to me geometrically either, especially for i=j. Isn't viscous force, by definition, a shear force? How can it produce a normal stress? Also why would Txy=Tyx? Why can't they be different...
  34. E

    Maxwell stress tensor coordinate system

    Hello, I am trying to understand the Maxwell Stress Tensor. Specifically, I would like to know if it is coordinate-system dependent (and if so, what the expressions are for the stress tensor in cylindrical and spherical coordinates). Griffiths gives the definition of the maxwell stress tensor...
  35. haushofer

    Geodesic equation via conserved stress tensor

    Hi, I have a question which was raised after reading the article "Derivation of the string equation of motion in general relativity" by Gürses and Gürsey. The geodesic equation for point particles can apparently be obtained as follows. First one takes the stress tensor of a point particle...
  36. D

    Stress Tensor vs Stress Matrix: What's the Difference?

    Hi,can u please tell me the difference between the Stress tensor & the Stress matrix...
  37. M

    Finding Shear Stress with a 2x2 Stress Tensor and Normal Vector

    Homework Statement If I have a 2x2 stress tensor and a normal vector n and want to find the shear stress. Is it the same which shear stress I find? I mean Pn = P dot n Pnn = Pn dot n Shear stress = Pnt = Pn - Pnnn This gives Shear stress with the same unit vectors as n, f.ex i an j But I can...
  38. T

    The Schwartzschild solution - why no Stress Tensor?

    I've been working my way through GR using mainly D'Inverno's Introducing Einstein's General Relativity, but with MTW as well and a couple of other books. I began to get a sense of what was going to happen and got a big surprise when I reached the Schwartzschild solution. The process goes...
  39. Q

    Shear stress in Energy-momentum Stress Tensor

    Hi all - first post at PF. As a 'science enthusiast' with no training in the tensor math of GR, was initially bewildered by the common assertion that still hypothetical 'dark energy' would act as a source of 'negative gravity' despite having positive energy density. Finally grasped that pressure...
  40. V

    Understanding of Maxwell's Stress Tensor

    Was a bit fuzzy as to whether this better fit HW or here, but since there really is no question associated with it, figured this made a bit more sense. I have a couple basic questions about the stress tensor: T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) +...
  41. V

    How Does the Stress Tensor Change at Point P Near Current-Carrying Wires?

    So I'm working on yet another problem, and have come to a minor stump. 2 wires run parallel with the Z axis in the xz plane, one with current I-1, other with current I-2. I need to determine the components of Maxwell's stress tensor at a field point P, where P is a point on the yz plane (x=0)...
  42. P

    Maxwell stress tensor in electrodynamics

    \hat{N}=\{\vec{E},\vec{D}\}+\{\vec{H},\vec{B}\}-\frac{1}{2}(\vec{D}\cdot\vec{E}+\vec{B}\cdot\vec{H})\hat{1} \hat{1} - unit tensor If I look \{\vec{E},\vec{D}\}. I know that \{\vec{E},\vec{D}\}=\{\vec{D},\vec{E}\}^* But when I can say that \{\vec{E},\vec{D}\}=\{\vec{D},\vec{E}\}? and when...
  43. D

    Application of Maxwell's stress tensor to linear pull type solenoids

    My professor recently tasked me with first understanding maxwell's stress tensor and then applying it to analyze the force produced by a linear pull type solenoid. However after having studied the stress tensor I am not sure that I can apply the stress tensor to a solenoid. I don't believe...
  44. T

    Stress tensor components question

    Homework Statement stress tensor in cartesian components. \sigma is the stress tensor. e_i are the basis vectors Homework Equations \sigma \cdot \sigma The Attempt at a Solution I tried to write out the components with a cartesian basis: \sigma=\sigma_{ij} (e_i \otimes e_j) But then I'm...
  45. H

    What is the stress tensor and how is it related to motion?

    I have a couple of questions about the stress tensor. I am not an engineering student, so this is the first time I have dealt with internal forces, stress, shears, and such. It is my understanding that the entries in the stress tensor are forces per unit area. I assume this means the total...
  46. Pengwuino

    Invariant stress tensor = Invariant force?

    So we know that \frac{d}{dt}(P_{mech} + P_{field}) = \oint_S {T_{\alpha \beta } n_\beta da} that is, the time rate of change of the momentum of a system plus the momentum of the electromagnetic fields is equal to the surface integral of the term with the Maxwell Stress Tensor where there is...
  47. X

    Plotting stress tensor in MATLAB, Parametric Surface

    First off, I'm not sure if MATLAB questions go here or not, or if they are even dealt with on this forum. If not more or remove I guess. I am no MATLAB wiz so this is mind-boggling to me. I am given a state of stress at a critical point of a component and told to plot a 3D parametric surface...
  48. Rasalhague

    What Is the Role of Space-Space Components in the Stress Tensor?

    I'm trying to understand the meaning of the components of the stress(-energy) tensor. Considering just the space-space components, is this right? T^{ij} = \frac{\mathrm{d} F^i}{\mathrm{d} A^j}, where F^i is the x^i[/tex]-component of force, i \neq 0, and \mathrm{d} A^j is an area element...
  49. T

    Maxwell Stress Tensor Homework: Find E & B Field Components

    Homework Statement x and y are nonconducting cylindrical shells. Both cylindrical shells are surrounding long wires that are carrying current. the x shell out of the page and the y shell into the page. x radius has a charge per unit length = to +\lambda y radius has a charge per unit length =...
  50. T

    How Do You Calculate the Maxwell Stress Tensor Between Cylindrical Shells?

    Homework Statement x and y are nonconducting cylindrical shells. Both cylindrical shells are surrounding long wires that are carrying current. the x shell out of the page and the y shell into the page. x radius has a charge per unit length = to +\lambda y radius has a charge per unit length =...
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