# Magnitude of a tensor

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 February 19, 2013, 09:06 Magnitude of a tensor #1 Member   M Mallikarjuna Reddy Join Date: Jul 2012 Posts: 91 Rep Power: 2 Dear all, I would like to share your opinion on my doubt. - The rate of strain tensor is written as(if i have not understand wrongly): E=symm(grad(U))=1/2[grad(U)+grad(U)^T].......................(1) - According to linear algebra the magnitude of any tensor S is: |S|=sqrt(2S:S) .........................(2) - According to Eq. (2) the magnitude of rate of strain tensor is: |E| = sqrt(2E:E). ..............................(3) = sqrt(2*(E_ij*E_ij))...................... (4) - In OpenFOAM notation the magnitude of rate of strain tensor can be written as: |E|=sqrt(2*symm(grad(U))).......................(5) But for the testing purpose i defined a Tensor Q as [1 0 0 0 0.8 0 0 0 0.5]. According to Eq. (2), it's magnitude is 1.944222. But the OpenFOAM is giving 1.3747727. So the definition of the magnitude of tensor according to OpenFOAM is: |S| = sqrt(S:S)=sqrt(S_ij*S_ij)...........................(6) Which contradicts Eq. (2). Can any one clarify my doubt and how to write rate of strain tensor in OpenFOAM notation. Thanks in advance. yours Mallikarjuna Reddy

 February 20, 2013, 09:54 #2 Senior Member   Fumiya Nozaki Join Date: Jun 2010 Location: Yokohama, Japan Posts: 110 Rep Power: 5 Hi, As you wrote, the magnitude of a tensor is defined in OpenFOAM as eq.(6). This is also described in the programmer's guide p-16. Fumiya

 February 20, 2013, 11:17 #3 Member   M Mallikarjuna Reddy Join Date: Jul 2012 Posts: 91 Rep Power: 2 Hi Fumiya Thanks for quick response. But in most of the books the shear rate is written as |E| = sqrt(2E:E) where rate of strain tensor is: E=symm(grad(U))=1/2[grad(U)+grad(U)^T] So according to OpenFOAM notation, the shear rate will be |E| = sqrt(E:E) = sqrt(magSqr(symm(fvc::grad(U)))) Could you please clarify whether this notation is correct or not. Thanks Mallikarjuna Reddy

 February 20, 2013, 16:32 #4 Senior Member   Mahdi Hosseinali Join Date: Apr 2009 Posts: 123 Rep Power: 6 I don't understand why the magnitude of tensor should be twice its double contraction? As a simple oppose example I can name vectors which are first order tensor and its magnitude would be sqrt(Ux^2+Uy^2+Uz^2). Still you may have shear stress right as: |E| = sqrt(E:E) = sqrt(2*magSqr(symm(fvc::grad(U))))

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