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How can i define Hook's Law interm of Tensor

  1. Oct 11, 2005 #1
    How can i define "Hook's Law interm of Tensor"

    i want to define hoos's law interm of tensor:confused:
    how cn i define it:surprised
    can you all friends help me?o:)
    i will be thanksfull to all of you
     
  2. jcsd
  3. Oct 12, 2005 #2

    PerennialII

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    Hi Rousha, welcome to PF!

    I'm checking I understood you right .... are you referring to the constitutive tensor of Hooke's law, the elasticity tensor [itex] E_{ijkl}[/itex]?
     
  4. Oct 12, 2005 #3
    need more help

    i want to know that how can i define relationship between stress tainsor and strain tensor and also asked that is the transformation law of tensor obey in this relationship
    thanx alot
     
  5. Oct 13, 2005 #4

    PerennialII

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    Write the generalized Hooke's law as

    [tex]
    \sigma^{rs}=E^{rsij}\epsilon_{ji}, r,s,i,j \in (1,2,3),
    [/tex]

    for relating the stress tensor [itex]\sigma[/itex] and the infinitesimal strain tensor [itex]\epsilon[/itex], where E is the elasticity tensor (by postulating the existence of a
    strain energy density has 21 independent coefficients).
    For homogeneous isotropic material the elasticity tensor, the generalized Hooke's aw can be expressed using the Lame coefficients as

    [tex]
    \sigma_{ij}=\lambda\delta_{ij}\epsilon_{kk}+2\mu\epsilon_{ji},
    [/tex]

    where the Lame coefficients are given as (by introducing the Young's
    modulus E and Poisson's ratio [itex]\nu[/itex] )

    [tex]
    \lambda= \frac{\nu E}{(1-2\nu)(1+\nu)}
    [/tex]

    [tex]
    \mu=E/2(1+\nu) .
    [/tex]

    The generalized Hooke's law in a general coordinate system can be written as (using the Lame constants again)

    [tex]
    \sigma^{pq}=\lambda I^{\epsilon}_{1} g^{pq}+2 \mu g^{ip} g^{jq} \epsilon_{ij},
    [/tex]

    where [itex]I^{\epsilon}_{1}[/itex] is the first invariant of [itex]trace(\epsilon)[/itex].
     
    Last edited: Oct 13, 2005
  6. Oct 20, 2005 #5
    thankx for ur reply:smile:
    ok i understand this terms and finally i want to know that what is the application of this relation
     
  7. Oct 21, 2005 #6

    PerennialII

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