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Isotropy: Updated Lagrangian Vs Total Lagrangian
Dear All,
I have seen it being mentioned that the material at an updated configuration in an Updated Lagrangian Formulation (say in FEM) may not remain isotropic, even if the reference configuration was isotropic. However, I have not seen any detailed discussion on the matter.
Hence, I would request your comments on:
- Whether this aspect is actually considered in an updated Lagrangian formulation, or is it ignored as a negligibly small effect?
- Does it mean that Total Lagrangian Formulation is "the" correct approach, and updated Lagrangian method is only an approximation?
Links to any references, wherein this matter is discussed are also welcome.
Thanks in advance,
Jayadeep
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Dear Jayadeep, Total and
Dear Jayadeep,
Total and updated Lagrangian are just two ways to look at the same conservation laws (mass, impuls, energy) - one uses material coordinates while the other uses space coordinates. Since material characteristics (e.g. isotropy) should be unchanged during changes of reference frames, I am pretty sure it does not change when you switch from updated to total Lagrangian formulation.
Regards, Andras
The updated configuration might not remain Isotropic
Dear Andras,
Thanks for the reply. However, as far as I understand, the updated configuration might not be isotropic. Since, I don't have a complete understanding of the matter, I shall give two points (and references) related to my initial question:
1. It can be shown that for a material, which is isotropic and fame-indifferent, the residual stress in the undeformed configuration must be hydrostatic [1, 2]. Since the state of stress in an updated configuration need not be hydrostatic, the undeformed (reference) configuration and updated configuration are not equivalent in this aspect.
2. Book by Bonet and Wood [3], clearly states that the spatial response characterised by the spatial elasticity tensor might be anisotropic, even when the reference configuration is isotropic. I feel the updated configuration is like a spatial configuration, and the elasticity tensor corresponding to this configuration, obtained by a push-forward operation performed on the referential form of the elasticity tensor, will not be isotropic. So my original question was whether such an operation is performed to get the elasticity tensor for the updated configuration, or is it a negligible effect so that the original elasticity tnsor can be used unchanged?
References:
[1] Gurtin M.E, Eliot Fried, and Lalit Anand, "The Mechanics and Thermodynamics of Continua ", Cambridge University Press.
[2] C.S. Jog, "Continuum Mechanics ", Alpha Science (or Narosa in India and neighboring countries)
[3] Javier Bonet, and Richard D. Wood, "Nonlinear Continuum Mechanics for Finite Element Analysis ", Cambridge University Press.
Regards,
Jayadeep
Hi Jayadeep, I have made
Hi Jayadeep,
I have made some research on the topic, and I have found the following things:
1, You were right about the spatial elasticity tensor being anisotropic even when the material counterpart is isotropic. Bonet clearly states that in his book, although he does not explain why. In an article of Peric [1] I have found that the properties of the spatial elastic tensor are strogly dependent on which strain measure you use in your material description. In case of Green-Lagrange strain, the spatial description involves the Truesdell rate of the Kirchhoff stress to be work conjugate. Your spatial elasticity tensor will then be non-constant and isotropic. If you use the Hencky (logarithmic) strain, the spatial description involves the Green-Naghdi rate of the Kirchhoff stress. The spatial elasticity tensor will then be non-constant and anisotropic. In both cases the involvement of the deformation gradient changes the properties of the (originally) isotropic and constant material elastic tensor.
2, The updated Langrangian formulation describes the problem exactly (same as total Lagrangian) if you involve the special features of the spatial elasticity tensor. The approximation comes from the fact that in many cases we assume the spatial elasticity tensor to be constant and isotropic. This assumption makes our formulation secon order accurate, though it saves a lot of computation. In metal plasticity, where the elastic strains are small, this approach gives the results within a reasonable accuracy. Since rate equations are rarely used in hyperelasticity, where are assumptions become invalid, one faces nearly no restrictions.
Regards, Andras
References:
[1]Peric: On consistent stress rates in solid mechanics: computational aspects
http://onlinelibrary.wiley.com/doi/10.1002/nme.1620330409/abstract