What is the tensor type for Green Lagrange strain tensor and 2nd 2nd piola kirchhoff tensor?
Stress and strain are called 2nd order tensor because they follow the transformation rule: sigma`= R * sigma * R' where R is transformation matrix.
Deformation gradient, rotation matrix and 1st piola kirchhoff are called two point tensor because they relate two configuration. For example deformation gradient is: F(iJ) = xi * XJ where * is dyadic or tensor product and XJ and xi are the element before and after deformation.
https://en.wikipedia.org/wiki/Two-point_tensor
What about Green Lagrange strain tensor and 2nd piola kirchhoff tensor?
which they do not change with element rotation
this book might be enlightening to you
ISBN-13: 978-0486661100
ISBN-10: 0486661105
Both Green strain (E) and Second Piola-Kirchoff stress are defined in the initial configuration, so they are not two point tensors.
All Tensors follow the transformation law, regardless. But when you look at material objectivity , the single point ( I don't know if that name exists) and two point tensors obey different equations.
❖₁ Generally 2-point tensors (or double tensors) Ƭ₂ are the direct generalization of (1-point) tensors Ƭ (≡ Ƭ₁) over maps ƒ:
• for ƒ = φ ≡ x̂ (direct map): φt ≡ xt: X ↦ x = x̂ (X,t) = φ (X,t);
• for ƒ = φ⁻ ≡ X̂ (inverse map): φ⁻t ≡ X̂t: x ↦ X = X̂ (x,t) = φ⁻ (x,t),
regarded as a function of:
• material position X ∈ RR ⊆ BR ⊂ ᴹ𝔼³ (mostly) or
• spatial position x ∈ Rt ⊂ Bt ⊂ ˢ𝔼³ (rarely),
and strictly are tensor-like geometric entities but not tensors with respect to the general meaning of tensors.
Extended generalization of Ƭ₂ is to:
• multi-point tensors (or multiple tensors) Ƭᶰ and
• 2-point spinors (or double spinors) Ƨ₂ and thus to multi-point spinors (or multiple spinors) ƧN.
❖₂ For a tensor type of a tensor or/& a tensor-like geometric entity, specifically for the T̳2PK & E̳GLV, just realize the objectivity requirements for tensors or/& tensor-like geometric entities.
From the (consequences of) principle of objectivity (or material frame indifference) of decomposable n-order tensor (or n-ad) to tensor product of n vectors:
*: ₸ ↦ ₸* = ⊗i=1n T̲i = T̲1 ⊗ T̲2 ⊗ ⋯ ⊗ T̲n,
•₀ for n = 0: T* = T {TT: obj. scalar - invariant};
•₁ for n = 1: T̲* = T̲ ▣ Q̳ = Q̳ᵀ ▣ T̲ {TT: obj. spatial vector};
•₂.₁ for n = 2: T̳* = Q̳ ▣ T̲ ▣ Q̳ᵀ {TT: obj. (1-point) spatial dyadic};
•₂.₂ for n = 2: T₂* = Q̳ ▣ T₂ = Q̳ᵀ ▣ T₂ {TT: obj. 2-point dyadic ~ "vector-like"};
•₂.₃ for n = 2: TM* = TM [TT: obj. (1-point) material dyadic ~ "scalar-like" - invariant],
where {₸|*|, T|*|,, T̲|*|,, T̳|*|,, T₂|*|,, TM|*|} = F|*| (x|*|,t|*|);
Q̳ = Q̳(t); ▣: contracted tensor product; TT: Tensor Type; obj.: objective
❖₃ Applying the above principle to Kinematics & Kinetics of elastic (continuum) media under large (finite) deformation:
• E̳GLV* = E̳GLV (∴ obj. material dyadic);
• T̳2PK* = T̳2PK (∴ obj. material dyadic),
where E̳GLV = ½ (C̳ - I̳) = ½ (F̳ᵀ ▣ F̳ - I̳) or EGLVIJ = ½ (CIJ - δIJ) = ½ (FIkFkJ - δIJ);
T̳2PK = J F̳⁻¹ ▣ T̳C ▣ F̳⁻ᵀ or T2PKIJ = J F⁻¹IkTCklF⁻¹lJ. ☐
₁,₂,₃⇒ Approximaterly one could say from above (❖₁,₂,₃) that, the theoretical & applied "use" of these [(1-point) material] tensors is due to the "privlege" of invariance under the principle of objectivity.
There is a huge & vast relative bibliography for more analytic-precise details about the nature of 2-point tensors.
* Relative 10-Bibliography *:
- [1] Eringen, A.C: Elastodynamics (1974).
- [2] Joshi, A.W.: Matrices and Tensors in Physics (1975).
- [3] Kollmann, F.G. & Hackenberg, H.P.: On the Algebra of Two Point Tensors on Manifolds with Applications in Nonlinear Solid Mechanics (1993).
- [4] Ruiz-Tolosa, J.R. & Castillo, E.: From Vectors to Tensors (2005).
- [5] Clayton, J.D.: Nonlinear Mechanics of Crystals (2010).
- [6] Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus (2010).
- [7] Luo, A.C.J.: Nonlinear Deformable-Body Dynamics (2011).
- [8] Marsden, J.E. & Hughes, T.J.R.: Mathematical Foundations of Elasticity (2012).
- [9] Clayton, J.D.: Differential Geometry And Kinematics Of Continua (2014).
- [10] Filho, E.d.S.S.: Tensor Calculus for Engineers and Physicists (2016).
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