this book might be enlightening to you

ISBN-13: 978-0486661100

ISBN-10: 0486661105

thanks Abhimanyu Kumar

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 (consequencess 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.

For more analytic-precise details about the nature of 2-point tensors see:

¹] Eringen: Elastodynamics, 1974;

²] Kollmann & Hackenberg: On the Algebra of Two Point Tensors on Manifolds with Applications in Nonlinear Solid Mechanics, 1993;

³] Clayton: Nonlinear Mechanics of Crystals, 2010;

⁴] Luo: Nonlinear Deformable-Body Dynamics, 2011;

⁵] Marsden & Hughes: Mathematical Foundations of Elasticity, 2012;

⁶] Clayton: Differential Geometry And Kinematics Of Continua, 2014.

Tensor is a mathematical object that can be used to describe physical properties Like Scalar and Vector. It is often used in physics and engineering applications. Magnitude, Direction and Plane of action is required to define a Tensor quantity completely. Tensors are associated with magnitude and two or more directions. For example, the stress of an elastic solid at a point depends upon two directions. One of the directions is given by the normal to the area, while other is that which that of the force on it is. The order (or rank) of a tensor is defined by the number of directions required to describe Tensor. Properties that require one direction (first order) can be described completely is First Order Tensors, two directions (second order) is second order tensors, and three directions (third order) is three order tensors.

The simplest representation is a 3x3 matrix. Multiply the first 3D vector (the force) by the matrix and the result is another 3D vector (the motion of the surface, also called the strain). If I want to understand the 4D tensors of general relativity, it is best to start with the 3D tensors of materials science (materials science is very important to understand tensors). Ironically, it is the engineers who study such tensors; many undergraduate physics student encounter tensors for the first time when they take an advanced knowledge in relativity, and they find difficult to understand. The best course of action for them is to back off and study some classical engineering mechanics examples in 3D.

Simply, Tensor is a multidimensional array of functions on a manifold that behaves in a certain way when the coordinates change but it is much more then that.Tensor is an object and special type of map. It provides a natural and concise mathematical framework for formulating and solving problems in physics such as elasticity and Fluid Mechanics.