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# Two-point Tensors

I am confused about the use of two-point tensors in elasticity. The deformation tensor F and first PK tensor are two point tensors, and the "two-point" property arises from use of two different Coordinate System When a continuum body is deformed, why it is necessary to move the Coordinate System as well? (or alternatively, why the coordinate system is attached to the body itself??, isn't it possible to use a general coordinate system which can represent the deformations and also account for the rigid body rotations of the continuum body?).

I suppose the sole purpose of using new coordinate system associated with the deformation is to separate the rigid body rotations during the deformation and the new CS provides rotation-free deformations. If my idea is not correct please suggest/reply with some simple

Please note: My question is about the coordinate systems, (not about coordinate values).

Thank you.

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## Comments

## Re: two-point tensor and Lagrange description

There are two separate issues here: two-point tensor and Lagrange desciption. While two-point tensor is not a must, a Lagrange system is usually required to properly describe a solid. A Lagrange system, in contrast to an Euler system, traces material points by its original (reference) position rather than its current position (so that a cooridnate system moves during deformation). The former is often adopted in solid mechanics while the later in fluid mechanics, because of the physics of the two types of materials.

Solids (especially elastic solids) differ from fluids in that they have "memory": a solid knows its original state to some extent, while a fluid only cares about its current state (with some exceptions). Therefore, to fully describe a solid, we need to specify a reference state, and measure the difference between its current state and the reference. A two-point tensor, the deformation gradient, is naturally involved to bridge the two states. Two point tensors can be avoided by introducing the right Cauchy-Green deformation tensor F'F (or Green strain), and the second PK stress, with both "legs" in the reference state. Actually, these tensors will remove the rigid-body rotations, which are included in F.

However, it is imposible to describe the deformation state of an elastic solid with Eulerian tensors only, defined in the current state. You need to tell the material where it was.

## Dear Wei, I have a

Dear Wei,

I have a question regarding to what you said.

If we consider a body just did rigid body rotation, then the deformation gradient F=R, R is the rotation. If we suppose there is only one frame (Euleran frame), then R is not a two-point tensor, R=RijEiEj, both Ei and Ej are in the same fame

however, if we still assume there exist a Lagrangian and a Eulerian frame, then F=R=Rij*ei*Ej is a two-point tensor?

Is this correct based on what you said?

## Two-point tensor, deformation gradient, rotation, deformation

I have described these ideas with some care in my class notes on finite deformation. Here is a list of points:

Deformation gradientYin a vector space U. That is, a linear combination of two segments in U is yet another segment in U, as defined by the usual addition of vectors.yin a vector space V. Note that U and V are two distinct vector spaces: we will not form linear combination of an element in U and an element in V.Fis a linear map that maps an element in U to an element in V,y=FY. The two elements represent the same set of material particles, respectively, in the reference state and in the current state.Geometric representation of the deformation gradientYin the reference state by a column, and represent the same set of material particlesyin the current state by another column. We represent the deformation gradientFby a matrix.Freprentsents a vector, corresponding to an edge of the parallelepiped.Green deformation tensorFTF. This tensor is the Green deformation tensor.My class notes on finite deformation develop these ideas, and also contain the development of the nominal stress tensor.

Also see a thread on linear algebra, tensors, and linear maps between linear spaces .