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Contents

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1 Conclusion and results

1.1 Different stress tensors

1.2 Deformation gradient tensor under rigid body transformation \(\tilde {Q}\)

1.3 Stress tensors under rigid body transformation \(\tilde {Q}\)

1.4 Conjugate pairs (Stress tensor/Strain tensor)

Let \(W\) be the current amount of energy stored in a unit volume as a result of the body undergoing deformation, then the time rate at which this energy changes will equal the stress tensor \(\mathbf {\tilde {B}}\) multiplied by the strain rate \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}\). Therefore

The following table gives the stress tensor \(\mathbf {\tilde {B}}\), the strain rate \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}\) and the strain \(\mathbf {\tilde {A}}\)

2 Overview of geometry and mathematical notations used

Position and deformation measurements are of central importance in continuum mechanics. Two methods are employed : The Lagrangian method and the Eulerian method.

In the Lagrangian method, the particle position and speed are measured in reference to a fixed stationary observer based coordinates systems. This is called the referential coordinates system where the observer is located. Hence in the Lagrangian method, the particle state is measured from a global fixed frame of reference.

In Eulerian methods,a frame of reference is attached locally to the area of interest where the measurement is to be made, and the particle state is measured relative to the local coordinates systems (also called the body coordinates system). In continuum mechanics the Lagrangian method is used and in fluid mechanics the Eulerian method is used, but it is also possible to attach a local frame of reference to the body itself and then convert these measurements back relative to the global frame of reference.

A coordinate transformation gives back the coordinates of a point on a body relative to the global fixed reference frame, given the coordinates of the same point as measured in the local reference frame. This transformation is given by

Where \(\mathbf {X}\) is the coordinate vector relative the global frame of reference, \(\mathbf {x}\) is the coordinate vector relative the local/body frame of reference and \(\mathbf {A}\) is the \(n\times n\) rotation matrix (where \(n=3\) for normal 3D space) that represents pure rotation, and \(\mathbf {d}\) is an n-dimensional vector that represents pure translation.

The following diagram illustrates these differences.

In general, the interest is in finding differential changes that occur when a body deformed. This mean measuring how a differential vector that represents the orientation of one point relative to another changes as a body deformed.

Considering the Lagrangian method from now on. Attention is now shifted to what happens when the body starts to deform. The global reference frame is selected, this is where all measurements are made with reference to.

Measurements made when the body is undeformed is distinguished from those measurements made when the body has deformed. Upper case \(X_{i}\) is used for the coordinates of a point on the body when the body is undeformed, and lower case \(x_{i}\) is used for the coordinates of the same point when measured in reference to this same global coordinate system but after the body has deformed.

Diagram below takes a snap shot of the system after 5 units of time and measures the deformation to illustrate the notation used.

Another way to represent the above is by using the same diagram to show both the undeformed and the deformed configuration as follows.

Let \(B\) be the undeformed configuration, referred to as the body \(state\). By state it is meant the set of independent variables needed to fully describe the forces and geometry of the body.

When the body is in the undeformed state \(B\), it is assumed to be free of internal stresses and that no traction forces act on it.

External loads are now applied to the body resulting in a change of state. The new state can be a result of only a deformation in the body shape, or due to only a rigid body translation/rotation, or it could be a result of a combination of deformation and rigid body motion.

The deformation will take sometime \(t\) to complete. However, in this discussion the interest is only in the final deformed state, which is called state \(b\). Hence no function(s) of time will be appear or be involved in this analysis.

The boundary conditions is assumed to be the same in state \(B\) and in state \(b\). This implied that if the solid body was in physical contact with some external non-moving supporting configuration, then after the deformation is completed, the body will remain in the same physical contact with these supports and at the same points of contact as before the deformation began.

This implies the body is free to deform everywhere, except that it is constrained to deform at those specific points it is in contact with the support. For the rigid body rotation, it is assumed the body with its support will rotate together.

A very important operator in continuum mechanics is the deformation tensor \(\mathbf {\tilde {F}}\). (A tensor can be viewed as an operator which takes a vector and maps it to another vector). This tensor allows the determination of the deformed differential vector \(dr\) knowing the undeformed differential vector \(dR\) as follows.

The tensor \(\mathbf {\tilde {F}}\) is a field tensor in general, which mean the actual value of the tensor changes depending on the location of the body where the tensor is evaluated. Hence it is a function of the body coordinates. Reference [4] gives simple examples showing how to calculate \(\mathbf {\tilde {F}}\) for simple cases of deformations in 2D. The appendix contains derivation of \(\mathbf {\tilde {F}}\) in the specific case of normal Cartesian coordinates.

2.1 Illustration of the polar decomposition of the deformation gradient tensor \(\tilde {F}\)

2.1.1 Polar decomposition applied to a vector

The effect of applying the deformation gradient tensor \(\mathbf {\tilde {F}}\) on a vector \(\mathbf {dR}\) can be considered to have the same result as the effect of first applying a stretch deforming tensor \(\mathbf {\tilde {U}}\) (Also called the deformation tensor) on \(\mathbf {dR}\), resulting in a vector \(\mathbf {dR}^{\ast }\), followed by applying a rotation deforming tensor \(\mathbf {\tilde {R}}\) on this new vector \(\mathbf {dR}^{\ast }\) to produce the final vector \(\mathbf {dr.}\)

Hence \(\mathbf {\tilde {F}}=\mathbf {\tilde {U}}\cdot \mathbf {\tilde {R}}\) and therefore

Using polar decomposition gives

This is called polar decomposition of \(\mathbf {\tilde {F}}\), and it is always possible to find such decomposition. In addition, this decomposition is unique for each tensor \(\mathbf {\tilde {F}}\).

2.1.2 Polar decomposition applied to an oriented area

An oriented area in the undeformed state is \(dA\ \mathbf {N}\) (Where \(\mathbf {N}\) is a unit normal to \(dA\)). This area becomes \(da\ \mathbf {n}^{\ast }\) after the application of the stretch tensor \(\mathbf {\tilde {U}}\). It is clear that rotation will not have an effect on the area \(da\) itself, but it will rotate the unit vector \(\mathbf {n}^{\ast }\) which is normal to \(da\) to become the unit vector \(\mathbf {n}\). This is illustrated in the diagram below.

Now that a brief description of the geometry and the important tensor \(\mathbf {\tilde {F}}\) is given above, discussion of the main topic of this paper will start.

3 Stress Measures

Before outlining the different stress measures, the different entities involved are described and illustrated.

Given the undeformed state \(B\), let \(P\) a point in \(B\) where its location in the deformed state becomes \(b\) (Lagrangian description). Let \(dA\) be a differential area at point \(P\) on the surface of \(B\) where \(\mathbf {dN}\) is a unit vector normal to this area in \(B\). After deformation, this differential area will is deformed to a new differential area \(da\) in the deformed state \(b\). Let \(\mathbf {dn}\) be the unit vector normal to \(da\) in \(b\).

Let \(\mathbf {df}\) be the differential force vector which represents the resultant of the total internal forces acting on \(da\) in the deformed state \(b\).

The following diagram illustrates the above.

3.1 Cauchy stress measure

The Cauchy stress measure \(\mathbf {\tilde {\tau }}\) is a measure of

It is called the true measure of stress. The followng follows from the above definition

Cauchy stress tensor is in general (in absence of body couples) a symmetric tensor.

3.2 First Piola-Kirchhoff or Piola-Lagrange stress measure

The above diagram shows that this stress \(\mathbf {\tilde {t}}\) can be regarded as

The following shows the derivation of this stress tensor. Starting by moving the vector \(\mathbf {df}\) (the result of internal forces in the deformed state) which acts on the deformed area \(da\) in a parallel transport to the image of \(da\) in the undeformed state, which will be the differential area \(dA\)

Hence in the undeformed state the following results

Given that

Which is a relationship derived from geometrical consideration [2], then from the above equation the following results

Since \(\mathbf {df}=\left ( da\ \mathbf {n}\right ) \cdot \tilde {\tau }\), then using the above equation gives

Comparing (1) to (2) gives

Hence

First Piola-Kirchhoff stress tensor in general is unsymmetrical.

3.3 Second Piola-Kirchhoff stress tensor

From the above diagram stress \(\tilde {s}_{1}\) can be regarded as

The stress measure \(\tilde {s}_{1}\) is similar to the first Piola-Kirchhoff stress measure, except that instead of parallel transporting the force \(\mathbf {df}\) from the deformed state to the undeformed state, a force vector \(\mathbf {d\hat {f}}\) is first created which is derived from \(\mathbf {df}\) and then parallel transport this new vector is made.

Everything else remains the same. The purpose of this is that the second Piola-Kirchhoff stress tensor will now be a symmetric tensor while the first Piola-Kirchhoff stress tensor was nonsymmetric.

Hence in the undeformed state (after parallel transporting \(\mathbf {d\hat {f}}\) to \(dA\)) the following relationship results

In the deformed state the following relation applies

As before, an expression for \(\mathbf {\tilde {s}}_{1}\) in terms of the Cauchy stress tensor \(\tilde {\tau }\) is now found.

Given that

Substituting the above in (3) gives

From (1) \(\mathbf {df}=\mathbf {\tilde {F}}^{T}\cdot \mathbf {d\hat {f}}\) , hence the above equation becomes

Therefore

Comparing (4) with (2) gives

Therefore the second Piola-Kirchhoff stress tensor is

The second Piola-Kirchhoff stress tensor is in general symmetric.

3.4 Kirchhoff stress tensor

Kirchhoff stress tensor \(\mathbf {\tilde {\sigma }}\) is a scalar multiple of the true stress tensor \(\mathbf {\tilde {\tau }}\). The scale factor is the determinant of \(\mathbf {\tilde {F}}\), the deformation gradient tensor.

Hence

\(\mathbf {\tilde {\sigma }}\) is symmetric when \(\mathbf {\tilde {\tau }}\) is symmetric which is in general the case.

3.5 \(\tilde {\Gamma }\) stress tensor

The \(\boldsymbol {\tilde {\Gamma }}\) stress tensor is a result of internal forces generated due to the application of the stretch tensor only. Hence this stress acts on the area deformed due to stretch only. Therefore this stress represents

Assuming these are called \(\mathbf {df}^{\ast }\), then applying this definition results in

When in the final deformed state the following relation applies before

The above means that the stretched state can be considered as a partial deformed state, and the final deformed state as the result of applying the rotation tensor on the stretched state. In the final deformed state the result of the internal forces is \(\mathbf {df}\) while in the stretched state, in which all the variables in that state are designated with a star *, the internal forces are called \(\mathbf {df}^{\ast }\)

Therefore

Equation (3) can be written as \(\mathbf {df}^{\ast }=\mathbf {df\cdot \tilde {R}}\). Substituting this into (1) gives

Substituting for \(\mathbf {df}\) in the above equation the expression for \(\mathbf {df}\) in (2) results in

But \(da\ \mathbf {n=\tilde {R}}\cdot \left ( da\ \mathbf {n}^{\ast }\right )\) hence the above equation becomes

Therefore

3.6 Biot-Lure \(\tilde {r}^{\ast }\) stress tensor

This stress measure exists in the undeformed state as a result of parallel translation of the \(\mathbf {df}^{\ast }\) forces generated in the stretched state back to the undeformed state and applying this force into the image of the stretched area in the undeformed state. Therefore this stress can be considered as

In a sense, it is one step more involved than the \(\boldsymbol {\tilde {\Gamma }}\) stress tensor described earlier. The following diagram illustrates the above.

From the above diagram an expression for the Biot-Lure stress tensor is now given

Now an expression for \(\mathbf {\tilde {r}}^{\ast }\) is found. Since \(\mathbf {df}^{\ast }=\mathbf {df}\cdot \mathbf {\tilde {R}}\), the above equation becomes

Given that \(\mathbf {df}=da\ \mathbf {n\cdot \tilde {\tau }}\), the above equation becomes

But \(da\ \mathbf {n}=J\ \left ( dA\ \mathbf {N}\right ) \cdot \mathbf {\tilde {F}}^{-1}\) hence the above equation becomes

By comparison it follows that

The stress tensor \(\mathbf {\tilde {r}}^{\ast }\) is un-symmetric when \(\mathbf {\tilde {\tau }}\) is symmetric which is in the general is the case.

3.7 Jaumann stress tensor \(\tilde {r}\)

This stress tensor is introduced to create a symmetric stress tensor from the Biot-Lure stress tensor as follows

No physical interpretation of this stress tensor can be made similar to the Biot-Lure stress tensor.

3.8 The stress tensor \(\tilde {T}^{\ast }\)

This stress tensor is defined in the rotated state without any stretch being applied before. The forces that act on the rotated area were parallel transported from the forces that were generated in the final deformed state. Hence this stress can be considered as

The following diagram illustrates this. Since rotation have been applied before stretch, then the polar decomposition of \(\mathbf {\tilde {F}}\) becomes

Where \(\mathbf {\tilde {V}}\) is the rotation tensor (which was called \(\mathbf {\tilde {R}}\) when it was applied after stretch), and \(\mathbf {\tilde {U}}\) is the stretch tensor.

The above diagram shows that

Since \(\mathbf {df}=da\ \mathbf {n\cdot \tilde {\tau }}\), the above equation becomes

But \(da\ \mathbf {n}=J\left ( dA\ \mathbf {N}^{\ast }\cdot \mathbf {\tilde {V}}^{-1}\right ) \) hence the above equation becomes

Therefore

\(\mathbf {\tilde {T}}^{\ast }\) is un-symmetric when \(\mathbf {\tilde {\tau }}\) is symmetric.

3.9 The stress tensor \(\tilde {T}\)

This stress tensor is introduced to create a symmetric stress tensor from the \(\mathbf {\tilde {T}}^{\ast }\) stress tensor as follows

No physical interpretation of this stress tensor can be made similar to the \(\mathbf {\tilde {T}}^{\ast }\) stress tensor.

4 Geometry and stress tensors transformation due to rigid body rotation

Now consideration is given to changes of the geometrical deforming tensors \(\mathbf {\tilde {F},\tilde {U}}\) and \(\mathbf {\tilde {V}}\) when the body is in its final deformed state and then subjected to a pure rigid body rotation \(\mathbf {\tilde {Q}}\), and to what happens to the various stress tensors derived above under the same \(\mathbf {\tilde {Q}}\).

Polar decomposition of \(\mathbf {\tilde {F}}\) is given by

where \(\mathbf {\tilde {F}}\) is the deformation gradient tensor and \(\mathbf {\tilde {U}}\) is the stretch before rotation \(\mathbf {\tilde {R}}\) tensor, and \(\mathbf {\tilde {R}}\) is the rotation tensor. The polar decomposition of \(\mathbf {\tilde {F}}\) is

where \(\mathbf {\tilde {V}}\) is the stretch after rotation \(\mathbf {\tilde {R}}\) tensor.

The effect of applying pure rigid body rotation \(\mathbf {\tilde {Q}}\) on \(\mathbf {\tilde {F},\tilde {U}}\) and \(\mathbf {\tilde {V} }\) is now determined.

In each of the following derivations the following setting is assumed to be in place: There is a body originally in the undeformed state \(B\) and loads are applied on the body. The body undergoes deformation governed by the deformation gradient tensor \(\mathbf {\tilde {F}}\) resulting in the body being in the final deformed state state \(b\) with a stress tensor \(\mathbf {\tilde {\tau }}\) at point \(p\). If the body is considered to be first under the effect of \(\mathbf {\tilde {U}}\) (stretch), then the new state will be called \(B^{\ast }\), and after applying the effect of \(\mathbf {\tilde {R}}\) (point to point rotation tensor), then the state will be called \(b\) (which is the final deformation state).

If however \(\mathbf {\tilde {R}}\) (rotation) is applied first, then the new state will also be called \(B^{\ast }\) and then when applying the stretch \(\mathbf {\tilde {V}}\) the state will becomes \(b\) (which is the final deformation state).

From state \(b\), which is the final deformation state, a pure rigid body rotation tensor \(\mathbf {\tilde {Q}}\) is applied to the whole body (with its fixed supports if any). Hence there will be no changes in the body shape, and the new state is called \(q\).

Is also possible to consider the change of state from state \(B\) to state \(q\) to be the result of a new deformation gradient tensor which is called \(\mathbf {\tilde {F}}_{q}\). The polar decomposition of \(\mathbf {\tilde {F}}_{q}\) can also be written as

or as

\(\mathbf {\tilde {F}}\) is compared to \(\mathbf {\tilde {F}}_{q}\), and \(\mathbf {\tilde {U}}\) is compared to \(\mathbf {\tilde {U}}_{q}\) and \(\mathbf {\tilde {V}}\) is compared to \(\mathbf {\tilde {V}}_{q}\) in order to see the effect of the rigid body rotation on these tensors.

4.1 Deformation tensors transformation (\(\tilde {F},\tilde {U}\) and \(\tilde {V}\)) due to rigid body rotation \(\tilde {Q}\)

4.1.1 Transformation of \(\tilde {F}\) (the deformation gradient tensor)

The above diagram show that

4.1.2 Transformation of \(\tilde {U}\) (the stretch before rotation \(\tilde {R}\) tensor)

Given that

Similarly,

Where \(\mathbf {\tilde {F}}_{q}\) \(=\mathbf {\tilde {Q}\cdot \tilde {F}}\), hence the above becomes

Using linear algebra it follows that \(\left ( \mathbf {\tilde {Q}\cdot \tilde {F}}\right ) ^{T}=\mathbf {\tilde {F}}^{T}\cdot \mathbf {\tilde {Q}}^{T}\). Therefore the above becomes

But \(\mathbf {\tilde {Q}}^{T}\mathbf {\cdot \tilde {Q}=\tilde {I}}\) since \(\mathbf {\tilde {Q}}\) is orthogonal. Hence the above becomes

Comparing (1) and (2) shows they are the same. Hence

Therefore

4.1.3 Transformation of \(\tilde {V}\) (The stretch after rotation \(\tilde {R}\) tensor)

Since

And \(\mathbf {\tilde {F}=\tilde {R}}\cdot \mathbf {\tilde {U}}\) by polar decomposition on \(\mathbf {\tilde {F}}\) the above can be written as

Applying polar decomposition on \(\mathbf {\tilde {F}}_{q}\) results in \(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {R}}_{q}\cdot \mathbf {\tilde {U}}_{q}\) and the above becomes

It was found earlier that \(\mathbf {\tilde {U}}_{q}=\mathbf {\tilde {U}}\), therefore the above becomes

Now the second form of polar decomposition on \(\mathbf {\tilde {F}}_{q}\) is utilized giving

Substituting (2) into the above equation results in

Substituting (1) into the above gives

Since \(\mathbf {\tilde {Q}\cdot \tilde {R}}\) is invertible (need to check), the above can be written as

But \(\left ( \mathbf {\tilde {Q}\cdot \tilde {R}}\right ) ^{-1}=\mathbf {\tilde {R}}^{T}\mathbf {\cdot \tilde {Q}}^{T}\) (Since \(\mathbf {\tilde {Q}\cdot \tilde {R}}\) is an orthogonal matrix. (check). Hence the above becomes

From polar decomposition it is known that \(\mathbf {\tilde {F}=\tilde {V}\cdot \tilde {R}}\), hence \(\mathbf {\tilde {V}=\tilde {F}\cdot \tilde {R}}^{-1}\), but \(\mathbf {\tilde {R}}^{-1}=\mathbf {\tilde {R}}^{T}\) since it is an orthogonal matrix, therefore

Substituting (4) in (3) gives

This above is how \(\mathbf {\tilde {V}}\) transforms due to rigid rotation \(\mathbf {\tilde {Q}}\).

Now that the transformation of \(\mathbf {\tilde {F},\tilde {U}}\) and \(\mathbf {\tilde {V}}\) was obtained, the next step is to find how each one of the stress tensors derived earlier transforms due to \(\mathbf {\tilde {Q}}\).

4.2 Stress tensors transformation due to rigid body rotation \(\tilde {Q}\)

4.2.1 Transformation of stress tensor \(\tilde {\tau }\) (Cauchy stress tensor)

The stress \(\mathbf {\tilde {\tau }}_{q}\) (Cauchy stress in state \(q\)) is calculated at the point \(b_{q}\). Since this is a rigid body rotation, the area \(da\) will not change, only the unit normal vector \(\mathbf {n}\) will change to \(\mathbf {n}_{q}\)

The tensor \(\mathbf {\tilde {Q}}\) maps the vector \(\mathbf {df}\) to the vector \(\mathbf {df}_{q}\)

But in state \(b\) (the deformed state), the Cauchy stress tensor is given by

Substituting (2) into (1) gives

Exchanging the order of \(\mathbf {\tilde {\tau }}\) and \(da\ \mathbf {n}\) and using the transpose of \(\mathbf {\tilde {\tau }}\) gives

Therefore since the tensor \(\mathbf {\tilde {Q}}\) maps the oriented area \(\left ( da\ \mathbf {n}\right ) \) to the oriented area \(\left ( da\ \mathbf {n}_{q}\right ) \) then

Or

Substituting (4) into (3) gives

However, the stress \(\mathbf {\tilde {\tau }}_{q}\) in state \(q\) is given by \(\mathbf {df}_{q}=\left ( da\ \mathbf {n}_{q}\right ) \cdot \mathbf {\tilde {\tau }}_{q}\), hence the above equation becomes

Or

This implies

Comparing the above transformation result with the deformation tensors transformation results in

4.2.2 Transformation of first Piola-Kirchhoff stress tensor

The transformation of the first Piola-Kirchhoff stress tensor \(\mathbf {\tilde {t}}\) is given below.

Earlier it was shown that

Which implies

However, it was found earlier that \(\mathbf {\tilde {\tau }}_{q}=\mathbf {\tilde {Q}\cdot \tilde {\tau }}\cdot \mathbf {\tilde {Q}}^{T}\) therefore the above becomes

Now an expression for \(\mathbf {\tilde {F}}_{q}^{-1}\) is found.

Since \(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {Q}\cdot \tilde {F}}\), then \(\mathbf {\tilde {F}}_{q}^{-1}=\left ( \mathbf {\tilde {Q}\cdot \tilde {F}}\right ) ^{-1}\), hence \(\mathbf {\tilde {F}}_{q}^{-1}=\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {Q}}^{-1}\). But since \(\mathbf {\tilde {Q}}\) is orthogonal, then \(\mathbf {\tilde {Q}}^{-1}=\mathbf {\tilde {Q}}^{T}\), therefore

Substituting the above in (1) gives

However, since \(\mathbf {\tilde {t}}=J\mathbf {\tilde {F}}^{-1}\mathbf {\cdot \tilde {\tau }}\) the above simplifies to

Hence

By examining how the geometrical tensors transform, results from before showed that \(\mathbf {\tilde {F}}_{q}\) \(=\mathbf {\tilde {Q}\cdot \tilde {F}}\) therefore

4.2.3 Transformation of second Piola-Kirchhoff stress tensor

The transformation of the second Piola-Kirchhoff stress tensor \(\mathbf {\tilde {s}}_{1}\) is given below.

From earlier

Hence

An expression for \(\mathbf {\tilde {F}}_{q}^{-1}\) is now found. Since \(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {Q}\cdot \tilde {F}}\), hence \(\mathbf {\tilde {F}}_{q}^{-1}=\left ( \mathbf {\tilde {Q}\cdot \tilde {F}}\right ) ^{-1}\), hence \(\mathbf {\tilde {F}}_{q}^{-1}=\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {Q}}^{-1}\). But since \(\mathbf {\tilde {Q}}\) is orthogonal, then \(\mathbf {\tilde {Q}}^{-1}=\mathbf {\tilde {Q}}^{T}\), hence

The above equation becomes

\(\mathbf {\tilde {F}}_{q}^{-T}=\left ( \mathbf {\tilde {F}}_{q}^{-1}\right ) ^{T}\) hence \(\mathbf {\tilde {F}}_{q}^{-T}=\left ( \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {Q}}^{T}\right ) ^{T}\), therefore

The above equation becomes

From earlier it was found that \(\mathbf {\tilde {\tau }}_{q}=\mathbf {\tilde {Q}\cdot \tilde {\tau }\cdot \tilde {Q}}^{T}\) Therefore the above equation becomes

Therefore

This is the same as \(\mathbf {\tilde {s}}_{1}\), hence

Since it was found earlier that \(\mathbf {\tilde {U}}_{q}=\mathbf {\tilde {U}}\) therefore

4.2.4 Transformation of Kirchhoff stress tensor \(\tilde {\sigma }\)

Since \(\mathbf {\tilde {\sigma }}\) is a scalar multiple of \(\mathbf {\tilde {\tau }}\) and from earlier it was found that \(\mathbf {\tilde {\tau }}\) is a conjugate pair with \(\mathbf {\tilde {V}}\) then it is concluded that

4.2.5 Transformation of \(\tilde {\Gamma }\) stress tensor

The transformation of the second \(\boldsymbol {\tilde {\Gamma }}\) stress tensor is shown below.

From earlier it is shown that

Hence

Since \(\mathbf {\tilde {\tau }}_{q}=\mathbf {\tilde {Q}\cdot \tilde {\tau }\cdot \tilde {Q}}^{T}\) the above becomes

But \(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {Q}}\cdot \mathbf {\tilde {F}}\) and using polar decomposition results in

hence

But \(\mathbf {\tilde {F}}=\mathbf {\tilde {R}}\cdot \mathbf {\tilde {U}}\ \) hence \(\mathbf {\tilde {F}}^{-1}=\left (\mathbf {\tilde {R}}\cdot \mathbf {\tilde {U}}\right )^{-1}=\mathbf {\tilde {U}}^{-1}\cdot \mathbf {\tilde {R}}^{-1}\), and the above becomes

Since \(\mathbf {\tilde {U}}_{q}=\mathbf {\tilde {U}}\) the above becomes

But \(\mathbf {\tilde {R}}^{-1}=\mathbf {\tilde {R}}^{T}\) therefore

Hence

Substituting (2) and (3) into (1) gives

Since \(\mathbf {\tilde {R}}\) and \(\mathbf {\tilde {R}}_{q}^{T}\) are orthogonal, the above reduces to

But \(\boldsymbol {\tilde {\Gamma }=\tilde {R}}^{T}\cdot \mathbf {\tilde {\tau }\cdot \tilde {R}}\) therefore

4.2.6 Transformation of Biot-Lure stress tensor \(\tilde {r}^{\ast }\)

The transformation of the Biot-Lure stress tensor \(\mathbf {\tilde {r}}^{\ast }\) is given below.

From earlier

Hence

But \(\mathbf {\tilde {\tau }}_{q}=\mathbf {\tilde {Q}\cdot \tilde {\tau }\cdot \tilde {Q}}^{T}\) therefore

Since \(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {Q}\cdot \tilde {F}}\) then \(\mathbf {\tilde {F}}_{q}^{-1}=\left ( \mathbf {\tilde {Q}\cdot \tilde {F}}\right ) ^{-1}=\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {Q}}^{-1}\) but \(\mathbf {\tilde {Q}}\) is orthogonal, hence \(\mathbf {\tilde {Q}}^{-1}=\mathbf {\tilde {Q}}^{T}\), hence \(\mathbf {\tilde {F}}_{q}^{-1}=\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {Q}}^{T}\). Therefore (2) can be written as

Now \(\mathbf {\tilde {R}}_{q}\) is resolved.

Since \(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {Q}\cdot \tilde {F}}\) and by polar decomposition \(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {R}}_{q}\mathbf {\cdot \tilde {U}}_{q}\) then

Substituting (4) into (3) gives

Since \(\mathbf {\tilde {F}=\tilde {R}\cdot \tilde {U}}\) the above becomes

From earlier \(\mathbf {\tilde {U}=\tilde {U}}_{q}\), therefore the above becomes

From (1) gives

And since \(\mathbf {\tilde {U}}_{q}\mathbf {=\tilde {U}}\) therefore

4.2.7 Transformation of Juamann stress tensor \(\tilde {r}\)

The transformation of the Juamann stress tensor \(\mathbf {\tilde {r}}\) is shown below.

From earlier

From (1) results

Since it was found that \(\mathbf {\tilde {r}}_{q}^{\ast }=\mathbf {\tilde {r}}^{\ast }\) then the above becomes

Therefore

Since \(\mathbf {\tilde {U}}_{q}\mathbf {=\tilde {U}}\) therefore

4.2.8 Transformation of \(\tilde {T}^{\ast }\) stress tensor

From earlier

Hence

But \(\mathbf {\tilde {\tau }}_{q}=\mathbf {\tilde {Q}\cdot \tilde {\tau }\cdot \tilde {Q}}^{T}\) and \(\mathbf {\tilde {V}}_{q}=\mathbf {\tilde {Q}\cdot \mathbf {\tilde {V}}\cdot \tilde {Q}}^{T}\) Therefore

Hence

Therefore

4.2.9 Transformation of \(\tilde {T}\) stress tensor

Since \(\mathbf {\tilde {T}=}\frac {\left ( \mathbf {\tilde {T}}^{\ast }+\mathbf {\tilde {T}}^{\ast T}\right ) }{2}\) and \(\mathbf {\tilde {T}}^{\ast }\) is conjugate pair with \(\mathbf {\tilde {U}}\) then

5 Constitutive Equations using conjugate pairs for nonlinear elastic materials with large deformations: Hyper-elasticity

Formulating the constitutive relation for a material seeks a formula that relates the stress measure to the strain measure. Therefore, using a specific stress measure, the correct strain measure must be used.

Therefore the problem at hand is the following: Given a stress tensor, one of the many stress tensors discussed earlier, how to determine the correct strain tensor to use with it?

To make the discussion general, the stress tensor is designated by \(\mathbf {\tilde {B}}\) and its conjugate pair, the strain tensor, by \(\mathbf {\tilde {A}}\).

The stress measure \(\mathbf {\tilde {B}}\) could be any of the stress measures discussed earlier, such as the Cauchy stress tensor \(\mathbf {\tilde {\tau }}\), the second Piola-kirchhoff stress tensor \(\mathbf {\tilde {s}}_{1}\). Now the strain tensor to use is determined. Let \(\left ( \mathbf {\tilde {B},\tilde {A}}\right )\) be the conjugate pair tensors.

Physics is used in finding of \(\mathbf {\tilde {A}}\) for each specific \(\mathbf {\tilde {B}}\)

Let the current amount of energy stored in a unit volume as a result of the body undergoing deformation be \(W\), then the time rate at which this energy changes will be equal to the stress multiplied by the strain rate. Hence

Where \(\colon \) is the trace matrix operator. This is the rule used to determine \(\mathbf {\tilde {A}}\).

On a stress-strain diagram the following is drawn

The strain measure \(\mathbf {\tilde {A}}\) (the conjugate pair for the stress measure \(\mathbf {\tilde {B}}\)) must satisfy the relation

For each stress/strain conjugate pair, the terms \(\frac {\partial W}{\partial t},\frac {\partial \mathbf {\tilde {A}}}{\partial t},\mathbf {\tilde {A}}\) are derived.

5.1 Conjugate pair for Cauchy stress tensor

In the deformed state, the stress tensor is the true stress tensor, which is the cauchy stress \(\mathbf {\tilde {\tau }}\), and the strain rate in this state is known to be [2]

Where \(\mathbf {\tilde {e}}\) is the velocity gradient tensor. It is shown in [2] that

Hence in the deformed state

In other words, the conjugate strain for the cauchy stress tensor is given by \(\mathbf {\tilde {A}}\) such that

\(\mathbf {\tilde {A}}\) should come out to be the Almansi strain tensor, which is

(check)

5.2 Conjugate pair for second Piola-kirchhoff stress tensor \(\tilde {s}_{1}\)

Pre dot multiplying \(\mathbf {\tilde {e}}\) by \(\mathbf {\tilde {I}=}\left ( \mathbf {\tilde {F}}^{-T}\cdot \mathbf {\tilde {F}}^{T}\right ) \) and post dot multiplying it with \(\mathbf {\tilde {I}=}\left ( \mathbf {\tilde {F}}\cdot \mathbf {\tilde {F}}^{-1}\right )\) which will make no change in the value, results in

Using the properties of \(\colon \) the above is written as

It was determined earlier that \(\mathbf {\tilde {s}}_{1}=J\ \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }}\cdot \mathbf {\tilde {F}}^{-T}\) hence \(\mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }\cdot \mathbf {\tilde {F}}^{-T}=}\frac {\mathbf {\tilde {s}}_{1}}{J}\) hence the above equation becomes

But \(\mathbf {\tilde {e}=\dot {F}\cdot \tilde {F}}^{-1}\) therefore

Therefore

This shows that \(\mathbf {\tilde {A}=}\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right )\), therefore \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}=\mathbf {\mathbf {\dot {F}}}^{T}\mathbf {\cdot \tilde {F}+\mathbf {\tilde {F}}}^{T}\mathbf {\mathbf {\cdot \dot {F}}}\)

Or

The advantage in using the second Piola Kirchhoff stress tensor instead of the Cauchy or the first Piola Kirchhoff stress tensor, is that with the second Piola Kirchhoff stress tensor, calculations are performed the reference configuration (undeformed state) where the state measurements are known instead of using the deformed configuration where state measurements are not known.

5.3 Conjugate pair for first Piola-kirchhoff stress tensor \(\tilde {t}\)

But \(\mathbf {\tilde {e}=\dot {F}\cdot \tilde {F}}^{-1}\) hence the above becomes

Using the property of \(\mathbf {\colon }\) \(A\colon B\cdot C\) can be written as \(A\cdot C^{T}\colon B\) hence applying this property to the above expression gives

Applying the property that \(A\cdot C^{T}\colon B\rightarrow C\cdot A\colon B^{T}\) to the above results in

It was found earlier that \(\mathbf {\tilde {t}=}J\ \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }}\) hence replacing this into the above gives

This shows that \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}=\frac {1}{J}\mathbf {\dot {F}}^{T}\) therefore

5.4 Conjugate pair for \(\tilde {\sigma }\) Kirchhoff stress tensor

Since \(\mathbf {\tilde {\sigma }}\) is a scaled version of \(\mathbf {\tilde {\tau }}\) where

It was found earlier that the strain tensor associated with \(\mathbf {\tilde {\tau }}\) is \(\frac {1}{2J}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right ) \) hence the strain tensor associated with \(\mathbf {\tilde {\sigma }}\) is \(\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {F}}^{T}\cdot \tilde {F}-\tilde {I}}\right )\)

Therefore

5.5 Conjugate pair for \(\tilde {r}^{\ast }\) Biot-Lure stress tensor

But \(\mathbf {\tilde {e}=\dot {F}\cdot \tilde {F}}^{-1}\) hence the above becomes

But \(\left ( \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {F}}\right ) =\mathbf {\tilde {I}}\). Using this the first \(\mathbf {\tilde {I}}\) in (1) above is replaced. Also \(\left ( \mathbf {\tilde {F}}\cdot \mathbf {\tilde {F}}^{-1}\right ) =\mathbf {\tilde {I}}\), and using this, the second \(\mathbf {\tilde {I}}\) in equation (1) above is replaced. Therefore (1) becomes

Switching the order of terms selected above by transposing them gives

Taking \(\mathbf {\mathbf {\tilde {F}}^{-T}}\) as common factor gives

But

Hence (2) becomes

But \(\frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot F}\right ) =\frac {d}{dt}\left ( \mathbf {\mathbf {\tilde {U}}}^{2}\right ) \) since \(\mathbf {\mathbf {\tilde {F}}}^{T}\mathbf {\cdot F=\mathbf {\tilde {U}}}^{2}\)

Therefore (3) becomes

But

Hence (4) becomes

But

From symmetry of \(\mathbf {\mathbf {U}}\) therefore

And (5) becomes

From property of \(\mathbf {\colon }\) the above can be written as

But from above, \(\frac {1}{2}\left ( \mathbf {\mathbf {\tilde {U}\cdot \dot {U}+\dot {U}\cdot U}}\right ) =\mathbf {\mathbf {\tilde {U}\cdot \dot {U}}}\) Hence

Using property of \(\mathbf {\colon }\) the term \(\mathbf {\mathbf {\tilde {U}}}\) is moved to the left of \(\mathbf {\colon }\) to obtain

But \(\mathbf {\mathbf {\tilde {F}}}^{-T}\cdot \mathbf {\mathbf {\tilde {U}=\tilde {R}}}\) hence the above becomes

But it was found earlier that \(\mathbf {\tilde {r}}^{\ast }=J\ \mathbf {\tilde {F}}^{-1}\mathbf {\cdot \tilde {\tau }}\cdot \mathbf {\tilde {R}}\)

Hence \(\dot {W}=\frac {1}{J}\mathbf {\tilde {r}}^{\ast }\mathbf {\colon \mathbf {\dot {U}}}\) Hence

Therefore \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}=\frac {1}{J}\mathbf {\mathbf {\dot {U}}}\) which results in

5.6 Conjugate pair for \(\tilde {r}\) Jaumann stress tensor

It was found earlier that \(\mathbf {\tilde {r}=}\frac {\left ( \mathbf {\tilde {r}}^{\ast }+\mathbf {\tilde {r}}^{\ast T}\right ) }{2}\) hence the conjugate pair for \(\mathbf {\tilde {r}}\) is \(\frac {\left ( \frac {1}{J}\mathbf {\mathbf {U}}+\frac {1}{J}\mathbf {\mathbf {U}}^{T}\right ) }{2}\)

Since \(\mathbf {\mathbf {\tilde {U}}}\) is symmetrical, therefore conjugate pair for \(\mathbf {\tilde {r}}\) is \(\frac {1}{J}\mathbf {\mathbf {U}}\) Hence

The same as strain tensor associated with the Biot-Lure stress.

6 Stress-Strain relations using conjugate pairs based on complementary strain energy

TO-DO for future work.

7 Appendix

7.1 Derivation of the deformation gradient tensor \(\tilde {F}\) in normal Cartesian coordinates system

In what follows the expression for the deformation gradient tensor \(\mathbf {\tilde {F}}\) is derived. This tensor transform one vector into another vector.

For simplicity it is assumed that the deformed and the undeformed states are described using the same coordinates system. In addition, it is assumed that this coordinates system is the normal Cartesian system with basis vectors \(\mathbf {i,j,k}\). Later these expression will be written in the more general case where the coordinate systems are different and use curvilinear coordinate. Other than using different notation, the derivation is the same in both cases.

Considering a point \(P\) in the undeformed state. This point will have coordinates \(\left ( X_{1},X_{2},X_{3}\right )\). When the body undergoes deformation, this point will be displaced to a new location. The image of this point in the deformed state is called the point \(p\). The coordinates of the the point \(p\) is \(\left ( x_{1},x_{2},x_{3}\right )\).

The coordinates \(x_{i}\) is function of the coordinates \(X_{j}\). These functions constitute the mapping between the undeformed shape and the deformed shape. These functions can be written in general as

Therefore by knowing the functions \(f_{i}\) the position of any point in the deformed state can be located if its position in the undeformed state is known. It is more customary to write the function \(f_{i}\) using the name of the coordinate itself. For example writing \(x_{1}=x_{1}\left ( X_{1},X_{2},X_{3}\right ) \) instead of \(x_{1}=f_{1}\left ( X_{1},X_{2},X_{3}\right ) \) as was done above.

However this can be a little confusing since it uses the letter \(x_{i}\) as function when on the RHS and a variable on the LHS. Hence here the choice was to use a new name for the mapping function.

From the above we the expression for a differential change in each of the 3 coordinates using the differentiation chain rule is determined as follows

Considering now a differential vector element \(\mathbf {dr}\) in the deformed state. This vector can be written as

Combining equations (1) and (2) gives

The above equation can be written in matrix form as follows

It is seen that the components of \(\mathbf {dr}\) can be obtained from the components \(\mathbf {dR}\) by pre-multiplying the components of \(\mathbf {dR}\) by the above \(3\times 3\) matrix. Hence this matrix acts as a transformation rule which maps one vector to another, it is a second order tensor, which is called the deformation gradient tensor \(\mathbf {\tilde {F}}\)

This relation can be written also in dyadic form as follows

To carry the multiplication on the RHS in the above equation, the normal dot product convention is followed using the following rules.

Performing the multiplication gives

The dot multiplication is simplified using the above mentioned rules to obtain

Simplifying gives

Collecting similar terms on the RHS gives

comparing the components of the vector on the LHS with those component of the vector on the RHS gives equation (1) as expected.

In addition to the matrix form and the dyadic form, the transformation from \(\mathbf {dR}\) to \(\mathbf {dr}\) can be expressed using indices notation as follows

7.2 Useful identities and formulas

A matrix \(\mathbf {A}\) is orthogonal if \(AA^{T}=I\) where \(I\) is the identity matrix.

If a matrix/tensor \(\mathbf {A}\) is orthogonal then \(\mathbf {A}^{-1}=\mathbf {A}^{T}\). In component form, \(a_{ij}^{-1}=a_{ji}\)

\(\left ( \mathbf {\tilde {A}\cdot \tilde {B}}\right ) ^{-1}=\mathbf {\tilde {B}}^{-1}\cdot \mathbf {\tilde {A}}^{-1}\)

\(\left ( \mathbf {\tilde {A}\cdot \tilde {B}}\right ) ^{T}=\mathbf {\tilde {B}}^{T}\cdot \mathbf {\tilde {A}}^{T}\)

\(\left ( \mathbf {\tilde {A}\cdot \tilde {B}}\right ) ^{-T}=\left ( \left ( \mathbf {\tilde {A}\cdot \tilde {B}}\right ) ^{T}\right ) ^{-1}=\left ( \mathbf {\tilde {B}}^{T}\cdot \mathbf {\tilde {A}}^{T}\right ) ^{-1}\)

\(\mathbf {\tilde {F}}=\mathbf {\tilde {R}\cdot \tilde {U}}\)

\(\mathbf {\tilde {F}}=\mathbf {\tilde {V}\cdot \tilde {R}}\)

\(\mathbf {\tilde {U}=\tilde {R}}^{T}\cdot \mathbf {\tilde {V}\cdot \tilde {R}}\)

\(\mathbf {\tilde {V}=\tilde {R}}\cdot \mathbf {\tilde {U}\cdot \tilde {R}}^{T}\)

\(\mathbf {\tilde {U}=}\left ( \mathbf {\tilde {F}}^{T}\cdot \mathbf {\tilde {F}}\right ) ^{\frac {1}{2}}\)

\(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {Q}}\cdot \mathbf {\tilde {F}}\)

8 References

1. Professor Atluri, SN lecture notes. MAE295 course. Solid mechanics. Winter 2006. University of California, Irvine.
2. Atluri, SN. "Alternative stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with applications to plates and shells". Printed in computers & structures Vol. 18, No. 1. pp 93-116. 1984
3. S.N.Atluri, A.Cazzani. "Rotations in computational solid mechanics". Published by Archives of computational methods in engineering. Vol 2.1. pp 49-138. 1995.
4. Nasser M. Abbasi, "Simple examples illustrating the use of the deformation gradient tensor". March 6,2006. http://12000.org/my_notes/deformation_gradient
5. Malvern. Introduction to the mechanics of a continuous mechanics.
6. class notes, for course EN175, Advanced mechanics of solid, Brown University. http://www.engin.brown.edu/courses/en175/Notes/Eqns_of_motion/Eqns_of_motion.htm