Diﬀerent known stress measures used in continuum mechanics during deformation analysis are derived and geometrically illustrated. The deformed solid body is subjected to rigid body rotation tensor . Expressions formulated showing how the deformation geometrical tensors , and are transformed under this rigid body motion. Each stress measure is analyzed under this rigid rotation.
For each stress tensor, the appropriate strain tensor used in the material stress-strain constitutive relation is derived analytically. The famous paper by Professor Satya N. Atluri  was used as the main framework and guide for all these derivations.
|Stress||Stress measure||Generally Symmetrical ?|
|The deformation gradient|
|Stretch before rotation|
|Stretch after rotation|
|Stress||based transformation||Transforms Similar to|
Let 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 multiplied by the strain rate . Therefore
The following table gives the stress tensor , the strain rate and the strain
|Stress tensor||Strain tensor rate||Strain tensor|
|Cauchy||Almansi strain tensor|
|Piola-Kirchhoﬀ||Green-Lagrange strain tensor|
|(For isotropic material only)|
|(for isotropic only)||(For isotropic material only)|
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 is the coordinate vector relative the global frame of reference, is the coordinate vector relative the local/body frame of reference and is the rotation matrix (where for normal 3D space) that represents pure rotation, and 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 is used for the coordinates of a point on the body when the body is undeformed, and lower case 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 be the undeformed conﬁguration, referred to as the body . 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 , 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 to complete. However, in this discussion the interest is only in the ﬁnal deformed state, which is called state . 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 and in state . 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 . (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 knowing the undeformed diﬀerential vector as follows.
The tensor 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  gives simple examples showing how to calculate for simple cases of deformations in 2D. The appendix contains derivation of in the speciﬁc case of normal Cartesian coordinates.
The eﬀect of applying the deformation gradient tensor on a vector can be considered to have the same result as the eﬀect of ﬁrst applying a stretch deforming tensor (Also called the deformation tensor) on , resulting in a vector , followed by applying a rotation deforming tensor on this new vector to produce the ﬁnal vector
Hence and therefore
Using polar decomposition gives
This is called polar decomposition of , and it is always possible to ﬁnd such decomposition. In addition, this decomposition is unique for each tensor .
An oriented area in the undeformed state is (Where is a unit normal to ). This area becomes after the application of the stretch tensor . It is clear that rotation will not have an eﬀect on the area itself, but it will rotate the unit vector which is normal to to become the unit vector . This is illustrated in the diagram below.
Now that a brief description of the geometry and the important tensor is given above, discussion of the main topic of this paper will start.
Before outlining the diﬀerent stress measures, the diﬀerent entities involved are described and illustrated.
Given the undeformed state , let a point in where its location in the deformed state becomes (Lagrangian description). Let be a diﬀerential area at point on the surface of where is a unit vector normal to this area in . After deformation, this diﬀerential area will is deformed to a new diﬀerential area in the deformed state . Let be the unit vector normal to in .
Let be the diﬀerential force vector which represents the resultant of the total internal forces acting on in the deformed state .
The following diagram illustrates the above.
The Cauchy stress measure 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.
The above diagram shows that this stress can be regarded as
The following shows the derivation of this stress tensor. Starting by moving the vector (the result of internal forces in the deformed state) which acts on the deformed area in a parallel transport to the image of in the undeformed state, which will be the diﬀerential area
Hence in the undeformed state the following results
Which is a relationship derived from geometrical consideration , then from the above equation the following results
Since , then using the above equation gives
Comparing (1) to (2) gives
First Piola-Kirchhoﬀ stress tensor in general is unsymmetrical.
From the above diagram stress can be regarded as
The stress measure is similar to the ﬁrst Piola-Kirchhoﬀ stress measure, except that instead of parallel transporting the force from the deformed state to the undeformed state, a force vector is ﬁrst created which is derived from 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 to ) the following relationship results
In the deformed state the following relation applies
As before, an expression for in terms of the Cauchy stress tensor is now found.
Substituting the above in (3) gives
From (1) , hence the above equation becomes
Comparing (4) with (2) gives
Therefore the second Piola-Kirchhoﬀ stress tensor is
The second Piola-Kirchhoﬀ stress tensor is in general symmetric.
Kirchhoﬀ stress tensor is a scalar multiple of the true stress tensor . The scale factor is the determinant of , the deformation gradient tensor.
is symmetric when is symmetric which is in general the case.
The 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 , 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 while in the stretched state, in which all the variables in that state are designated with a star *, the internal forces are called
Equation (3) can be written as . Substituting this into (1) gives
Substituting for in the above equation the expression for in (2) results in
But hence the above equation becomes
This stress measure exists in the undeformed state as a result of parallel translation of the 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 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 is found. Since , the above equation becomes
Given that , the above equation becomes
But hence the above equation becomes
By comparison it follows that
The stress tensor is un-symmetric when is symmetric which is in the general is the case.
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.
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 becomes
Where is the rotation tensor (which was called when it was applied after stretch), and is the stretch tensor.
The above diagram shows that
Since , the above equation becomes
But hence the above equation becomes
is un-symmetric when is symmetric.
This stress tensor is introduced to create a symmetric stress tensor from the stress tensor as follows
No physical interpretation of this stress tensor can be made similar to the stress tensor.
Now consideration is given to changes of the geometrical deforming tensors and when the body is in its ﬁnal deformed state and then subjected to a pure rigid body rotation , and to what happens to the various stress tensors derived above under the same .
Polar decomposition of is given by
where is the deformation gradient tensor and is the stretch before rotation tensor, and is the rotation tensor. The polar decomposition of is
where is the stretch after rotation tensor.
The eﬀect of applying pure rigid body rotation on and 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 and loads are applied on the body. The body undergoes deformation governed by the deformation gradient tensor resulting in the body being in the ﬁnal deformed state state with a stress tensor at point . If the body is considered to be ﬁrst under the eﬀect of (stretch), then the new state will be called , and after applying the eﬀect of (point to point rotation tensor), then the state will be called (which is the ﬁnal deformation state).
If however (rotation) is applied ﬁrst, then the new state will also be called and then when applying the stretch the state will becomes (which is the ﬁnal deformation state).
From state , which is the ﬁnal deformation state, a pure rigid body rotation tensor 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 .
Is also possible to consider the change of state from state to state to be the result of a new deformation gradient tensor which is called . The polar decomposition of can also be written as
is compared to , and is compared to and is compared to in order to see the eﬀect of the rigid body rotation on these tensors.
The above diagram show that
Where , hence the above becomes
Using linear algebra it follows that . Therefore the above becomes
But since is orthogonal. Hence the above becomes
Comparing (1) and (2) shows they are the same. Hence
And by polar decomposition on the above can be written as
Applying polar decomposition on results in and the above becomes
It was found earlier that , therefore the above becomes
Now the second form of polar decomposition on is utilized giving
Substituting (2) into the above equation results in
Substituting (1) into the above gives
Since is invertible (need to check), the above can be written as
But (Since is an orthogonal matrix. (check). Hence the above becomes
From polar decomposition it is known that , hence , but since it is an orthogonal matrix, therefore
Substituting (4) in (3) gives
This above is how transforms due to rigid rotation .
Now that the transformation of and was obtained, the next step is to ﬁnd how each one of the stress tensors derived earlier transforms due to .
The stress (Cauchy stress in state ) is calculated at the point . Since this is a rigid body rotation, the area will not change, only the unit normal vector will change to
The tensor maps the vector to the vector
But in state (the deformed state), the Cauchy stress tensor is given by
Substituting (2) into (1) gives
Exchanging the order of and and using the transpose of gives
Therefore since the tensor maps the oriented area to the oriented area then
Substituting (4) into (3) gives
However, the stress in state is given by , hence the above equation becomes
Comparing the above transformation result with the deformation tensors transformation results in
The transformation of the ﬁrst Piola-Kirchhoﬀ stress tensor is given below.
Earlier it was shown that
However, it was found earlier that therefore the above becomes
Now an expression for is found.
Since , then , hence . But since is orthogonal, then , therefore
Substituting the above in (1) gives
However, since the above simpliﬁes to
By examining how the geometrical tensors transform, results from before showed that therefore
The transformation of the second Piola-Kirchhoﬀ stress tensor is given below.
An expression for is now found. Since , hence , hence . But since is orthogonal, then , hence
The above equation becomes
hence , therefore
The above equation becomes
From earlier it was found that Therefore the above equation becomes
This is the same as , hence
Since it was found earlier that therefore
Since is a scalar multiple of and from earlier it was found that is a conjugate pair with then it is concluded that
The transformation of the second stress tensor is shown below.
From earlier it is shown that
Since the above becomes
But and using polar decomposition results in
But hence , and the above becomes
Since the above becomes
Substituting (2) and (3) into (1) gives
Since and are orthogonal, the above reduces to
The transformation of the Biot-Lure stress tensor is given below.
Since then but is orthogonal, hence , hence . Therefore (2) can be written as
Now is resolved.
Since and by polar decomposition then
Substituting (4) into (3) gives
Since the above becomes
From earlier , therefore the above becomes
From (1) gives
And since therefore
The transformation of the Juamann stress tensor is shown below.
From (1) results
Since it was found that then the above becomes
But and Therefore
Since and is conjugate pair with then
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 and its conjugate pair, the strain tensor, by .
The stress measure could be any of the stress measures discussed earlier, such as the Cauchy stress tensor , the second Piola-kirchhoﬀ stress tensor . Now the strain tensor to use is determined. Let be the conjugate pair tensors.
Physics is used in ﬁnding of for each speciﬁc
Let the current amount of energy stored in a unit volume as a result of the body undergoing deformation be , then the time rate at which this energy changes will be equal to the stress multiplied by the strain rate. Hence
Where is the trace matrix operator. This is the rule used to determine .
On a stress-strain diagram the following is drawn
The strain measure (the conjugate pair for the stress measure ) must satisfy the relation
For each stress/strain conjugate pair, the terms are derived.
In the deformed state, the stress tensor is the true stress tensor, which is the cauchy stress , and the strain rate in this state is known to be 
Where is the velocity gradient tensor. It is shown in  that
Hence in the deformed state
In other words, the conjugate strain for the cauchy stress tensor is given by such that
should come out to be the Almansi strain tensor, which is
Pre dot multiplying by and post dot multiplying it with which will make no change in the value, results in
Using the properties of the above is written as
It was determined earlier that hence hence the above equation becomes
This shows that , therefore
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.
But hence the above becomes
Using the property of can be written as hence applying this property to the above expression gives
Applying the property that to the above results in
It was found earlier that hence replacing this into the above gives
This shows that therefore
Since is a scaled version of where
It was found earlier that the strain tensor associated with is hence the strain tensor associated with is
But hence the above becomes
But . Using this the ﬁrst in (1) above is replaced. Also , and using this, the second in equation (1) above is replaced. Therefore (1) becomes
Switching the order of terms selected above by transposing them gives
Taking as common factor gives
Hence (2) becomes
Therefore (3) becomes
Hence (4) becomes
From symmetry of therefore
And (5) becomes
From property of the above can be written as
But from above, Hence
Using property of the term is moved to the left of to obtain
But hence the above becomes
But it was found earlier that
Therefore which results in
It was found earlier that hence the conjugate pair for is
Since is symmetrical, therefore conjugate pair for is Hence
The same as strain tensor associated with the Biot-Lure stress.
TO-DO for future work.
In what follows the expression for the deformation gradient tensor 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 . 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 in the undeformed state. This point will have coordinates . 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 . The coordinates of the the point is .
The coordinates is function of the coordinates . 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 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 using the name of the coordinate itself. For example writing instead of as was done above.
However this can be a little confusing since it uses the letter 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 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 can be obtained from the components by pre-multiplying the components of by the above 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
This relation can be written also in dyadic form as follows
Performing the multiplication gives
The dot multiplication is simpliﬁed using the above mentioned rules to obtain
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 to can be expressed using indices notation as follows
A matrix is orthogonal if where is the identity matrix.
If a matrix/tensor is orthogonal then . In component form,