The concept of objectivity in science means that qualitative and quantitative descriptions of physical phenomena remain unchanged when the phenomena are observed under a variety of conditions. For example, physical processes (e.g. material properties) are invariant under changes of observers; that is, it is possible to reconcile observations of the process into a single coherent description of it.
Contents
- Euclidean transformation
- Displacement
- Velocity
- Acceleration
- Objectivity for higher order tensor fields
- Example for a second order tensor
- Example for a scalar field
- Deformation gradient
- Cauchy stress tensor
- PiolaKirchhoff stress tensors
- Objective rates
- Invariance of material response
- Cauchy elastic materials
- Isotropic Cauchy elastic materials
- References
Euclidean transformation
Physical processes can be described by an observer denoted by
Consider an event in Euclidean space characterized by the pairs
By introducing a vector
The one-to-one mapping connection of the pair
Displacement
A physical quantity such as displacement should be invariant relative to a change of observer. Consider one event recorded by two observers; for
Any spatial vector field
is said to be objective, since
Velocity
Because
The velocity can be obtained by differentiating the above expression:
By reorganizing the terms in the above equation, one can obtain:
where
is a skew tensor representing the spin of the reference frame of observer
From the above expression, one can conclude that velocity is not objective because of the presence of the extra terms
A time-independent rigid transformation such as:
respects this condition.
Acceleration
The material time derivative of the spatial velocity
which can be rewritten as the following:
Just like the spatial velocity, the acceleration is not an objective quantity for a general change of observer (Holzapfel 2000). As for the spatial velocity, the acceleration can also be made objective by constraining the change of observer. One possibility would be to use the time-independent rigid transformation introduced above.
Objectivity for higher-order tensor fields
A tensor field of order
Example for a second-order tensor
Introducing a second order tensor
Example for a scalar field
The general condition of objectivity for a tensor of order
Physically, this means that a scalar field is independent of the observer. Temperature is an example of scalar field and it is easy to understand that the temperature at a given point in a room and at a given time would have the same value for any observer.
Deformation gradient
The deformation gradient at point
where
From the above equation, one can conclude that the deformation gradient
Cauchy stress tensor
The Cauchy traction vector
This demonstrates that the Cauchy stress tensor is objective.
Piola–Kirchhoff stress tensors
The first Piola–Kirchhoff stress tensor
where
Using identities developed previously, one can write:
This proves that the first Piola–Kirchhoff stress tensor is not objective. Similarly to the deformation gradient, this second order tensor transforms like a vector.
The second Piola–Kirchhoff stress tensor
Tensors
Objective rates
It was shown above that even if a displacement field is objective, the velocity field is not. An objective vector
Objectivity rates are modified material derivatives that allows to have an objective time differentiation. Before presenting some examples of objectivity rates, certain other quantities need to be introduced. First, the spatial velocity gradient
where
Substituting
Substituting the above result in the previously obtained equation for the rate of an objective vector, one can write:
where the co-rotational rate of the objective vector field
and represents an objective quantity. Similarly, using the above equations, one can obtain the co-rotational rate of the objective second-order tensor field
This co-rotational rate second order tensor is defined as:
This objective rate is known as the Jaumann–Zaremba rate and it is often used in plasticity theory. Many different objective rates can be developed. Objective stress rates are of particular interest in continuum mechanics because they are required for constitutive models, expressed in terms of time derivatives of stress and strain, to be frame-indifferent.
Invariance of material response
The principal of material invariance basically means that the material properties are independent of the observer. In this section it will be shown how this principle adds constraints to constitutive laws.
Cauchy-elastic materials
A Cauchy-elastic material depends only on the current state of deformation at a given time (Holzapfel 2000). In other words, the material is independent of the deformation path and time.
Neglecting the effect of temperature and assuming the body to be homogeneous, a constitutive equation for the Cauchy stress tensor can be formulated based on the deformation gradient:
This constitutive equation for another arbitrary observer can be written
The above is a condition that the constitutive law
Isotropic Cauchy-elastic materials
Here, it will be assumed that the Cauchy stress tensor
In order to find the restriction on
A constitutive equation that respects the above condition is said to be isotropic (Holzapfel 2000). Physically, this characteristic means that the material has no preferential direction. Wood and most fibre-reinforced composites are generally stronger in the direction of their fibres therefore they are not isotropic materials (they are qualified as anisotropic).