Constitutive equation - Wikipedia
The concepts and equations introduced in chapters 3 to 5, within the framework of To summarize, constitutive relations are required for two reasons: the deformation field, i.e., σ = f(F), then this can describe the elastic deformation of a solid. A constitutive equation is typically a phenomenological mathematical model used to describe the relationship between stress and deformation. Although. the effect that different constitutive equations have on the resulting spatial pattern. . equation of the Maxwell model, describing the relationship between the.
But the constitutive relation or equation of state which relates the state variables and the kinematic variable depends on the material. However, since the value of the kinematic variable, say the displacement or the deformation gradient, depends on the configuration used as reference, the constitutive relation also depends on the reference configuration or more particularly on the value of the state variables in the configuration used as reference.
Hence, just because the constitutive relation for two particles are different does not mean that they are different materials, the difference can arise due to the use of different configurations being used as reference. Consequently, mathematically, we say that two particles in a body belong to the same material, if there exist a configuration in which the density and temperature of these particles are same and with respect to which the constitutive equations are also same.
In other words, what we are looking at is if the value of the state variables evolve in the same manner when two particles along with their neighborhood are subjected to identical motion fields from some reference configuration in which the value of the state variables are the same.Warren Lecture Series - Holm Altenbach (Sept. 14, 2018)
A body that is made up of particles that belong to the same material is called homogeneous. If a body is not homogeneous it is inhomogeneous. Now, say we have a body, in which different subsets of the body have the same constitutive relation only when different configurations are used as reference, i. Any body with residual stresses 10 like shrink fitted shafts, biological bodies are a couple of examples of bodies that fall in this category. One school of thought is to classify these bodies also as inhomogeneous, we subscribe to this definition simply for mathematical convenience.
Having seen what a homogeneous body is we are now in a position to understand what an isotropic material is.
Consider an experimentalist who has mathematically represented the reference configuration of a homogeneous body, i. Now, say without the knowledge of the experimentalist, this reference configuration of the body is deformed or rotated.
Then, the question is will this deformation or rotation be recognized by the experimentalist? Theoretically, if the experimentalist cannot identify the deformation or rotationthen the functional form of the constitutive relations should be the same for this deformed and initial reference configuration.
This set of indistinguishable deformation or rotation forms a group called the symmetry group and it depends on the material as well as the configuration that it is in. If the symmetry group contains all the elements in the orthogonal group 11 then the material in that configuration is said to possess isotropic material symmetry.
If the symmetry group does not contain all the elements in the orthogonal group, the material in that configuration is said to be anisotropic.
There are various classes of anisotropy like transversely isotropic, orthorhombic, etc. Here we like to emphasize on some subtleties.
Firstly, we emphasize that the symmetry group of a material depends on the configuration in which it is assessed. Thus, the material in a stress free configuration could be isotropic but the same material in uniaxially stressed state will not be isotropic. Secondly, we allow the body to be deformed because it has been shown [ 10 ] that certain deformations superposed on an uniaxially extended body does not alter the state of the body. Thirdly, unlike in the restriction due to objectivity, in this case only the body is rotated or deformed virtually, not its surroundings.
Even though like in the restriction due to objectivity the rotation or deformation is virtual, it has to maintain the integrity of the body and satisfy the balance laws, i. Now, let us mathematically investigate the restriction material symmetry imposes on the constitutive relation.
Of course, for this case the restriction is similar to that obtained for objectivity 6. However, for isotropic material the symmetry group is the set of all orthogonal tensors only. To further elucidate the difference between the restriction due to objectivity and material symmetry, consider a material whose response is different along a direction M identified in the reference configuration.
Now due to objectivity we require 6. Due to material symmetry we require 6. Thus, it immediately transpires that restriction due to material symmetry 6. Thus, the result that for isotropic materials the response would be same in all directions and consequently there are no preferred directions.
If the material response along one direction is different, then it is called as transversely isotropic and if its response along three directions are different, it is called orthotropic. Thus, restriction due to objectivity, has reduced the number of variables in the function from 9 to 3 and the number of unknown functions from 6 to 3.
Next, let us see if we can further reduce the number of variables that the function depends upon or the number of functions themselves. Since, in an elastic process there is no dissipation of energy, this reduces the number of unknown functions to be determined to just one and thus Cauchy stress is given by 136.
Notice that here the stored energy function is the only function that needs to be determined through experimentation. In practice, the materials undergo a non-dissipative process only when the relative displacements are small, resulting in the components of the displacement gradient being small.
Also, it is important to reemphasis that this strain energy function contains the two components we need in all strain energy functions, namely a measure of deformation the principal stretch and constants that are determined experimentally.
If we apply the equations for deriving the Cauchy, 1st PK and the 2nd PK stress from a strain energy function derived in terms of principal stretcheswe can either differentiate the strain energy function analytically or in MATLAB. If we do this analytically for the 2nd PK stress we obtain: For the principal Cauchy stress we have: Two classic forms of the strain energy function may be derived from the Ogden model and are known as the Mooney-Rivlin model and the neo-Hookian model.
These models are been generally applied to analyze rubber and assume incompressibility. Incompressibility for deformation defined in terms of prinicpal stretch is easy to calculate since the volume ratio is defined as: Now, let us consider how these may be defined in terms of the invariants.
Recall that the right Cauchy deformation tensor is defined in terms of the stretch tensor U as: The principal stretches are the eigenvalues of U, so U may be written in terms of the principal stretches as: Thus, we define C in terms of principal stretches as: The first invariant of C is the trace of C sum of the diagonal termsso this gives: However, recall the incompressibility constraint.
In this case we can substitute for l3 with the following: If we substitute the above relationship obtained from the incompressibility constraint into the 2nd Invariant from the principal stretches, we obtain: The above relationship for the second invariant derived using the incompressibility constraint is equivalent to the second term in the Mooney-Rivlin strain energy function. Thus, we can write the final form of the Mooney-Rivlin strain energy function as: Since we have already established that the first three terms in the parentheses constitute the first invariant of the right Cauchy deformation tensorwe have the neo-Hookian strain energy function written as: Another model for isotropic hyperelasticity is the Varga model.
The above isotropic strain energy functions could be derived as special cases of the Ogden strain energy function. There are other widely used strain energy functions that are not derived from the Ogden strain energy function. We detail these strain energy functions next. The first of these is the Yeoh strain energy function.
It is written as: The second is the Arruda and Boyce strain energy function, that was dervied based on statisical models of chain orientations in polymers. The first three terms are written as: The last specific strain energy function we will present for isotropic materials is that of the Blatz-Ko model. The model was formulated for elastomeric foams which are compressible.
Thus, unlike the specific forms of the strain energy functions presented previously, the Blatz-Ko model is for compressible materials. This model is written as: Because of the random nature of the microstructure, the isotropic strain energy functions fit this material behavior well.
Although these strain energy functions have been adopted to biological tissues, in some cases fitting data very well, many biological tissues are not isotropic, due to perferred orientations in the microstructure. Thus, we have to look to anisotropic strain energy functions to characterize these tissues.
In general, anisotropic strain energy functions based on invariants assume some type of embedded fiber. This corresponds with many biologic soft tissues, that have for example collagen fibers embedded in them.
To start with, we are interested in how much the embedded fibers within a matrix stretch.
BME Constitutive Equations: Elasticity
If we consider a unit vector a0, then the vector in the deformed configuation that results from a0 is obtained using the deformation gradient tensor Fij as: However, since a0 is a unit vector, then a as a vector really represents how much a0 has been stretched during deformation. Thus, if we take the square of the length of a, this will give the square of the stretch ratio l, as: However, if we substitute for a with the deformation gradient times a0we obtain: The center F terms are simply the right Cauchy deformation tensor, so we have: This shows that the fiber stretch depends on the initial fiber direction a0 and the right Cauchy deformation tensor C.
If we assume that the strain energy function is transversely isotropic, then the strain energy function which is also referred to as the Helmholtz free-energy function should depend on both the right Cauchy deformation tensor and the fiber orientation. We can therefore write the transversely isotropic strain energy function formally as: Just as a transversely isotropic linear elastic material has more constants than an isotropic linear elastic material, so does a transversely isotropic strain energy function depend on more invariants than an isotropic strain energy function.
In fact, the transversely isotropic strain energy function actually depends on five invariants. The first three invariants are the 1st, 2nd and 3rd invariants of the right Cauchy deformation tensor. The fourth and fifth invariants depend on both the right Cauchy deformation tensor and the initial fiber direction vector. The 4th and 5th invariants describe the anisotropy arising from a preferred fiber direction.
These 4th invariant is written as: