BME Constitutive Equations: Elasticity
the effect that different constitutive equations have on the resulting spatial pattern. . equation of the Maxwell model, describing the relationship between the. the stress-strain relationships in the three principal orthogonal directions of the region, while the power law equation can only be used to describe material. 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.
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.
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: The 5th invariant is written as: With five invariants, we write the general transversely isotropic strain energy function as: Similar to the isotropic strain energy function, we can derive the 2nd Piola-Kirchoff stress tensor from the transversely isotropic strain energy function by taking the derviative of the strain energy function with respect to Conly this time we have to sum the derivative for five invariants: The first three terms of the 2nd Piola-Kirchoff stress is the same as for the isotropic strain energy function.
The last two terms we derive as: The derivative of W with respect to I4 represents the contribution of the fibers, while the derivative with respect to I5 represents the interaction of the fiber and the matrix.
This type of strain energy function is especially relevant to matrices with two types of embedded fibers.
This occurs in biological tissues for example in arterial walls and the intervertebral disk. In this case, we introduce another direction vector for the second fiber named g0. This direction vector g0 has the same properties as the direction vector a0namely: To accomodate the additional degree of anisotropy, we need to use eight invariants. These include the three invariants from the isotropic strain energy function and two invariants from the transversely isotropic strain energy function.
In addition, we now have two invariants from the second fiber family giving 7 invariants and 1 more invariant that accounts for the interaction in deformation between the fiber families giving a total of 8 invariants. Again, we may expand the expression for the 2nd PK stress from the transversely isotropic strain energy function to obtain the following expression for 2nd PK stress from an orthotropic strain energy function: The derivatives for the first five invariants have already been developed.
The derivatives for the 6th and 7th invariants with respect to the right Cauchy deformation tensor C are the same as the 4th and 5th invariant derivative, respectively, with a0 being replaced by g0: This leaves us to determine the derivative of C with respect to the 8th and last invariant, which represents the interaction between the two fiber families.
We note that the first term is the dot product of a0 and g0 and does not depend on C. We note that the second term is similar to both the 4th and 6th invariant, with one a being replaced by a g or vice versa. Thus when we take the derivative we have: Thus, we have the 2nd PK stress from the general orthotropic strain energy function gives: Relationships between Linear and Non-linear Elasticity A question that arises is whether there is a relationship between linear and non-linear elasticity.
One piece of common ground is that we may write a strain energy function for both linear and non-linear elastic materials. The other relationship can be gleaned by considering the following. In both linear and non-linear elasticity, if we differentiate the strain energy function with respect to the strain, we obtain the stress.
In the case of small deformation the Cauchy stress, in the case of large deformation the 2nd PK stress. If we take a look at Hooke's law for a linear elastic material, it becomes clear that if we differentiate the strain energy function twice with respect to the strain we obtain the Hooke's law constants: It is clear for linear elasticity that Cijkl will be constant irregardless of deformation.
It turns out that if we differentiate any strain energy function with respect to either the finite strain tensor or the right Cauchy deformation tensor we obtain elastic "constants". However, for the case of a nonlinear material, these elastic "constants" will differ depending on the level of deformation. 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. 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.
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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.
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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. Hence, it is of interest to see the implications of this approximation on a general representation for Cauchy stress 6. In chapter 3 section 3.