BME/ME 456 Biomechanics
Large Deformation Mechanics
I. Overview
Other than bone tissue, all biological tissues can in general undergo large deformation in physiologic activity. Large deformation is typically defined as anything greater 3 to 5% strain. When the deformation is small, we can use the small deformation linear strain tensor and the Cauchy stress tensor. This is because the change in volume is assumed to be negligible. However, once we have deformations greater than 5% strain, we must use the appropriate stress and strain tensors to account for large deformation. We must also use the appropriate constitutive relations that account for large deformation. Most soft tissues, including tendons, ligaments, skin and blood vessels fall into the category of large deformation. In this section, we give an overview of large deformation strain measures, stress measures, and example constitutive equations for nonlinear elastic materials. Many soft tissues can be modeled as nonlinear elastic, if we assume pseudo-elasticity.
II. Definition of the Deformation Gradient Tensor
The first step in defining large deformation strain measures is to define the relationship between what is known as the reference, initial or undeformed configuration of a body, and the deformed configuration of the body. The reference or undeformed configuation is the condition of the body in 3D space before loads have been applied to it. The deformed configuration is the location and shape of the body after loads have been applied to it. It is important to note that the body may undergo rigid body motion in addition to strain when loades are placed on it. An illustration of the relationship between the initial (note that I will use initial, reference, and undeformed configuration interchangeably throughout this section) and deformed configuration is shown below:
Note that we have defined a vector in 3D space x' in the reference configuration of the body and a vector x in the deformed configuration of the body. We note that the relationship between the two position vectors in space is the displacement vector, as shown below:
By vector addition, we can directly write the relationship between the position vectors in the initial and deformed configuration:
Next, let us consider vectors that describe material orientation in each configuration. These vectors essentially describe an infinitesimal piece of material in the body. We will denote these vectors as dx' in the reference configuration and dx in the deformed configuration. If we use the chain rule, we can directly write a mapping between the material orientation vector in the initial and deformed configuration as:
The above equation gives a relationship between a material vector in the undeformed and deformed configuration. We define the mapping itself as the deformation gradient tensor:
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where Fij is the deformation gradient tensor. We know by the rules of index notation that F is a second order tensor, since it has two independent indices. It is also important to note that F IS NOT Symmetric. Finally, let us take the displacement equation relating the initial and deformed configuration. If we take the derivative of the entire equation we obtain:
We note that the first term on the right hand side is actually a second order identity tensor, represented by the kroneckor delta as:
We can thus also define the deformation gradient tensor as the kroneckor delta plus a displacement gradient:
There are other properties of F that are important for us to derive information about strains, stresses and constitutive equations for large deformation mechanics. First, as with all second order tensors, we can calculate three invariants of the deformation gradient tensor. These are defined as follows:
Since F is a one to one mapping from the undeformed to the deformed configuration, then the inverse mapping must exist:
The following relationships involving F are useful because they allow us to map area and volume between the deformed and the undeformed configurations. First, let us consider mapping a volume from the undeformed configuration to the deformed configuration. This is done using the 3rd invariant or determinant of the deformation gradient tensor as:
To map an area from the initial configuration into the deformed configuration, consider the drawings of area change given below:
The mapping between areas in the reference and deformed configuration is written with the normal vectors as:
III. Strain and Deformation Measures for Large Deformation Mechanics
Although the deformation gradient tensor defined above is one measure of how a body changes under load, it cannot be used directly for strain characterization because it contains rigid body motions. For strain and deformation characterization, we would like a measure that does not contain rigid body motions. To define a strain measure, we will try to measures the change in length squared in a material vector in going from one configuration to the deformed configuration. We note that this measures should be independent of rigid body rotation. We note that the length squared of a vector is merely the dot product with itself. This can be written for the initial configuration as:
we can similarly write this for the deformed configuration as:
A strain measures should tell us how much a length of material has changed. In other words, it should be a mapping that tells us how much a piece of material is squeezed when going from the initial configuration to the deformed configuration. Thus, we write the following definition for a strain tensor Eij:
We note, however, that we can replace the material vector dx' using the deformation gradient tensor definition. We can then rewrite the above equation as:
Let us take a look at the first term in the above equation. We note the following holds:
However, since the k index is repeated in the above equation we can replace it with i for an index and nothing changes. This gives us back the original strain equation. Now, let us consider the second terms in the original strain equation. We note that we can rewrite this term as:
With the kroneckor delta, dij, the non-repeated index becomes the index on the term multiplied by the kroneckor delta. Using the above equation, we can rewrite the strain equation as:
from which we can directly write the strain tensor in terms of the deformation gradient tensor as by matching the strain E to the quantities in the parentheses:
As we noted previously, the strain tensor gives us deformations independent of rigid body motion. Noting that we haven't made any assumptions about the magnitude of the deformation, the strain tensor given above is thus exact for any size deformation. The above strain tensor is known as the Green-Lagrange strain tensor. It is also a second order tensor because it has two independent indices.
An obvious question is how the Green-Lagrange strain tensor compares to the small deformation strain tensor that we did make assumptions to derive. To answer this question, we begin by substituting the displacement equation for the deformation gradient tensor to obtain:
If we expand the terms in parentheses we obtain
We obtain the above because when a quantity is multiplied by the kronecker delta we obtain the same quantity back, with the free index from the kronecker delta being exchanged with the repeated index on the quantity it is multiplying. We see that the large deformation strain tensor contains a quadratic term. This means that all large deformation analyses are nonlinear. Recall that the small deformation strain tensor is defined as:
Thus, we see by making an assumption that the deformation is small, we drop the quadratic terms from the Green-Lagrange strain tensor and make the resulting strain definition linear. It is also critical to note that the gradients in the Green Lagrange strain tensor are defined with respect to the initial configuration. Thus, all the strain measures are calculated with respect to the initial configuration.
A final note on large deformation measures. An often used quantity to define constitutive equations for large deformation is the right Cauchy deformation tensor. This is defined as:
If we use the right Cauchy deformation tensor, we can rewrite the Green-Lagrange strain tensor as:
As a final note on the right Cauchy stress tensor, we note that this tensor is symmetric and positive definite. Thus, this tensor will have three real eigenvalues. The square root of these eigenvalues are denoted as the principals stretch of the material.
IV. Large Deformation Stress Measures
Just as we must define new strain measures to account for large deformation, we must define new stress measures for large deformation. In addition, these stress measures must be appropriate to use with the strain measures that we defined in the previous section. We have encountered one stress measure already in our discussion of small deformation elasticity, the Cauchy stress tensor. This stress is basically defined as force/(unit deformed area).The strain measure that is appropriate to use with the Cauchy stress tensor is the small deformation strain tensor. The problem with using the Cauchy stress tensor for analyzing materials undergoing large deformation is that we generally do not know the area in the deformed configuration. Thus we need to define a stress measure that we can use in the reference configuration.
The first principle we utilize when trying to derive a stress tensor with respect to the reference configuration is that the stress tensor in the reference configuration area should give the same force as the Cauchy stress tensor defined in the deformed configuation. Recalling that the traction is a force on the surface and is the product of the stress and the normal vector to the surface, we can define the total force in the deformed configuration as:
where dP is the total force, sij is the Cauchy stress tensor, nj is the normal vector to the surface of the deformed configuration, and ds is the surface area in the deformed configuration. Noting that we would like the same force developed by the new stress tensor, T, in the reference configuration with normal N and area ds', we define:
where we denote the new stress tensor as the 1st Piola-Kirchoff stress tensor. Now recalling the mapping between reference and deformed configuration surface areas, we can write the definition of the force in the deformed configuration as:
Now that we have the expressions for the total force consistent in terms of the normal and area, we can substract the two expressions and set the result equal to zero:
Since the term in the parentheses must be zero, we obtain the relationship between the 1st Piola-Kirchoff stress and the Cauchy stress:
We can also directly give the Cauchy stress in terms of the 1st Piola-Kirchoff stress:
Thus, the 1st Piola-Kirchoff stress gives us the actual force rendered in the undeformed surface area.
There are two difficulties with using the 1st Piola-Kirchoff stress. First, it is not energetically appropriate to use with the Green-Lagrange strain tensor. That is, the 1st Piola-Kirchoff stress tensor multiplied by the Green-Lagrange strain will not produce the same strain energy density result as the Cauchy stress multiplied by the small deformation strain tensor. Second, the 1st Piola-Kirchoff stress tensor is not symmetric. This makes it more difficult to use with numerical analyses like the finite element method. Thus, we need to look further for an appropriate stress tensor.
An additional step we can take is not to derive a stress tensor based on the force dP in the deformed configuration, but rather map the force dP back into the undeformed configuration using the inverse of the deformation gradient tensor. Doing this we obtain:
If we define another stress tensor, S, that gives the total force in the undeformed configuration on the area in the undeformed configuration we obtain:
Now let us transform the Cauchy stress such that the force it produces is in the undeformed configuration:
As before, we can also substitute for the normal vector in the deformed surface area to obtain:
We can now consistently compare the third stress tensor S, which is the 2nd Piola-Kirchoff stress tensor, to the Cauchy stress tensor as:
which gives us the relationship of the 2nd Piola-Kirchoff stress tensor to the Cauchy stress tensor:
from which we can obtain the inverse relationship:
The number of indices and different stress tensors being thrown around can be quite confusing. However, there is a simple fact to keep in mind that tells you what configuration you are in. If you have the deformation gradient itself multiplied by a given quantity, the resulting quantity will be in the deformed configuration. If you have the inverse of the deformation gradient multiplying a quantity, you will be in the initial configuration. In other words, the deformation gradient maps forward to the deformed configuration, while the inverse deformation gradient maps backwards to the initial configuration.
We now have derived the 2nd Piola Kirchoff stress tensor which is symmetric and energetically consistent with the Green-Lagrange strain. In other words, the strain energy density calculated using the 2nd Piola Kirchoff stress tensor with the Green-Lagrange strain will be the same as that calculated with the Cauchy stress tensor and the small deformation strain tensor:
The advantage is that the expression on the right is calculated in the reference configuration while that on the left is calculated in the deformed configuration.
One difficulty with all these stress tensors is how to envision their physical meaning. In truth, the physical meaning of the 2nd PK stress is hard to interpret. It is mainly used as a vehicle to solve the large deformation problem, after which the Cauchy stress is computed from the 2nd PK stress. Nonetheless, the diagram below gives the physical comparison between the three stress tensors:
V. Constitutive Equations for Nonlinear Elastic Materials under Large Deformations: Hyperelasticity
In this section we will only give a brief introduction to nonlinear elastic materials under large deformations, materials more generally known as hyperelastic. Many soft tissues under cyclic loading may be represented as hyperelastic materials, in a from referred to as pseudo-elastic materials. The stress-strain curve of hyperelastic materials is shown below:
Notice that the stress and strain quantities used are the 2nd Piola Kirchoff stress and the Green-Lagrange strain tensor. This serves as a reminder that to be consistent we have to derive the constitutive equations in terms of these quantities.
Next, we recall that any elastic material may be represented by a strain energy function. For Hookean (linear) elastic materials this takes the following form:
This strain energy function is defined in terms of the small deformation strain tensor because for small deformations the Green-Lagrange strain tensor and the small deformation tensor are equivalent. If we take the derivative of the above expression with respect to strain, we obtain Hooke's law for stress:
From the above equation, it can be seen that if we take a second derivative with respect to strain we obtain the elastic coefficients:
The same principles illustrated above hold for nonlinear materials as well:
1. The first derivative of the strain energy function with respect to the strain (GL strain) gives the stress (2nd PK):
2. The second derivative of the strain energy function with respect to the strain gives the tangential stiffness for the current deformation:
As a final note, in many cases for hyperelastic materials the strain energy function may be defined in terms of invariants of the right Cauchy deformation tensor. This is especially true if the material is isotropic. In that case, we need to apply the chain rule to determine the stress as:
where the relationship between E and C is used to calculate the second term on the right hand side.
Let us consider a simple strain energy function W for a neo-hookean material:
where a1 is a constant and I is the first invariant of the right Cauchy deformation tensor. Recall that the first invariant of the right Cauchy deformation tensor is:
Thus, the strain energy function is given explicitly as:
Let's calculate the first term of the 2nd PK stress tensor. We do this using the relationship given above, modified using the chain rule. For the first term we have:
For the derivative of the right Cauchy deformation tensor with respect to the strain we have:
Which gives us the final result:
We know have the basic ground rules to understand large deformation mechanics as it applies to soft tissues. We can use this knowledge to better understand structure function relationships for soft tissues and mechanically mediated adaptation of soft tissues.
