BME/ME 456 Biomechanics

Small Deformation Elasticity

I. Overview

            Over most of the rest of the course we will discuss tissue structure function, where the function portion denotes primarily mechanical behavior. To this end, we need to define the governing equations for solid mechanics and the role material properties, or more generally, constitutive laws, play in these equation. Constitutive laws are the primary way that tissue function is characterized. Tissue structure is most often related to constitutive laws. Toward this end, we will begin with a review of small deformation elasticity. This theory of solid mechanics is most relevant to characterizing bone tissue. However, it also provides a good beginning foundation to study the more advanced nonlinear and viscoelastic constitutive laws we will use to characterize soft tissue behavior. To begin, we will cover indicial notation.

II. Indicial Notation

A. What Indices Mean

            The use of indicial notation to represent elasticity equations is in general a more advanced topic for continuum mechanics. However, because we will cover constitutive equations for biological tissues through out most of the course, the ability to understand indicial notation will make covering these topics much more efficient. I will introduce the basis of indicial notation in the section. Most everyone should now be familiar with the concept of scalars and vectors. A scalar respresents a quantity that has no directional compoment, for instance temperature. A vector is a quantity that has directional components, like the forces and moments we talked about in the previous sections. If we write a scalar we have:

                                                  T

A vector we represent as:

            Indicial notation is really a short hand method to write scalars, vectors and higher order quantities known as tensors, without having to write the full blown representation as we did for the vector above. In addition by specifying rules about how combinations of indices indicate summation, we can use indicial notation to compactly right very complicated equations. Indicial notation was developed by Albert Einstein and is also called Einstein notation.

            We begin with example of how indicial notation may be used to represent scalars and vectors. A scalar is actually not assigned any index. Thus, a scalar is written just as we did above with temperature: T. A vector on the other hand, is given a single index, for example the force vector above in vector notaton would be written in indicial notation as:

where the three bar symbol means "is defined as" and is not equivalent to an equal sign. In this case, the index on F is i. However, it is important to note the indices are really place holders. If we write the above equation with a j index instead of an i, it means the same thing:

Using F with the single subscript i or j means the same thing.

Another important application of indicial notation is to compactly represent coordinate systems. Where before we would denote a three-dimensional (3D) coordinate system using x,y and z, we now represent it using x1,x2, and x3. The correspondence is:

We can then write x as a coordinate system vector using indicial notation as:

            Based on our work with scalars and vectors, we can generate a formula that tells us how many separate quantities and indexed symbol represents. If we denote d as the dimension of the space in which we work, eg 2D or 3D, and n the number of independent or non-repeated indices (we will see why this is important soon), then the total number of individual quantities q can be calculated as:

Thus, if we are working in 2D with a vector and denote it as then we have two quantities since 2^1 = 2. In this case we have:

If we have a scalar we have 0 indices and q = 1 regardless of dimension since d^0 = 1. This raises another important point, all indices have a range, and the range is from 1 to the number of dimensions. Thus, for 2D the index i ranges from 1 to 2 and for 3D the index i ranges from 1 to 3, or i = 1,2, and 3.

            In solid mechanics, we deal with quantities beyond scalars and vectors. These quantities include stress, strain and consitutitve coefficients. In general, these quantities are called tensors. The number of quantities a tensor contained follows the dimension formula. For example, stress is a tensor with two indices written as: . There are two independent indices, i and j. Thus, for 3D stress we have q = 3^2 or 9 quantities. In this case, i and j both range from 1 to 3. They way in which the range is specificed is to set i = 1, and let j = 1,2,3, then set i = 2, and let j = 1,2,3. This is the same way we would write a nested do or for loop:

do i = 1,3
   do j = 1,3
   enddo
enddo

Since stress has two indices, we say that it is a 2nd order tensor. A quantity with 4 indices is a 4th order tensor. A vector is a 1st order tensor and a scalar is a 0th order tensor. If we write out all the quantities of the 2nd order stress tensor we have:

Each index refers to an axis in the same way as the coordinate system designation, where x = 1, y = 2, and z = 3. Thus, the corresponding x, y and z stresses for the above indices are:

B.         Repeated Indices and Summation

            One of the most powerful aspects of indicial notation is the ability to represent very complicated equations in a very efficient manner. Although at first the use of indicial notation seems complex and confusing, once you are able to understand the basic rules (such as those presented above and in what follows) indicial notation actually becomes a clearer way to present complicated equations than writing them out explicitly.

            A fundamental rule of indicial notation is that any repeated indices represent a sum of the terms containing the repeated indices. For example, hydrostatic pressure is explicitly defined as one third of the sum of the normal components of stress:

Following the rule that repeated indices indicated summation over the range of the indices (remember that range is equal to the dimension of the problem, for 2D i = 1,2 and for 3D i = 1,2,3), we can rewrite the above explicit representation of hydrostatic pressure using index notation as:

Note that hydrostatic pressure is a scalar, ie it has no directional component. This leads to another critical point: The number of independent or non-repeated indices gives the tensorial order of the quantity. Let us look at the above expression. The indices ii are repeated, so they are not independent. There are no other indices, so therefore the number of non-repeated or independent indices is zero and the quantity is a scalar. The same situation occurs when we have the dot product of two vectors, u and v. We know that the resulting quantity is a scalar. Using indicial notation we write the dot product as:

We can see that the indicial notation also gives a scalar from 1 equation.

Let's next consider the situation where we have the product between a second order tensor and a vector. This situation occurs for example when we define a traction (force) boundary condition for an elasticity problem. The tractions is given in index notation as:

where s is the 2nd order stress tensor and n is the normal vector for the boundary. We note in the above that j is a repeated index while i is a non-repeated or independent index. Therefore, for each i, we sum over the j indices. This gives us three separate equations, hence the result is a vector

In fact, we define the traction vector t as:

The above equations raise another critical point when performing dot product like operations using index notation. The independent or non-repeated indices for a product are transferred to the resulting quantity. In the example above, i is the independent index that is transferred on the resulting traction quantity t. Note that the same rule applies no matter what order tensors are involved. Consider Hooke's law for an elastic material, if we denote the stress , the strain , and the elastic constants ,
then we may write Hooke's law in indicial form as:

Note that in this case both the stress and strain are 2nd order tensors (both have two independent indices) while the constitutive or elastic constant tensor is 4th order because it has 4 indices. Note that the other rules for indicial notation are followed in this case. First, ij remain as independent indices so these are transferred to the stress tensor. Second, there are actually nine equations represented by the above Hooke's law, since we have two independent indices ij, so the number of equations is 3^2 = 9. To give an example of what the above Hooke's law is, we write out the first equation:

Note that the above equation is analogous to four nested do loops:

do i = 1,3
   do j = 1,3
        do k = 1,3
             do l = 1,3

              sij  =  sij  +   C(i,j,k,l)*e(k,l)

              enddo
          enddo

          s(i,j)  =  sij

      enddo
enddo

where we sum over k and l but i and j remain independent indices. It is also important to note that repeated indices are often called dummy indices, since they can be replaced by any other letter index other than the independent index. Thus, our indicial rendition of Hooke's law can also be written as:

and it would have exactly the same meaning as the Hooke's law equation with kl.

III. Equilibrium (Balance of Linear Momentum)

            Just as we have balance of forces on a rigid body for static equilibrium, so the interior forces in a deformable body must balance for it to be in static equilibrium. If we recall that stress is defined as force per unit area, then we have the following balance of forces in the x or x1 (for indicial notation) direction for an infinitesimal piece of material:

where s11 is the normal stress on the plane perpendicular to the x1 axis, s21 is the shear stress on the plane perpendicular to x2 axis acting in the x1 direction, and s31 is the shear stress on the plane perpendicular to the x3 axis acting in the x1 direction. In general, sij represents a stress acting on the plane perpendicular to the xi axis in the direction of the xj axis. Thus, s11 is acting on the plane perpendicular to the x1 axis in the x1 direction. In the fgirue above, b is a general body force. Only the b1 component of the body force enters into the balance of forces in the x direction. We can now write the balance of force in the x direction noting two facts. First, stress is defined as force per unit area so to get the force we must multiply the stress by the area of the plane on which it acts. Thus the area of the plane perpendicular to the x1 axis is , the area of the plane perpendicular to x2 is , and the plane perpendicular to x3 has an area of . Next we note the the positive and negative forces in the same direction are separated by , , or . s11 acts at a location of . If we add all the forces that are acting including the x1 component of the body force b we obtain:

where it is important to note that all multiplication of terms is represented by an asterisk *.   We next divide the above equation through by the quantity **, the volume of the infinitesimal cube to obtain:

Next, because we are now using principles of continuum mechanics, we let the size of the incremental cube in the limit shrink towards zero. Note that the cube volume does not become zero, but approaches zero. In this case, the quantities involving forces acting in different directions turn out to be partial derivatives of the stresses with respect to the direction of the denominator. For example,

Using this defintion for all terms, we obtain the equilibrium equation in the x1: direction as:

Now, let us consider the balance of forces in the x2 direction:

Again, if we write the balance of forces we obtain:

If we again divide by the infinitesimal cube volume and then in the limit let the cube volume approach zero we have:

Finally, if we look at the balance of forces in the x3 direction we have:

We can write the equation for force balance in the x3 direction as:

We again divide by the infinitesimal volume and let the volume approach zero to obtain:

Thus, we can see that if we balance internal forces on an infinitesimal piece of material inside a body, we obtain three equations that define the stress state, assuming we have a body force active:

Note that we have nine unknowns, namely the components of the 2nd order stress tensor s, but only three governing equations. The body force b is typically given. We can rewrite the above equation much more compactly using indicial notation as:

note that i is a repeated index so we sum over i. j is an independent index so each j represents a separate equation which is why we arrive at 3 equations as above.

IV       Equilibrium: Balance of Angular Momentum

            We saw in the above section that balance of linear momentum, ie balance of forces, on an infinitesimal scale within the body led us to three equilibrium equations defining balance of body forces and internal stresses. This is analogous to balance of linear momentum for the rigid body models we previously considered. This raises the question of whether there is a balance of angular momentum (balance of moments) for the infinitesimal scale inside a body. It turns out there is such a principle. Again, we look at stresses acting on the faces of an infinitesimal cube inside a body. The normal stresses do not cause a moment since they have a moment arm of zero. This leaves only the shear stresses for us to consider. Consider the moments generated on the planes perpendicular to the x1 and the x3 axes:

The total moment associated with the stress s13 is:

where the first two delta x terms are the area of the plane and the last term is the moment arm. We can now balance the moments in the x2 plane due to these shear stresses. It is important to note that on the innfinitesimal cube there are no externally applied moments (there are some continuum theories that assume moments at this level, but they are not widely used and are beyond the scope of these notes). Thus the balance of moments in the x2 direction leads to:

We see that the final result leads to the conclusion that there are only three indepedent shear stresses not six, and that the shear stresses are symmetric as:

Thus, applying the balance of forces and balance of moment equations to stresses on an infinitesimal cube inside the body has led us to three equilibrium equations for six independent stresses, accounting for the fact that the shear stresses are symmetric.

It is important to note that for the equilibrium equations no assumptions are made regarding the deformation magnitude. The only explicit assumption we make is that the stress tensor we use is the Cauchy stress tensor, which is defined in the current or deformed configuration of the body. Although other stress tensors are useful if we have large deformations, they are indistinguishable from the Cauchy stress tensor for small deformation. Therefore, for small deformation elasticity we will use the Cauchy stress tensor.

            An important boundary condition relevant to the stress equilibrium equations is the stress traction boundary condition. If we assume a tetrahedron within the material and again balance forces upon the tetrahedron, we can come up with three equations relating an external traction vector t on a boundary with normal n to the internal Cauchy stress tensor s. The full equations are shown below:

Can you see the pattern of repeated indices in the above equation? Can you also see the indepedent indices and the fact that the above equations each give one component of a vector? Knowing this, we can rewrite the above equation in indicial notation as:

Here we can see why the work on rigid body models and optimization from the first section is important in the realm of tissue mechanics. The traction in the above boundary condition comes in general from two sources on musculoskeletal tissues: 1) muscle forces through tendon attachments and 2) contact forces between joints. Thus, knowing muscle forces is extremely important to solving the stress equilibirum partial differential equations through the stress traction boundary conditions. Using these equations together with the equations we will discuss next allows us to estimate tissue stress and strain, and see how tissues adapt to stress and strain and also understand tissue structure function relationships.

V         The Small Strain Tensor

            The balance of internal forces led to the stress equilibrium equations. The balance of internal angular momentum led us to the fact that shear stresses are symmetric. We also had the stress traction boundary conditions. These equations in and of themselves are the governing equations for stress in any body. However, in and of themselves these equations are also indeterminate, like the rigid body models we considered for muscle forces are indeterminate. We have six independent stresses but only three governing equations. We thus must find a way to develop unique solutions to the stress equilibrium equations to allow us to determine stress in a body. In classic continuum mechanics, this is done by expanding the set of equations to account in addition to stress also for deformation. We next derive considerations for the small strain tensor.

            Consider the deformation of an infinitesimal cube (once more!) in the x1 direction:

If we in the limit allow both and both to approach zero, then we arrive at the following strain displacement relationship:

This corresponds to our notion that strain is defined as the change in length over length. Note that this is an approximation of strain for small deformations. Thus, it is our definition of strain that renders this discussion for the small deformation part of small deformation elasticity. We can also do the same change in length for the x2 and x3 directions to arrive at:

Next, let us consider a pure shear deformation:

where the total deformation is the sum of q1 and q2, and can be written as:

Note that q1 is defined to equal q2,q1 = q2.  Now we use some trigonometry. We note that the tangent of q1 and the tangent of  q2 can be written as:

If we assume that the angle of deformation is small (again referring back that this section is small deformation elasticity), we know that the tangent of a small angle is equivalent to the angle itself, assuming we define the angle in radians. Thus, we may write the above as:

We can thus rewrite the angular deformation from above as:

Note that we have added an extra definition e12. This is the definition of tensorial shear strain that is one half g12 that is the definiton of engineering shear strain. If we taken the limit as all delta quantities go to zero, then we have the following definition of the shear strain:

by definition shear strains are symmetric as we see below:

The definition of the other shear strains follows directly as:

Given the definition of normal strains, and the shear strains given above, we can write a compact defintion of strain in indicial notation as:

The strain displacement equations we defined above give us nine additional equations, one for each component of the strain tensor. However, they also introduce an additional 12 unknowns, 9 strain tensor components plus three unknown displacements: u1, u2, and u3. Thus, our tally is as follows:

We can see at best that we have 12 equations but 21 unknowns. We therefore need additional equations to make the problem determinate. In addition, we don't have a link between the stress quantities and the strain and displacement quantities. We will address both of these issues in the next section.

VI         Constitutive Stress Strain Relations aka Material Properties

            The final link in the continuum mechanics approach to determine the state of stress, strain and displacement in a body are the constitutive relations, commonly known as the stress strain relationships because they give stress as a function of strain. Not only do these equations provide a link between stress and strain, they give us an equal number of equations to unknowns. For a general complex materials, the stress at any given point in the body at any given time may be a function of the previous stress history, the strain, and the strain rate. This can be written in general functional form as:

At their most complex, biological tissues can encompass any of these complicated dependenences because of damage, viscoelasticity, large deformation etc. The key point of using constitutive equations is finding the simplest one that can give you the accuracy you need. Bone and many soft tissues under repeated loading may be characterized as elastic. Elasticity implies certain material characterisitics. First and foremost among these is the attribute of storage and release of energy. When we load an elastic material, it stores internally the energy that is equivalent to the energy of the external loads. An elastic material does not dissipate energy, and thus when we release the load all of the energy is released. The loading and unloading curve of the stress strain relationship follow the same path:

The area underneath the stress strain curve is the stored energy. Bone is a good example of simplifying stress strain relationships. Although bone damages, and exhibits viscoelastic behavior, for many cases of physiologic loading we may simplify bone as a linear elastic material that undergoes small deformations (ie less than 2 to 3% strain; we now also have the elastic in small deformation elasticity). Thus, not only does bone pretty much only store energy under many physiologic situations, the stress in the material only depends on the strain itself and furthermore depends linearly on strain. This is illustrated below:

The energy stored for a linear elastic material can easily be calculated as since the stress times the strain is a square box (shown in dotted lines on the above figure) and the area of the triangle within the square box is one half the area of the square box.

            Now that we know the characteristics of an elastic material, we need to know how we can derive the stress strain relationship for an elastic material. We noted that an elastic material always stores energy. Therefore, the relationship between stress and strain should be some function of the stored energy. In fact, an elastic material is generally defined as one for which a strain energy function or potential W exists. When we take the derivative of this potential with respect to strain we obtain the stress:

The form above gives a material typically called a hyperelastic material. We will come back to this form because it is generally used to characterize soft tissues. However, we can see that the above form can be used to calculate a property known as a tangent stiffness tensor, if we taken the second derivative of W:

The above equation tells us that the current stiffness of any elastic material is the second derivative of W with respect to the current strain. We also know that the current stiffness for an elastic material is the relationship between stress and strain at a given strain:

Where the parentheses denote that the current stiffness is a function of strain. Of course you may be asking what any of this has to do with reality. Well for bone reality (and any linear elastic material undergoing small deformations) we know that the stiffness is constant regardless of the amount of strain (until it breaks or yields) as seen in the linear elastic stress strain curve. Thus, we note that the stiffness is a constant, and we can apply the above relationships to determine the general linear elastic stress strain or constitutive equation as:

The above equation gives us the final piece of the puzzle. Note that since the k and l indices are repeated in the above equation, we sum over those indices. Stress and strain have two indices so they are 2nd order tensors while the constitutive tensor has 4 indices so it is a 4th order tensor. We write out explicitly how the above equation translates into s11:

Also, note that for a linear elastic material the constitutive tensor C is independent of deformation. Since stress is a 2nd order tensor, we note that the constitutive equation gives us 9 equations but does not introduce any more unknowns. Thus, we end up with 21 unknowns (not counting symmetric quantities) and 21 equations when we include stress equilibrium equations, strain displacement equations, and constitutitive equations. We summarize in the table below:

If we consider symmetry as a constraint, then the number of strain displacement and constitutive equations are reduced to 6 each. In that case we have 15 independent unknowns and 15 independent equations.

VII       Further notes on the Constitutive Tensor C

             We now have the complete set of equations to solve for the stress, strain, and displacement in a body. This is important for tissue mechanics for a number of reasons including computing if stress or strain levels are high enough to cause failure or damage to tissues, determining the state of stress and strain in a tissue and if this will lead to adaptation, and determining how tissue structure can feedback stress and strain information to cells that can in turn alter tissue structure. One way that we characterize tissue structure function is through the constitutive matrix. It is clear that the constitutive matrix determines the relationship between stress and strain. It is important to point out, however, that the constitutive tensor is the only quantity that we must have experimental data to construct. The constitutive tensor is solely a function of the material. Thus, in biological tissues, it is a function of the arrangement of all the biomolecules that cells produce.

            The constitutive tensor as it stands has 81 constants, since 3^4 = 81. However, by symmetry considerations of stress and strain, ie that , we can immediately reduce the number of independent quantities to 36. This is because the symmetric stress tensors eliminates 3 out of 9 equations and since each equation has 9 quantities that is a reduction of 27 constants. In the remaining six equations, 3 of each of the constants in size equations are not independent because of symmetry in the strain tensor. That reduces another 3x6 = 18 constants. Thus, because of symmetry in stress and strain tensors we have 81 - (18+27) = 36 independent constants left. Now we note that because we derive the constitutive tensor from an energy function, whether we take the derivative first with respect to eij or ekl we should get the same constitutive tensors:

The above symmetry reduces the total constants by another 15 to give us a final total of 21 independent constants. Thus, for linear elastic materials, the stress strain behavior can be represented by 21 numbers at the most. These may be reduced even further if certain material symmetries are present. These material symmetries are a function of tissue structure for biological tissues.

VIII.     Material Symmetries for the Linear Elastic Constitutive Tensor C

            In section VII we noted that the number of independent constants in the material constitutive tensor for an elastic material could be reduced to 21 due to the symmetry of stress and strain and the fact that an energy potential exists to describe the material behavior. Note that these considerations depended on the stress and strain definitions which are material independent (note: we used an argument for strain symmetry for the small deformation strain tensor, but the same arguments hold for the large deformation strain tensor), and the existence of a potential which indicates that the material is elastic. In this section we note that the number of independent constants may be further reduced based on symmetry behavior of the material. This is the point where tissue structure function relationships start to play important roles. We can fairly directly link tissue structure, especially for bone, to the type of material symmetry we obtain when we test bone. Before proceeding, we can show that we may write the material constitutive tensor in a more compact matrix form. First consider the first equation for the s11 normal stress. We have:

Because we have symmetry such that Cijkl = Cijlk, as well as symmetry in strain ekl = elk, we can rewrite the above as:

Note that because of symmetry considerations we have reduced the number of constants to six. If we note that the engineering shear strain is equal to twice the tensorial shear strain, we can write the following matrix compact form of the linear elastic constitutive relation:

The sym denotes a symmetric matrix. This is the fully anisotropic stiffness matrix. This stiffness matrix is indicative of a material that has a different stiffness in different directions. If a given material has three planes of symmetry, then we may reduce the number of unknown constants from 21 to 9. This can be written as:

This type of symmetry is most often used to represent trabecular bone effective properties at the zeroth level. It is also used sometimes to represent cortical bone properties at the zeroth level, or macroscopic level.

The next higher level of symmetry occurs when a material has one stiffness longintudinally and has a second stiffness in every direction radially. This symmetry is known as transversely isotropic, and is often used to represent cortical bone. The matrix for a transversely isotropic material is shown below:

Finally, we note that if the material has basically infinite planes of symmetry, and the stiffness is equal in all directions, we may represent that material as isotropic, with only two independent constants. The constitutive matrix is then:

where we note that the two independent constants are C1111 and C1122. More often than not, biological tissues are represented as either orthotropic or transversely isotropic, although simplications are often made in analyses to represent tissues as isotropic due to lack of available experimental data.

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