Assemble member stiffness matrices to obtain the global stiffness matrix. 7. Analyse plane truss form triangulated patterns. A hinge connection Now the force-displacement relation for the truss member may be written as, u. L. AE. F = ( ) .. The missing columns and rows in matrices and are filled with zeroes. Thus,. 1. useful estimates of the normal stress due to bending for loadings that included shear, so too ferential equation for the transverse displacement, v(x) of the beam at every point . The key to resolving our predicament is revealed by the form of the equation Thus, the second column of our stiffness matrix can be filled in. The Strain-Displacement Relations displacement changes through the material, and is the strain at),(yx. .. Any expression of the form.
Figure 1 below, illustrates a unit cube of material with forces acting on it in three dimensions. By dividing by the surface area over which the forces are acting, the stresses on the cube can be obtained. These components form a second rank tensor; the stress tensor Figure 1.
Figure 1 Tensor math allows us to solve problems that involve tensors. For example, let's say you measure the forces imposed on a single crystal in a deformation apparatus. It is easy to calculate the values in the stress tensor in the coordinate system tied to the apparatus. However you may be really interested in understanding the stresses acting on various crystallographic planes, which are best viewed in terms of the crystallographic coordinates.
Tensor math allows you to calculate the stresses acting on the crystallographic planes by transforming the stress tensor from one coordinate system to another. Another familiar tensor property is electrical permittivity which gives rise to birefringence in polarized light microscopy. You are probably familiar with the optical indicatrix which is an ellipsoid constructed on the three principle refractive indices. The refractive index in any given direction through the crystal is governed by the dielectric constant Kij which is a tensor.
The dielectric constants "maps" the electric field Ej into the electric displacement Di: Were k0 is the permitivity of a vacuum. Di can be calculated from Ej as follows: Rank of a Tensor Tensors are referred to by their "rank" which is a description of the tensor's dimension.
A zero rank tensor is a scalar, a first rank tensor is a vector; a one-dimensional array of numbers. A second rank tensor looks like a typical square matrix. Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all second rank tensors.
A third rank tensor would look like a three-dimensional matrix; a cube of numbers.
The Stress Tensor
Piezoelectricity is described by a third rank tensor. A fourth rank tensor is a four-dimensional array of numbers. The elasticity of single crystals is described by a fourth rank tensor. Tensor transformation As mentioned above, it is often desirable to know the value of a tensor property in a new coordinate system, so the tensor needs to be "transformed" from the original coordinate system to the new one.
As an example we will consider the transformation of a first rank tensor; which is a vector.
This can be done by noting the angle between each axis of the new coordinate system and each axis of the new coordinate system; altogether there will be 9 angles, three of which are illustrated in Figure 2: If we utilize Einstein's summation convention, we can leave out the summation symbol and get: There is a similar process for transforming a second rank tensor, but calculating a formula for the transformation by the same means that we transformed the vector above would be quite laborious.
There is a more convenient shortcut. Just as the dielectric constants "maps" the electric field Ej into the electric displacement Di, we can imagine a second rank tensor Tkl that takes Ql and produces Pk in a given coordinate system: It is abbreviated as: The Stress Tensor Stress is defined as force per unit area. If we take a cube of material and subject it to an arbitrary load we can measure the stress on it in various directions figure 4.
These measurements will form a second rank tensor; the stress tensor. Therefore, it is important to be aware of which sign convention is being used. How does the addition of the strain displacement relationship affect our ability to solve the mechanics problem. Well, now we have introduced 12 more unknowns, the nine strains eij and the three displacements ui.
We have introduced, however, only 9 additional equations, namely the strain-displacement equations. Definition of Reference and Deformed Configurations, and the Relationships between Them We explicitly made the assumption in deriving the small strain tensor that strains were "small" for both normal and shear strain. While this assumption is valid for bone, it is not valid for soft tissues. Of course, one may ask why bother with assuming small deformation at all. As we will see in this section, large deformation, while valid for all deformation situations, is much more complex and introduces nonlinearity into the problem, called geometric nonlinearity.
Reference Initial and Deformed Configurations 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: Relationship between a point in the Reference and Deformed Configuration 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, which represent the locations of a point in the reference configuration and a point in the deformed configuration, 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: Relationship between a material vector in the Reference and Deformed Configuration The above results tells us how a point displaces from the reference to the deformed configuration.
However, we would also like to know how a piece of material, again, an infinitesimal piece, is stretched and rotated as the body moves from the reference to the deformed configuration.
The infinitesimal material vector in the reference configuration, dx', is shown in red in the reference configuration. The material vector after it has been stretched and rotated, dx, is shown in red in the deformed configuration below: Since the length of dx' can change when going to the deformed configuration as well as its orientation, we can say that dx' deforms into dx.
The question then becomes how to relate dx in the deformed configuration into dx' of the reference configuration. This can readily be done through the chain rule 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: We know by the rules of index notation that F is a second order tensor, since it has two independent indices.
We can also write the deformation gradient tensor in matrix format as: Relating the Displacement Vector and Deformation Gradient Tensor We now have two entities that relate quantities in the reference configuration to the deformed configuration: The next logical question is whether these quantities themselves have any connection.
First, let us begin with the displacement vector that relates a position in the reference configuration to a position in the deformed configuration: The above equation actually represents the relationship between the three coordinates in a reference and deformed configuration: Now let us differentiate each on of the above equations with repect to ,: Let's first look at and group the left hand side of the differentiated displacement relationship.
If we group all these terms, we end up with the deformation gradient tensor. Now, let's take a look at the first term on the right hand side in each of the above equations. If we group these terms in a matrix, we obtain: Let us look more carefully at each entry in the above matrix.
Obviously, if we differentiate something with respect to itself we get 1. Thus, the diagonal of the above matrix will all have ones. For all the off axis terms, we are differentiating a one coordinate position with respect to one of the two other coordinate positions.
Since one coordinate position e. This leaves us with the identity matrix, which is equivalent to the kronecker delta 2nd order tensor we defined in the section on mathematical preliminaries: Finally, we consider the term on the right hand side of the differentiated displacement equation.
We can also write this in a matrix from as: