In continuum mechanics, we view the body—the structure (solid or fluid) that is of our interest—as a continuous region that lives in the three-dimensional space .
The word continuous means, we assume the body to be made up of infinitely many, infinitely small points, with no space in-between (like the cloud of paint seen above), rather than discrete, finitely small atoms, with an atomistic space in-between.
The space is special in many ways (mathematically, that is), and allows us to set up and solve problems in mechanics in a rigorous manner. To keep things interesting and concise, we will replace some definitions with layman, more intuitive explanations.
1. Vector Space
a. is a vector space
The elements in a vector space are called vectors. Addition and scalar multiplication of vectors are closed (we always stay inside the space) and follow certain axioms (rules). In other words, we can safely compute addition and scalar multiplication of vectors because they are well-defined operations.
A vector space always contains the zero vector . We can arbitrarily assign a point in our physical world to be the origin. This allows us to identify points that make up the body as vectors in (called position vectors in continuum mechanics).
b. has a dimension of 3
We can find three vectors that are linearly independent (quite literally, independent from one another) and span (cover) the entire space of . The set of such vectors is called a basis and these vectors are called basis vectors.
Using basis vectors, we can write any vector as their linear combination, i.e. there always exist scalars such that
The representation of in terms of the basis is unique. There is no other set of scalars such that will also give us back .
What does this mean? As long as we have 3 vectors that form a basis, we can locate any point in the body and describe mathematically what is happening physically at a point and its nearby points. For example, we can talk about a point in terms of its x, y, and z coordinates (Cartesian coordinates). We can also talk about it in terms of radius, latitude, and longitude (spherical coordinates).
The fact that the representation is unique allows us to write as an array of numbers:
It is crucial to understand that the scalar components only mean something under the basis . In other words, it is important to first agree on which basis will represent the space . (Is it x, y, and z? Or radius, latitude, and longitude?)
If we choose a different basis , the vector will remain the same but the scalar components that represent will be different. We write,
The numbers , again, only mean something under the basis .
Later, we will show how the two sets of components and (two coordinate systems) are related to each other. We will see that the two are related linearly through the directional cosines.
c. comes with a norm and an inner product
An inner product is a function that maps a vector to a scalar. This function meets a set of requirements and, as a result, we can use numbers to describe the “geometry” of the vector space. In particular, we can talk about the length (norm) of a vector and the angle between two vectors. (On some vector spaces, we can’t define an inner product or a norm. We should be glad that both exist in .)
Let denote the standard basis for . The three vectors are defined to align with the Cartesian coordinate axes in the positive directions. The components of in terms of the standard basis (themselves) are as follows:
Then, we can define an inner product between two vectors and as follows:
Equation 1.1. Inner product
This does meet the properties of an inner product, and is called the inner product (or the Euclidean inner product) of and .
We can always use an inner product to define a norm, a vector-to-scalar function that measures the length of a vector. (Not all norms come from an inner product.)
Using the inner product, we define the norm of the vector as follows:
Equation 1.2. Norm
This does meet the properties of a norm, and is called the norm (or the Euclidean norm) of . We see that,
i.e. the standard basis vectors have a length of 1 in the norm.
Now that we have an inner product and a norm, we can define the angle between two vectors and . We let be the number that satisfies the equation,
Equation 1.3. Angle
We say that and are orthogonal, or perpendicular, if . From equation 1.3, we see that and are orthogonal when they form an angle of (assuming neither is the zero vector).
Any two of the standard basis vectors are orthogonal. Because they are pairwise orthogonal and have a unit length (“normalized”), we say that the standard basis is an orthonormal basis.
We defined the inner product and the norm after agreeing to use the standard basis. However, we can show that their outputs, i.e. the numbers that we get, will be the same under any orthonormal basis.
2. Coordinate Transformations
Recall that is a vector space of dimension 3, so three linearly independent vectors will span the entire space. For example, we can use the standard basis that aligns with the Cartesian coordinate system. The standard basis is an orthonormal basis, i.e. the basis vectors are pairwise orthogonal (in the sense of inner product) and have a unit length (in the sense of norm).
As long as three vectors are linearly independent, they will form a basis of . What is the benefit of considering basis vectors that are pairwise orthogonal?
a. Transforming vectors
Consider a vector . We can use the standard basis to write that,
But sometimes, it is easier to describe a structure’s behavior (e.g. displacement, reaction force) in a different coordinate system, i.e. a different basis.
Let denote a second orthonormal basis for . Note, A stands for axial, T for transverse, and S for the axis that is orthogonal to A and T.
Then, in this basis,
We would like to know how to find the components under the new basis from the components , or vice versa. To do so, we first note that,
Equation 2.1. Identity
since both sides of the equation represent the same vector .
The idea is to take the inner product of both sides of equation 2.1 with the new basis vectors. For example, if we take the inner product with , we get,
Since is an orthonormal basis, the equation above simply becomes,
Similarly, by taking the inner product with and with , we get,
The idea of using consistent nodal forces to approximate a distributed load relies on inner products. In some sense, nodal forces represent one coordinate system, and distributed loads represent another.
We can combine these three equations into a single matrix equation:
Hence, we can change the components of from one coordinate system to a new one by applying a linear transformation :
Note that, by the angle formula (equation 1.3), we can interpret each inner product as the cosine of the angle between two coordinate axes and :
We call these cosines, the directional cosines.
We can also show that is an orthogonal matrix. This means is invertible and its inverse is easily given by the transpose matrix, i.e. .
Hence, we can find from :
b. Transforming matrices
A matrix can be represented by nine components. Again, we would like to know how these components will change under a new basis (basis for matrices ). This would be useful for computing an element stiffness matrix and for describing the strains and stresses within the structure. While it is possible to come up with a basis and an inner product for matrices and repeat what we just did for vectors, there is an easier way to arrive at the result.
We use the fact that a matrix linearly maps vectors in to vectors in . In other words, allows the equation,
From equation 2.3, we know how the components of a vector change under a new basis for . Let and denote the components of and in the new basis. Substitute these relations into the equation above (recall, is orthogonal) to find that,
Because this equation holds for all vectors and (that satisfied the original equation), we can conclude that , the components of the matrix in the new basis, is given by,
Notice the similarity and dissimilarity between the transformation rules for vectors (equation 2.3) and for matrices (equation 2.5).
a. Element stiffness matrix of 2D and 3D truss elements
Consider the element stiffness matrix for a 2D truss element. Locally (in axial and transversal directions), the element stiffness matrix is very easy to describe:
To study how all 2D truss elements interact with each other (and possibly with other types of elements), we need to convert this local element stiffness matrix to a global one. For each 2D truss element, we get to apply a change of coordinates to two nodes (i.e. two vectors in ). Therefore, the transformation matrix is a block diagonal matrix as shown below:
Hence, the components of the element stiffness matrix in the global coordinate system (Cartesian) are given by,
We can show that the product simplifies to,
This element stiffness matrix is the same as that given in Topics in Computational Mechanics: Part 1, with and .
How will the element stiffness matrix for a 3D truss element look like in the global coordinate system?
b. Axial strain in 2D and 3D truss elements
We define the axial strain to be the relative change in the length of a truss. Locally, this is very easy to describe using axial displacements (1 and 2 are node indices):
Recall the relation between axial displacement and the global displacements,
Hence, we can find the axial strain from the global displacements using directional cosines: