Mechanics of Continua and Structures
We describe the motion of a continuum by describing the motion of all its constituent material particles. In order to describe the motion of all the particles we need to have names for all of them. However, since there are an \(\infty\) of particles we need an infinity of names.
Material particle=Material point
Consider a time invariant configuration (spatial arrangement of all the continuum particles) of the solid that we will call the reference configuration. Consider four particles at the four corners of a beam. For now let us name them as Alex, Wei, and Sara. In the reference configuration the material particle Alex is at $(0,0)$, the material particle Wei is at $(5,0)$, and the material particle Sara is at $(5,1)$. Since, the co-ordinates that the mat. pts. have in the reference configuration is always fixed, we can use these co-ordinates as the mat. pts. names. So, instead of saying
'’the material particle Alex,’’ we can say
'’the material particle $(0,0)$.’’
We can also make statement’s such as
The above statement is to be interpretede as
At some later time, say $\tau=1$, the material particles Alex, Wei, and Sara move to new points in space. At $\tau=1$ Alex is still at $(0,0)$, Wei is at $(6,2)$, and Sara is at $(4,3)$.
Now consider the following statement
The above statement is equivalent to the statements
We use the symbol $\mathcal{P}$ to designate an arbitrary material particle. We use the symbol $X$ to designate an arbitrary material point, or the material point corresponding to $\mathcal{P}$.
Reference point space and reference vector space The reference configuration is assumed to be contained in the (finite dimensional affine) point space $\mathcal{E}_{\rm Ref}$. Let $\mathbb{E}_{\rm Ref}$ be $\mathcal{E}_{\rm Ref}$’s associated vector space. It is real and finite dimensional. We select the finite set of vectors $\boldsymbol{E}_i\in \mathbb{E}_{\rm Ref}$ as the basis vectors for $\mathbb{E}_{\rm Ref}$. We definite an inner product on $\mathbb{E}_{\rm Ref}$ by setting $\boldsymbol{E}_i \cdot \boldsymbol{E}_j=\delta_{ij}$.
Reference position vector=Material vector=material particle The position vector corresponding to Sara (the material particle $(5,1)$ or the material point $(5,1)$) is $5\boldsymbol{E}_1+\boldsymbol{E}_2$. This vector is called Sara’s reference position vector. Hence the statement
can also be made as
Other names for material point are material co-ordinates and Lagrangian co-ordinates We use the symbols $X_1$, $X_2$, $\ldots$ to designate an arbitrary material particl’e Lagrangian co-ordinates. We use the symbol $\boldsymbol{X}$ to denote an arbitrary
The collection of all materials points in $\mathcal{E}{\rm Ref}$ is called the material body $\mathcal{B}{\rm Ref}$. The collection of all materials vectors in $\mathbb{E}{\rm Ref}$ too is called the material body $\boldsymbol{\mathcal{B}}{\rm Ref}$.
Let us go back to the statement
In the above statement the co-ordinates $(4,3)$ are called Sara’s spatial co-ordinates or Eulerian co-ordinates at the time instance $\tau=1$. So, we can now make the following equivalent statements.