Geometry

We let the configuration of the rod be described by a function $\boldsymbol y(x)$, where $x\in(0,\ell)$ is the reference interval. The reference interval has the sole purpose of labeling the material points of the body, and it may be also chosen to be $(0,1)$. The only reason why we assign some length $\ell$ to the interval is that we want $x$ to have the same dimension of $\bm y$, that of a length.

We introduce the following unit vector fields

where $\bm e\times$ denotes the counterclockwise rotation.

Partwise and pointwise equilibrium

We denote by $\bm f$ and $m$ the internal force and the internal couple. Imposing equilibrium of the part $(0,x)$ we obtain

Differentiation with respect to $x$ yields

Taking the scalar product of the second equation with $\mathbf e$ we arrive at

Virtual work and strains

Under construction

We suppose that the current configuration of the rod consists in the specification of a placement $\mathbf y$ and of a director field $\mathbf d$. The physical interpretation of these fields is specified when we write the expression of the work done by the applied forces concomitant with a variation of the current configuration:¹:

The virtual work performed by the forces and couples acting on a part $(a,b)$ is

Integrating by parts

If, in particular, the variation is a rigid displacement, then there exists a vector $\mathbf v$ and a scalar $\theta$ such that

In this case,

Thus, we can take

as strains. Thus, the Principle of Virtual Work takes the form:

The equivalence between

Constitutive equations

The standard constitutive equations of rod theory are better written by introducing the following splitting of the displacement, the strain, and internal force:

We define ¹

It is easily verified that $k=\mathbf t'\cdot\mathbf n$. We have

The internal work can be written as

This motivates the strain energy