Chaire d'Analyse Appliquée                    The DNA modelling COURSE

Session 3

Balance Laws

We now turn from the kinematics of rod models to a consideration of the balance laws, which will allow us to determine which configurations are possible equilibrium shapes.

The stresses exerted by the material on one side of the cross-section at s acting on the material on the other side of the cross-section can be averaged to yield a net force ${\bf n}(s)$(of the material on the side $s^+$ acting on the material on the side$s^-$).

There is an arbitrary sign convention here, and as long as it is treated consistently it doesn't matter which convention you take.

Similarly the first moment of the stresses acting across the cross-section can be averaged to yield a net moment ${\bf m}(s)$ (taken with the same sign convention as ${\bf n}(s)$).

As before ${\bf n}(s)$ and ${\bf m}(s)$ denote vectors, while $n_i(s)$ and$m_i(s)$ will denote components of these vectors with respect to the variable basis $\{\d_i(s)\}$.

It will be of importance to recall the elementary fact that because the frame $\{\d_i(s)\}$ is not constant, the component of the derivative ${\bf m}'(s)$ is not the derivative of the component$m_i'(s)$. Rather using the kinematics of the basis $\{\d_i(s)\}$, it can be seen that there is an additional cross-product term. Explicitly

$${\bf m}'=\left(\sum_i m_i\d_i\right)'= \sum_i \left( m_i'\d_i + m_i\d_i'\right)$$ $$= \sum_i \left( m_i'\d_i + m_i(\u\times\d_i)\right).$$  
 

or

$${\bf m}'\cdot\d_j=m_j' + (\u\times{\bf m})\cdot\d_j$$

 
 
 

so that the $j$th component of ${\bf m}'$ is the derivative of the $j$th component plus the $j$th component of the cross-product$(\u\times{\bf m})$. The coordinate free equilibrium equations are:

$${\bf n}'(s)={\bf f}\,,$$

and

$${\bf m}'(s)+{\bf r}'(s)\times {\bf n}(s)={\bf \tau},$$

 
 
 

where ${\bf f}(s)$ is a distributed external force acting on the rod, and ${\bf \tau}(s)$ is a distributed external torque acting on the rod.

For the most part we will assume that the external force and torque loadings vanish. But possible interesting case would be weight due to a gravitational field, or a distributed force loading arising from self-contact of the rod. That is the rod at arc-length $s_1$ touches (and pushes) the rod at arc-length $s_2$.

Constitutive Relations

Thus far our concept of a rod has implicitly used the assumption that the body is long and slender, in order that the kinematics yields a sensible description of the body. But we have not said anything about the properties of the material. How do we tell the difference between a long, slender body made of steel, of rubber, of chewing gum, of a viscous fluid such as treacle, or a segment of a DNA molecule? The differences are quantified by specifying the constitutive relations of the rod. That is the material properties are determined within the mathematical model by specifying relations that give the components of the stresses $m_i$ and $n_i$ in terms of the strains$u_i$ and $v_i$ (or vice versa).

When selecting definitions of strains and stresses, it is important i) that the strains fully determine the configuration of the system, ii) the stresses allow a full determination of the balance laws, iii) the strains determine the stresses.

However in the constitutive model it is perfectly possible that the stresses could depend upon, for example, time derivatives of the strains (which are called viscous or plastic models), or time history of the strains (memory effect models), or space derivatives of the strains (which are called strain-gradient models).

Here we shall consider exclusively the case where the stresses are related to the strains through an algebraic constitutive relation, i.e. no derivatives or integrals of any type. In fact this assumption is precisely our definition of the model being elastic.

We first give some examples of simple elastic constitutive relations, and (somewhat) later discuss how general the constitutive relations might reasonably be, and how reasonable the simple choices might be.

The choice of appropriate constitutive relations is one of the most crucial issues in using rod models to describe physical phenomena.

In the case of modelling DNA the issue is (in my opinion) not at all well understood.

Perhaps the simplest choice of constitutive relation is a diagonal, linear (or at least affine) one. For each i (with no sum)

$$m_{i}=K_{i}(u_{i}-\hat{u}_{i})\,,\qquad n_{i}=A_{i}(v_{i}-\hat{v}_{i})\,.$$

 
 
 

Here, as before, the $\hat{u}_{i}(s)$ and $\hat{v}_{i}(s)$ are prescribed functions of s determining the reference shape. And the$K_{i}(s)$ and $A_{i}(s)$ are prescribed functions, which for reasons we shall see later, are usually assumed to be strictly positive.

Because of the form of the shift in the constitutive relations

$$m_{i}=K_{i}(u_{i}-\hat{u}_{i})\,,\qquad n_{i}=A_{i}(v_{i}-\hat{v}_{i})\,.$$

we see that the stresses in the reference state vanish, i.e. the reference state is unstressed.

More generally for positive coefficients $K_{i}(s)$ and $A_{i}(s)$ the stress-strain law can be inverted (here trivially) to yield

$$u_i = \hat{u}_{i} + m_{i}/K_{i}\,,\qquad v_i=\hat{v}_i + n_i/A_i.$$

 
 
 

In particular, invertibility implies (among other things) that when the reference state is stress-free it is the only unstressed state.

We will also use the fact that the constitutive laws are hyper-elastic, which means precisely that there is a scalar-valued function$W(u_i,v_i;s)$ of the six strains with the property that the constitutive relations can be written in the form of partial derivatives of $W$

$$m_i = W_{u_i}\,,\qquad n_i = W_{v_i}.$$

The linear diagonal constitutive laws described above are hyper-elastic with the quadratic strain-energy density function

$$\frac{1}{2} \sum_i \left\{ K_{i}(u_{i}-\hat{u}_{i})^2 + A_i(v_i-\hatv_i)^2\right\}$$

The elastic energy of the rod with strains $u_i(s)$ and $v_i(s)$ can be defined as

$$\int_0^L W(u_i,v_i;s)\, ds\,$$

and $W$ is called the strain-energy density function.

In the diagonal, linearly elastic case we can see that the unstressed reference configuration is also the unique minimum energy configuration.

Much of the time we will be concerned with the case of inextensible, unshearable elastic rods. As discussed earlier the inextensibility and unshearability conditions are expressed by specifying the values of the strains $v_i$ to always be their reference values $\hat v_i$(which are usually assumed to be the triple $(0,0,1)$).
For such rods the force${\bf n}$ is a basic unknown, with no related constitutive relation, while the bending and twist strains $u_i$ and components of the moments $m_i$are still related through constitutive relations, for example of the affine form:

$$m_{i}=K_{i}(u_{i}-\hat{u}_{i})\,,\qquad i=1,\,2,\,3\, ,$$

where the $K_{i}(s)\gt$ and the $\hat u_i$ are as before. We will also talk about hyper-elastic, inextensible, unshearable rods. For the affine constitutive relation on the previous page, the elastic strain energy of the rod is

$$\int_0^L \frac{1}{2} \sum_i K_{i}(u_{i}-\hat{u}_{i})^2\ ds$$

Equilibrium conditions

We can now describe the system of field equations (which for rods are ordinary differential equations) that must be solved to determine the equilibria of elastic rods. They are made up of the coupled system formed by the balance laws taken with the kinematic equations. $${\bf n}'(s)={\bf 0}\,,$$ $${\bf m}'(s)+{\bf r}'(s)\times {\bf n}(s)={\bf 0}.$$ $${\bf r}'=\v,\qquad \d_i'=\u\times\d_i$$

 
 
 

plus some combination of constitutive relations, or unshearability or inextensibility constraints.

In fact when the balance laws are written in terms of components wrt the frame $\{\d_i\}$, they (apparently) decouple from the kinematic equations.
Balance laws become a closed system of 6 first-order ODE
 
 
 

$$n_i' + \epsilon_{ijk} u_j n_k = 0\,\qquadm_i' + \epsilon_{ijk} (u_j m_k + v_j n_k) =0,$$

 
 
 

after six of the twelve quantities $m_i$, $n_i$, $u_i$ and $v_i$ are eliminated using the constitutive relations

$$m_{i}=K_{i}(s)(u_{i}-\hat{u}_{i}(s))\,,\qquadn_{i}=A_{i}(s)(v_{i}-\hat{v}_{i}(s))\,.$$

(if the rod is inextensible and unshearable the unknowns $n_i$ must be three of the variables retained).

Resulting equations are nonlinear with quadratic nonlinearities (because the constitutive relations were assumed to be linear). When the constitutive laws are homogeneous (or autonomous) i.e. have no explicit $s$ dependence, it is possible to find simple (and not so simple) closed form solutions of the balance laws, and then with known strains to integrate the kinematic equations to find explicit expressions for the associated centerline and director frame. See Exercise Session 1. In general, both for non-homogeneous and homogeneous constitutive relations, it is necessary to solve the system numerically. Particularly because a typical rod problem involves two-point boundary conditions, i.e. some of the variables are given specified variables at $s=0$, and some at $s=L$ say. It should be recalled (or realized) that the numerical solution of initial value problems, i.e. problems where all the unknowns have specified values at one value of the independent variable $s$(usually $s=0$), for smooth equations such as the above, and on a bounded interval, is essentially trivial given todays computers and knowledge of algorithms. In contrast the numerical solution of two-point boundary value problems is still nontrivial (although of course the appropriate numerical methods are for the most part very well understood). The other important observation is that even though the balance laws, and the kinematic equations appear to naturally decouple, the boundary conditions typically do not decouple, so that there is effectively little simplification. For example perhaps the simplest boundary value problem for an elastic rod is the strut, where with a standard fixed basis $\i_i$ of $R^3$,the appropriate boundary conditions are

$${\bf r}(0)=0\quad\d_i(0)=\i_i, \quad{\bf m}(L)=0\quad {\bf n}(L)=-\lambda\i_3$$

These do not reduce to an initial value problem for the stress components $m_i$ and $n_i$ because to know the components $n_i(L)$given the boundary condition ${\bf n}(L)=-\lambda\i_3$ it would be necessary to know the relation between the directors $\d_i(L)$ and the basis vectors $\i_i$. Thus the problem is implicitly coupled. (We remark parenthetically that the boundary conditions

$${\bf r}(0)=0\quad\d_i(0)=\i_i, \quad{\bf m}(L)=0\quad {\bf n}(L)=-\lambda\d_3$$

do lead to a decoupled system for the stress components. This is a physically interesting, much studied problem with a so-called follower load. Its full solution behaviour is actually much more complicated than the strut.) Another physically interesting set of boundary conditions that we will study in our preparation for modelling DNA are

$${\bf r}(0)=0\quad {\bf r}(L)\cdot\i_1={\bf r}(L)\cdot\i_2=0, \quad {\bf r}(L)\cdot\i_3=\delta$$ $$\d_i(0)=\d_i(L)=\i_i.$$

 
 
 

which have no boundary conditions whatsoever on the stress components.

Note that for an inextensible rod, there can be no solution for$\delta \gt L$

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