## Abstract

In a recent paper by Bosi *et al*. (2014 *Proc. R. Soc. A* **470**, 20140232. (doi:10.1098/rspa.2014.0232)) an ingenious deformable arm scale was designed and developed. The scale's operation relies on the presence of Eshelby-like forces. In this paper, we gain new insight into the operation of the arm scale by using a material (configurational) force balance and by exploiting an analogy to a system of two pendula.

## 1. Introduction

In a recent paper, Bosi *et al*. [1] present a novel measuring scale featuring an elastic rod of length *ℓ* that is free to move inside a frictionless sleeve which is inclined at an angle *α* to the vertical. Weights *P*_{1} and *P*_{2} are attached to the respective ends of the lamella and, assuming that one of the weights and the slope of the tangent at the ends of the rod are known, the second weight can be determined from the relation
*et al*. [2,3] on Eshelby-like forces in continua.

The discussion in [1] on the mechanics of the scale features a detailed variational formulation based on Euler's elastica. Here, we find it illuminating to consider a complementary (and equivalent) formulation using the balance laws for the elastica supplemented by a material momentum balance law from O'Reilly [4]. The latter law was introduced recently into the literature on the mechanics of rods and it serves to help formulate a unified treatment for a variety of shock, adhesion and phase transformation problems in one-dimensional continua. The balance law includes the jump condition (see (2.7)_{1}) that is obtained from variational principles and, additionally, a local form (see (2.2)_{1}) of the material momentum balance law. With the help of (2.2)_{1} and (2.7)_{1}, we find that (1.1) is simply a consequence of conservation of a contact material force *C* and are easily able to generalize (1.1) to the cases where the self-weight of the rod is considered (see (4.2)) or applied moments act at the ends of the rod (see (6.4)). The conservations we discuss also illuminate connections between (1.1), (4.2) and (6.4) and the conservation law for a terminally loaded elastica discussed in Love's classic text [5], eqn. (7) in Sect. 262.

After collecting background on the balance laws in §2, we turn to solving for the material momentum fields in §3 and exploring the consequence of our analysis in §4. Where feasible, we retain as much of the notation and conventions of [1] as possible but include the self-weight of the rod. Our analysis shows how the reaction forces at the points of the rod where it enters and leaves the sleeve can be attributed to the contact material force C, how C can be easily varied by changing the axial force at the ends of the rod, and how C is conserved throughout the rod when the self-weight is neglected. We then turn in §5 to an analogue model for the elastica arm scale that features a pair of pendula, a perfectly plastic impact condition for one pendulum and a launch mechanism for the other pendulum that is prescribed by an energy conservation. The system of pendula also serves to illuminate a method to solve the complicated boundary-value problem associated with the elastica arm scale. We remark in closing this introduction that the method we discuss is also applicable to adhesion problems featuring the elastica.

## 2. Background on the balance laws

The elastica is a perfectly flexible, inextensible curve of length *ℓ* whose material points are defined using an arc-length coordinate *s*. As can be seen from figure 1, the position vector of a point on the curve is denoted by **r**, and the unit tangent to the curve can be defined by the derivative of **r** with respect to *s*:
*θ* represents the angle subtended by **r**′ with the **E**_{2} axis and the prime denotes the partial derivative with respect to *s*. The rod will be assumed to have a uniform mass density *ρ* per unit length of *s*.

We are now in a position to recall the standard balance laws for forces and moments for the elastica. In this paper, we supplement this pair of laws with a static material momentum balance law (or material force balance) from [4]. In the context of Euler's theory of the elastica, the local forms of the balances of material forces, forces and moments are
**n** is the contact force, *ρ***f** is the assigned body force per unit length, **m**=*M***E**_{3} is the bending moment, *b* is the assigned material body force and C is the contact material force. The force *b* is such that (2.2)_{1} is identically satisfied. In the sequel, we shall assume the elastica is homogeneous.

On denoting the strain energy function per unit length by *ρψ*=*ρψ*(*θ*′), the force C is prescribed as
*ρψ* and **m**, both of which feature the flexural rigidity EI:
*ρψ* can easily be expressed as a function of *M* and EI.

At a point of discontinuity *s*=*χ*, the following jump conditions hold:
_{2}, **F**_{χ} represents a singular force (source of linear momentum) acting at *s*=*χ* and *B*_{χ} represents a source of material momentum at *s*=*χ*. Examples of the former include point contact forces and examples of the latter include dry adhesion energy and energy dissipation phenomena that feature in models for falling chains.

In variational formulations for the equations of motion for the elastica, such as the one used in [1], a variation of an arc-length parameter typically leads to (2.7)_{1} and the material momentum balance law is not mentioned. Here, we are following Gurtin and Maugin, among others, in using this balance law (cf. [7–11] and references therein). Among others, the balance law often yields conserved quantities (integrals of motion) and provides added insight on the manner in which the upstream and downstream conditions contribute to the jump condition. For the elastica arm scale, we take this opportunity to emphasize that without the jump condition (2.7)_{1} provided by the material momentum balance law the problem would be indeterminate. An additional benefit of this balance law is that the local form (2.2)_{1} yields the conservation law that is central to our analysis.

## 3. The deformable arm scale

To analyse the arm scale, it is convenient to consider three segments. The left freely hanging section *s*∈[0,*a*_{1}), the right freely hanging section *s*∈(*a*_{2},*ℓ*] and the section inside the smooth guide of length *ℓ**:
*a*_{1} or *a*_{2}. The guide or sleeve is inclined at an angle *α* to the vertical.

### (a) The freely hanging segment *s*∈[0,*a*_{1})

The first section of the rod we consider extends from the free end at *s*=0 to the start of the guide at *s*=*a*_{1}. The boundary conditions on this section are
_{2,3}, we find that
*b*, it is straightforward to find the following energy conservation law from (2.2)_{1}:
**r** on the rod. Thus, in the case where gravity is ignored, the material force C is constant throughout this segment of the rod:

### (b) The freely hanging segment *s*∈(*a*_{2},*ℓ*]

The second segment of the rod of interest is terminally loaded at one end and extends to the sleeve at the other. For this portion of the rod, the boundary conditions are

### (c) The segment *s*∈[*a*_{1},*a*_{2}] of the rod in the smooth sleeve and the points of discontinuity

When the rod enters and exits the sleeve, it is straightforward to show that the slope of the rod is continuous:
*s*=*a*_{1} and *s*=*a*_{2}, we assume that singular forces, **F**_{a1} and **F**_{a2}, act on the rod. In addition for the segment of the rod in the frictionless guide, the assigned force acting on the rod can be decomposed into a gravitational force and a normal force λ(*s*)**E**_{1} (cf. figure 2).

The balance of linear momentum for the portion of the rod in the sleeve reads
*θ*=0 for this section of the rod, the balance of angular momentum reduces to
**n** is tangent to the rod. We can now revisit the balance of linear momentum and solve for the normal force acting on the rod:
*s* and the following material force *b* needs to be supplied to satisfy (2.2)_{1}:

At *s*=*a*_{1}, we assume a vanishing singular supply *B*_{a1}=0 along with a singular force **F**_{a1} acts. Thus,
**F**_{a1} is an unknown reaction force. Noting that C is continuous at *s*=*a*_{1}, we use the jump condition (3.17)_{2} to solve for **n**(*a*^{+}_{1})⋅**E**_{2}:
*s*=*a*_{2} closely parallels the case for *s*=*a*_{1}. Again we prescribe *B*_{a2}=0 and assume that a singular force **F**_{a2} acts at *s*=*a*_{2}. The jump condition associated with the material force balance yields continuity of C, and so we find
**n** experiences jumps at *s*=*a*_{1} and *s*=*a*_{2}. However, C does not and this continuity serves to determine the jump in **n**. Continuity of C and **r** implies that the conserved quantity *s*=*a*_{1} and *s*=*a*_{2}.

## 4. The material force C, the reactions **F**_{a1} and **F**_{a2}, and the operationof the arm scale

With the help of (3.5), (3.10) and (3.15), we are now in a position to examine the distributions of the material force C and the conserved quantity **r**′ from (2.1), *P*_{1}, *α*, the length of the sleeve, *ρg*, the difference in heights between the ends of the rod and measurements of *θ*(0) and *θ*(*ℓ*), *P*_{2} can be determined.

We can use the jump conditions **F**_{a1}+[[**n**]]_{a1}=**0** and **F**_{a2}+[[**n**]]_{a2}=**0** to determine the reaction forces
*M*^{2}(*a*^{+}_{2})/2EI and *M*^{2}(*a*^{−}_{1})/2EI, which are the axial components of **F**_{a1} and **F**_{a2}, are called Eshelby-like forces. Here, and as displayed in figure 2, we have shown how they manifest in reaction forces and how they can be explicitly attributed to the material force C.

## 5. A pair of pendula

To gain a different appreciation for the dramatic change in strain energy that occurs at *s*=*a*_{1} and *s*=*a*_{2} in the elastica arm scale, we ignore the self-weight of the elastica and consider a pair of pendula. The dimensionless time variable *τ* and important instances for the pendula are identified as follows:
*s* ranges from 0→*ℓ*, *τ* ranges from 0→1. Here, we are modifying the classic kinetic analogue for a single elastica that is discussed in [5] to incorporate the unusual boundary conditions at *s*=*a*_{1,2}.

One of the pendula is analogous to the section *ω*_{1}
*ω*_{2},
*ℓ*_{1} as Pendulum I. If the dimensional measure *t* of time is given by *t*=*βτ*, where *β* is a constant, then the lengths of Pendulum I and Pendulum II are

Using figure 5, we are now in a position to discuss the analogue model for the elastic arm scale. Consider Pendulum I and assume that it is released from rest with *ϕ*_{1}(0)=*θ*(0)+*α*. The pendulum falls as shown in figure 5*b* and eventually collides with a surface in a perfectly plastic collision (cf. figure 5*c*) wherein it loses all its kinetic energy. After a period (*a*_{2}−*a*_{1})/*ℓ* of no motion, Pendulum II, which is at rest inclined at an angle *ϕ*_{2}=*α* to the vertical, is launched with a speed (d*ϕ*_{2}/d*τ*)(*τ*^{+}_{2})>0 (cf. figure 5*d*). The resulting motion of Pendulum II persists until *τ*=1 where it eventually comes to a state of instantaneous rest (cf. figure 5*e*,*f*). The counterpart of the material force C in this problem is the total energy of the individual pendula:
*ϕ*_{2}/d*τ*)(*τ*^{+}_{2})

The kinetic analogy also sheds light on solving the boundary-value problem associated with the elastic arm (whose self-weight is ignored). For a given loading *P*_{1} and *P*_{2} on the elastica arm scale and a given length *a*_{2}−*a*_{1} of sleeve, *ω*_{1} and *ω*_{2} can be computed, and the phase portraits for the both pendula can be constructed (cf. figure 6). Now the solution shown in figure 7 starts with a chosen *ϕ*_{1}(0). This value of *ϕ*_{1}(0) then determines the time of flight *τ*_{1}=*a*_{1}/*ℓ* to the impact event. This time of flight then prescribes the allowable time of flight 1−*a*_{2}/*ℓ* for Pendulum II. The initial speed (d*ϕ*_{2}/d*τ*)(*τ*^{+}_{2})>0 is prescribed by (5.10) and then it remains to verify that *ϕ*_{2} must transition to a state of instantaneous rest in 1−*a*_{2}/*ℓ* units of dimensionless time at a value of *ϕ*_{2} given by (5.11). Typically, for a given loading *P*_{1} and *P*_{2} on the elastica scale arm and a given length *a*_{1}−*a*_{2} of sleeve, it is necessary to iterate the values of *ϕ*_{1}(0) and *a*_{1} so as to find a solution that satisfies (5.10) and (5.11) given the time of flight 1−*a*_{2}/*ℓ* for Pendulum II.

An example of a solution for given prescribed values of *ω*_{1}, *ω*_{2} and *a*_{2}−*a*_{1} is shown in figure 7*a*. The configuration of the elastic rod corresponding to this solution is constructed in figure 7*b* after integrating (2.1) to determine the position vector of the centreline **r**. In computing the solution numerically, we found that
*τ*=*τ*_{1}=*a*_{1}/*ℓ*, an energy
*τ*=*τ*_{2}=*a*_{2}/*ℓ*, an energy
*c* confirm conservation of energy during the pendulum motions while the pendula are moving—although the analogy we have presented breaks down when the pendula are stationary during the time interval *τ*∈(*τ*_{1},*τ*_{2}).

When one of the terminal loads is zero, then it is possible to use a single pendulum to develop an analogue model for the deformable scale. In this case, say if *P*_{1}=0, then (5.11) implies that *θ*(*ℓ*)+*α*=90°. For Pendulum II, we have *e*_{2}=0, *ϕ*_{2}(1−*a*_{2}/*ℓ*)=*α* and *ϕ*_{2}(1)=90°. Thus, it is possible to quickly arrive at a closed form expression for *a*_{2}/*ℓ* from (5.9)_{2}:^{1}
*K*(*x*) is a complete elliptic integral of the first kind, and *F*(*x*,*m*) is an elliptic integral of the first kind. The solutions *a*_{2}/*ℓ* to (5.15) are shown in figure 8. Observe that for *P*_{2}*ℓ*^{2}/EI>*K*^{2}(1/2), the scale can be operated at all values of *α*∈(0,90°). In addition, the more vertical the sleeve (i.e. the smaller the value of *α*), the more sensitive the measurement *a*_{2}/*ℓ* is to changes in *ℓ*(∂*f*/∂*P*_{2}) that can be calculated from (5.15). Finally, we note that for 0<*P*_{2}*ℓ*^{2}/EI<*K*^{2}(1/2), the range of values of *α* for which the scale operates is limited.

## 6. Conclusion

We have examined Bosi *et al*.'s [1] recently developed elastica arm scale using a material force balance from O'Reilly [4]. This balance law features a contact material force C which is the one-dimensional counterpart of Eshelby's energy-momentum tensor and is a necessary complement to the balances of linear and angular momentum. The balance law quickly yields the conservation of an energy **F**_{a1} and **F**_{a2}, and we were also able to use the conservation of

The effects of terminal moments on the governing equation (4.2) for the arm scale are also easily inferred using the conservation of **M**_{1}=*M*_{1}**E**_{3} is applied at *s*=0 and another moment **M**_{2}=*M*_{2}**E**_{3} acts at *s*=*ℓ* of the arm scale. These moments introduce bending strains at the ends of the rod so that the boundary conditions (3.2)_{2} and (3.7)_{2} now become
*P*_{1}, *P*_{2}, *M*_{1} and *M*_{2}. That is, the scale can be used to measure an applied moment or an applied force.

The analysis we have presented shows that C is conserved throughout the rod when the self-weight is neglected. For the pair of hanging segments of the rod this conservation is equivalent to a classical conservation law that can be found in [5]. However, how the guide affects this conservation requires the use of (2.7) which is not found in classic works. The conservation of C was used in conjunction with the kinetic analogy to develop a two pendulum analogue model for the elastica arm scale. This facilitates a transparent discussion of the boundary-value problem for the determination of the shape of the deformed elastica featured in the arm scale. The corresponding kinetic analogy for the case where terminal moments are present at the end of the arm scale would feature a non-zero initial velocity (d*ϕ*_{1}/d*τ*)(0) and a non-zero final velocity (d*ϕ*_{2}/d*τ*)(1).

## Acknowledgements

I am grateful to Carmel Majidi for his helpful comments on an earlier draft of the paper and to two anonymous reviewers for their constructive comments which served to improve the final version.

- Received October 13, 2014.
- Accepted December 2, 2014.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.