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.
In a recent paper, Bosi et al.  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 P1 and P2 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 1.1 The device, which is sketched in figure 1, is referred to as an ‘elastica arm scale’ and the inspiration for its design can be traced to the papers by Bigoni et al. [2,3] on Eshelby-like forces in continua.
The discussion in  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 . 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 , 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  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: 2.1 Here, θ represents the angle subtended by r′ with the E2 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 . In the context of Euler's theory of the elastica, the local forms of the balances of material forces, forces and moments are 2.2 In these local forms, n is the contact force, ρf is the assigned body force per unit length, m=ME3 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 2.3 and the assigned body force for the homogeneous rod is 2.4 As discussed in , the prescription for C is related to Eshelby's prescription in  of the energy momentum tensor for continua. For the elastica, we have the following classic constitutive relations for ρψ and m, both of which feature the flexural rigidity EI: 2.5 where 2.6 Note that ρψ can easily be expressed as a function of M and EI.
At a point of discontinuity s=χ, the following jump conditions hold: 2.7 The first of these conditions is associated with the balance of material forces and the second is associated with a force balance. The corresponding jump condition associated with the moment balance does not contribute significantly to the analysis and is not mentioned further (cf.  for further details). In (2.7)2, Fχ represents a singular force (source of linear momentum) acting at s=χ and Bχ 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 , 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,a1), the right freely hanging section s∈(a2,ℓ] and the section inside the smooth guide of length ℓ*: 3.1 Thus, in determining the equilibrium configuration of the deformed rod, it suffices to determine either a1 or a2. The guide or sleeve is inclined at an angle α to the vertical.
(a) The freely hanging segment s∈[0,a1)
The first section of the rod we consider extends from the free end at s=0 to the start of the guide at s=a1. The boundary conditions on this section are 3.2 where the unit vector , which points downward, has the representation 3.3 With the help of the balance laws for linear and angular momentum (2.2)2,3, we find that 3.4 From one of these results, expressions for the material force C can be computed from (2.3) and (3.2): 3.5 where . Because the rod is homogeneous, after computing b, it is straightforward to find the following energy conservation law from (2.2)1: 3.6 We note that is the gravitational potential energy for the material point located at r on the rod. Thus, in the case where gravity is ignored, the material force C is constant throughout this segment of the rod: . This conservation is equivalent to the conservation law presented in Love [5, eqn. (7) in Sect. 262].
(b) The freely hanging segment s∈(a2,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 3.7 We parallel the developments in the previous section and compute that 3.8 From these results and the balance of material forces, we again find a conservation law: 3.9 For this segment of the rod, the material force has the representations 3.10 where . As in the previous section, if gravity is ignored then C is conserved along this segment of the rod where .
(c) The segment s∈[a1,a2] 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: 3.11 However, these results in no way imply that the curvature of the rod is continuous at these points. At s=a1 and s=a2, we assume that singular forces, Fa1 and Fa2, 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)E1 (cf. figure 2).
The balance of linear momentum for the portion of the rod in the sleeve reads 3.12 As θ=0 for this section of the rod, the balance of angular momentum reduces to 3.13 That is, 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: 3.14 Furthermore, the contact material force C is simply 3.15 In contrast to the other two segments of the rod, C decreases linearly with increasing s and the following material force b needs to be supplied to satisfy (2.2)1: 3.16 Paralleling the developments in the previous segments of the rod, we again find that the energy is conserved for this segment of the rod.
At s=a1, we assume a vanishing singular supply Ba1=0 along with a singular force Fa1 acts. Thus, 3.17 It is important to observe here that Fa1 is an unknown reaction force. Noting that C is continuous at s=a1, we use the jump condition (3.17)2 to solve for n(a+1)⋅E2: 3.18 The analysis at the exit point s=a2 closely parallels the case for s=a1. Again we prescribe Ba2=0 and assume that a singular force Fa2 acts at s=a2. The jump condition associated with the material force balance yields continuity of C, and so we find 3.19 It should be clear from (3.18) and (3.19) that the axial component of the force n experiences jumps at s=a1 and s=a2. However, C does not and this continuity serves to determine the jump in n. Continuity of C and r implies that the conserved quantity is continuous at s=a1 and s=a2.
4. The material force C, the reactions Fa1 and Fa2, 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 in the rod. A summary of the results is presented in figure 3. The conservation of provides the equation governing the operation of the arm scale. To see this, we use the expressions for C mentioned earlier to find that 4.1 On substituting for r′ from (2.1), , and , it is straightforward to show that 4.2 This equation is the extension of (1.1) when the self-weight of the rod is included and is the operating principle for the arm scale: given P1, 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), P2 can be determined.
We can use the jump conditions Fa1+[[n]]a1=0 and Fa2+[[n]]a2=0 to determine the reaction forces 4.3 Both of these forces are related to the bending moment (and bending strain) in the rod. In , the terms M2(a+2)/2EI and M2(a−1)/2EI, which are the axial components of Fa1 and Fa2, 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=a1 and s=a2 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: 5.1 Observe that as 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  to incorporate the unusual boundary conditions at s=a1,2.
One of the pendula is analogous to the section of the elastica. In the absence of self-weight, the equation governing this section is (from (3.4)) 5.2 Thus, we can consider the motion of an analogous simple pendulum which oscillates about its downward equilibrium with a dimensionless frequency ω1 5.3 The equations of motion of this pendulum, which is shown in figure 4, are 5.4 The other pendulum models the segment of rod . For this segment of the rod, we have (from (3.8)) 5.5 Thus, we can consider the motion of an analogous simple pendulum which oscillates about its downward equilibrium with a dimensionless frequency ω2, 5.6 and whose equations of motion are described by 5.7 We refer to this pendulum as Pendulum II and its counterpart of length ℓ1 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 5.8
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 ϕ1(0)=θ(0)+α. The pendulum falls as shown in figure 5b and eventually collides with a surface in a perfectly plastic collision (cf. figure 5c) wherein it loses all its kinetic energy. After a period (a2−a1)/ℓ of no motion, Pendulum II, which is at rest inclined at an angle ϕ2=α to the vertical, is launched with a speed (dϕ2/dτ)(τ+2)>0 (cf. figure 5d). The resulting motion of Pendulum II persists until τ=1 where it eventually comes to a state of instantaneous rest (cf. figure 5e,f). The counterpart of the material force C in this problem is the total energy of the individual pendula: 5.9 We assume that the value of the total energies of both pendula are identical when they are in motion. This assumption prescribes (dϕ2/dτ)(τ+2) 5.10 In addition, the equality of the energies also implies that 5.11 This identity is the counterpart to (1.1) for the arm scale.
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 P1 and P2 on the elastica arm scale and a given length a2−a1 of sleeve, ω1 and ω2 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 ϕ1(0). This value of ϕ1(0) then determines the time of flight τ1=a1/ℓ to the impact event. This time of flight then prescribes the allowable time of flight 1−a2/ℓ for Pendulum II. The initial speed (dϕ2/dτ)(τ+2)>0 is prescribed by (5.10) and then it remains to verify that ϕ2 must transition to a state of instantaneous rest in 1−a2/ℓ units of dimensionless time at a value of ϕ2 given by (5.11). Typically, for a given loading P1 and P2 on the elastica scale arm and a given length a1−a2 of sleeve, it is necessary to iterate the values of ϕ1(0) and a1 so as to find a solution that satisfies (5.10) and (5.11) given the time of flight 1−a2/ℓ for Pendulum II.
An example of a solution for given prescribed values of ω1, ω2 and a2−a1 is shown in figure 7a. The configuration of the elastic rod corresponding to this solution is constructed in figure 7b after integrating (2.1) to determine the position vector of the centreline r. In computing the solution numerically, we found that 5.12 For the impact at τ=τ1=a1/ℓ, an energy 5.13 is lost by Pendulum I. However for the launch at τ=τ2=a2/ℓ, an energy 5.14 is transferred to Pendulum II. The plots of the total energies of the pendula shown in figure 7c 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∈(τ1,τ2).
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 P1=0, then (5.11) implies that θ(ℓ)+α=90°. For Pendulum II, we have e2=0, ϕ2(1−a2/ℓ)=α and ϕ2(1)=90°. Thus, it is possible to quickly arrive at a closed form expression for a2/ℓ from (5.9)2:1 5.15 where 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 a2/ℓ to (5.15) are shown in figure 8. Observe that for P2ℓ2/EI>K2(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 a2/ℓ is to changes in . This result is evident from figure 8 or ℓ(∂f/∂P2) that can be calculated from (5.15). Finally, we note that for 0<P2ℓ2/EI<K2(1/2), the range of values of α for which the scale operates is limited.
We have examined Bosi et al.'s  recently developed elastica arm scale using a material force balance from O'Reilly . 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 , which has several representations 6.1 Our analysis has shown how continuity of C enables a solution of the tangential components of the reaction forces Fa1 and Fa2, and we were also able to use the conservation of to establish the governing equation (4.2) for the arm scale when the self-weight of the rod was included.
The effects of terminal moments on the governing equation (4.2) for the arm scale are also easily inferred using the conservation of and this enables the scale to be used as a torque measurement device. To elaborate, suppose a moment M1=M1E3 is applied at s=0 and another moment M2=M2E3 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 6.2 However, it is still straightforward to show that 6.3 is conserved. Indeed, evaluating at the ends of the rod and setting the two values equal, we find that (4.2) now would read 6.4 Thus, the scale can be used to determine one of the four possible unknowns from the set P1, P2, M1 and M2. 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 . 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 (dϕ1/dτ)(0) and a non-zero final velocity (dϕ2/dτ)(1).
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.