Rheological basis of skeletal muscle work loops
RRheology of tunable materials
Khoi D. Nguyen and Madhusudhan Venkadesan ∗ Department of Mechanical Engineering & Materials Science, Yale University, New Haven, CT, USA
Materials with tunable mechanical properties are ubiquitous in biology and engineering, suchas muscle, the cytoskeleton, soft robotic actuators, and magnetorheological fluids. Theirfunctionality arises through an external signal that modulates intrinsic mechanical properties, likeneural spikes in muscle. However, current rheological methods assume approximately invariant me-chanical properties during measurement, and thus cannot be used for characterizing the rheologyof tunable materials. Here, we develop a geometric framework for characterizing tunable materialsby combining classical oscillatory rheology that uses Lissajous force-length loops to characterizenon-tunable materials, with the technique of work loops that is used to study muscle’s workoutput under varying stimulation. We derive the force-length loop under varying stimulationby splicing Lissajous loops obtained under constant stimulation. This splicing approach capturesthe force-length loop shapes in muscle that are not seen in non-tunable materials, and yields anondimensional parametrization for the space of possible responses of tunable materials.
Keywords: oscillatory rheology, Lissajous figures, work loops, muscle, tunable rheology
Rheology, or how materials deform under forces,is a central consideration for the study of biolog-ical and engineering materials. A vast number ofthese materials may be considered tunable becausetheir functionality arises from modulation of theirrheological properties by an external stimulus. Anextensive toolkit is available for characterizing ma-terials whose rheology is invariant during the mea-surement duration, but no similar methods areavailable for tunable materials because their me-chanical properties are actively modulated. Thetunability often arises from energy-consuming mi-croscopic processes, and many tunable materialsfunction as actuators. For example, a stimulatedmuscle consumes metabolic energy to provide actu-ation, but is also considerably stiffer than a relaxedmuscle. Industrial materials that are responsive tolight, magnetic fields, and electric fields or heat also possess dual-capability as actuators and rheo-logical mechanical interfaces. Considerable atten-tion has been paid to the actuation capabilities ofthese tunable materials, but little is known abouthow to characterize their rheology under varyingstimulation. Oscillatory rheology
The technique of oscillatory rheology is widelyused to characterize materials whose properties areinvariant during measurement.
It generalizes in-tuitive notions of stiffness and damping by char-acterizing the material’s force (or stress) responseto sinusoidal length (or strain) oscillations of dif-ferent frequencies and amplitudes. The Lissajousfigure of force versus length provide a graphical ∗ Email for correspondence: [email protected] signature of the material’s rheology. The figuresare approximately elliptic loops for small amplitudeoscillations, but more complicated and non-ellipticfor large-amplitude oscillations (figure 1a).The response to small-amplitude oscillations aremodeled and understood using the complex modu-lus E ( ω ) that depends on the frequency ω of thelength oscillation. The in-phase and out-of-phasecomponents of the measured force divided by the im-posed length amplitude ∆ L yields the storage mod-ulus E ( ω ) and loss modulus E ( ω ), or the complexmodulus E ( ω ) = E ( ω ) + i E ( ω ). The Lissajousfigure at a specific frequency is the sum of a linewith slope E and an ellipse of area π ∆ L E , result-ing in a vertically sheared ellipse (figure 1b, Supple-ment § S1). Non-elliptic shapes for large amplitudeoscillations are understood as length-dependence ofthe moduli or as higher order terms of a Fourierexpansion whose linear terms are E and E (Sup-plement § S2).
Although current approaches can accommodate avast number of complex materials, tunable materialssubject to varying stimulation are not part of theframework. For example, oscillatory tests have beenused to measure the complex modulus of tunablematerials, but only when the external stimulus washeld constant.
As a result, the rheology ofmaterials such as muscle under varying stimulationcannot be explained using small or large-amplitudeoscillatory rheology.
Muscle work loops
The work loop technique is similar to oscillatoryrheological testing and was developed to study mus-cle under conditions that mimic periodic locomo-tory movements.
Lissajous loops are generatedby measuring forces during length oscillations, but a r X i v : . [ c ond - m a t . s o f t ] M a y a c d Katydid wing muscle Rabbit latissimus dorsi
Cockroach leg extensors 178 and 179--+ = + - -Tunablematerial Load c e ll Stimuli LengthoscillationsLinear spring b F o r c e Lengthsmall amplitudelarge amplitudePedal mucus of a terrestrial slug +-+ -+ +Ideal force generator+Non-tunablematerial
Load c e ll Lengthoscillations L E E Linear damper-- F o r c e Length F o r c e Length F o r c e Length F o r c e Length F o r c e Length
FIG. 1.
Force-length loops in non-tunable andtunable materials. a , Oscillatory rheology of pedalmucus of a terrestial slug (
Limax maximus ) for smalland large amplitudes. Loops are scaled and shifted forcomparison. b , The force-length loop in the linear set-ting is the sum of a sloped line (elastic component) anda clockwise horizontal ellipse (viscous component). c ,Work loops of muscle, measured under sinusoidal lengthoscillation and phasic external stimulation, for a wingmuscle of a katydid ( Neoconocephalus triops ) , rabbit latissimus dorsi , and cockroach leg extensor muscles178 and 179. The yellow dots and yellow shading in-dicate discrete neural impulses and continuous electricalstimulation, respectively. d , Hypothetical loops for ide-alized tunable materials. Positive and negative mechan-ical work output are shaded green and red, respectively. unlike for non-tunable materials an external stim-ulus is phasically applied and removed during thelength oscillations (figure 1c, stimulation depictedas yellow dots or shading). Typically, electrical cur-rent is used to directly stimulate the muscle or thenerve supply to the muscle, which releases intracellu-lar calcium ions that activate binding sites on actinand leads to muscle contraction. Work loop mea-surements are performed at a select few frequencies,unlike the frequency sweeps in rheological testing,but produce loops that may exhibit multiple self-intersections and bear little resemblance to measure-ments on linear or nonlinear non-tunable materials(figure 1c). Therefore, we turn to the extensive lit-erature on muscle work loop as the inspiration forour framework.
Sign convention:
Following the muscle work loopliterature, increasing length is positive but positiveforces imply the opposite sense, namely contraction.Therefore, positive work or a counter-clockwise loop is when the material produces energy to do workon the environment, and negative work or clockwiseloops occur when the material dissipates energy inevery cycle. (‘+’ and ‘ − ’ regions in figure 1). Activation and external stimulus:
The experimenterapplies a “stimulus” but “activation” is an internalvariable that more directly affects rheology. For ex-ample, the stimulus is external temperature and ac-tivation is the material’s internal temperature, orthe stimulus is an external current or neural spikefrequency and activation is the intracellular calciumconcentration.
Idealized tunable materials
Even in simple models of idealized tunable ma-terials, phasic activation can result in complicatedloop shapes, including reversal of the loop direc-tion and self-intersections (figure 1d). For example,loops constructed by phasically varying the stiffnessand the neutral length of a Hookean spring, or thedamping of a linear viscous damper can exhibit self-intersections and loop reversals (figure 1d). Fur-thermore, many tunable materials such as musclecan also actively contract and produce forces. So,in addition to the elastic and viscous components,we consider an idealized force generator that main-tains a constant force as a function of its activation.Not surprisingly, the ideal force generator can ex-hibit work-producing counter-clockwise loops.
Construction by splicing Lissajous figures
We model the effect of varying activation asswitching between different rheological materialsand examine the shapes of Lissajous loops that canarise under this model. This approach is consistentwith materials such as muscle, which are like tra-ditional rheological materials under constant excita-tion and their Lissajous loops do not exhibit self-intersections or other features that only emerge un-der time-varying activation. To illustrate this, con-sider a tunable material that behaves like two differ-ent linear materials at different activations: greaterstorage modulus, loss modulus, and contractile forceat a highly activated state A compared with a lowor deactivated state D . Graphically, these manifestas a more inclined, wider, and vertically offset el-lipse for A compared with D (figure 2a). In thismodel, periodically changing the activation from D to A and back to D would result in switching be-tween the two ellipses. Additionally, assume thatthe time taken to switch and settle into the newstate is negligible compared to the period of the os-cillation. Then, switching the activation results in aspliced loop with two self-intersections and net pos-itive work although each of the A and D ellipses areindividually dissipative (figure 2a).We can generalize the method of constructinga spliced response to nonlinear materials by usingknown or empirical measurements of the force re-sponse at constant activations (figure 2b,d). Whenthe material is phasically activated during φ A ≤ ωt ≤ φ D , we construct the force response F ( t ) bysplicing the forces F D ( t ) and F A ( t ) correspondingto the low and high activations, respectively, andexpressed as, F ( t ) = ( F A ( t ) for φ A ≤ ωt ≤ φ D ,F D ( t ) otherwise. (1)Splicing is graphically viewed as jumping fromone loop to the other at the instant that activationchanges (figure 2a). Generically, there are two tar-get points to jump to, but only one of them willcontinue to traverse the loop in a direction that isconsistent with its loss modulus, thus fully definingthe spliced Lissajous loop. Recall that a positive lossmodulus implies a dissipative material and clockwiseloops, whereas a negative loss modulus would implyan energy-producing active material with counter-clockwise loops. Although passive materials only ex-hibit clockwise looks, active materials such as mus-cle can have negative loss and storage moduli. However, even with loops arising from passive rhe-ologies, splicing can produce complicated Lissajousshapes with self-intersections and counter-clockwiseloops. Importantly, the geometric underpinning ofthe splicing method enables generalization to nonlin-ear rheology where the basis loops have non-ellipticshapes (figure 2b,d).The splicing construction assumes that the timefor switching between the two rheologies and settlinginto the new periodic response is negligible com-pared to the time period of the oscillation. Thus,the spliced loop may be understood as a singular pe-riodic orbit of a piecewise smooth dynamical systemwith two switching planes (figure 2c,d). The switch-ing planes P D and P A are defined by the phases φ D and φ A when the activation is changed from highto low or vice versa . At constant activation D or A ,the dynamics of the material when subjected to peri-odic length oscillations are governed by the compos-ite functions ( R D ◦ R D ) and ( R A ◦ R A ), respectively(figure 2c), which map initial conditions on the plane P A to the plane P D and back onto P A again. Theoscillatory responses at a constant activation, D or A , correspond to stable periodic orbits of the twodynamical systems. The splicing of those two pe-riodic orbits by instantaneously switching betweenthem at the planes P D and P A (figure 2c,d) yieldsa periodic solution to the spliced dynamical system ca F o r c e A - e lli p s e Length D - e l l i p s e imposed motion L e n g t h V e l o c i t y F o r c e db Leng t h time A c t i v a t i on φ A φ D F o r c e F A ( t ) F D ( t ) F ( t ) L ( t ) a ( t ) + − − J A J D R A R D R D R A P A P A P D P D P A P D FIG. 2. a, Splicing the A-ellipse and D-ellipse resultsin loops constructed from linear rheologies. b, Splicinggeneralizes to nonlinear rheologies, illustrated here usingnumerical simulations of a nonlinear muscle model, byjumping between known force responses at constant ac-tivation. c,d, Geometric viewpoint of splicing as a piece-wise smooth dynamic system with jumps in rheology attwo phases. ( J D ◦ R A ◦ J A ◦ R D ). The instantaneous jumps J A and J D (figure 2c) between slowly varying trajec-tories resembles approaches from geometric singularperturbation theory for constructing relaxation os-cillations in multiple-timescale systems. Our sin-gular construction of the spliced periodic orbit is in-formative of a real material’s response when the re-laxation time upon changing the activation is shorterthan the time period of the length oscillation. Welater assess the robustness of this assumption withrespect to muscle work loop data.
Shape-space of spliced loops
Under the hypothesis that the singular loop un-derlies the response of real materials, we proceed toinvestigate the parameters that affect the shape ofthe spliced loop. For this, we consider the linear set-ting where the material under constant activation isfully described by a constant force term, and thestorage and loss moduli. The nonlinear setting withstorage and loss moduli of higher harmonics is a di-rect extension of the linear case (Supplement § S2).The oscillatory force response F ( t ) of a tunable ma-terial that is phasically activated between A and D ,is given in terms of the constant force terms F A and F D that represent active contraction, the stor-age moduli E A and E D , the loss moduli E A and E D ,and length amplitude ∆ L , according to, F ( t ) = F A + ∆ L ( E A sin ωt + E A cos ωt ) , for ωt ∈ [ φ A , φ D ] ,F D + ∆ L ( E D sin ωt + E D cos ωt ) , otherwise . (2)Novel Lissajous loops emerge from phasic changesin rheology between the D and A states. Therefore,it is the difference between the two rheologies thataffects the shape (figure 3a). We subtract the deac-tivated force response F D ( t ) from F ( t ), and derivenondimensional expressions using the length scale∆ L , force scale ( F A − F D ), and timescale 1 /ω .In terms of the nondimensional phase φ = ωt , theactivated nondimensional force response f A ( φ ) =( F A ( t ) − F D ( t )) / ( F A − F D ) is expressed using thedifference in moduli ∆ e = ∆ L ( E A − E D ) / ( F A − F D ) and ∆ e D = ∆ L ( E A − E D ) / ( F A − F D ) as, f A ( φ ) = 1 + ∆ e sin φ + ∆ e cos φ. (3)We recast the modulus parameters ∆ e and ∆ e in terms of the nondimensional work w and w thatis performed by the material as a result of switch-ing the storage and loss moduli respectively, and anadditional work w as a result of switching the idealforce component between F D and F A . Thus, thenondimensional response f ( φ ) is, f ( φ ) = F ( t ) − F D ( t ) F A − F D = w w ‘ A + ‘ D sin φ + w w ‘ D − ‘ A β cos φ,φ ∈ [ φ A , φ D ]0 , otherwise , (4)where the shape factor β = R φ D φ A cos φ dφ representsthe partial area of the ellipse traversed in switchingthe rheology, and ‘ A = ‘ ( φ A ) and ‘ D = ‘ ( φ D ) arethe lengths at activation and deactivation, respec-tively. The nondimensional work w for the elasticmodulus, w for the loss modulus, and w for theideal force term are, w = − ∆ e Z ‘ D ‘ A ‘d‘ = 12 ∆ e ( ‘ A − ‘ D ) , (5) w = − ∆ e Z φ D φ A cos φdφ = − ∆ e β (6) w = − Z ‘ D ‘ A d‘ = ‘ A − ‘ D , (7) and the nondimensional and dimensional net workper loop are respectively, w n = I f d‘ = w + w + w , and (8) W n = w n ∆ L ( F A − F D ) − πE D ∆ L . (9) -1.0 1.0 - . . w /w w / w a b F ( t ) L ( t )( φ ) f ( φ ) dimensionalnondimensional w n = . w n = . w n = − . FIG. 3.
Shape-space of spliced loops. a,
Exampleto illustrate the influence of nondimensionalization onloop shape. b, Two work ratios define the shape-spaceof spliced loops. The loops are scaled to have the samewidth and height and shown here for one activation pro-tocol ( φ A = π/ φ D = 5 π/ w n are also shown. Expressing the force response, and thus the loopshape, using work ratios aids in the interpretationof the response under varying activation in termsof the material’s functionality as a force actuator,elastic body, or viscous damper (figure 3b). It de-lineates the activation-dependent material proper-ties from the effect of the stimulation protocol thatare captured by the parameters ( ‘ D + ‘ A ) /
2, and( ‘ D − ‘ A ) /β . Furthermore, multiple loop shapes havethe same net work because it is a sum of multiplework contributions ( w n contour lines in figure 3b).Thus, our characterization of shape and work lendquantitative interpretation to the long-held hypoth-esis in muscle work loops that the shape of the loopgoverns how the net work is done and thus impor-tant for the functional use of the muscle by theanimal. Muscle work loops and slow activation
The spliced loop, when fit to measured work loopdata generate experimentally testable predictions,namely the basis loop shapes that would give riseto the measured loop under phasic stimulation. Toillustrate such an experiment, the steady periodicresponse of muscle is first measured at two differentactivation levels that would correspond to the leastand highest activation experienced by the muscleduring a work loop measurement. A hypothesizedwork loop is constructed by splicing the basis loopsand compared against the measured loop in a phasicactivation experiment. However, muscle does not ac-tivate instantaneously when stimulated, and neitherdo its rheological properties or force output changeinstantaneously upon activation. In many settings,the release of calcium ions and development of forcetakes around 10–50 ms but the time period of oscilla-tion during animal locomotion is far slower, around100–200 ms. But this may not always be the casedepending upon the muscle and animal, especiallyfor invertebrate muscles.To examine the influence of delays between an ex-ternal stimulation u ( t ) and activation a ( t ), we per-formed a numerical study for four specific measuredwork loops with considerably different shapes. Ourchoice of muscles span invertebrates and vertebrates.The question is whether simple basis loops can giverise to the measured loop shapes using our splic-ing approach. For this preliminary exercise, the ba-sis loops were found by visual examination and notexact parameter estimation. The stimulation u ( t )is modeled as a phasic pulse that is equal to unitywhen the muscle is receiving a neural input and zerootherwise. Following standard practice in the lit-erature, our model enforces a first-order lag with atime-constant τ a between the stimulation u ( t ) andactivation a ( t ). To accommodate smoothly vary-ing activation we use a simple model of linear inter-polation to find the force at intermediate activationsbetween the D and A ellipses. The force f ( t ), as afunction of the activation a ( t ) that develops due tothe phasic stimulation u ( t ) and activation time con-stant τ a is, f ( t ) = a ( t ) f A ( t ) + (1 − a ( t )) f D ( t ) , (10) τ a da ( t ) dt + a ( t ) = u ( t ) . (11)We find that many features of the measured loopshapes are captured by simple, elliptic basis loops(figure 4). Importantly, essential features of the loopshape are robust to finite lags in activation, and theactivation time-constant τ a needed for fitting themeasured data are much smaller then the oscilla-tion time-period T . Although future experimentsare necessary to test the veracity of these hypoth-esized basis loops, the splicing approach provides asingle framework in which to view vastly differentloop shapes and muscle types, and generate testablepredictions. For example, the activated basis loopfor the cockroach leg muscle 178 requires a negativestorage modulus. This may indicate failure of thesplicing approach, in which case that muscle may beendowed with new activation phenomena that arenot captured by treating activation as a simple in-terpolation variable. Alternatively, it may be a valid a b c Rabbit latissimus dorsi
Cockroach legextensor 178 Cockroach legextensor 179 d + -+ - - τ a = 0 . Tτ a = 0 . Tτ a = 0 . T τ a = 0 . Tτ a = 0 . Tτ a = 0 . Tτ a = 0 . Tτ a = 0 . Tτ a = 0 . Tτ a = 0 . Tτ a = 0 . Tτ a = 0 . T + Katydid wing muscle ww = 0 . ww = − . ww = 0 . ww = 0 . ww = − . ww = − . ww = − . ww = − . FIG. 4.
Muscle work loops versus spliced loopswith smooth activation dynamics.
The activationtime constant τ a is varied as a fraction of loop’s period T . Work ratios are calculated for the singular loop with τ a = 0. Yellow dots represent neural spikes acting as ex-ternal stimuli and bold yellow lines represent continuousstimulation. a, Wing muscle of a katydid. b, Rabbit latissimus dorsi muscle. c-d, Leg extensor muscles 178and 179 of a cockroach. hypothesis because some insect muscles indeed havenegative loss modulus when activated, and similarfeatures may arise for the storage modulus. In thismanner, the splicing approach enables the applica-tion of rheological modeling to generate predictionsfor tunable materials that are subjected to varyingactivation. Success of the splicing approach providesa means to understand the effect of tuning the rhe-ology, and its failure may point to novel materialproperties or activation dynamics. Conclusion
We have shown how oscillatory rheological mea-surements at constant activation can be used to con-struct a force-length Lissajous loop under changingrheological conditions in tunable materials. We thenderived a deconstruction of possible loop shapes interms of work ratios associated with the elastic, vis-cous, and active force production components of thematerial. This approach yields force versus lengthloops under varying stimulation that exhibit self-intersections and loop reversals although the be-havior under constant stimulation follows the rulesof classical rheology. In this manner, our frame-work incorporates current understanding from rhe-ology for characterizing tunable materials and gen-erates testable predictions whose failure may pointto potentially novel material properties. Finally, thespace of loop shapes provides a tool for the designof stimulation protocols that enable the same tun-able material to variably perform as motors, springs,dampers, or combinations thereof.
Acknowledgments:
Funding support from theRaymond and Beverly Sackler Institute for Biologi-cal, Physical and Engineering Sciences at Yale andNIH training grant T32EB019941.
Author Contributions:
K.D.N. and M.V. con-ceived the research, designed the research, analyzedand interpreted the results, and wrote the paper.
Competing interests:
The authors declare nocompeting interests. [1] Sawicki, G. S., Robertson, B. D., Azizi, E. &Roberts, T. J. Timing matters: tuning the mechan-ics of a muscle–tendon unit by adjusting stimulationphase during cyclic contractions.
Journal of Exper-imental Biology , 3150–3159 (2015).[2] Nishikawa, K. C., Monroy, J. A. & Tahir, U. Musclefunction from organisms to molecules.
Integrativeand Comparative Biology , 194–206 (2018).[3] Nguyen, K. D., Sharma, N. & Venkadesan, M.Active viscoelasticity of sarcomeres. Frontiers inRobotics and AI (2018).[4] Gardel, M. L., Kasza, K. E., Brangwynne, C. P.,Liu, J. & Weitz, D. A. Mechanical response of cy-toskeletal networks. In
Biophysical Tools for Biolo-gists, Volume Two: In Vivo Techniques , vol. 89 of
Methods in Cell Biology , chap. 19, 487 – 519 (Aca-demic Press, 2008).[5] Kollmannsberger, P. & Fabry, B. Linear and nonlin-ear rheology of living cells.
Annual Review of Ma-terials Research , 75–97 (2011).[6] de Gennes, P.-G. Un muscle artificiel semi-rapide. Comptes Rendus de l’Acad´emie des Sciences-SeriesIIB-Mechanics-Physics-Chemistry-Astronomy ,343–348 (1997).[7] Hines, L., Petersen, K., Lum, G. Z. & Sitti, M. Softactuators for small-scale robotics.
Advanced Mate-rials , 1603483 (2017).[8] Majidi, C. Soft-matter engineering for soft robotics. Advanced Materials Technologies , 1800477 (2019).[9] De Vicente, J., Klingenberg, D. J. & Hidalgo-Alvarez, R. Magnetorheological fluids: a review. Soft Matter , 3701–3710 (2011).[10] Chen, D. T., Wen, Q., Janmey, P. A., Crocker, J. C.& Yodh, A. G. Rheology of soft materials. Annu.Rev. Condens. Matter Phys. , 301–322 (2010).[11] Tschoegl, N. W. The phenomenological theory of lin-ear viscoelastic behavior: an introduction (Springer-Verlag Berlin Heidelberg, 1989), 1 edn.[12] Hyun, K. et al.
A review of nonlinear oscillatory shear tests: Analysis and application of large ampli-tude oscillatory shear (LAOS).
Progress in PolymerScience , 1697–1753 (2011).[13] Josephson, R. K. Mechanical power output fromstriated muscle during cyclic contraction. Journalof Experimental Biology , 493–512 (1985).[14] Ahn, A. N. How muscles function – the work looptechnique.
Journal of Experimental Biology ,1051–1052 (2012).[15] Menzel, A. M. Tuned, driven, and active soft mat-ter.
Physics Reports , 1–45 (2015).[16] Lissajous, J. A. M´emoire sur l’´etude optique desmouvements vibratoires.
Ann. Chim. Phys. Ser. 3 , 147 (1857).[17] Ewoldt, R. H., Hosoi, A. & McKinley, G. H. Newmeasures for characterizing nonlinear viscoelastic-ity in large amplitude oscillatory shear. Journal ofRheology , 1427–1458 (2008).[18] Machin, K. & Pringle, J. W. S. The physiologyof insect fibrillar muscle iii. the effect of sinusoidalchanges of length on a beetle flight muscle. Pro-ceedings of the Royal Society of London. Series B.Biological Sciences , 311–330 (1960).[19] Kawai, M. & Brandt, P. W. Sinusoidal analysis: ahigh resolution method for correlating biochemicalreactions with physiological processes in activatedskeletal muscles of rabbit, frog and crayfish.
Jour-nal of Muscle Research & Cell Motility , 279–303(1980).[20] James, R. S., Young, I. S., Cox, V. M., Gold-spink, D. F. & Altringham, J. D. Isometric andisotonic muscle properties as determinants of workloop power output. Pfl¨ugers Archiv , 767–774(1996).[21] Ahn, A. N. & Full, R. J. A motor and a brake: twoleg extensor muscles acting at the same joint manageenergy differently in a running insect.
Journal ofExperimental Biology , 379–389 (2002).[22] Palmer, B. M.
A Strain-Dependency of Myosin Off-Rate Must Be Sensitive to Frequency to Predict theB-Process of Sinusoidal Analysis , 57–75 (SpringerNew York, New York, NY, 2010).[23] Todorov, E. On the role of primary motor cortex inarm movement control.
Progress in Motor ControlIII , 125–166 (2003).[24] Grasman, J. Relaxation Oscillations , 1475–1488(Springer New York, New York, NY, 2011).[25] Ahn, A. N., Meijer, K. & Full, R. J. In situ musclepower differs without varying in vitro mechanicalproperties in two insect leg muscles innervated bythe same motor neuron.
Journal of ExperimentalBiology , 3370–3382 (2006).[26] Zajac, F. E. Muscle and tendon: properties, models,scaling, and application to biomechanics and motorcontrol.
Critical Reviews in Biomedical Engineering , 359–411 (1989). upplementary NotesRheology of tunable materials Khoi D. Nguyen and Madhusudhan Venkadesan ∗ Department of Mechanical Engineering & Materials Science, Yale University, New Haven, CT, USA
S1 Linear viscoelastic response as a sheared ellipse
The objective is to show that the linear viscoelastic response of a material to a sinusoidal lengthis geometrically a sheared ellipse. The force F ( t ) attributed to the rheological response is F ( t ) = ∆ L ( E sin ωt + E cos ωt ) (S1.1)where L ( t ) = ∆ L sin ωt is the length input, ω is angular frequency, and t is time. The horizontalellipse due to the loss modulus E is the plot of L E cos( ωt ) against L ( t ). The total response F ( t ) due to the addition of the storage modulus) is the same as a shear transformation of thisellipse as given by, (cid:20) f ( t ) L ( t ) (cid:21) = (cid:20) E (cid:21) (cid:20) L E cos( ωt ) L ( t ) (cid:21) . (S1.2)The 2x2 matrix is the shear transformation with E as a shear factor. The linear viscoelasticresponse is therefore a sheared ellipse. S2 Expansion with higher harmonics:
The generalization of splicing the constant-activation basis loops to include higher harmonics isa straightforward extension of splicing ellipses by using a Fourier series for the force response.The force response to sinusoidal length perturbations as given by equation (2) in the main textgeneralizes to F ( t ) = F A + ∆ L P k ( E A,k sin( kωt ) + E A,k cos( kωt )) , for ωt ∈ [ φ A , φ D ] F D + ∆ L P k ( E D,k sin( kωt ) + E D,k cos( kωt )) , otherwise (S2.1)for index k over the set of positive integers and where each higher harmonic introduces fouradditional moduli. Subtracting the deactivated force response and normalizing by lengthscale ∆ L and force scale F A − F D result in two difference of moduli of the k th harmonic:∆ e k = ∆ L ( E A,k − E D,k ) / ( F A − F D ) and ∆ e k = ∆ L ( E A,k − E D,k ) / ( F A − F D ). Expanding ∗ [email protected] a r X i v : . [ c ond - m a t . s o f t ] M a y quations (4)–(7) from the main text to include the higher harmonics results in f ( φ ) = P k (cid:16) w k w l D − l A α k sin kφ + w k w l D − l A β k cos kφ (cid:17) , for φ ∈ [ φ A , φ D ]0 , otherwise . (S2.2) w k = − ∆ e k φ D Z φ A sin kφ cos φ dφ = − ∆ e k α k (S2.3) w k = − ∆ e k φ D Z φ A cos kφ cos φ dφ = − ∆ e k β k (S2.4) w = − Z ‘ D ‘ A d‘ = ‘ A − ‘ D , (S2.5)where α k = R φ D φ A sin kφ cos φ dφ and β k = R φ D φ A cos kφ cos φ dφ are shape parameters. The terms w k and w k are the work by storage forces and loss forces of the k th harmonic, respectively. Thenet work now includes contributes from all higher harmonics as w n = w + P k ( w k + w k ) . (S2.6)The relation between nondimensional net work w nn