Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active C1 beam elements in a nonlinear finite element (FE) code.

References

References
1.
Trivedi
,
D.
,
Rahn
,
C. D.
,
Kier
,
W. M.
, and
Walker
,
I. D.
,
2008
, “
Soft Robotics: Biological Inspiration, State of the Art, and Future Research
,”
Appl. Bionics Biomech.
,
5
(
3
), pp.
99
117
.
2.
Majide
,
C.
,
Shepherd
,
R. F.
,
Kramer
,
R. F.
,
Whitesides
,
G. M.
, and
Wood
,
R. J.
,
2013
, “
Influence of Surface Traction on Soft Robot Undulation
,”
Int. J. Rob. Res.
,
32
(
13
), pp.
1577
1584
.
3.
Trivedi
,
D.
, and
Rahn
,
C. D.
,
2014
, “
Model-Based Shape Estimation for Soft Robotic Manipulators: The Planar Case
,”
ASME J. Mech. Rob.
,
6
(
2
), p.
021005
.
4.
Love
,
A. E. H.
,
1944
,
The Mathematical Theory of Elasticity
,
4th ed.
,
Dover
,
New York
.
5.
Reissner
,
E
.,
1972
, “
On One-Dimensional Finite-Strain Beam Theory: The Plane Problem
,”
J. Appl. Math. Phys.
,
23
(
5
), pp.
795
804
.
6.
Reissner
,
E
.,
1973
, “
On One-Dimensional Large-Displacement Finite-Strain Beam Theory
,”
Stud. Appl. Math.
,
52
(
2
), pp.
87
95
.
7.
Reissner
,
E
.,
1981
, “
On Finite Deformation of Space-Curved Beams
,”
J. Appl. Math. Phys.
,
32
(
6
), pp.
734
744
.
8.
Antman
,
S. S.
,
1972
, “
The Theory of Rods
,”
Handbuch der Physik
, Vol.
Via/2
,
Springer
,
Berlin
, pp.
641
703
.
9.
Atluri
,
S. N.
,
Iura
,
M.
, and
Vasudevan
,
S.
,
2001
, “
A Consistent Theory of Finite Stretches and Finite Rotations, in Space-Curve Beams of Arbitrary Cross-Section
,”
Comput. Mech.
,
27
(
4
), pp.
271
281
.
10.
Simo
,
J. C.
,
1985
, “
A Finite Strain Beam Formulation. The Three-Dimensional Dynamic Problem—Part I
,”
Comput. Methods Appl. Mech. Eng.
,
49
(
1
), pp.
55
70
.
11.
Timoshenko
,
S.
, and
Woinowsky-Krieger
,
S.
,
1959
,
Theory of Plates and Shells
,
2nd ed.
,
McGraw-Hill
,
New York
.
12.
Reissner
,
E
.,
1944
, “
On the Theory of Bending of Elastic Plates
,”
J. Math. Phys.
,
23
(1–4), pp.
184
191
.
13.
Reissner
,
E
.,
1950
, “
On a Variational Theorem in Elasticity
,”
J. Math. Phys.
,
29
(1–4), pp.
90
95
.
14.
Atluri
,
S. N.
,
1979
, “
On the Rate Principle for Finite Strain Analysis of Elastic and Inelastic Nonlinear Solids
,”
Recent Research on Mechanical Behavior of Solids
,
University of Tokyo Press
,
Tokyo, Japan
, pp.
79
107
.
15.
Atluri
,
S. N.
,
1983
, “
Alternate Stress and Conjugate Strain Measures, and Mixed Variational Formulations Involving Rigid Rotations, for Computational Analysis of Finitely Deformed Solids, With Application to Plates and Shell-I Theory
,”
Comput. Struct.
,
18
(
1
), pp.
93
116
.
16.
Reddy
,
J. N.
,
2004
,
Mechanics of Laminated Composite Plates and Shells: Theory and Analysis
,
2nd ed.
,
CRC Press
,
Boca Raton, FL
.
17.
Murakami
,
H.
, and
Yamakawa
,
J.
,
2000
, “
Development of One-Dimensional Models for Elastic Waves in Heterogeneous Beams
,”
ASME J. Appl. Mech.
,
67
(
4
), pp.
671
684
.
18.
Marsden
,
J. E.
, and
Hughes
,
T. J. R.
,
1983
,
Mathematical Foundations of Elasticity
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
19.
Truesdell
,
C.
, and
Toupin
,
R. A.
,
1960
, “
The Classical Field Theories
,”
Handbuck der Physik
, Vol.
III-1
, Springer-Verlag, Berlin, pp.
226
790
.
20.
Cartan
,
É.
,
1928
,
Leçons sur la Géométrie des Espaces de Riemann
,
Gauthiers-Villars
, Paris, France.
21.
Cartan
,
É.
,
1986
,
On Manifolds With an Affine Connection and the Theory of General Relativity
,
A.
Magnon
and
A.
Ashtekar
, eds.,
Bibiliopolis
,
Napoli, Italy
.
22.
Frankel
,
T.
,
2012
,
The Geometry of Physics, an Introduction
,
3rd ed.
,
Cambridge University Press
,
New York.
23.
Murakami
,
H
.,
2016
, “
Integrability Conditions in Nonlinear Beam Kinematics
,”
ASME
Paper No. IMECE2016-65293.
24.
Wang
,
W.
,
Lee
,
J.-Y.
,
Rodrigue
,
H.
,
Song
,
S.-H.
,
Chu
,
W.-S.
, and
Ahn
,
S.-H.
,
2004
, “
Locomotion of Inchworm-Inspired Robot Made of Smart Soft Composite (SSC)
,”
Bioinspiration Biomimetics
,
9
(
4
), p.
046006
.
25.
Green
,
A. E.
, and
Zerna
,
W.
,
1968
,
Theoretical Elasticity
,
2nd ed.
,
Dover Publications
,
New York
.
26.
Washizu
,
K
.,
1964
, “
Some Considerations on a Naturally Curved and Twisted Slender Beam
,”
J. Math. Phys.
,
43
(1–4), pp.
111
116
.
27.
O'Neill
,
B.
,
1997
,
Elementary Differential Geometry
,
2nd ed.
,
Academic Press
,
New York
.
28.
Do Carmo
,
M. P.
,
1976
,
Differential Geometry of Curves and Surfaces
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
29.
Murakami
,
H.
, and
Yamakawa
,
J.
,
1998
, “
Torsional Wave Propagation in Retrofitted Reinforced Concrete Columns
,”
Int. J. Solids Struct.
,
35
(
20
), pp.
2617
2637
.
30.
Timoshenko
,
S. P.
,
1921
, “
On the Correction Factor for Shear of the Differential Equation for Transverse Vibrations of Bars of Uniform Cross-Section
,”
Philos. Mag.
,
41
(245), pp.
744
746
.
31.
Rios
,
O.
,
Ono
,
T.
, and
Murakami
,
H.
,
2016
, “
Development of Active Mechanical Models for Flexible Robots to Duplicate the Motion of Inch Worms and Snakes
,”
ASME
Paper No. IMECE2016-65550.
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