arxiv: 2602.17002 · v6 · submitted 2026-02-19 · 💻 cs.CE · math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

A Total Lagrangian Finite Element Framework for Multibody Dynamics: Part I -- Formulation

Zhenhao Zhou , Ganesh Arivoli , Dan Negrut

Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:16 UTC · model grok-4.3

classification 💻 cs.CE math-phmath.MP

keywords total Lagrangianfinite element methodmultibody dynamicsfinite deformationengineering jointsconstraint formulationdeformation gradientdeformable bodies

0 comments

The pith

A Total Lagrangian finite element framework derives equations of motion for finite-deformation multibody systems coupled by engineering joints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unified finite element approach cast in the total Lagrangian description to simulate collections of deformable bodies that undergo large shape changes and interact through standard engineering joints. It starts from a compact kinematic description and a deformation-gradient formulation, then adds an interface that accepts any constitutive model and a systematic way to build constraint equations for joints, contact, and external loads. If the approach holds, modelers can treat flexible components and rigid-like assemblies within the same discretization without switching formulations when deformations become finite.

Core claim

The framework combines a compact kinematic representation, a deformation-gradient-based formulation, an element-agnostic constitutive interface, and a systematic constraint-construction machinery for coupling deformable bodies through engineering joints. Within this setting the equations of motion are derived for collections of deformable bodies that carry external loads, frictional contact forces, and constraint reaction forces. The same setting accepts field forces applied at points, over surfaces, or throughout volumes and directly supports material models such as Mooney-Rivlin, Neo-Hookean, and Kelvin-Voigt.

What carries the argument

The Total Lagrangian finite-element formulation with deformation-gradient kinematics and systematic joint-constraint construction, which supplies the equations of motion and reaction forces for arbitrarily connected deformable bodies.

If this is right

Equations of motion for any number of deformable bodies become available once their individual finite-element meshes and joint constraints are assembled.
Frictional contact and distributed body forces enter the formulation without requiring separate modules.
Material models can be swapped through the constitutive interface without altering the kinematic or constraint machinery.
Field loads applied pointwise, on surfaces, or volumetrically are treated uniformly inside the same weak-form statement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same constraint-construction machinery could be reused for unilateral contact or inequality constraints without changing the core derivation.
Because the formulation stays total Lagrangian, remeshing or updated-Lagrangian corrections become unnecessary even after very large accumulated rotations.
Integration with existing rigid-body multibody codes would require only replacement of the rigid-body inertia blocks by the deformable tangent matrices produced here.

Load-bearing premise

Standard continuum mechanics and ordinary finite-element discretization remain consistent when applied to multiple deformable bodies linked by engineering joints under finite strain.

What would settle it

A closed-form analytic solution or high-resolution benchmark for a simple two-body system joined by a hinge and undergoing large rotation and deformation that produces constraint violation or incorrect reaction forces when simulated with the framework.

read the original abstract

We present a Total Lagrangian finite element framework for finite-deformation multibody dynamics. The framework combines a compact kinematic representation, a deformation-gradient-based formulation, an element-agnostic constitutive interface, and a systematic constraint-construction machinery for coupling deformable bodies through engineering joints. Within this setting, we derive the equations of motion for collections of deformable bodies and formulate their response in the presence of external loads, frictional contact forces, and constraint reaction forces. The framework accommodates field forces applied pointwise, over surfaces, or throughout volumes, and supports material models of practical interest, including Mooney-Rivlin, Neo-Hookean, and Kelvin-Voigt. A companion paper discusses the GPU-accelerated implementation of the framework outlined herein and reports on numerical experiments and benchmark results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a straightforward formulation paper that assembles total-Lagrangian FE kinematics with joint constraints and common materials, but leaves all checks to the companion paper.

read the letter

Colleague, the main takeaway is that this paper gives a clean total Lagrangian finite element setup for finite-deformation multibody dynamics. It pulls together a compact kinematic representation, deformation-gradient constitutive response, an element-agnostic material interface, and a systematic way to build Lagrange-multiplier constraints for engineering joints. They derive the equations of motion that cover external loads, frictional contact, and reaction forces, and they list support for Mooney-Rivlin, Neo-Hookean, and Kelvin-Voigt models. The constraint-construction rules look practical for coupling deformable bodies without custom per-joint code. That combination is new enough to be worth noting, even if each piece draws from existing continuum mechanics and multibody literature. The derivations follow standard principles, so there is no obvious circularity or invented physics in the outline. The element-agnostic constitutive layer is a useful engineering detail that could simplify mixing element types. The main soft spot is that this is explicitly Part I. All numerical verification, stability checks under large deformation, and GPU implementation are deferred to the companion paper. Without those results it is hard to judge whether the constraint handling or contact formulation stays consistent when finite strains interact with joints. The central assumption—that standard FE discretization and continuum principles carry over without hidden inconsistencies—is reasonable on paper but remains untested here. This work is aimed at computational mechanics groups that already run flexible multibody codes and want a unified way to add joints and contact. Readers who need the formulation details before trying to implement or extend it will get value. I would send it to peer review. The framework is grounded in established methods and the integration is specific enough that referees can usefully check the constraint machinery and kinematic choices.

Referee Report

1 major / 3 minor

Summary. The manuscript presents a Total Lagrangian finite element framework for finite-deformation multibody dynamics. It integrates a compact kinematic representation, a deformation-gradient-based formulation, an element-agnostic constitutive interface, and systematic constraint construction for coupling deformable bodies via engineering joints. The paper derives the equations of motion for collections of deformable bodies including external loads, frictional contact forces, and constraint reaction forces. It supports field forces applied pointwise, over surfaces or volumes, and material models such as Mooney-Rivlin, Neo-Hookean, and Kelvin-Voigt. Numerical verification and implementation details are deferred to a companion paper.

Significance. If the formulation proves internally consistent, the work supplies a unified total-Lagrangian setting for multibody dynamics that accommodates finite deformations and standard engineering joints without ad-hoc kinematic reductions. The element-agnostic constitutive interface and systematic constraint machinery could streamline modeling of large-deformation coupled systems in applications such as soft robotics and vehicle dynamics. Because the manuscript is explicitly Part I (formulation only), its significance will ultimately rest on whether the companion paper demonstrates stability and accuracy on representative benchmarks.

major comments (1)

The derivation of the equations of motion (likely in the section following the kinematic description) relies on the principle of virtual work in the total Lagrangian setting. The manuscript should explicitly verify that the linearization of the constraint terms does not introduce spurious coupling between deformation and rigid-body modes when the deformation gradient is evaluated at quadrature points inside each element.

minor comments (3)

The abstract refers to a 'systematic constraint-construction machinery' for engineering joints; the main text would benefit from an explicit algorithm or table listing the constraint equations and associated Jacobians for the most common joint types (revolute, spherical, prismatic).
Notation for the first and second Piola-Kirchhoff stress tensors and their variations should be introduced once in a dedicated nomenclature section or early in the kinematics section to prevent later ambiguity.
The description of frictional contact forces would be clearer if the authors stated whether the contact formulation is node-to-surface or surface-to-surface and how the gap function is defined in the reference configuration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the framework's potential, and recommendation for minor revision. We address the single major comment below and will incorporate the requested verification into the revised manuscript.

read point-by-point responses

Referee: The derivation of the equations of motion (likely in the section following the kinematic description) relies on the principle of virtual work in the total Lagrangian setting. The manuscript should explicitly verify that the linearization of the constraint terms does not introduce spurious coupling between deformation and rigid-body modes when the deformation gradient is evaluated at quadrature points inside each element.

Authors: We agree that an explicit verification of this property strengthens the presentation. In the total-Lagrangian formulation the constraints act on the current configuration through the deformation map, while the deformation gradient appears only inside the constitutive response evaluated at quadrature points. Consequently, the virtual work of the constraint forces depends on nodal positions (and their variations) but not on the internal quadrature-point values of F. For any rigid-body motion the deformation gradient equals the identity at every quadrature point and its variation vanishes identically; the resulting constraint Jacobian and its linearization therefore reduce exactly to the standard rigid-body case with no additional deformation-coupling terms. We will add a short analytical remark (or dedicated paragraph) immediately after the derivation of the equations of motion that demonstrates this cancellation explicitly, confirming the absence of spurious coupling. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from standard total-Lagrangian continuum mechanics and constraint principles

full rationale

The manuscript is explicitly a formulation paper (Part I) that assembles the equations of motion, external loads, frictional contact, and constitutive response directly from well-established total-Lagrangian kinematics, deformation-gradient constitutive laws, and Lagrange-multiplier joint constraints. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the derivation follows standard continuum-mechanics principles once the kinematic representation and constraint-construction rules are accepted. The companion paper is reserved for numerical verification, confirming that this part contains no internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard continuum mechanics and finite-element assumptions without introducing new free parameters or invented entities in the abstract description.

axioms (2)

domain assumption Standard assumptions of finite-strain continuum mechanics hold for the deformable bodies
The Total Lagrangian formulation and deformation-gradient approach rely on these background principles.
domain assumption Engineering joints can be represented through systematic constraint equations that are compatible with the finite-element discretization
The constraint-construction machinery assumes this compatibility without further justification in the abstract.

pith-pipeline@v0.9.0 · 5433 in / 1306 out tokens · 30709 ms · 2026-05-15T21:16:36.798362+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

deformation gradient F(u;t)=N(t)H(u,v,w) ... strain measures ... right Cauchy-Green C:=F^TF and Green-Lagrange E:=1/2(C-I)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

engineering joints ... primitive constraints DP1, DP2, DIST, CD ... revolute joint ... 5 scalar constraints

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Total Lagrangian Finite Element Framework for Multibody Dynamics: Part II -- GPU Implementation and Numerical Experiments
cs.CE 2026-04 unverdicted novelty 5.0

The authors implement a GPU-accelerated total Lagrangian FE framework for multibody dynamics with implicit backward-Euler stepping, augmented Lagrangian constraints, and a two-stage parallelization strategy, reporting...

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

A Total Lagrangian Finite Element Framework for Multibody Dynamics: Part II -- GPU Implementation and Numerical Experiments

Zhou, Z., Zhang, R., Arivoli, G., Negrut, D.: A Total Lagrangian Finite Ele- ment Framework for Multibody Dynamics: Part II – GPU Implementation and Numerical Experiments (2026). https://arxiv.org/abs/2604.10357

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

John Wiley & Sons, Chichester, UK (2000)

Belytschko, T., Liu, W.K., Moran, B.: Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, Chichester, UK (2000)

work page 2000
[3]

Cambridge University Press, Cambridge, UK (2016)

Bonet, J., Gil, A.J., Wood, R.D.: Nonlinear Solid Mechanics for Finite Element Analysis: Statics. Cambridge University Press, Cambridge, UK (2016)

work page 2016
[4]

Nonlinear Dynamics111(15), 13753–13779 (2023)

Peng, Q., Li, M.: Comparison of finite element methods for dynamic analysis about rotating flexible beam. Nonlinear Dynamics111(15), 13753–13779 (2023)

work page 2023
[5]

Multibody System Dynamics1(3), 339–348 (1997)

Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody System Dynamics1(3), 339–348 (1997)

work page 1997
[6]

ASME Journal of Mechanical Design123, 606–613 (2001)

Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate for- mulation for beam elements: Theory. ASME Journal of Mechanical Design123, 606–613 (2001)

work page 2001
[7]

International Journal for Numerical Methods in Engineering122(20), 5744–5772 (2021)

Zhang, J.: A direct jacobian total lagrangian explicit dynamics finite element algorithm for real-time simulation of hyperelastic materials. International Journal for Numerical Methods in Engineering122(20), 5744–5772 (2021)

work page 2021
[8]

Cambridge University Press, Cambridge, England (2020)

Shabana, A.A.: Dynamics of Multibody Systems, 5th edn. Cambridge University Press, Cambridge, England (2020)

work page 2020
[9]

In: Acm Siggraph 2012 Courses, pp

Sifakis, E., Barbic, J.: Fem simulation of 3d deformable solids: a practitioner’s guide to theory, discretization and model reduction. In: Acm Siggraph 2012 Courses, pp. 1–50 (2012)

work page 2012
[10]

Computational Mechanics69(3), 639–660 (2022)

Stickle, M.M., Molinos, M., Navas, P., Yag¨ ue,´A., Manzanal, D., Moussavi, S., Pas- tor, M.: A component-free lagrangian finite element formulation for large strain elastodynamics. Computational Mechanics69(3), 639–660 (2022)

work page 2022
[11]

Journal of Computational and Nonlinear Dynamics17(10), 101008 (2022) https://doi.org/10.1115/1.4055140

Kissel, A., Taves, J., Negrut, D.: Constrained multibody kinematics and dynamics in absolute coordinates: A discussion of three approaches to representing rigid body rotation. Journal of Computational and Nonlinear Dynamics17(10), 101008 (2022) https://doi.org/10.1115/1.4055140 . 101008

work page doi:10.1115/1.4055140 2022
[12]

Computational Mechanics31(1), 49–59 (2003)

Betsch, P., Steinmann, P.: Constrained dynamics of geometrically exact beams. Computational Mechanics31(1), 49–59 (2003)

work page 2003
[13]

Nonlinear Dynamics31(2), 167–195 (2003) 42

Sugiyama, H., Escalona, J.L., Shabana, A.A.: Formulation of three-dimensional joint constraints using the absolute nodal coordinates. Nonlinear Dynamics31(2), 167–195 (2003) 42

work page 2003
[14]

Cambridge University Press, Cambridge, UK (2008)

Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd edn. Cambridge University Press, Cambridge, UK (2008). https: //doi.org/10.1017/CBO9780511755446

work page doi:10.1017/cbo9780511755446 2008
[15]

Butterworth-Heinemann, Oxford, UK (2005)

Zienkiewicz, O., Taylor, R.: The Finite Element Method for Solid and Structural Mechanics, 6th edn. Butterworth-Heinemann, Oxford, UK (2005)

work page 2005
[16]

Nonlinear Dynamics35(4), 313–329 (2004)

Garc´ ıa-Vallejo, D., Mayo, J., Escalona, J.L., Dom´ ınguez, J.: Efficient evaluation of the elastic forces and the jacobian in the absolute nodal coordinate formulation. Nonlinear Dynamics35(4), 313–329 (2004)

work page 2004
[17]

Multibody system dynamics 18(3), 375–396 (2007)

Maqueda, L.G., Shabana, A.A.: Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams. Multibody system dynamics 18(3), 375–396 (2007)

work page 2007
[18]

Nonlinear Dynamics82, 451–464 (2015)

Orzechowski, G., Fra¸ czek, J.: Nearly incompressible nonlinear material models in the large deformation analysis of beams using ANCF. Nonlinear Dynamics82, 451–464 (2015)

work page 2015
[19]

International Journal for Numerical Methods in Engineering79(11), 1309–1331 (2009) https://doi.org/10.1002/nme

Geuzaine, C., Remacle, J.-F.: Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering79(11), 1309–1331 (2009) https://doi.org/10.1002/nme. 2579

work page doi:10.1002/nme 2009
[20]

ACM Transactions on Mathematical Software41(2) (2015) https://doi.org/10.1145/ 2629697

Si, H.: TetGen, a delaunay-based quality tetrahedral mesh generator. ACM Transactions on Mathematical Software41(2) (2015) https://doi.org/10.1145/ 2629697

work page 2015
[21]

Technical Report MBS96-1-UIC, University of Illinois at Chicago (1996)

Shabana, A.A.: An Absolute Nodal Coordinate Formulation for the large rota- tion and deformation analysis of flexible bodies. Technical Report MBS96-1-UIC, University of Illinois at Chicago (1996). Technical report (ANCF original formu- lation)

work page 1996
[22]

Berzeri, M., Shabana, A.A.: Development of simple models for the elastic forces in the absolute nodal coordinate formulation. J. of Sound and Vibration235(4), 539–565 (2000)

work page 2000
[23]

Nonlinear Dynam- ics50, 249–264 (2007) https://doi.org/10.1007/s11071-006-9155-4

Garc´ ıa-Vallejo, D., Mikkola, A.M., Escalona, J.L.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dynam- ics50, 249–264 (2007) https://doi.org/10.1007/s11071-006-9155-4

work page doi:10.1007/s11071-006-9155-4 2007
[24]

Journal of Computational and Nonlinear Dynamics8(3), 031016 (2013)

Gerstmayr, J., Sugiyama, H., Mikkola, A.: Review on the absolute nodal coordi- nate formulation for large deformation analysis of multibody systems. Journal of Computational and Nonlinear Dynamics8(3), 031016 (2013)

work page 2013
[25]

Journal of Compu- tational and Nonlinear Dynamics17(8), 080803 (2022) https://doi.org/10.1115/ 1.4054113

Otsuka, K., Makihara, K., Sugiyama, H.: Recent advances in the Absolute Nodal 43 Coordinate Formulation: Literature review from 2012 to 2020. Journal of Compu- tational and Nonlinear Dynamics17(8), 080803 (2022) https://doi.org/10.1115/ 1.4054113

work page 2012
[26]

Multibody System Dynamics (2023) https://doi.org/10.1007/s11044-023-09890-z

Shabana, A.A.: An overview of the ANCF approach, justifications for its use, implementation issues, and future research directions. Multibody System Dynamics (2023) https://doi.org/10.1007/s11044-023-09890-z

work page doi:10.1007/s11044-023-09890-z 2023
[27]

Cambridge University Press, Cambridge, UK (1987)

Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge, UK (1987)

work page 1987
[28]

Europhysics Letters (EPL)42(5), 511–516 (1998) https://doi.org/10.1209/epl/ i1998-00281-7

Brilliantov, N.V., P¨ oschel, T.: Rolling friction of a viscous sphere on a hard plane. Europhysics Letters (EPL)42(5), 511–516 (1998) https://doi.org/10.1209/epl/ i1998-00281-7

work page doi:10.1209/epl/ 1998
[29]

Powder Technology71(3), 239–250 (1992)

Tsuji, Y., Tanaka, T., Ishida, T.: Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technology71(3), 239–250 (1992)

work page 1992
[30]

Mindlin, R.: Compliance of elastic bodies in contact. J. of Appl. Mech.16(1949)

work page 1949
[31]

Journal of Applied Mechanics20, 327–344 (1953)

Mindlin, R., Deresiewicz, H.: Elastic spheres in contact under varying oblique forces. Journal of Applied Mechanics20, 327–344 (1953)

work page 1953
[32]

Journal of Computational and Nonlinear Dynamics11(4), 044502 (2016)

Fleischmann, J., Serban, R., Negrut, D., Jayakumar, P.: On the importance of displacement history in soft-body contact models. Journal of Computational and Nonlinear Dynamics11(4), 044502 (2016)

work page 2016
[33]

Computer Methods in Applied Mechanics and Engineering171(3–4), 419–444 (1999) https://doi.org/10.1016/S0045-7825(98)00219-9

Ortiz, M., Stainier, L.: The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering171(3–4), 419–444 (1999) https://doi.org/10.1016/S0045-7825(98)00219-9

work page doi:10.1016/s0045-7825(98)00219-9 1999
[34]

Journal of optimization theory and applications4(5), 303–320 (1969)

Hestenes, M.R.: Multiplier and gradient methods. Journal of optimization theory and applications4(5), 303–320 (1969)

work page 1969
[35]

Academic Press, San Diego, CA (2014)

Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, San Diego, CA (2014)

work page 2014
[36]

Nocedal, J., Wright, S.J.: Numerical Optimization vol. 39. Springer, New York, NY (1999)

work page 1999
[37]

Multibody System Dynamics2, 307–332 (1998) https://doi.org/10.1023/A:1008072517368

Shabana, A.A.: Computer implementation of the Absolute Nodal Coordinate Formulation for flexible multibody dynamics. Multibody System Dynamics2, 307–332 (1998) https://doi.org/10.1023/A:1008072517368

work page doi:10.1023/a:1008072517368 1998
[38]

In: Proceeding of Multibody Dynamics ECCOMAS Thematic Conference, 44 Madrid, Spain (2005)

Gerstmayr, J., Shabana, A.A.: Efficient integration of the elastic forces and thin three-dimensional beam elements in the absolute nodal coordinate formula- tion. In: Proceeding of Multibody Dynamics ECCOMAS Thematic Conference, 44 Madrid, Spain (2005)

work page 2005
[39]

venant–kirchhoff material model in large strain applications

Bonet, J.,et al.: Limitations of the st. venant–kirchhoff material model in large strain applications. International Journal of Solids and Structures248, 111618 (2022) https://doi.org/10.1016/j.ijsolstr.2022.111618

work page doi:10.1016/j.ijsolstr.2022.111618 2022
[40]

Computer Methods in Applied Mechanics and Engineering416, 116368 (2023) https://doi.org/10.1016/j.cma.2023.116368

˙Zur, K.K., Firouzi, N., Rabczuk, T., Zhuang, X.: Large deformation of hyper- elastic modified timoshenko–ehrenfest beams under different types of loads. Computer Methods in Applied Mechanics and Engineering416, 116368 (2023) https://doi.org/10.1016/j.cma.2023.116368

work page doi:10.1016/j.cma.2023.116368 2023
[41]

Applied Sciences10(4), 1240 (2020) https://doi.org/10.3390/ app10041240

Luo, Y., Peng, B.: Benchmark problems of hyper-elasticity analysis in eval- uation of fem. Applied Sciences10(4), 1240 (2020) https://doi.org/10.3390/ app10041240

work page 2020
[42]

Journal of Applied Physics 11(9), 582–592 (1940) https://doi.org/10.1063/1.1712836

Mooney, M.: A theory of large elastic deformation. Journal of Applied Physics 11(9), 582–592 (1940) https://doi.org/10.1063/1.1712836

work page doi:10.1063/1.1712836 1940
[43]

Rivlin, R.S.: Large elastic deformations of isotropic materials. I. Fundamental concepts. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences240(822), 459–490 (1948) https://doi.org/ 10.1098/rsta.1948.0002 45

work page doi:10.1098/rsta.1948.0002 1948