# Mass Spring Models of Amorphous Solids

## Abstract

**:**

## 1. Introduction

## 2. Elastic Moduli

## 3. Mass Spring Models

- Compute forces from springs ${F}_{i}=-{k}_{i}\Delta {L}_{i}$.
- Apply ${F}_{i}^{\mu}=(1-|D\left|\right){F}_{i}$ as a regular force
- Additionally accumulate $0.5D{F}_{i}{L}_{i}$ as ${J}_{acc}$.

- Redistribute the accumulated force as${F}^{*}={J}_{acc}/\left({L}_{i}b\right)$to all the springs connected with the node (by applying it on both nodes the spring connects), where b denotes the number of these springs.

## 4. Accuracy of Random MSM Models

- ${x}_{\alpha \beta}$—distance between nodes $\alpha $ and $\beta $
- ${\overline{F}}_{ab}$—force acting through the spring $ab$
- ${B}_{\alpha \beta}$—bond value

## 5. MSM Tuning

- (a)
- forces acting on inner MSM nodes sum to zero,
- (b)
- stress is isotropic.

#### 5.1. Constant kL

#### 5.2. kL-Tuning

- compute forces
- update velocities
- update positions

#### 5.3. ikL-Tuning

## 6. Effects on Young’s Modulus

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Klapetek, P.; Charvátová Campbell, A.; Buršíková, V. Fast mechanical model for probe–sample elastic deformation estimation in scanning probe microscopy. Ultramicroscopy
**2019**, 201, 18–27. [Google Scholar] [CrossRef] [PubMed] - Cristoforetti, A.; Masè, M.; Bonmassari, R.; Dallago, M.; Nollo, G.; Ravelli, F. A patient-specific mass-spring model for biomechanical simulation of aortic root tissue during transcatheter aortic valve implantation. Phys. Med. Biol.
**2019**, 64, 085014. [Google Scholar] [CrossRef] [PubMed] - Quillen, A.C.; Martini, L.; Nakajima, M. Near/far side asymmetry in the tidally heated Moon. Icarus
**2019**, 329, 182–196. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Quillen, A.C.; Zhao, Y.; Chen, Y.; Sánchez, P.; Nelson, R.C.; Schwartz, S.R. Impact excitation of a seismic pulse and vibrational normal modes on asteroid Bennu and associated slumping of regolith. Icarus
**2019**, 319, 312–333. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sahputra, I.; Alexiadis, A.; Adams, M. A Coarse Grained Model for Viscoelastic Solids in Discrete Multiphysics Simulations. ChemEngineering
**2020**, 4, 30. [Google Scholar] [CrossRef] - Vicente, G.S.; Buchart, C.; Borro, D.; Celigüeta, J.T. Maxillofacial surgery simulation using a mass-spring model derived from continuum and the scaled displacement method. Int. J. Comput. Assist. Radiol. Surg.
**2009**, 4, 89–98. [Google Scholar] [CrossRef] - Kot, M.; Nagahashi, H.; Szymczak, P. Elastic moduli of simple mass spring models. Vis. Comput.
**2015**, 31, 1339–1350. [Google Scholar] [CrossRef] - Chen, H.; Lin, E.; Jiao, Y.; Liu, Y. A generalized 2D non-local lattice spring model for fracture simulation. Comput. Mech.
**2014**, 54, 1541–1558. [Google Scholar] [CrossRef] - Goehring, L.; Mahadevan, L.; Morris, S.W. Nonequilibrium scale selection mechanism for columnar jointing. Proc. Natl. Acad. Sci. USA
**2009**, 106, 387–392. [Google Scholar] [CrossRef] [Green Version] - Aydin, A.; Degraff, J. Evoluton of Polygonal Fracture Patterns in Lava Flows. Science
**1988**, 239, 471–476. [Google Scholar] [CrossRef] - Hofmann, M.; Anderssohn, R.; Bahr, H.A.; Weiß, H.J.; Nellesen, J. Why Hexagonal Basalt Columns? Phys. Rev. Lett.
**2015**, 115, 154301. [Google Scholar] [CrossRef] [PubMed] - Maurini, C.; Bourdin, B.; Gauthier, G.; Lazarus, V. Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: Experiments and numerical simulations using a two-dimensional variational approach. Int. J. Fract.
**2013**, 184. [Google Scholar] [CrossRef] - Gauthier, G.; Lazarus, V.; Pauchard, L. Shrinkage star-shaped cracks: Explaining the transition from 90 degrees to 120 degrees. EPL
**2010**, 89, 26002. [Google Scholar] [CrossRef] [Green Version] - Goehring, L. Evolving fracture patterns: Columnar joints, mud cracks and polygonal terrain. Philos. Trans. Ser. A Math. Phys. Eng. Sci.
**2013**, 371, 20120353. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Frouard, J.; Quillen, A.; Efroimsky, M.; Giannella, D. Numerical Simulation of Tidal Evolution of a Viscoelastic Body Modelled with a Mass-Spring Network. Mon. Not. R. Astron. Soc.
**2016**, 458, stw491. [Google Scholar] [CrossRef] - Kot, M.; Nagahashi, H. Mass Spring Models with Adjustable Poisson’s Ratio. Vis. Comput.
**2017**, 33, 283–291. [Google Scholar] [CrossRef] - Golec, K.; Palierne, J.F.; Zara, F.; Nicolle, S.; Damiand, G. Hybrid 3D mass-spring system for simulation of isotropic materials with any Poisson’s ratio. Vis. Comput.
**2019**. [Google Scholar] [CrossRef] [Green Version] - Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed.; Cambridge University Press: New York, NY, USA, 2007. [Google Scholar]
- Liu, T.; Bargteil, A.W.; O’Brien, J.F.; Kavan, L. Fast Simulation of Mass-Spring Systems. ACM Trans. Graph.
**2013**, 32. [Google Scholar] [CrossRef] - Love, A.E.H. A Treatise on the Mathematical Theory of Elasticity; Cambridge University Press: Cambridge, UK, 1906. [Google Scholar]
- Baudet, V.; Beuve, M.; Jaillet, F.; Shariat, B.; Zara, F. Integrating Tensile Parameters in 3D Mass-Spring System; Technical Report RR-LIRIS-2007-004, LIRIS UMR 5205 CNRS/INSA de Lyon/Université Claude Bernard Lyon 1/Université Lumiére Lyon 2/École Centrale de Lyon; 2007; Available online: https://link.springer.com/article/10.1007/s11548-008-0271-0 (accessed on 1 December 2020).
- Lloyd, B.A.; Szekely, G.; Harders, M. Identification of Spring Parameters for Deformable Object Simulation. IEEE Trans. Vis. Comput. Graph.
**2007**, 13, 1081–1094. [Google Scholar] [CrossRef] - Ostoja-Starzewski, M. Lattice models in micromechanics. Appl. Mech. Rev.
**2002**, 55, 35–60. [Google Scholar] [CrossRef] - Chen, H.; Lin, E.; Liu, Y. A novel Volume-Compensated Particle method for 2D elasticity and plasticity analysis. Int. J. Solids Struct.
**2014**, 51, 1819–1833. [Google Scholar] [CrossRef] [Green Version] - Chen, H.; Liu, Y. A Nonlocal Lattice Particle Framework for Modeling of Solids. In ASME International Mechanical Engineering Congress and Exposition; American Society of Mechanical Engineers: New York, NY, USA, 2016; p. V001T03A001. [Google Scholar] [CrossRef]
- Golec, K. Hybrid 3D Mass Spring System for Soft Tissue Simulation. Ph.D. Theses, Université de Lyon, Lyon, France, 2018. [Google Scholar]
- Hardy, R.J. Formulas for determining local properties in molecular-dynamics simulations—Shock waves. J. Chem. Phys.
**1982**, 76, 622–628. [Google Scholar] [CrossRef] - Zimmerman, J.A.; WebbIII, E.B.; Hoyt, J.J.; Jones, R.E.; Klein, P.A.; Bammann, D.J. Calculation of stress in atomistic simulation. Model. Simul. Mater. Sci. Eng.
**2004**, 12, S319. [Google Scholar] [CrossRef] - Sheinman, M.; Broedersz, C.P.; MacKintosh, F.C. Nonlinear effective-medium theory of disordered spring networks. Phys. Rev. E
**2012**, 85, 021801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Banks, M.K.; Hazel, A.L.; Riley, G.D. Quantitative Validation of Physically Based Deformable Models in Computer Graphics. In Workshop on Virtual Reality Interaction and Physical Simulation; Andrews, S., Erleben, K., Jaillet, F., Zachmann, G., Eds.; The Eurographics Association: Genoa, Italy, 2018. [Google Scholar] [CrossRef]

**Figure 3.**A cube compressed in x direction by $10\%$ (x-borders frozen). First row $\nu =0.49$, second $\nu =0.25$, third $\nu =0$, fourth $\nu =-0.99$. First column: total force on springs, second: direct component, third: dispersive component. Red colour indicates expansive force, blue compressive.

**Figure 4.**The strain component ${\u03f5}_{22}$ in a uniformly compressed solid modeled with mass spring models (MSM) with constant k, same for all springs. Dotted lines represent theoretical values. Solid lines indicate the strain after material has relaxed to the equilibrium compressed state.

**Figure 5.**Same as Figure 4, but for ${\sigma}_{22}\left[\frac{{k}_{0}}{{a}_{0}}\right]$. Dotted lines represent stress in uniformly compressed material. Solid lines indicate stress after the material has relaxed to its new equilibrium state.

**Figure 7.**Same as Figure 4, but for kl–tune MSM.

**Figure 8.**Same as Figure 5, but for kl–tune MSM.

**Figure 10.**Same as Figure 5, but for ikl–tune MSM.

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Kot, M.
Mass Spring Models of Amorphous Solids. *ChemEngineering* **2021**, *5*, 3.
https://doi.org/10.3390/chemengineering5010003

**AMA Style**

Kot M.
Mass Spring Models of Amorphous Solids. *ChemEngineering*. 2021; 5(1):3.
https://doi.org/10.3390/chemengineering5010003

**Chicago/Turabian Style**

Kot, Maciej.
2021. "Mass Spring Models of Amorphous Solids" *ChemEngineering* 5, no. 1: 3.
https://doi.org/10.3390/chemengineering5010003