## Abstract

We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory, the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material is added to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion.

## Keywords

Accretion Surface growth Nonlinear elasticity Residual stress Foliations Material metric## Mathematics Subject Classification

74B20 74A05 74A10 53Z05## Notes

### Acknowledgements

This research was partially supported by NSF—Grant Nos. CMMI 1130856 and ARO W911NF-16-1-0064.

## Supplementary material

## References

- Abi-Akl, R., Abeyaratne, R., Cohen, T.: Kinetics of Surface Growth with Coupled Diffusion and the Emergence of a Universal Growth Path. arXiv:1803.08399 (2018)
- Arnowitt, R., Deser, S., Misner, C.W.: Dynamical structure and definition of energy in general relativity. Phys. Rev.
**116**(5), 1322 (1959)MathSciNetCrossRefzbMATHGoogle Scholar - Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal.
**63**(4), 337–403 (1976)MathSciNetCrossRefzbMATHGoogle Scholar - Ben Amar, M., Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids
**53**, 2284–2319 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - Bilby, B.A., Bullough, R., Smith, E.: Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc. R. Soc. Lond. A
**231**(1185), 263–273 (1955)MathSciNetCrossRefGoogle Scholar - Camacho, C., Neto, A.L.: Geom. Theory Foliations. Springer, Berlin (2013)Google Scholar
- do Carmo, M.: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser, Boston (1992); ISBN 1584883553Google Scholar
- Eckart, C.: The thermodynamics of irreversible processes. 4. The theory of elasticity and anelasticity. Phys. Rev.
**73**(4), 373–382 (1948)MathSciNetCrossRefzbMATHGoogle Scholar - Epstein, M.: Kinetics of boundary growth. Mech. Res. Commun.
**37**(5), 453–457 (2010)CrossRefzbMATHGoogle Scholar - Epstein, M., Maugin, G.A.: Thermomechanics of volumetric growth in uniform bodies. Int. J. Plast.
**16**, 951–978 (2000)CrossRefzbMATHGoogle Scholar - Ganghoffer, J.-F.: Mechanics and thermodynamics of surface growth viewed as moving discontinuities. Mech. Res. Commun.
**38**(5), 372–377 (2011)CrossRefzbMATHGoogle Scholar - Garikipati, K., Arruda, E.M., Grosh, K., Narayanan, H., Calve, S.: A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics. J. Mech. Phys. Solids
**52**(7), 1595–1625 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - Golgoon, A., Sadik, S., Yavari, A.: Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges. Int. J. Non-Linear Mech.
**84**, 116–129 (2016)CrossRefGoogle Scholar - Golovnev, A.: ADM Analysis and Massive Gravity. arXiv:1302.0687 (2013)
- Goriely, A.: The Mathematics and Mechanics of Biological Growth, vol. 45. Springer, Berlin (2017)zbMATHGoogle Scholar
- Kadish, J., Barber, J., Washabaugh, P.: Stresses in rotating spheres grown by accretion. Int. J. Solids Struct.
**42**(20), 5322–5334 (2005)CrossRefzbMATHGoogle Scholar - Klarbring, A., Olsson, T., Stalhand, J.: Theory of residual stresses with application to an arterial geometry. Arch. Mech.
**59**(4–5), 341–364 (2007)MathSciNetzbMATHGoogle Scholar - Kondo, K.: Geometry of Elastic Deformation and Incompatibility. In: Kondo, K (Ed.) Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, volume 1, Division C, pp. 5–17. Gakujutsu Bunken Fukyo-Kai (1955a)Google Scholar
- Kondo, K.: Non-Riemannian geometry of imperfect crystals from a macroscopic viewpoint. In: Kondo, K. (Ed.) Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, volume 1, Division D-I, pp. 6–17. Gakujutsu Bunken Fukyo-Kai (1955b)Google Scholar
- Lychev, S., Kostin, G., Koifman, K., Lycheva, T.: Modeling and optimization of layer-by-layer structures. In: Journal of Physics: Conference Series, vol. 1009, p. 012014. IOP Publishing (2018)Google Scholar
- Manzhirov, A.V., Lychev, S.A.: Mathematical modeling of additive manufacturing technologies. In: Proceedings of the World Congress on Engineering, volume 2 (2014)Google Scholar
- Marsden, J., Hughes, T.: Mathematical Foundations of Elasticity. Dover, New York (1983)zbMATHGoogle Scholar
- Metlov, V.: On the accretion of inhomogeneous viscoelastic bodies under finite deformations. J. Appl. Math. Mech.
**49**(4), 490–498 (1985)CrossRefzbMATHGoogle Scholar - Naumov, V.E.: Mechanics of growing deformable solids: a review. J. Eng. Mech.
**120**(2), 207–220 (1994)CrossRefGoogle Scholar - Ong, J.J., O’Reilly, O.M.: On the equations of motion for rigid bodies with surface growth. Int. J. Eng. Sci.
**42**(19), 2159–2174 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys.
**51**, 032902 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - Poincaré, H.: Science and Hypothesis. Science Press, Berlin (1905)zbMATHGoogle Scholar
- Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech.
**27**, 455–467 (1994)CrossRefGoogle Scholar - Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids
**22**(7), 1546–1587 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - Sadik, S., Yavari, A.: On the origins of the idea of the multiplicative decomposition of the deformation gradient. Math. Mech. Solids
**22**, 771–772 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - Sadik, S., Angoshtari, A., Goriely, A., Yavari, A.: A geometric theory of nonlinear morphoelastic shells. J. Nonlinear Sci.
**26**(4), 929–978 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - Schwerdtfeger, K., Sato, M., Tacke, K.-H.: Stress formation in solidifying bodies. Solidification in a round continuous casting mold. Metall. Mater. Trans. B
**29**(5), 1057–1068 (1998)CrossRefGoogle Scholar - Segev, R.: On smoothly growing bodies and the Eshelby tensor. Meccanica
**31**(5), 507–518 (1996)CrossRefzbMATHGoogle Scholar - Skalak, R., Farrow, D., Hoger, A.: Kinematics of surface growth. J. Math. Biol.
**35**(8), 869–907 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - Sozio, F., Yavari, A.: Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies. J. Mech. Phys. Solids
**98**, 12–48 (2017)MathSciNetCrossRefGoogle Scholar - Sozio, F., Sadik, S., Shojaei, M.F., Yavari, A. : Nonlinear mechanics of thermoelastic surface growth. In: preparation (2019)Google Scholar
- Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 3rd edn. Publish or Perish, Inc, New York (1999)zbMATHGoogle Scholar
- Takamizawa, K.: Stress-free configuration of a thick-walled cylindrical model of the artery—an application of Riemann geometry to the biomechanics of soft tissues. J. Appl. Mech.
**58**, 840–842 (1991)CrossRefGoogle Scholar - Takamizawa, K., Matsuda, T.: Kinematics for bodies undergoing residual stress and its applications to the left ventricle. J. Appl. Mech.
**57**, 321–329 (1990)CrossRefGoogle Scholar - Tomassetti, G., Cohen, T., Abeyaratne, R.: Steady accretion of an elastic body on a hard spherical surface and the notion of a four-dimensional reference space. J. Mech. Phys. Solids
**96**, 333–352 (2016)MathSciNetCrossRefGoogle Scholar - Wang, C.-C.: Universal solutions for incompressible laminated bodies. Arch. Ration. Mech. Anal.
**29**(3), 161–192 (1968)MathSciNetCrossRefzbMATHGoogle Scholar - Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci.
**20**(6), 781–830 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - Yavari, A.: Compatibility equations of nonlinear elasticity for non-simply-connected bodies. Arch. Ration. Mech. Anal.
**209**(1), 237–253 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal.
**205**(1), 59–118 (2012a)MathSciNetCrossRefzbMATHGoogle Scholar - Yavari, A., Goriely, A.: Weyl geometry and the nonlinear mechanics of distributed point defects. Proc. R. Soc. A
**468**, 3902–3922 (2012b)MathSciNetCrossRefzbMATHGoogle Scholar - Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids
**18**(1), 91–102 (2013a)MathSciNetCrossRefGoogle Scholar - Yavari, A., Goriely, A.: Nonlinear elastic inclusions in isotropic solids. Proc. R. Soc. A
**469**, 20130415 (2013b)MathSciNetCrossRefzbMATHGoogle Scholar - Yavari, A., Goriely, A.: The geometry of discombinations and its applications to semi-inverse problems in anelasticity. Proc. R. Soc. A
**470**, 20140403 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - Yavari, A., Goriely, A.: The twist-fit problem: finite torsional and shear eigenstrains in nonlinear elastic solids. Proc. R. Soc. A
**471**, 20150596 (2015)CrossRefGoogle Scholar - Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys.
**47**, 042903 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - Zurlo, G., Truskinovsky, L.: Printing non-Euclidean solids. Phys. Rev. Lett.
**119**(4), 048001 (2017)CrossRefGoogle Scholar - Zurlo, G., Truskinovsky, L.: Inelastic surface growth. Mech. Res. Commun.
**93**, 174–179 (2018)CrossRefGoogle Scholar