Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1813–1863

# Nonlinear mechanics of accretion

• Fabio Sozio
• Arash Yavari
Article

## 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.

## References

1. 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)
2. Arnowitt, R., Deser, S., Misner, C.W.: Dynamical structure and definition of energy in general relativity. Phys. Rev. 116(5), 1322 (1959)
3. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4), 337–403 (1976)
4. Ben Amar, M., Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53, 2284–2319 (2005)
5. 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)
6. Camacho, C., Neto, A.L.: Geom. Theory Foliations. Springer, Berlin (2013)Google Scholar
7. do Carmo, M.: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser, Boston (1992); ISBN 1584883553Google Scholar
8. Eckart, C.: The thermodynamics of irreversible processes. 4. The theory of elasticity and anelasticity. Phys. Rev. 73(4), 373–382 (1948)
9. Epstein, M.: Kinetics of boundary growth. Mech. Res. Commun. 37(5), 453–457 (2010)
10. Epstein, M., Maugin, G.A.: Thermomechanics of volumetric growth in uniform bodies. Int. J. Plast. 16, 951–978 (2000)
11. Ganghoffer, J.-F.: Mechanics and thermodynamics of surface growth viewed as moving discontinuities. Mech. Res. Commun. 38(5), 372–377 (2011)
12. 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)
13. 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)
14. Golovnev, A.: ADM Analysis and Massive Gravity. arXiv:1302.0687 (2013)
15. Goriely, A.: The Mathematics and Mechanics of Biological Growth, vol. 45. Springer, Berlin (2017)
16. Kadish, J., Barber, J., Washabaugh, P.: Stresses in rotating spheres grown by accretion. Int. J. Solids Struct. 42(20), 5322–5334 (2005)
17. Klarbring, A., Olsson, T., Stalhand, J.: Theory of residual stresses with application to an arterial geometry. Arch. Mech. 59(4–5), 341–364 (2007)
18. 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
19. 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
20. 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
21. 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
22. Marsden, J., Hughes, T.: Mathematical Foundations of Elasticity. Dover, New York (1983)
23. Metlov, V.: On the accretion of inhomogeneous viscoelastic bodies under finite deformations. J. Appl. Math. Mech. 49(4), 490–498 (1985)
24. Naumov, V.E.: Mechanics of growing deformable solids: a review. J. Eng. Mech. 120(2), 207–220 (1994)
25. 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)
26. Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51, 032902 (2010)
27. Poincaré, H.: Science and Hypothesis. Science Press, Berlin (1905)
28. Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27, 455–467 (1994)
29. Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids 22(7), 1546–1587 (2016)
30. 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)
31. Sadik, S., Angoshtari, A., Goriely, A., Yavari, A.: A geometric theory of nonlinear morphoelastic shells. J. Nonlinear Sci. 26(4), 929–978 (2016)
32. 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)
33. Segev, R.: On smoothly growing bodies and the Eshelby tensor. Meccanica 31(5), 507–518 (1996)
34. Skalak, R., Farrow, D., Hoger, A.: Kinematics of surface growth. J. Math. Biol. 35(8), 869–907 (1997)
35. Sozio, F., Yavari, A.: Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies. J. Mech. Phys. Solids 98, 12–48 (2017)
36. Sozio, F., Sadik, S., Shojaei, M.F., Yavari, A. : Nonlinear mechanics of thermoelastic surface growth. In: preparation (2019)Google Scholar
37. Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 3rd edn. Publish or Perish, Inc, New York (1999)
38. 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)
39. Takamizawa, K., Matsuda, T.: Kinematics for bodies undergoing residual stress and its applications to the left ventricle. J. Appl. Mech. 57, 321–329 (1990)
40. 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)
41. Wang, C.-C.: Universal solutions for incompressible laminated bodies. Arch. Ration. Mech. Anal. 29(3), 161–192 (1968)
42. Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20(6), 781–830 (2010)
43. Yavari, A.: Compatibility equations of nonlinear elasticity for non-simply-connected bodies. Arch. Ration. Mech. Anal. 209(1), 237–253 (2013)
44. Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205(1), 59–118 (2012a)
45. Yavari, A., Goriely, A.: Weyl geometry and the nonlinear mechanics of distributed point defects. Proc. R. Soc. A 468, 3902–3922 (2012b)
46. Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids 18(1), 91–102 (2013a)
47. Yavari, A., Goriely, A.: Nonlinear elastic inclusions in isotropic solids. Proc. R. Soc. A 469, 20130415 (2013b)
48. Yavari, A., Goriely, A.: The geometry of discombinations and its applications to semi-inverse problems in anelasticity. Proc. R. Soc. A 470, 20140403 (2014)
49. Yavari, A., Goriely, A.: The twist-fit problem: finite torsional and shear eigenstrains in nonlinear elastic solids. Proc. R. Soc. A 471, 20150596 (2015)
50. Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903 (2006)
51. Zurlo, G., Truskinovsky, L.: Printing non-Euclidean solids. Phys. Rev. Lett. 119(4), 048001 (2017)
52. Zurlo, G., Truskinovsky, L.: Inelastic surface growth. Mech. Res. Commun. 93, 174–179 (2018)