# zbMATH — the first resource for mathematics

Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. (English) Zbl 1120.74690
Summary: We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.
We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.
We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.

##### MSC:
 74N20 Dynamics of phase boundaries in solids
LAPACK
Full Text:
##### References:
 [1] Abeyaratne, R.; Knowles, J.K., On the driving traction on a surface of a strain discontinuity in a continuum, J. mech. phys. solids, 38, 345-360, (1990) · Zbl 0713.73030 [2] Abeyaratne, R.; Knowles, J.K., Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. appl. math., 51, 5, 1205-1221, (1991) · Zbl 0764.73013 [3] Abeyaratne, R.; Knowles, J.K., Kinetic relations and the propagation of phase boundaries in solids, Arch. ration. mech. anal., 114, 119-154, (1991) · Zbl 0745.73001 [4] Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.L.; Demmel, J.W.; Dongarra, J.J.; DuCroz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Sorensen, D., LAPACK User’s guide, (1999), Society for Industrial and Applied Mathematics. · Zbl 0934.65030 [5] Artemev, A.; Jin, Y.; Khachaturyan, A., Three-dimensional phase field model of proper martensitic transformation, Acta mater., 49, 7, 1165-1177, (2001) [6] Ball, J.M., James, R.D., 2005. Metastability in martensitic phase transformations. In preparation. [7] Bhattacharya, K., Microstructure of martensite, (2003), Oxford University Press Oxford [8] Christian, J.W., The theory of transformations in metals and alloys, (1975), Pergamon New York [9] Dondl, P.W.; Zimmer, J., Modeling and simulation of martensitic phase transitions with a triple point, J. mech. phys. solids, 52, 9, 2057-2077, (2004) · Zbl 1078.74037 [10] Engl, H.W.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0711.34018 [11] Eshelby, J.D., 1956. Solid State Physics, vol. 3. Academic Press, New York, pp. 17-144. [12] Eshelby, J.D., The elastic energy – momentum tensor, J. elasticity, 5, 321-335, (1975) · Zbl 0323.73011 [13] James, R.D. , 2005. Unpublished micrographs. [14] Killough, M.G., 1998. A diffuse interface approach to the development of microstructure in martensite. Ph.D. Thesis, New York University. [15] Kloucek, P.; Luskin, M., Computational modeling of the martensitic transformation with surface energy, Math. comput. model., 20, 101-121, (1994) · Zbl 0813.73064 [16] Kunin, I.A., 1982. Elastic media with microstructure. Springer Series in Solid-State Sciences, vol. 26. Springer, Berlin. · Zbl 0527.73002 [17] Lefloch, P., Propagating phase boundaries: formulation of the problem and existence via the glimm method, Arch. ration. mech. anal., 123, 153-197, (1993) · Zbl 0784.73010 [18] Lei, Y.; Friswell, M.I.; Adhikari, S., A Galerkin method for distributed systems with non-local damping, Int. J. solids struct., 43, 3381-3400, (2006) · Zbl 1121.74376 [19] Olson, G.B., Roitburd, A.L., 1992. Martensitic nucleation. In: Martensite. ASM, pp. 149-174 (Chapter). [20] Porter, D.; Stirling, D.G.S., Integral equations, (1990), Cambridge University Press Cambridge [21] Purohit, P.K., 2001. Dynamics of phase transitions in strings, beams and atomic chains. Ph.D. Thesis, California Institute of Technology. [22] Silling, S.A., Reformulation of elasticity theory for discontinuities and long-range forces, J. mech. phys. solids, 48, 175-209, (2000) · Zbl 0970.74030 [23] Silling, S.A., Kahan, S., 2004. Peridynamic modeling of structural damage and failure. In: Conference on High Speed Computing, Gleneden Beach, Oregon, USA. [24] Silling, S.A.; Zimmermann, M.; Abeyaratne, R., Deformation of a peridynamic bar, J. elasticity, 73, 173-190, (2003) · Zbl 1061.74031 [25] Truskinovsky, L., 1993. Kinks versus shocks. In: Fosdick, R., Dunn, E., Slemrod, M. (Eds.), Shock Induced Transitions and Phase Structures in General Media, IMA, vol. 52. Springer, Berlin. · Zbl 0818.76036 [26] Truskinovsky, L.; Vainchtein, A., Kinetics of martensitic phase transitions: lattice model, SIAM J. appl. math., 66, 2, 533-553, (2005) · Zbl 1136.74362 [27] Wang, Y.Z., Chen, L.Q., Khachaturyan, A.G., 1994. Computer simulation of microstructure evolution in coherent solids. In: Solid-Solid Phase Transformations. The Minerals Metals and Material Society, pp. 245-265 (Chapter). [28] Weckner, O.; Abeyaratne, R., The effect of long-range forces on the dynamics of a bar, J. mech. phys. solids, 53, 705-728, (2005) · Zbl 1122.74431 [29] Zimmermann, M., 2002. Phase transformations in the one dimensional peridynamic theory. Unpublished private communication.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.