A theory of hyperelasticity of multi-phase media with surface/interface energy effect

SummaryIn addition to the classical governing equations in continuum mechanics, two kinds of governing equations are necessary in the solution of boundary-value problems for the stress fields in multi-phase hyperelastic media with the surface/interface energy effect. The first is the interface constitutive relation, and the second are the discontinuity conditions of the traction across the interface, namely, the Young-Laplace equations. In this paper, the interface consitutive relations are presented in terms of the interface energy in both Lagrangian and Eulerian descriptions within the framework of finite deformation, and the expressions of the interface stress for an isotropic interface are given as a special case. Then, by introducing a fictitious stress-free configuration, a new energy functional for multi-phase hyperelastic media with interface energy effect is proposed. The functional takes into account the interface energy and the interface stress-induced ``residual'' elastic field, which reflects the intrinsic physical properties of the material. All field equations, including the generalized Young-Laplace equation, can be derived from the stationary condition of this functional. The present theory is illustrated by simple examples. The results in this paper provide a theoretical framework for studying the elastostatic problems of multi-phase hyperelastic bodies that involve surface/interface energy effects at finite deformation.

[1]  Ray W. Ogden,et al.  Nonlinear Elastic Deformations , 1985 .

[2]  C. Horgan,et al.  Void nucleation and growth for a class of incompressible nonlinearly elastic materials , 1989 .

[3]  Jianmin Qu,et al.  Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films , 2005 .

[4]  Harald Ibach,et al.  The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures , 1997 .

[5]  M. M. Carroll.  Finite strain solutions in compressible isotropic elasticity , 1988 .

[6]  Zvi Hashin,et al.  Large isotropic elastic deformation of composites and porous media , 1985 .

[7]  Ray W. Ogden,et al.  Elastic surface—substrate interactions , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  Bernard Nysten,et al.  Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy , 2004 .

[9]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[10]  Bhushan Lal Karihaloo,et al.  Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress , 2005 .

[11]  R. Gomer,et al.  Structure and Properties of Solid Surfaces , 1953 .

[12]  J. Willard Gibbs,et al.  The scientific papers of J. Willard Gibbs , 1907 .

[13]  C. Horgan,et al.  The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials , 1986 .

[14]  Yibin Fu,et al.  Nonlinear elasticity : theory and applications , 2001 .

[15]  J. W. Gibbs,et al.  Scientific Papers , 2002, Molecular Imaging and Biology.

[16]  M. Gurtin,et al.  Addenda to our paper A continuum theory of elastic material surfaces , 1975 .

[17]  J. Aboudi,et al.  Micromechanical Modeling of the Finite Deformation of Thermoelastic Multiphase Composites , 2000 .

[18]  Hanchen Huang,et al.  Are surfaces elastically softer or stiffer , 2004 .

[19]  E. Müller,et al.  The Use of Classical Macroscopic Concepts in Surface Energy Problems , 1953 .

[20]  W. Haiss,et al.  Surface stress of clean and adsorbate-covered solids , 2001 .

[21]  Robert C. Cammarata,et al.  Surface and interface stress effects on interfacial and nanostructured materials , 1997 .

[22]  Fritz John,et al.  Plane strain problems for a perfectly elastic material of harmonic type , 1960 .

[23]  R. Cammarata,et al.  Effects and measurement of internal surface stresses in materials with ultrafine microstructures , 1991 .

[24]  R Shuttleworth,et al.  The Surface Tension of Solids , 1950 .

[25]  S. Biwa,et al.  Nonmonotonic cavity growth in finite, compressible elasticity , 1997 .

[26]  C. Horgan.  Void nucleation and growth for compressible non-linearly elastic materials: An example , 1992 .

[27]  B. R. Seth.  GENERALIZED STRAIN MEASURE WITH APPLICATIONS TO PHYSICAL PROBLEMS , 1961 .

[28]  D. Haughton.  Inflation and bifurcation of thick-walled compressible elastic spherical shells , 1987 .

[29]  A. Saúl,et al.  Elastic effects on surface physics , 2004 .