Dilatational and distortional behavior of cracks in magnetoelectroelastic materials

Abstract Dilatation and distortion are the two basic modes of deformation in continuum mechanics. They occur at different time and position depending on the size scale under consideration. At the macroscopic scale, dilatational waves are said to arrive prior to distortional waves at a given location. Macro-plastic deformation caused by distortion off to the side of a macro-crack precedes that of macro-fracture caused by dilatation ahead of a macro-crack. Micro-plasticity and micro-cracking can introduce non-linearity to the macroscopic material behavior. Such effects can be modeled directly as geometric defects (dislocations and/or micro-cracks) or indirectly by introducing effective material coefficients in the constitutive relations. Material anisotropy and/or inhomogeneity, however, do not separate hydrostatic tension (or compression) state of stress from that of shear on the octahedral planes. The latter is the basis of the von-Mises yield criterion. Stated in terms of the strain energy density function, linear separation into one part for dilatation and another for distortion can be made only if the material is linearly isotropic and homogeneous. Recent trends of miniaturization of electronic components requires a better knowledge of how the micro-structure would affect the material response due to mechanical stress as well as other sources of disturbances. Continuum mechanics results need to be interpreted over a wider range of time and size scales, and a better understanding of their limitations need to be achieved. In what follows, the stationary values of the strain energy density function d W /d V will be shown to yield an interpretation of the local dilatational and distortional behavior of anisotropic (or non-homogeneous) materials with or without cracks. This quantity is not only attractive from the view point of mathematics and mechanics because it is positive definite, it also makes physical sense for non-linear materials where the quotient d W /d V could still be used to separate dilatation from distortion. Application of the minimum local energy density function should be distinguished from the global energy minimization method used in molecular dynamics which is applied to determine the equilibrium configuration of atoms. The response of a line crack in a magnetoelectroelastic (MEE) material will be used for discussion in order to illustrate the character of multi-scaling. Expressed differently is that the time and size scales for the transfer of magnetic, electric and elastic energies may not be the same. An arbitrary limiting process that assumes vanishing crack segments created by all forms of energy may not be justified at the local scale levels. What has been assumed at the global scale for isotropic and homogeneous situations may not be valid at the local scale levels where anisotropy and non-homogeneity are the rule rather then the exception. Numerical results are obtained for the composite BaTiO 3 –CoFe 2 O 4 . They show how magnetic and electric poling normal to a line crack could greatly affect the interplay between volume change and shape change of the local continuum elements at both the macroscopic and microscopic scale level. The micro-energy-density function is found to be three orders of magnitude higher than that at the macroscopic scale. The volume fraction of the inclusions made of BaTiO 3 also comes into play for both Mode I and II mechanical loading. Potential micro-crack bifurcation is predicted in a region very close to the crack tip while macro-crack initiation occurs in a straight line for the MEE material. The information is relevant for developing macro- and micro-crack models where quantities and situations of secondary importance can be weeded out.