2
Mesoscale level in the mechanics of
materials
2.1On the definitions of scale levels:micro-and mesomechanics
The following scale levels are usually recognized in the analysis of the material behavior:•Macroscale(or a specimen scale),of the order of more than1mm.Material behavior at this scale level is analyzed using continuum mechanics methods.
•Micro-and mesoscale(or a microstructure scale),between1 m and1mm.Material behavior at this scale level falls into the area of materials science,and is analyzed using methods of both physics and mechanics of materials,including micromechanics and fracture mechanics.
•Nano-and atomistic scales,less than1 m.Material behavior at this scale level falls into the area of the physics of materials.
The question arises as to what is the difference between‘mesolevel’and‘microlevel’in the mechanics of materials.
The term‘mesomechanics’has gained relatively wide acceptance after it was intro-duced by Panin and co-workers(Panin,1998)as the name for a new area of research (‘physical mesomechanics’),the main purpose of which has been defined as the develop-ment of the theoretical basis of material improvement,using experimental and theoretical studies of physical processes in loaded materials at the mesolevel.
Needleman(Needleman,2000)defined‘mesoscale continuum mechanics’as‘interme-diate between direct atomistic and an unstructured continuum description of deformation processes’.As a characteristic feature of this scale level,it was noted that‘size matters’at the mesoscale level.
Panin(Panin,1998)suggested a more detailed classification,with meso I and meso II levels(defined as related to the rotation modes inside structural elements of deformation,
Computational Mesomechanics of Composites: Numerical analysis of the effect of microstructures of composites on their strength and damage resistance L. Mishnaevsky Jr ©2007John Wiley&Sons,Ltd. ISBN: 978-0-470-02764-6
14Size effects
and to the self-correlated rotations of many structural elements,respectively).The meso I level corresponds to the level‘inside microstructural elements’(grains),while the meso II level is related to the‘conglomerates’of microstructural elements.Actually,Needleman’s ‘mesoscale continuum mechanics’corresponds to both Panin’s meso I and II levels,and to both micromechanics and mesomechanics in the above classification.
Kocks(cited by Estrin,1999)stated that‘“mesoscopic”should refer to cases where the structural scale is,say,of the order of100 m’(scale between‘microscopic’,associated with the microscope and micrometer,and‘macroscopic’).
Mishnaevsky Jr and Schmauder(Mishnaevsky Jr and Schmauder,2001)defined the mesolevel in the material structure as a range of scale levels which are two to three orders of magnitude greater than defects of structure(which varied in the10−9–10−5m scale range)and one to three orders of magnitude smaller than a specimen or workpiece in the following.The levels of the description can be related to the methods of the control of material properties:whereas the improvement at macrolevel can be done by modifying the specimen construction,the improvement of the strength and reliability of materials at micro-and mesolevel is carried out by heat treatment,metal working,impregnation, powder metallurgy methods,etc.
Following Mishnaevsky Jr(Mishnaevsky Jr,2005b),one may state that the mesome-chanics of materials studies quantitatively the interaction and synergistic effects of many elements of microstructures(as inclusions,voids,shear bands,microcracks,etc.,or generally the heterogeneity of materials)on the strength and mechanical properties of materials,whereas micromechanics deals with the effects of single elements and averaged parameters of microstructures on the mechanical behavior of putational mesomechanics seeks to create the necessary numerical tools for the analysis of the mechanical behavior and degradation of materials,which allows the computational anal-ysis of the interaction between many elements of microstructures and microstructure evolution,and can serve as a basis for the material design.
2.2Size effects
Needleman(Needleman,2000)defined‘mesoscale’as a scale level at which‘size matters’.The influence of geometrical size of components on the mechanical behavior and strength of materials was observed in numerous experiments,carried out on metals (Fleck et al.,1994),ceramics(Xu and Rowcliffe,2002),concretes(Bazant and Yavari, 2005)and polymers(Tjernlund,2005).The common observation is that‘smaller is stronger’:the smaller a specimen or an element of the microstructure,the higher its strength.
2.2.1Brittle and quasi-brittle materials
The interest of the research community in the size effect in materials was first aroused in connection with the testing of materials for civil engineering use.The necessity to predict the strength of concretes for large-scale structures on the basis of testing much smaller specimens led to intensive scientific efforts in this area.
Several explanations for the size effect in concretes,ceramics and other quasi-brittle materials have been suggested.The oldest model of the size effect is based on Weibull’s
