Homogenization and micromechanics 
Homogenization and micromechanics are mathematical theories that allow one to "upscale" the microstructure of complex materials, in order to provide macroscopic or "homogenized" constitutive relations and material properties. This is very useful because frequently we are not interested in the variation of stress, displacement, velocity, etc. over short lengthscales such as a micro or nanometre if the entire system is of the order of say a meter. A good example is the application of such theories to composite media, which are made from distinct materials (often known as phases), combined in such a way as to optimize some physical property such as Youngs modulus, bulk modulus, thermal conductivity, viscosity, etc. Furthermore the composite may often exhibit certain behaviour, not exhibited by the phases from which it is comprised, such as induced anisotropy. This is particularly important in fibre reinforced composites and laminated plates. Here are two pictures of the microstructure of some composite media: the first is a Copper Cobalt composite and the second is a silicon carbidetitanium alloy.
Experimental studies prove extremely expensive and it is therefore of great interest to model these materials mathematically. The subject areas devoted to such studies are those of homogenization and micromechanics. Over the last few decades, these subjects have grown immensely, leading to a much greater understanding of materials that are used extensively in engineering applications including the automotive, aerospace and energy sectors. More recently, homogenization has been applied in order to study and model the effective constitutive behaviour of biological tissues such as bone, tendon and skin.
Within the Waves group we have developed a number of new techniques of homogenization and applied these and other methods to a number of materials. These can be classified as dynamic (using wave techniques) and static methods. Below we refer to publications associated with statics methods, those associated with dynamic techniques can be see on the metamaterials page under the theme of multiple scattering.
Within the Waves group we have developed a number of new techniques of homogenization and applied these and other methods to a number of materials. These can be classified as dynamic (using wave techniques) and static methods. Below we refer to publications associated with statics methods, those associated with dynamic techniques can be see on the metamaterials page under the theme of multiple scattering.
 G. Rus, J. Melchor, N. Bochud, L. Peralta and W.J. Parnell (2019) (open access)
"Damage prediction via nonlinear ultrasound: a micromechanical approach"
Ultrasonics 93, 145155.  W.D. Rowley, W.J. Parnell, I.D. Abrahams, S.R. Voisey, J. Lamb and N. Etaix, N. (2018) (open access)
"Deepening subwavelength acoustic resonance via metamaterials with universal broadband elliptical microstructure"
Applied Physics Letters 112, 251902  C. CalvoJurado and W.J. Parnell, (2018)
"Induced fields in isolated elliptical inhomogeneities due to imposed polynomial fields at infinity"
Int. J. Computer Math.  Geer, S., Berger, J.R., Parnell, W.J. and Mustoe, G.G.W. (2017)
"A Comparison of Discrete Element and Micromechanical Methods for Determining the Effective Elastic Properties of Geomaterials"
Computers and Geotechnics. 87, 19  CalvoJurado, C. and Parnell, W.J. (2017) (open access)
" The influence of twopoint statistics on the HashinShtrikman bounds for threephase composites"
J. Comp. Applied Math. 318, 354365.  D. Joyce, W.J. Parnell, R.C. Assier and I.D. Abrahams (2017) (open access)
Proc. Roy. Soc. A 473:20170080
 Joyce, D. and Parnell, W.J. (2017)
"The Newtonian potential inhomogeneity problem: nonuniform eigenstrains in cylinders of nonuniform cross section"
J. Eng. Mathematics (50 years special issue)  Parnell, W.J. (2016) (open access)
"The Eshelby, Hill, moment and concentration tensors for ellipsoidal inhomogeneities in the Newtonian potential problem and linear elastostatics"
J. Elast. 164  CalvoJurado, C., & Parnell, W. J. (2016)
"The influence of twopoint statistics on the Hashin–Shtrikman bounds for three phase composites"
J. Comp. Appl. Mathematics 03770427  Parnell, W.J. and CalvoJurado, C. (2015)
"On the construction of the HashinShtrikman bounds for transversely isotropic twophase linear elastic fibre reinforced composites"
J. Eng. Mathematics 95, 295323  CalvoJurado, C. and Parnell, W.J. (2015)
"The HashinShtrikman bounds on the effective thermal conductivity of a transversely isotropic twophase composite"
J. Math. Chemistry 53, 828843
 Williams, T.D. and Parnell, W.J. (2014) (open access)
"Effective antiplane elastic properties of an orthotropic solid weakened by a periodic distribution of cracks"
Quart. J. Mech. Appl. Math. 67, 311342
 De Pascalis, R., Parnell, W.J. and Abrahams, I.D. (2013)
"Predicting the nonlinear pressurevolume curve of an elastic microsphere composite"
J. Mech. Physics Solids 61, 11061123  Willoughby, N., Parnell, W.J., Hazel, A., Abrahams, I.D. (2012)
"Homogenization methods to approximate the effective response of random fibrereinforced composites"
Int. J. Solids Structures 49, 1421–1433
 Parnell, W.J., Abrahams, I.D. and BrazierSmith, P.R. (2010)
"Effective properties of a composite halfspace: exploring the relationship between homogenization and multiple scattering theories"
Quart. J. Mech. Appl. Math. 63, 145175
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