Preliminary Studies On Willis-form Linear Elasticity Equations

In the early 1980's,Professor Willis proposed a set of linear elasticity equations for weak inhomogeneous materials,which are characterized as velocity coupling terms.However,these equations had not drawn much attention until the upsurge of designing metamaterials by using transformation elastodynamics.People found that only the Willis equations have the property of form-invariance.However,the Willis equations are very abstract and lack of clear physical meanings and experimental evidences,which impedes the progress of relevant researches.This thesis aims at clarifying the physical meanings of the Willis elasticity equations,and tries to give self-consistent explanations from views of transformation method,differential method and energy principles.For this purpose,based on clearly distinguishing the mechanical quantities defined in the configurations before and after the infinitesimal elastic deformation,the displacement coupling Willis-form equations are derived by the energy principle and the differential method,respectively.It reveals that the displacement coupling terms are related to the gradient of pre-stresses.Then,the velocity coupling Willis-form equations are also derived by using the homogenization method and the energy-conservation principle.It is found that the velocity coupling terms only compensate the error of kinetic energy density due to the homogenization,and do not represent physical dissipations.In addition,a general from of Willis-form equations are also obtained by using the perturbation method,which contain both the displacement coupling terms and the velocity coupling terms in nonlocal forms.In order to verify the 1D displacement coupling Willis-form elasticity equations,this thesis firstly proves that they have the time-synchronization property for the spatial transformation from the virtual space to the physical space,which is a necessity condition for a correct elastodynamic equation.Finally,it proves the 1D static displacement coupling Willis-form elasticity equations can correctly describe the incremental behavior of a rotational spring.This was also verified by experimental results.In addition,it points out that only the Willis-form constitutive equation can simultaneously explain the stress-stiffening and the spin-softening effect of a rotational spring.This thesis just presents some preliminary results on the Willis-form elasticity equations.It summarizes some pending theoretical and experimental works,and points out many potential applications.