Constitutive Laws of Smart Materials

In the case of smart materials, Hooke s law has to be substituted or amended by a constitutive law that couples the mechanical properties of the material with other physical properties such as electric, magnetic, or thermal entities. 

Among the large variety of smart materials discussed today, piezoelectric and magnetostrictive materials can be described by linearised constitutive laws that are given below. Other widely used material types, such as electrostric-tive or shape memory materials, exhibit strongly nonlinear behaviour, the modelling of which may become quite demanding. 

Piezoelectric Materials. In the constitutive law of piezoelectrics, a coupling between strain e, stress cr, electric displacement D and electric field E exists as follows  

d and e are the matrices of piezoelectric coupling and dielectric constants, respectively. 

Magnetostrictive Materials. Substituting magneto-mechanics for electromechanics, mechanical strain e and stress cr are coupled with magnetic field intensity H and flux density B as follows  

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While only some of the FE codes include the fully coupled constitutive laws of smart materials, many formulations exist however for piezoelectric materials [10]. 

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