Materials with memory

In continuum physics, materials with memory, also referred as materials with hereditary effects are a class of materials whose constitutive equations contains a dependence upon the past history of thermodynamic, kinetic, electromagnetic or other kind of state variables.

Historical notes
The study of these materials arises from the pioneering articles of Ludwig Boltzmann and Vito Volterra,  in which they sought an extension of the concept of an elastic material. The key assumption of their theory was that the local stress value at a time $t$ depends upon the history of the local deformation up to $t$. In general, in materials with memory the local value of some constitutive quantity (stress, heat flux, electric current, polarization and magnetization, etc.) at a time $t$ depends upon the history of the state variables (deformation, temperature, electric and magnetic fields, etc.). The hypothesis that the remote history of a variable has less influence than its values in the recent past, was stated in modern continuum mechanics as the fading memory principle by Bernard Coleman and Walter Noll. This assumption was implicit in the pioneer works: when restricted to cyclic hystories, it traces back to the closed cycle principle stated by Volterra, which leads to a constitutive relation of integral convolution type. In the linear case, this relation takes the form of a Volterra equation.

Constitutive relations of materials with memory
In the linear case, this relation takes the form of a Volterra equation:
 * $$\boldsymbol{T}(\boldsymbol{x},t)=\boldsymbol{G}_0(\boldsymbol{x})\boldsymbol{E}(\boldsymbol{x},t)+\int_0^{+\infty}\!\!\!\!\boldsymbol{G}^\prime(\boldsymbol{x},s)\boldsymbol{E}(\boldsymbol{x},t-s)\mathrm{d}s$$

Articles

 * , published also as.
 * (online version ).
 * . In this paper, written only few years after the discovery of the effect itself, Dario Graffi proposes a theory of the Luxemburg effect based on Volterra's theory of hereditary phenomena.
 * (online version ).
 * (online version ). In this paper, Graffi introduces the free energy now called Graffi–Volterra free energy after him.
 * . "Memory and dissipation in solids" is a short survey on the determination of the response functions of materials with memory by acoustical measurements.
 * . "On the principle of fading memory" is a note on fading memory principle.
 * . "The work of Vito Volterra on hereditary phenomena and some of their consequences" is an ample technical survey paper on the research work of Vito Volterra on hereditary phenomena in mathematical physics.
 * . An introduction to modern rational continuum mechanics by one of its leading contributors.
 * (online version ). G. Del Piero, L. Deseri (1997); "", Arch. Rational Mech. Anal. 138, 1–35.
 * . "The work of Vito Volterra on hereditary phenomena and some of their consequences" is an ample technical survey paper on the research work of Vito Volterra on hereditary phenomena in mathematical physics.
 * . An introduction to modern rational continuum mechanics by one of its leading contributors.
 * (online version ). G. Del Piero, L. Deseri (1997); "", Arch. Rational Mech. Anal. 138, 1–35.

Books

 * (also available as e-book with ISBN 978-1-4614-1692-0).
 * (available also as e-book with ISBN 978-94-007-0034-5).
 * (see also ).
 * (see also : the JFM reviews refer to the original 1930 English edition).
 * (available also as e-book with ISBN 978-94-007-0034-5).
 * (see also ).
 * (see also : the JFM reviews refer to the original 1930 English edition).
 * (see also ).
 * (see also : the JFM reviews refer to the original 1930 English edition).
 * (see also ).
 * (see also : the JFM reviews refer to the original 1930 English edition).