  It has been known that the Lifshitz theory of the Casimir force comes into
conflict with the measurement data if the response of conduction electrons in
metals to electromagnetic fluctuations is described by the well tested
dissipative Drude model. The same theory is in a very good agreement with
measurements of the Casimir force from graphene whose spatially nonlocal
electromagnetic response is derived from the first principles of quantum
electrodynamics. Here, we propose the spatially nonlocal phenomenological
dielectric functions of metals which lead to nearly the same response, as the
standard Drude model, to the propagating waves, but to a different response in
the case of evanescent waves. Unlike some previous suggestions of this type,
the response functions considered here depend on all components of the wave
vector as is most natural in the formalism of specular reflection used. It is
shown that these response functions satisfy the Kramers-Kronig relations. We
derive respective expressions for the surface impedances and reflection
coefficients. The obtained results are used to compute the effective Casimir
pressure between two parallel plates, the Casimir force between a sphere and a
plate, and its gradient in configurations of the most precise experiments
performed with both nonmagnetic (Au) and magnetic (Ni) test bodies. It is shown
that in all cases (Au-Au, Au-Ni, and Ni-Ni test bodies) the predictions of the
Lifshitz theory found by using the dissipative nonlocal response functions are
in as good agreement with the measurement data, as was reached previously with
the dissipationless plasma model. Possible developments and applications of
these results are discussed.
