# M03-D in SI units: a definition, not a material calibration

For a named specimen at one fixed geometry, let q be displacement in metres,
v velocity in m/s, M effective moving mass in kg, C viscous coefficient in N s/m,
K0 baseline stiffness in N/m, F applied force in N, m a dimensionless state,
beta dimensionless stiffness coupling, tau recovery time in s and alpha_E
writing coefficient in J^-1:

 dq/dt=v
 M dv/dt=F-Cv-K0(1+beta m)q
 dm/dt=alpha_E C v^2(1-m)-m/tau.

C v^2 is a power in watts; multiplying by alpha_E gives a rate in s^-1.
For nonnegative alpha_E,C and positive tau, the state interval [0,1] is invariant.
This is the stiffening candidate for beta>0. Substituting a negative sign for an
elastomer softening experiment would define a different constitutive hypothesis.

Choose a displacement scale Q>0, time t*=sqrt(M/K0), and force F*=K0 Q.
With s=t/t*, y=q/Q, p=v t*/Q, f=F/F*, the equations become
 y'=p; p'=f-cbar p-(1+beta m)y;
 m'=abar cbar p^2(1-m)-m/taubar,
where cbar=C/sqrt(M K0), abar=alpha_E K0 Q^2 and taubar=tau/t*.
This matches the normalized M03-D at L=1. General normalized L is a declared
frequency ratio; it is not automatically a length in metres. A geometry-dependent
K0 must be separately measured or derived for each specimen and boundary condition.

The included synthetic example uses M=0.02 kg, K0=8 N/m, C=0.096 N s/m,
Q=0.005 m, beta=1.5, alpha_E=500 J^-1 and tau=0.75 s. It corresponds to
cbar=0.24, abar=0.1, taubar=15. None of these numbers was measured from an
elastomer. SI and normalized adaptive integrations agree within 4.1e-11 in
normalized state coordinates over 30 normalized time units.

Energy E=M v^2/2+K0(1+beta m)q^2/2 satisfies
 dE/dt=Fv-Cv^2+K0 beta q^2 (dm/dt)/2.
The last term records changing stiffness. The energetic cost and physical
mechanism for writing state are still missing; this is not a thermodynamic closure.

## Calibration needed before a prediction
Use calibration-template.csv to record specimen ID, geometry, fixture compliance,
effective mass, force and displacement calibration, temperature, strain range,
rate, sample timing, uncertainty and parameter-estimation dataset IDs. Fit K0 and
C from appropriate small-amplitude measurements. Define how m is observed;
beta and state scale can otherwise be confounded. Estimate alpha_E and tau on
calibration-only trajectories spanning write and recovery regimes, check practical
identifiability, then freeze all choices before untouched specimens. A probe may
itself write memory. Do not treat fitting the same trajectories as prediction.

## Frequency ratio correction
For an imposed sinusoidal displacement of fixed amplitude and constant C,
cycle-average power is proportional to f^2. In the small-write, negligible-recovery
limit, a tenfold frequency increase gives a tenfold state increase at matched
cycles or a hundredfold increase at matched duration. These are conditional limits,
not universal material predictions. Recovery, saturation, frequency-dependent C,
strain-rate effects and heating can change or erase them.

output/frequency-ratio-sensitivity.csv illustrates finite retention and saturation
using the averaged equation m'=r(1-m)-m/tau, initially m=0. Its solution is
m(T)=r/(r+1/tau) [1-exp(-(r+1/tau)T)]. This approximation does not solve the
periodic elastomer response or establish a full-cycle constitutive law. E02-F
must compare calibrated rivals and use specimen-level uncertainty. A null in E02-F
does not automatically retire M03-D without a justified mapping between them.