I just need to know the correct expression:

When narrowing down constitutive equations for the mechanics solids in continuum mechanics, one has in the very general case a Cauchy stress $\mathbf{T}$ as a result of the deformation history $\chi(\mathbf{x},t)$, written as a functional involving integrals over time and the body $\mathcal{B}$:

$$ \mathbf{T}(\mathbf{x}_0,t)=\int_\mathcal{B} \int_0^t f(\chi(\mathbf{x},\tau)-\chi(\mathbf{x}_0,\tau)) \ \mathrm d\tau \ \mathrm d \mathbf{x} $$

where $f$ is the constitutive function (Peridynamics is an example).

Then, principles of material modelling are invoked to reduce the functional freedom of the material model. One generally replaces the time integral by an internal variable $\mathbf{v}$,

$$ \mathbf{T}(\mathbf{x}_0,\mathbf{v})=\int_\mathcal{B} g(\mathbf{\chi(x)}-\chi(\mathbf{x}_0),\mathbf{v}) \mathrm d \mathbf{x}\\ \dot{\mathbf{v}}(t,\mathbf{v})=h(...) $$

the evolution of which is prescribed by another constitutive function $h$.

I would like to know if this reduction has a name. It is probably some kind of "principle of", like the "principles of determinism" or "principle of local action".


The replacement of the whole history by some (finite dimensional) internal variables is called state space formulation.


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