The Relation Between Mindlin's Strain Gradient and Eringen's Nonlocal Linear Elasticity Theories

Here are some definitions which I have in my mind about the hierarchy of field theories of any kind. Please read them and correct me if there is any misunderstanding. Then I shall raise two questions based on this model of thought about the Mindlin's strain gradient and Eringen's nonlocal linear elasticity theories.

Definition 1. Two field theories are the same if and only if the field equations of one can be obtained from the other.

Definition 2. A field theory is a subset of the other if and only if its field equations are a special case of the other or can be obtained from it in an exact or approximate sense.

As an example, classical linear elasticity field equations can be derived as special cases of both nonlocal and strain gradient theories. So classical elasticity is a subset of these two non-classical theories.

Remark About Definition 2

Note that the field equations include the governing PDE and BCs so considering just one set of these equations does not suffice. There are cases where the PDE of two theories coincide but their BCs are not the same.

With these definitions in mind, here are two questions which step out of mind about these non-classical theories.

Questions

1. Can the field equations, including PDE and BCs, of strain gradient theory proposed by Mindlin be derived from those of the nonlocal theory of Eringen or viceversa?

2. Is it true that Mindlin's strain gradient theory is a nonlocal theory with finite nonlocality range or neighbourhood?