Not sure exactly what you are asking, but maybe this helps.

You defined stretching as being associated with tension. But there is also the opposite "compression" situation where you shorten a material with force or pressure.

Conventionally, bending can be thought of as a combination of the two. A typical beam sticking out of a brick wall with a weight hanging on the free end would have tension in the top half and compression in the bottom half. You could, if you wanted, also pull on the beam away from the wall. The effects would sum mathematically and the overall result would be reducing the compressive stress on the lower half but increasing the tensile stress on the top half.

Whether the beam is thick or thin does not matter in my example above. But maybe I am missing your point.

Not sure exactly what you are asking, but maybe this helps.

You defined stretching as being associated with tension. But there is also the opposite "compression" situation where you shorten a material with force or pressure.

Conventionally, bending can be thought of as a combination of the two. A typical beam sticking out of a brick wall with a weight hanging on the free end would have tension in the top half and compression in the bottom half. You could, if you wanted, also pull on the beam away from the wall. The effects would sum mathematically and the overall result would be reducing the compressive stress on the lower half but increasing the tensile stress on the top half.

Whether the beam is thick or thin does not matter in my example above. But maybe I am missing your point.

An infinitessimally thin membrane would not be able to support a bending moment because some finite thickness is required as a moment arm of sorts for the tension and compression. Aluminum foil might be a good practical example.