Both linear and nonlinear elastic materials will elastically return to an “unloaded” state after loading (without permanent deformations), but the relationship between stress and strain is different in them. It’s linear for linear elastic material (hence the name) and more complex in a nonlinear case.

For a detailed explanation you may refer to the following link:

https://enterfea.com/difference-between-linear-and-nonlinear-elastic-material/#:~:text=Both%20linear%20and%20nonlinear%20elastic,complex%20in%20a%20nonlinear%20case.

hi..your main question is the difference between nonlinear elastic and elastic plastic material..nonlinear elastic - the stress - strain curve is nonlinear in nature under the applied load but upon unloading it fully recovered to original state ..the elastic plastic material - the stress - strain curve is same nonlinear in nature but it wont come back to original upon loading..mean there will be a permanent deformation. In this case, the response of material to the loading is linear in nature up to the ultimate (up to this stage there is a linear relationship between stress and strain - there is a constant of proportionality between the stress and strain) but after reaching to the ultimate state there will be a continuous deformation (this deformation is called plastic deformation)..if you want to understand more details of plastic behaviour of soil, read the text book 'theory of plasticity of soils' by Chen and Martin..ok all the best in your research..

The structure. Molecular chains versus a polycrystalline aggregate. e.g rubber versus metal.

There are in fact two questions in your post - difference between non-linear elastic and elastic-plastic materials and what Abaqus says about the J-integral. The first question was answered above: non-linear elastic means non-linear relation between stress and strain, which is followed upon loading and unloading, so that there is no permanent deformation after full unloading; elastic-plastic means non-linear relation upon loading and in most cases linear relation upon unloading leaving permanent deformation. Regarding the J-integral, what they mean is that it is strictly speaking applicable to non-linear elastic material. However, due to similarity between non-linear elastic and elastic-plastic behaviour upon loading, it is used for elastic-plastic materials modelled with e.g. R-O. The results are correct as far as there is no unloading in any integration point within the contours of integration (which is never the case I am afraid when you do not used deformation plasticity, such as R-O, but use tabulated stress-plastic strain input for e.g. isotopic hardening)

Andrey P Jivkov is right, your question mixes different material models. All given answers tell that linear and nonlinear elasticity is modelled by a bijective relation between stress and strain so that loading and unloading is represented by the same function and no permanent strain remains after complete unloading.

Plastic deformation leads to permanent plastic strain. More important, plasticity is formulated incrementally with distinction between loading, neutral and unloading increments. There is no functional relation between stress and strain.

The deformation theory of plasticity tries to formulate a nonlinearly elastic function for loading and another such function for unloading. This is a simplified theory, which only serves for analytical solutions for restricted load histories.

RO stands for the Ramberg-Osgood function, which is often chosen as approximation of the nonlinear stress strain curve. It is sometimes used with deformation plasticity. Other approaches use tabulated stress strain curves. But remember, there is generally no closed stress strain function in plasticity.