On the truth of integral and differential constitutive forms in strain-driven nonlocal theories with bi-Helmholtz kernels for nanobeam analysis

• This is the first attempt to provide theoretical proofs for the consistency of strain-driven nonlocal theories. • Both IFs and DFs are ill-posed in strain-driven pure and mixed nonlocal theories. • Both IFs and DFs are well-posed and equivalent in strain-driven local/nonlocal theories. • The presence of CBCs is closely tied to IFs and the kernels, but not to DFs. • Verification examples are presented for illustration. Considering various strain-driven nonlocal theories that apply bi-Helmholtz kernels, prior studies pointed out the ill-posedness of differential forms (DFs). However, a perplexing question persists regarding the clear demonstration of true consistencies of integral forms (IFs). In addition, the role of constitutive boundary conditions (CBCs) in IFs is not well understood. To address these problems, we re-visit the existence of CBCs and evaluate the consistencies of both DFs and IFs. In this study, we conduct a comprehensive analysis for Eringen’s nonlocal theory, Eringen’s local/nonlocal theory, nonlocal strain-gradient theory, mixture nonlocal strain-gradient theory and local/nonlocal strain-gradient theory with bi-Helmholtz kernels. Our findings indicate that both IFs and DFs in strain-driven pure and mixed nonlocal theories are indeed ill-posed due to overabundance of boundary conditions. Conversely, IFs and DFs in strain-driven local/nonlocal theories, where classical and higher-order terms in IFs are represented in mixed local/nonlocal forms, are well-posed. Furthermore, for well-posed systems, CBCs can be directly derived from IFs and are essential for both IFs and DFs, being explicit in DFs but implicit in IFs. The presence of CBCs is closely tied to IFs and kernels, but is unrelated to DFs. Result verification is provided through illustrative examples of nanobeam analysis. The research presented here is the first attempt to provide theoretical proofs to illustrate the true consistencies of strain-driven nonlocal theories from both differential and integral perspectives to bridge the theoretical gap.