The Noll lemma for stress representation, originally derived by Noll in the 1950s [1, 2], establishes a rigorous connection between the force vector (or force state) in discrete particle systems at the micro- or mesoscale and the continuum Cauchy stress tensor. This result provides a foundational bridge between particle dynamics formulations, such as peridynamics, and continuum mechanics. The classical form of Noll’s lemma was developed for ordinary particle systems, such as bond-based peridynamics, in which pairwise bond forces are assumed to be collinear with the bond vector. This assumption is consistent with molecular dynamics models governed by central-force interactions. However, it is not generally valid for mesoscale particle systems, where interactions may include non-central forces and the bond force is not necessarily aligned with the bond direction. Examples include micropolar peridynamics, extended bond-based peridynamics, and cohesive peridynamics. In such cases, a direct application of the original lemma can lead to a non-symmetric Cauchy stress tensor, which is incompatible with the local balance of angular momentum in classical continuum mechanics. This limitation indicates that the original formulation of Noll’s lemma is insufficient for general particle interactions. In this work, we develop a generalized form of Noll’s lemma that extends its applicability to general pairwise particle formulations, specifically, ordinary and non-ordinary bond-based peridynamics. The proposed framework accommodates bond forces with both normal and tangential (shear) components at the micro- or meso-scale, and yields a symmetric continuum-scale stress tensor consistent with angular momentum balance.
