Materialpunkts

Materialpunkt (plural Materialpunkte; in English often material point or material point particle) is an idealization used in physics and continuum mechanics to represent a finite amount of matter with mass but negligible size. It is a basic building block for describing the motion of continua by tracking parcels of matter rather than extended bodies. Each material point carries properties such as mass, velocity, density, internal energy, and, in continuum models, stress measures according to the chosen constitutive description.

In a continuum description, a material point is identified by reference coordinates X in the initial configuration and a present position x(X,t) that evolves in time. The velocity field is v(X,t) = ∂x/∂t and the acceleration is a = ∂v/∂t. The motion of material points is governed by Newton’s second law: m a equals the sum of external forces and interactions. In this framework, the material description (following individual points) is contrasted with the Eulerian description, which focuses on fields defined at fixed spatial locations. The material derivative D/Dt, defined as ∂/∂t + v · ∇, describes rates observed as one follows a particular material point.

Material points serve as carriers of state in numerical methods and analytical models. They may carry density ρ(X,t), momentum, and stress σ(X,t); their evolution is coupled to a deformation map F = ∂x/∂X and to constitutive laws for elastic, plastic, or viscous behavior. The concept underpins methods such as the Material Point Method (MPM) and, in a broader sense, particle-based approaches like SPH. Limitations include the idealization of zero size and the need to interpolate between points in simulations; it cannot capture sub-point structure directly.