Morphology of the Deformation Curve of Elastoplastic Materials

Abstract

A hierarchical variational generalization of the Reinsch principle is proposed for constructing an adaptive interpolationspline architecture from a finite experimental sample of the deformation curve of an elastoplastic material. The approach separates three interrelated levels. At the first level, stationarity of the integral part of the variational functional determines the admissible structure of local solutions, including their fundamental functions and variationally consistent boundary and contact conditions. The C² class is adopted as the preferred maximum local smoothness class, while information-insufficient segments are represented by the highest admissible lower class according to the hierarchy C²→C¹→C⁰. At the second level, for each fixed architecture, minimization of the discrete part determines the phenotypic coefficients of the local fundamental solutions from the experimental sample. At the third level, DOE (Design of Experiments) is used to organize and compare the successive architectural states of the deterministic genealogy generated by Geometry Builder; no global search over all admissible spline architectures is performed. The genealogical construction starts from a full-range C² model and develops through greedy multiplication. At each iteration, the segment containing the experimental point of the global maximum Euclidean point-to-curve defect is selected as the parent segment. This point becomes the mandatory new internal boundary and the common experimental point of the two child segments. Their highest admissible local classes are determined by information capacity, whereas insufficient approximation accuracy leads to subsequent multiplication rather than artificial reduction of the local class. Previously accepted boundaries are neither relocated nor removed, so the architecture evolves monotonically. Construction terminates either when both the distributed root-mean-square tolerance and the geometric point-to-curve tolerance are satisfied or when no further information-supported multiplication is possible and the information limit is reached. The terminal morphology is defined exclusively by the number of spline segments and the coordinates of their internal boundaries and is therefore distinguished from the complete computational architecture, which additionally contains local smoothness classes, fundamental solutions, phenotypic coefficients, and boundary and contact conditions. A theorem onmorphological stability is formulated, and numerical examples demonstrate automatic construction of different deformation-curve morphologies without prescribing the number of segments in advance. The resulting morphology constitutes an independent geometric result and a fixed basis for the subsequent construction of an approximation template and identification of physical material parameters by the Material Builder module within the Computational Platform for Machine Recognition of Elastoplastic Materials.