Theorems of Linear Elasticity extended to Gradient Elasticity and their Applications

• Applicant: Professor Dr.-Ing. Holm Altenbach
• Subject Area: Mechanics ; Mechanical Properties of Metallic Materials and their Microstructural Origins
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 426324717

The classical theory of elasticity is an integral part of the daily routine of engineers. It was placed on a firm theoretical foundation between the beginning of the 19th century and the mid-20th century. Its development can be considered complete.Unfortunately, its scope is limited: It is size insensitive, it contains singularities in the stresses and displacements when discontinuities appear in the boundary data, and can not include boundary and surface energies. Thus, it is limited to typical engineering applications. For the description of micro-components or phenomena in the micron- and nanometer range it is only partially suitable.A natural extension of classical elasticity is the strain gradient elasticity, in which higher derivatives of the displacement field appear. It has been shown in numerous papers that the limitations of classical elasticity theory can be overcome with gradient expansion without blurring the usual separation between structure- and material properties, as is the case with alternative nonlocal theories. Unfortunately, it has not yet been possible to develop a complete solid foundation for gradient elasticity as it exists for classical elasticity theory.This is not a purely academic matter. The increasing miniaturization of components and the targeted development of micro-structured materials require us to go beyond the classical theory of elasticity. Furthermore, by removing the singularities of classical elasticity, we are able to apply a number of criteria (e.g. fracture and flow criteria), which are usually formulated in the Cauchy stresses, also in the vicinity of boundary discontinuities. This significantly increases the applicability of the elasticity theory.In the proposed project, the well-established theoretical foundations of classical elasticity are to be expanded to strain gradient elasticity. For this purpose, a generalizing axiomatic theory has been worked out, about 2/3 of which have already been transferred to the gradient theory. We try to complete this transfer, which is the core of the work of the German project partner. The Russian project partner is concerned with specific applications. For example, uniqueness theorems for boundary value problems with pure displacement or pure stress boundary conditions are applied in homogenization. With them, for example, the Eshelby fundamental solution of an elliptical inclusion in an infinite matrix can be extended. Another application are transversely isotropic fiber-reinforced composites, for which both a scale transition and the specific properties of the stiffness tensor are to be investigated. Finally, the de Saint-Venant principle for gradient elasticity will be investigated in beam experiments.

• DFG Programme: Research Grants
• International Connection: Russia
• Partner Organisation: Russian Science Foundation, until 3/2022
• Co-Investigator: Privatdozent Dr.-Ing. Rainer Glüge
• Cooperation Partner: Professor Francesco del Isola, Ph.D., until 3/2022