Reinhold Kienzler

Comparison of various linear plate theories in the light of a consistent second-order approximation:

The uniform-approximation technique in combination with the pseudo-reduction technique is applied in order to derive consistent theories for isotropic and anisotropic plates. The approach has already been used to assess and validate theories established in the literature, e.g., the theories of Reissner and Zhilin. In this contribution, we also present a comparison with the theories of Vekua, Ambartsumyan, Steigmann and Reddy’s third-order theory. The current paper is a corrected and extended version of the one originally published in Kienzler and Schneider (Comparison of various linear plate theories in the light of a consistent second-order approximation. In Pietraszkiewicz, W and Górski, J (eds), Shell Structures: Theory and Applications. London: Taylor & Francis Group, 2014, pp. 109–112). In addition we put special emphasis on the derivation and validation of Ambartsumyan’s general and simplified theories.

An Algorithm for the Automatisation of Pseudo Reductions of PDE Systems Arising from the Uniform-approximation Technique

One way to develop theories for the elastic deformation of two- or one-dimensional structures (like, e.g., shells and beams) under a given load is the uniform-approximation technique (see [2] for an introduction). This technique derives lower-dimensional theories from the general three-dimensional boundary value problem of linear elasticity by the use of series-expansions. It leads to a set of power series in one or two characteristic parameters, which are truncated after a given power, defining the order of the approximating theory. Finally, a so-called pseudo reduction of the resulting PDE system in the unknown displacement coefficients is performed, as the last step of the derivation of a consistent theory. The aim is to find a main differential equation system (at best a single PDE) in a few main variables (at best only one) and a set of reduction differential equations, which express all other unknown variables in terms of the variables of the main differential equation system, so that the original PDE system is identically solved by inserting the reduction equations, if the main variables are a solution of the main differential equation system. To find a valid pseudo reduction by inserting the PDEs of the original system into each other is a complicated and very time-consuming task for higher-order theories. Therefore, an structured algorithm seeking all possibilities of valid pseudo reductions (to a given number of PDEs in a given number of variables) is presented. The key idea is to reduce the problem to finding a solution of a linear equation system, by treating each product of different powers of characteristic parameters with the same variable as formally independent variables. To this end, all necessary equations, which can be build from the original PDE system, have to be identified and added to the system a-priori.

Influences of the spray deposition process on the properties of copper and copper alloys

Abstract The microstructure and the mechanical properties of spray deposited materials depend critically on the conditions that are present during deposition and may be modified through changes of the processing parameters. Within the scope of the special research program at the University of Bremen, investigations of the influences of various process and production related parameters on the spray deposition behaviour and the resulting material properties of copper based materials have performed. In order to achieve a fundamental analysis, the investigations were at first focused on pure copper to eliminate influences of alloying elements. Based on these results, investigations of several copper alloys, especially Cu–Sn, Cu–Ni and Cu–Cr with different compositions have been started. The present paper gives an overview of the microstructure development of sprayed pure copper and selected copper alloys in dependence on the process parameters and the process technology supplemented by results of numerical process simulation.

On a new method for calculating stress-intensity factors of multiple edge cracks

Stress-intensity factors for multiple edge cracks in bars and beams have been calculated using geometrical considerations to provide quick and easy-to-use estimates. The research concentrates on symmetric double and triple edge crack configurations. The results have been verified by FE calculations. Furthermore, the analytical estimate has been improved based on numerical results.

Direct Approach Versus Consistent Approximation

Relations between plate theories resulting from the direct approach and the consistent approximation are established and the resulting equations are compared. By introducing a scalar measure for the thickness strain, both theories can be reconciled within a consistent second-order approximation.

Teaching Nonlinear Mechanics: An Extensive Discussion of a Standard Example Feasible for Undergraduate Courses

Most courses on mechanics of materials use a linearized (second-order) buckling analysis of a simple elastic system as an introductory example. We propose to start with a third-order buckling analysis instead, to enable the students to understand the crucial load-response diagrams from the beginning of the course. We present an extensive mathematical discussion of an extended standard introductory example, leading to an easy-to-implement plotting routine for load-response diagrams. The resulting diagrams are interpreted in physical terms. An implementation of the plotting algorithm using Maplesoft MapleTM is attached.

Teaching Fracture Mechanics Within the Theory of Strength-of-Materials

Some elements of fracture mechanics, such as energy-release rates and stress-intensity factors, might be examined not on the basis of continuum theories, but on the basis of the much simpler theories of strength-of-materials. Thus it becomes possible to teach fracture mechanics in undergraduate courses for engineers.

Dimensioning of thick-walled spherical and cylindrical pressure vessels

In this contribution, we revisit the rather classical problem of Lamé and provide a novel and easy way to plot the stress distributions and the overall absolute maximum von Mises stress for arbitrary parameters in only two diagrams. We also provide a maximum hoop stress formula for combined loading and an extensive discussion covering the accuracy of dimensioning via the maximum hoop stress instead of the maximum von Mises stress, as well as the accuracy of the classical approximative hoop stress formulas.

Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure

Emphasis in modern day efforts in mechanics of materials is increasingly directed towards integration with computational materials science, which itself rests on solid physical and mathematical foundations in thermodynamics and kinetics of processes. Practical applications demand attention to length and time scales which are sufficiently large to preclude direct application of quantum mechanics approaches; accordingly, there are numerous pathways to mathematical modelling of the complexity of material structure during processing and in service. The conventional mathematical machinery of energy minimization provides guidance but has limited direct applicability to material systems evolving away from equilibrium. Material response depends on driving forces, whether arising from mechanical, electromagnetic, or thermal fields. When microstructures evolve, as during plastic deformation, progressive damage and fracture, corrosion, stress-assisted diffusion, migration or chemical/thermal aging, the associated classical mathematical frameworks are often ad hoc and heuristic. Advancing new and improved methods is a major focus of 21st century mechanics of materials of interfaces and evolving microstructure. Mathematics Subject Classification (2010): 74xx. Introduction by the Organisers The workshop Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure attracted about 50 participants with broad geographic representation from Europe and the United States. This workshop was a well balanced blend of researchers with backgrounds in mathematics, mechanics and materials science. 798 Oberwolfach Report 17/2016 The organizers successfully recruited a significant number of younger representatives of the mentioned research communities. One of the most pressing current trends in materials discovery, design and development involves the intersection of applied mathematics and computational/data sciences with mechanics of materials and computational materials science to enhance understanding and to produce improved next generation computational methods and materials modelling tools. This places increased emphasis on the predictive nature of computational mechanics pertaining to realistic material nanostructures and microstructures. Material complexity is high today’s leading materials are hierarchical, having characteristics of structure at multiple length scales to satisfy a complex set of performance requirements and constraints. This is true for materials in electronic devices, in automotive and aerospace applications, multilayers for electronics and MEMS applications, interfaces for catalysis and chemical separations, and numerous application domains. This state of affairs motivated the central theme of this workshop, namely to explore new and emerging mathematical approaches to describe interface properties and to quantify evolving microstructures. Owing to the mutual dependence of properties on various scales, improved methods (both physically and mathematically based) must be developed to describe correctly equilibrium and nonequilibrium microstructure evolution at multiple length scales and time scales. There are serious unresolved physical and mathematical issues in applying phase field theory to realistic microstructures, up-scaling atomistic simulations to relevant time and length scales, how to address the complexity of potential energy landscapes that govern interface structure and microstructure rearrangements in materials undergoing deformation, damage, and phase/structure transitions, mathematical methods to manage complexity of many body structure and defect fields in real materials (big data and inverse modelling/data analytics), and mathematical approaches to representing grain boundary and phase interfaces in materials with an eye towards field theories that facilitate up-scaling. Based on the above outline of current and highly relevant topics and the experience gained in organizing preceding workshops on mechanics of materials, the following main topics were suggested: • Phase field modeling and relations to realistic microstructures at multiple scales, as well as other approaches to similar field equations. • Coarse graining atomistic simulations to longer length and time scales to support materials development. • Mathematical methods for convergence in homogenization of materials with microstructure (e.g., Γ-convergence). • Thermodynamics and kinetics of evolving microstructure, including novel mathematical approaches to explore complex energy landscapes of interfaces and heterostructures. • Methods for reducing the order of continuum descriptions and mathematical techniques and data sciences approaches to address the ”big data” Mechanics of Materials 799 aspect of complex hierarchy of material structure and its relation to properties. • Mathematical representation of evolving interfaces to facilitate field theory approaches that bridge with atomistic and continuum levels of treatment. The sequence and duration of the sessions were defined on Monday morning. They were moderated by session chairs and each consisted of 1-2 overview lectures (3040 minutes each, including discussion), along with several short “thought piece” contributions (2-3 presentations, each 15 minutes). Ample time was devoted to discussion, both during and following presentations. The format of sessions (subject to hard stops for lunch and dinner), including coffee breaks, gave the flexibility to maximize productive discussion. Reports from the session chairs were summarized and discussed extensively on Friday. The organizers regard this particular workshop as among the most successful in the topical area of Mechanics of Materials in recent memory, for several reasons. First, a number of young participants were involved and highly active in presentations and discussions, representing the next generation of blending applied mathematics with mechanics of materials. Second, the discussions were detailed and animated from the outset, with many useful points and counterpoints discussed. We believe that this workshop has launched many potentially fruitful couplings of researchers in Europe and the USA, and has defined some specific target areas as goals for mathematics, including extended methods that can consider convergence characteristic of homogenization methods for evolving microstructure, include viscoplastic, dissipative cases, extreme value properties/responses in addition to mean responses, methods for constrained optimization of nonconvex energy potentials with relevance to shear banding and formation of laminate microstructures, and enhancement of mathematical approaches for modelling the role interfaces. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Mechanics of Materials 801 Workshop: Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure

On influence surfaces in material space

Based on reciprocity relations in material space, influence surfaces are established for the evaluation of the energy changes between interacting defects. In this way, the change of the driving force on a crack tip due to changes of the position, size and orientation of neighboring defects can be evaluated. The method enables to assess defect configurations in an elastic material in a straight-forward manner.