In the damage analysis of material with complex microstructures, a direct numerical simulation (DNS) could produce huge computational costs. Alternatively, the multiscale modeling is a more effective method by using less degrees of freedom to balance the computational accuracy and cost. In this paper, a new multiscale method for damage analysis is proposed based on the framework of the variational effective model for elastoplastic problems originally presented in author’s previous work. The key idea is to construct variational statement for the free energy and the dissipation potential of a coarse scale model by relating the free energy and the dissipation potential of a fine scale model. In this way, the damage evolution can be directly simulated at the coarse scale solution, resulting in significantly higher computational efficiency compared to the DNS in fine scale. Compared with previous work, a new multiscale computational scheme is derived for damage evolution. In addition, the relaxation damage model is used to effectively avoid ill-posed boundary value problems in damage analysis without using gradient enhanced or integration techniques. Four numerical examples demonstrate the effectiveness of the proposed method in both 2D and 3D problems by comparing its computational accuracy and cost with reference solutions from DNS and FE2 algorithm.

In this work, we suggest a framework for modeling evolution of phase-fractions in elastoplastic materials associated with phase transformations. The model is based on variational principles for inelastic materials. We restrict ourselves to infinitesimal strains and isotropic materials. In our work we employ the so-called dissipation distance, which describes an immediate phase transition in time via an underlying probability matrix. The volume fractions of the various phases are represented by Young measures to obtain a time continuous microstructure evolution. A minimum principle is presented to govern the initiation of new phases. The model is verified employing a two-dimensional benchmark test implemented by the Finite Element Method.

The construction sector significantly contributes to global greenhouse gas emissions, primarily from cement production. To mitigate this, a strategy focusing on reusing structural components from existing reinforced concrete structures is being explored. This study highlights challenges and presents initial results in designing new structures from reused elements. The objective is to develop methods for designing load-bearing structures using available elements from demolished buildings, categorized in a construction kit. The challenge is to find a structure that meets load-bearing capacity and architectural demands under the constraints of available elements. The feasibility of integrating existing foundations into the design process is investigated. Non-destructive measurements and simulations characterize the foundation and soil properties, while methods for strengthening or adjusting the foundation are developed. The design process considers the constraints of the foundation and construction kit, with the coordinated arrangement of reused elements and connection types controlling stress distribution. The structural reliability of the proposed structure is assessed, quantifying the effect of uncertainties related to individual elements. Modulare Strukturen aus wiederverwendeten Bauteilen: Herausforderungen bei der Nutzung von Bestandsgr & uuml;ndungenDie Bauwirtschaft tr & auml;gt durch die Zementproduktion erheblich zu den globalen Treibhausgasemissionen bei. Zur Reduktion soll eine Strategie zur Wiederverwendung von Stahlbetonbauteilen vorgestellt werden. In diesem Beitrag wird die Entwicklung von Methoden f & uuml;r den Tragwerksentwurf unter Verwendung verf & uuml;gbarer Bauteile, die aus abzurei ss enden Geb & auml;uden entnommen wurden und in einem Baukasten-System kategorisiert sind, pr & auml;sentiert. Die zentrale Herausforderung besteht darin, eine Struktur zu finden, die eine ausreichende Tragf & auml;higkeit aufweist und den architektonischen Anforderungen gen & uuml;gt. Hierbei sollen bestehende Fundamente in den Entwurfsprozess integriert werden. Dazu werden zerst & ouml;rungsfreie Messungen mit Simulationen kombiniert, um die Eigenschaften von Fundament und Boden zu charakterisieren. Weiterhin wird die M & ouml;glichkeit einer Verst & auml;rkung durch Biozementierung untersucht. Im optimierungsgesteuerten Entwurfsprozess werden die Fundamente und verf & uuml;gbaren Bauteile als Randbedingungen ber & uuml;cksichtigt, wobei der Kraftfluss durch die gezielte Anordnung der Bauteile und entsprechender Verbindungstypen gesteuert werden kann. Die Zuverl & auml;ssigkeit der abgeleiteten Tragstrukturen wird durch nichtlineare Simulationen bewertet und Unsch & auml;rfen werden bez & uuml;glich des Bauteilzustands quantifiziert.

A general model is formulated for elasto-plastic materials undergoing linear kinematic hardening to describe microstructure evolution associated with phase transformations. Using infinitesimal strain theory, the model is based on variational principles for inelastic materials. In our work we combine the so-called dissipation distance, which describes an immediate phase transition in time via an underlying probability matrix. In addition, the volume fractions of the newly emerging phases are represented by Young measures to obtain a time continuous microstructure evolution. The model is verified employing a two-dimensional benchmark test implemented by the Finite Element Method (FEM).

The compaction mechanisms of SiO2 glass under pressure include under certain conditions a specific reduction of the elastic moduli and a complex inelastic behavior whose nature is not yet fully understood. In our work we establish a variational framework describing the evolution of SiO2 glass under hydrostatic pressure. Based on a previous work that presents a model for multi-phase transformations in inelastic materials, we assume isothermal conditions during a compaction process and interpret the typical sigmoidal stress response as indicator of a binary phase transformation. During the process, two volume fractions coexist macroscopically and microstructures such as shear bands or disclination pattern develop in between. We restrict our approach to resolve only the volume fractions, not the corresponding microstructures. Nevertheless, the resulting model is shown to match experimental findings very well. Numerical examples successfully illustrate the relationship between the changes in the elastic moduli and the corresponding change in volume with respect to pressure.

ABSTRACT In this proceeding, a phase‐field model to describe the evolution of size, shape, and composition of volcanic crystals is introduced. It is built on top of an existing model for idealized spherical crystals but uses the phase‐field approach to investigate arbitrary initial shapes and considers the diffusion of diffusion elements as well as heat. Furthermore, it also takes the development of the dislocation density into account, which makes it possible to make predictions about the stability of the crystals. In this proceeding, the effects of different initial configurations and the behavior of the model under instability are investigated.

Constitutive material modeling is an important basis for mechanical analysis. Machine learning methods based on neural networks have been extensively used for the discovery of material constitutive laws. Recently, Flaschel et al. (Comput Methods Appl Mech Eng 381:113852, 2021) proposed a new “unsupervised” framework that does not need any stress labels, which are usually difficult to measure in experiments. However, there is no work on the application of the unsupervised machine learning models for multi-regional constitutive models of heterogeneous materials. To address this issue, this paper develops a new unsupervised multi-regional constitutive learning model. In the implementation of this machine learning model, two indicators, i.e., the residual nodal force indicator and the free energy indicator, are first defined to identify nonhomogeneous interfaces or material distributions. Then, the combined identification of multi-regional constitutive models is conducted by adding a switch function to the extended strain invariants layer of the neural network. Finally, multi-regional constitutive models can be trained in one neural network at the same time. The effectiveness of the proposed model is verified through numerical simulation examples including an application to constitutive modeling of soft biological tissues undergoing regional damage.

A new thermoelastic model is introduced to reveal equivalent mechanical and thermal properties of randomly oriented (RO), agglomerated carbon nanotube (CNT) inclusions within a matrix material. Thereafter, a bioinspired FG-CNTR-TPMS material model is established through three typical triply periodic minimal surfaces (TPMS) microstructures reinforced with CNTs and functionally graded (FG) schemes. The free vibration behavior of macro-scale plates made from FG-CNTR-TPMS materials under thermal effects and material temperature dependencies is then devised. A new higher-order shear deformation (HSDT) five-variable plate theory incorporated with isogeometric analysis (IGA) is proposed to show its reliability and efficiency. Various material conditions have been thoroughly studied, emphasizing the influence of CNT reinforcement states and environment temperatures. While increasing the CNT volume fraction (fr) greatly improves the plate frequencies, the temperature rise (Delta T) leads to an opposite influence. These properties can amplify or weaken the effects of porosity distributions on the plate's natural frequencies both beneficial and unfavorable aspects. Notably, the outstanding elastic modulus of IWP-type TPMS enlarges the initial thermal stress and ratio between mechanical and thermal stiffness, causing greater impacts on plate behaviors in the thermal environment. In some exceptional cases, FG-CNTR-TPMS plates with P-type can exceed isotropic plates in stiffness-to-weight ratios, which is an extraordinary characteristic of porous structures. In various scenarios of CNT agglomeration, this natural phenomenon shows noticeable reductions in plate frequencies up to 40% when considering temperature changes. Bridging the gap between TPMS-based lattice structures and CNTreinforced composites, this study contributes to advancing the knowledge of advanced bio-inspired materials. The findings from this work can revolutionize their potential applications in various engineering areas, particularly biomedical devices, energy storage, flexible electronics, advanced textiles, and soft robotics, where lightweight, high-strength, and temperature-resistant structural components are critical.

The compositional record in minerals formed by element partitioning is the basis of many tools used in geochemistry and petrology such as geothermometry, geobarometry and geochronology. Compositional resetting in response to changes in ambient conditions is considered largely in the context of chemical (such as diffusion, dissolution, precipitation or growth) or mechanical deformation processes. Here we develop a model to show that lattice strain resulting from atomic size mismatch during chemical exchange may lead to recrystallization of grains and thus, erasure of previous records. The model includes the processes of diffusion, solidification and melting, elastic deformation and dislocation motion. Analysis of the parameters of the model shows that crystals undergoing chemical change may be divided into two groups. One group of crystals is stable and grows continuously while the other group of crystals may begin to shrink because the dislocation density, resulting from internal lattice strain due to element partitioning, reaches a critical value. These grains eventually recrystallize. The results of this study have implications for the understanding of closure behavior of geothermometers, geobarometers and geochronometers as well as for the interpretation of results of diffusion chronometry of rocks. Such processes are likely to play an important role in the process of nucleation of minerals. Coupled with models of mechanical evolution, the results of this study carry implications for the evolution of grain size, rheology and ultimately, the behavior of lithospheric plates.

Beim R & uuml;ckbau eines Geb & auml;udes verbleiben die Gr & uuml;ndungselemente oft im Boden, da ihre Entfernung und Wiederverwendung technisch und wirtschaftlich herausfordernd ist. Eine Wiederverwendung vor Ort bietet jedoch & ouml;kologische und & ouml;konomische Vorteile, da Ressourcen geschont, Emissionen reduziert und Bauzeiten verk & uuml;rzt werden. Voraussetzung hierf & uuml;r ist eine detaillierte Charakterisierung der Bestandsfundamente hinsichtlich ihrer strukturellen Integrit & auml;t, m & ouml;glicher Sch & auml;den und der verbleibenden Tragf & auml;higkeit. Einschr & auml;nkungen ergeben sich insbesondere durch die unver & auml;nderliche Bewehrungsanordnung, die die Lastaufnahme f & uuml;r neue Bauwerke begrenzt. Gleichzeitig kann eine Verst & auml;rkung oder Erg & auml;nzung der Bestandsfundamente erforderlich sein, was oft mit hohen Treibhausgas-Emissionen verbunden ist. Die Biozementierung mittels bakterieller Calciumcarbonat-Ausf & auml;llung bietet eine nachhaltige Alternative, deren mechanische Eigenschaften und Verbundwirkung mit Bestandsgr & uuml;ndungen jedoch noch erforscht werden m & uuml;ssen. Die Nutzung von Abwasser zur Biozementierung durch Ureolyse sowie der alternative Pfad der Denitrifikation sind im Hinblick auf die Reduktion der Kosten und eine Steigerung der Effizienz vielversprechend, jedoch bisher weitestgehend unerforscht. Dieser Beitrag beschreibt zwei Teilprojekte des SFB 1683 ,,Interaktionsmethoden zur modularen Wiederverwendung von Bestandstragwerken": B04, das sich mit der Charakterisierung von Gr & uuml;ndungen mittels inverser Identifikationsmethoden befasst, und A04, das bio-basierte Anpassungen von Bestandsgr & uuml;ndungen f & uuml;r eine nachhaltige Wiederverwendung untersucht. Beide Projekte sind Teil der Interaktionskette 1 ,,Zirkul & auml;re modulare Tragwerke aus wiederverwendeten Bauteilen" und leisten einen Beitrag zur Optimierung von Tragwerksplanung, Fundamentverst & auml;rkung und Zuverl & auml;ssigkeitsbewertung. Characterization and sustainable improvement of existing foundationsWhen deconstructing a building, foundation elements often remain in the ground, as their removal and reuse are technically and economically challenging. However, on-site reuse offers ecological and economic benefits. A prerequisite for this is a detailed characterization of the existing foundations in terms of their structural integrity, potential damage, and remaining load-bearing capacity. Limitations arise, particularly due to the immutable reinforcement arrangement, which restricts the load-bearing capacity for new structures. Strengthening or supplementing the existing foundations may be necessary, which is often associated with high greenhouse gas emissions. Bio-cementation through bacterial calcium carbonate precipitation offers a sustainable alternative, but its mechanical properties and bonding effect with existing foundations still need to be investigated. The use of wastewater for bio-cementation through ureolysis, as well as the alternative path of denitrification, are promising with regard to cost reduction and efficiency improvement, but have so far been largely unexplored. This paper describes two sub-projects of the SFB 1683: B04, which deals with the characterization of foundations using inverse identification methods, and A04, which investigates bio-based adaptations of existing foundations for sustainable reuse.

Based on Hamilton's principle of stationary action, we present a holistic variational formulation for material modeling including dissipative evolution. To this end, we recall the definition of the action as path integral of the momentum vector. Reformulation of the action and inserting the 1st and 2nd Law of Thermodynamics yield an extended Hamilton functional. We show that the stationarity conditions yield well-known expressions as well as new conditions in an extended nested time domain. Introducing an asymptotic two-scale approach transforms the expressions in the nested time domain back to the physical time. Hereby, we receive usual differential equations, e.g., heat conductivity equation, diffusion equation, and Biot equation, and the constitutive laws for, e.g., temperature, entropy, and chemical potential, all from one holistic stationarity principle. Moreover, the formulation in the nested time domain produces additional, virtual conditions that naturally lead to the concept of dissipation distances. Due to its variational origin, our approach yields in a consistent manner a coupled space-time formulation.

While most previous developed metamaterials only consider a single physical effect, we introduce a novel class of electro-mechanical metamaterials, which allows a direct controllable reduction of the total stress by applying an electric field counteracting the mechanical stress. The solution of the resulting minimization problem yields a relation involving the eigenvalues of the mechanical stress tensor. Additionally, we evaluate the constrained cases allowing only tensile or compressive stresses, respectively, and consider the plane stress problem. We show numerical results for all cases and discuss, to what extent a stress reduction is possible.

In our previous contributions we established a multiscale, multiphase material model for the simulation of cancellous bone with the novel idea of including the full coupling of mechanical, electric and magnetic effects, which could be used for example, for the early detection of osteoporosis. While our calculations have already shown promising results, our previous approach lacks very important aspects, strongly limiting the applicability of our findings. In this paper we extend our base model by considering the effect of a surrounding medium on our bone specimen, using improved boundary conditions and differentiating between cortical bone, bone marrow and spongy bone to better reflect the physiological properties of bone. We show numerical results and compare our calculations to our previous modeling.

A matter of fact is that extremal principles have been introduced in mechanics in more (Euler, Lagrange) or less (Hamilton) than 200 years ago. One may also observe an impact of thermodynamic extremal principles based on maximum dissipation due to all the entropy production expressed in several disciplines. According fields are theory of communication, statistical mechanics and later physics of earth since already 70 years. The current paper offers some (historical) overview on several applications. “Ziegler’s principle” is an implementation of the maximum entropy production going out to the dissipation and yielding a maximum dissipation. The goal of this paper is now the implementation of this extremal principle performed along an algebraic concept. Such a concept can be extended to a system with several internal variables as outlined by Coleman and Gurtin in context with the Gibbs (free) energy.

The prevalence of osteoporosis is about one in three for women and one in five for men over the age of fifty, making it the most common bone disease worldwide. The disease is characterized by a reduction of the amount of volume percent of cortical bone, weakening the bone and increasing the likelihood of fractures. Modeling and numerical simulations can be used to better understand effects observed in measurements and thus the workings of human bones. Furthermore, they can be used to support the development of new diagnostic tools such as sonography. For this purpose, in previous contributions we developed a two-scale bone model that considers mechanical, electric and magnetic effects. To connect the scales, we resorted to the finite element square method FE 2 ${\text{FE}}^{2}$ . For diagnostics, the main quantity of interest is the resulting magnetic field strength, which can be used to draw conclusions about the bone health. In this contribution, we investigate the influence of the used microstructure model in detail. We created different representative volume elements (RVEs), which are for example, randomly orientated, anisotropic, or differ in shape or mesh resolution. In addition, we consider the use of different shape functions in the finite element calculation. We compare our findings with previous results, which were obtained using only regular RVEs. We investigate the extent, to which the microscale results and the overall macroscale simulation results are affected by the choice of the RVE.

Model-free data-driven computational mechanics, first proposed by Kirchdoerfer and Ortiz, replace phenomenological models with numerical simulations based on sample data sets in strain-stress space. In this study, we integrate this paradigm within physics-informed generative adversarial networks (GANs). We enhance the conventional physics-informed neural network framework by implementing the principles of data-driven computational mechanics into GANs. Specifically, the generator is informed by physical constraints, while the discriminator utilizes the closest strain-stress data to discern the authenticity of the generator's output. This combined approach presents a new formalism to harness data-driven mechanics and deep learning to simulate and predict mechanical behaviors.

For the modelling of complex materials, internal variables are usually introduced which characterize the microstructural state. Then, evolution equations describe the change of the internal variables due to varying external loading conditions. These equations can be derived, for instance, on the basis of variational principles. The consideration of characteristic observations, such as the preservation of the volume during a change in the microstructural state, can significantly improve the accuracy of the evolution equations. We present a Hamilton principle that provides a unique way to derive evolution equations that obey holonomic constraints and opens up new possibilities for their algorithmic treatment. This is demonstrated for isochoric finite plasticity and phase transformation based on Backward-Euler time discretization. The models presented are efficient and are characterized by simple implementation compared to the exponential map, for example, without suffering a loss of accuracy due to unfulfilled constraints.

AbstractThe evolution of microstructures can be observed in various engineering and natural materials, such as special alloys, silica glasses or soils. These macroscopic and microscopic effects are often controlled by thermal and chemical changes or extern forces. In addition, several phases are usually involved in these processes, whose energy potentials result in a non‐convex total energy. A theory is introduced that can be applied to a wide range of elasto‐plastic materials with kinematic hardening. In order to obtain a general model, a relaxed free energy in small strain theory is defined using variational principles for inelastic materials. This energy includes the dissipation of the substance as so‐called dissipation distance to facilitate a time‐stepped incremental representation. Furthermore, the phase volume fractions are contained in the weighted sum of the individual energies and are expressed here using Young measures. In the centre of the model, transition rates with underlying ordinary differential equations (ODEs) are calculated, which allow access to the phase evolution of the system. The initial results and the behaviour of this theory are explained using numerical simulations created with the programming language Julia.

Given a set of inelastic material models, a microstructure, a macroscopic structural geometry, and a set of boundary conditions, one can in principle always solve the governing equations to determine the system's mechanical response. However, for large systems this procedure can quickly become computationally overwhelming, especially in three-dimensions when the microstructure is locally complex. In such settings multi-scale modeling offers a route to a more efficient model by holding out the promise of a framework with fewer degrees of freedom, which at the same time faithfully represents, up to a certain scale, the behavior of the system. In this paper, we present a methodology that produces such models for inelastic systems upon the basis of a variational scheme. The essence of the scheme is the construction of a variational statement for the free energy as well as the dissipation potential for a coarse scale model in terms of the free energy and dissipation functions of the fine scale model. From the coarse scale energy and dissipation we can then generate coarse scale material models that are computationally far more efficient than either directly solving the fine scale model or by resorting to FE2 type modeling. Moreover, the coarse scale model preserves the essential mathematical structure of the fine scale model. An essential feature for such schemes is the proper definition of the coarse scale inelastic variables. By way of concrete examples, we illustrate the needed steps to generate successful models via application to problems in classical plasticity, included are comparisons to direct numerical simulations of the microstructure to illustrate the accuracy of the proposed methodology.