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Extremal principles can generally be divided into two rather distinct classes. There are, on the one hand side, formulations based on the Lagrangian or Hamiltonian mechanics, respectively, dealing with time dependent problems, but essentially resting on conservation of energy and thus being not applicable to dissipative systems in a consistent way. On the other hand, there are formulations based essentially on maximizing the dissipation, working efficiently for the description of dissipative systems, but being not suitable for including inertia effects. Many at-tempts can be found in the literature to overcome this split into incompatible principles. How-ever, essentially all of them possess an unnatural appearance. In this work, we suggest a solution to this dilemma resting on an additional assumption based on the thermodynamic driving forces involved. Applications to a simple dissipative structure and a material with varying mass demonstrate the capability of the proposed approach.

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.

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 FE-square 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.

Model-free data-driven computational mechanics, first proposed by Kirchdoerfer and Ortiz, replaces phenomenological models with numerical simulations based on sample data sets in strain-stress space. Recent literature extended the approach to inelastic problems using structured data sets, tangent space information, and transition rules. From an application perspective, the coverage of qualified data states and calculating the corresponding tangent space is crucial. In this respect, material symmetry significantly helps to reduce the amount of necessary data. This study applies the data-driven paradigm to elasto-plasticity with isotropic hardening. We formulate our approach employing Haigh-Westergaard coordinates, providing information on the underlying material yield surface. Based on this, we use a combined tension-torsion test to cover the knowledge of the yield surface and a single tensile test to calculate the corresponding tangent space. The resulting data-driven method minimizes the distance over the Haigh-Westergaard space augmented with directions in the tangent space subject to compatibility and equilibrium constraints.

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., Comment: 9 pages + 6 pages supplemental material

A variational model of pressure-dependent plasticity employing a time-incremental setting is introduced. A novel formulation of the dissipation potential allows one to construct the condensed energy in a variationally consistent manner. For a one-dimensional model problem, an explicit expression for the quasiconvex envelope can be found which turns out to be essentially independent of the original pressure-dependent yield surface. The model problem can be extended to higher dimensions in an empirical manner. Numerical simulation exhibit well-posed behavior showing mesh-independent results., Comment: Submitted to Cont. Mech. Thermodyn

The excavation process in mechanized tunneling can be improved by reconnaissance of the geology ahead. A nondestructive exploration can be achieved in means of seismic imaging. A full waveform inversion approach, which works in the frequency domain, is investigated for the application in tunneling. The approach tries to minimize the difference of seismic records from field observations and from a discretized ground model by changing the ground properties. The final ground model might be a representation of the geology. The used elastic wave modeling approach is described as well as the application of convolutional perfectly matched layers. The proposed inversion scheme uses the discrete adjoint gradient method, a multi-scale approach as well as the L-BFGS method. Numerical parameters are identified as well as a validation of the forward wave modeling approach is performed in advance to the inversion of every example. Two-dimensional blind tests with two different ground scenarios and with two different source and receiver station configurations are performed and analyzed, where only the seismic records, the source functions and the ambient ground properties are provided. Finally, an inversion for a three-dimensional tunnel model is performed and analyzed for three different source and receiver station configurations.

Modeling of cancellous bone has important applications in the detection and treatment of fatigue fractures and diseases like osteoporosis. In this paper, we present a fully coupled multiscale approach considering mechanical, electrical and magnetic effects by using the multiscale finite element method and a two-phase material model on the microscale. We show numerical results for both scales, including calculations for a femur bone, comparing a healthy bone to ones affected by different stages of osteoporosis. Here, the magnetic field strength resulting from a small mechanical impact decreases drastically for later stages of the disease, confirming experimental research.

When they are damaged or injured, soft biological tissues are able to self-repair and heal. Mechanics is critical during the healing process, as the damaged extracellular matrix (ECM) tends to be replaced with a new undamaged ECM supporting homeostatic stresses. Computational modeling has been commonly used to simulate the healing process. However, there is a pressing need to have a unified thermodynamics theory for healing. From the viewpoint of continuum damage mechanics, some key parameters related to healing processes, for instance, the volume fraction of newly grown soft tissue and the growth deformation, can be regarded as internal variables and have related evolution equations. This paper is aiming to establish this unified framework inspired by thermodynamics for continuum damage models for the healing of soft biological tissues. The significant advantage of the proposed model is that no \textit{ad hoc} equations are required for describing the healing process. Therefore, this new model is more concise and offers a universal approach to simulate the healing process. Three numerical examples are provided to demonstrate the effectiveness of the proposed model, which is in good agreement with the existing works, including an application for balloon angioplasty in an arteriosclerotic artery with a fiber cap., Comment: 42 pages, 17 figures, 3 tables, submit to Journal of the Mechanics and Physics of Solids

Model-free data-driven computational mechanics replaces phenomenological constitutive functions by numerical simulations based on data sets of representative samples in stress-strain space. The distance of strain and stress pairs from the data set is minimized, subject to equilibrium and compatibility constraints. Although this method operates well for non-linear elastic problems, there are challenges dealing with history-dependent materials, since one and the same point in stress-strain space might correspond to different material behaviour.In recent literature, this issue has been treated by including local histories into the data set. However, there is still the necessity to include models for the evolution of specific internal variables. Thus, a mixed formulation of classical and data-driven modeling is obtained. In the presented approach, the data set is augmented with directions in the tangent space of points in stress-strain space. Moreover, the data set is divided into subsets corresponding to different material behaviour. Based on this classification, transition rules map the modeling points to the various subsets. The approach will be applied to non-linear elasticity and elasto-plasticity with isotropic hardening.

Healing of soft biological tissue is the process of self-recovering or self-repairing the injured or damaged extracellular matrix (ECM). Healing is assumed to be stress-driven, with the objective of returning to a homeostatic stress metrics in the tissue after replacing the damaged ECM with new undamaged one. However, based on the existence of intrinsic length-scales in soft tissues, it is thought that computational models of healing should be non-local. In the present study, we introduce for the first time two gradient-enhanced con-stitutive healing models for soft tissues including non-local variables. The first model combines a continuum damage model with a temporally homogenized growth model, where the growth direction is determined according to local principal stress directions. The second one is based on a gradient-enhanced healing model with continuously recoverable damage variable. Both models are implemented in the finite-element package Abaqus by means of a user sub-routine UEL. Three two-dimensional situations simulating the healing process of soft tissues are modeled numerically with both models, and their application for simulation of balloon angioplasty is provided by illustrating the change of damage field and geometry in the media layer throughout the healing process.

We propose an adaptive polygonal finite element formulation for collapse plastic analysis of solids. The article contributes into four crucial points: 1) Wachspress shape functions at vertex and bubble nodes handled at a primal-mesh level; 2) plastic strain rates and dissipation performed over a dual-mesh level; 3) a new adaptive primal-mesh strategy driven by the L^2 -norm-based indicator of strain rates; and 4) a spatial decomposition structure obtained from a so-called polytree mesh scheme. We investigate both purely cohesive and cohesive-frictional materials. We prove numerically that the present method performs well for volumetric locking problem. In addition, the optimization formulation of limit analysis is written by the form of second-order cone programming (SOCP) in order to exploit the high efficiency of interior-point solvers. The present method retains a low number of optimization variables. This convenient approach allows us to design and solve the large-scale optimization problems effectively. Numerical validations show the excellent performance of the proposed method., Comment: This manuscript needs to be improved furthermore

In this paper we propose a canonical variational framework for rate-independent phenomenological geometrically linear gradient plasticity with plastic spin. The model combines the additive decomposition of the total distortion into non-symmetric elastic and plastic distortions, with a defect energy contribution taking account of the Burgers vector through a dependence only on the dislocation density tensor Curl(p) giving rise to a non-symmetric nonlocal backstress, and isotropic hardening response only depending on the accumulated equivalent plastic strain. The model is fully isotropic and satisfies linearized gauge-invariance conditions, i.e., only true state-variables appear. The model satisfies also the principle of maximum dissipation which allows to show existence for the weak formulation. For this result, a recently introduced Korn's inequality for incompatible tensor fields is necessary. Uniqueness is shown in the class of strong solutions. For vanishing energetic length scale, the model reduces to classical elasto-plasticity with symmetric plastic strain sym(p) and standard isotropic hardening., Comment: 5 figures