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Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the so-called energy form of the problem is applied to the primary variable as an additional constraint for training. The proposed approach only required up to first-order derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulation-based PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids to show the capability of deep learning for predicting the solution in a complex environment under different boundary conditions. The performance of the proposed PINN model is checked against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steady-state diffusion problem). We concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and FEM for a fast and accurate design of composite materials in future developments.

Joining by plastic deformation is recognized as a promising technique to produce laminate metal composites. In this technique, metallic layers are joined together through a metallurgical bond created by severe plastic deformation. Obviously, the accurate design of these processes requires adequate numerical tools which take into account the most important factors affecting the bond formation. In this paper, first, the mechanism of joining by plastic deformation is outlined. Then, based on the microscopic description of solid-state welding, an extended version of a cohesive zone element is introduced to describe both bonding and debonding processes in finite element modeling. In this element, the bond strength between metallic layers is calculated based on the governing parameters of the bonding process such as pressure, plastic deformation and surface cleanness. Subsequently, the mathematical formulation of the introduced element and its finite element implementation are presented. Finally, a numerical example is given to demonstrate the applicability of the proposed method in cold roll bonding processes. The element formulation introduced in this paper can be adopted by other finite element subroutines to simulate the process of bonding and debonding in engineering applications.

Grain boundary (GB) characteristics play an important role in the determination and prediction of material behavior, especially when it comes to nanocrystalline metals and ceramics. The main goal of this work is to develop a general interface model to accurately incorporate grain boundary sliding as well as intergranular fracture as two main phenomena in characterizing the grain boundary. To gain a deeper insight into the behavior of different grain boundaries, molecular dynamics (MD) simulations are utilized for mode I and mode II loadings. By adding the unloading path to the MD simulations it was possible to differentiate between different active mechanisms at the GB. Current MD investigations motivate a model which accounts for anisotropic plasticity and damage within the grain boundary to capture the complex interface behavior. Therefore, a two-surface formulation is utilized in which damage and plasticity at the interface are coupled in a thermodynamically consistent way. The parameters for the introduced interface model are determined using the MD simulations based on an embedded atom potential. Finally, the calibrated interface model is implemented into a cohesive zone (CZ) element. In order to show the applicability of the proposed interface model, several numerical studies are carried out. A volume element is selected which depicts a point in an arbitrary polycrystalline material at the macroscale. The results of these studies reveal interesting behaviors of the selected volume element which can be used, e.g., to determine the parameters of a continuum damage model at the macroscale.

The paper aims to present a finite element model for the bonding process of metals at the microscale. To accomplish this, first, the mechanism of joining by plastic deformation at the microscopic level is explained. Then, based on the film theory of bonding, a finite element model is developed, which enables to simulate the bonding process between metallic layers subjected to large plastic deformation. The model presented in this paper takes into account the most important physical micro-mechanisms taking place during the bond formation process, i.e. (1) the breakage of the brittle oxide layer above the metallic surfaces, (2) the decohesion process occurring between the oxide layer and the metal substrate, (3) the extrusion of the substrate into the created cracks under large plastic deformations, and (4) the bond formation in between the fractured oxide layers. In addition, an extended version of a cohesive zone model is proposed to describe the bond formation between the metal surfaces. Finally, it is shown that the model can be used to provide a description regarding bond strength evolution. In this context, the effects of influencing factors, such as the degree of deformation and the thickness of the oxide layer, are numerically investigated. The presented finite element model can be regarded as a useful tool to characterize the key factors in joining processes such as roll bonding and cold forging.

The mechanical behavior of coating systems is studied by means of a novel numerical modeling approach. The damage and fracture evolution within the coating system are modeled employing cohesive zone (CZ) regions in addition to an elasto-plastic model. The focus of this paper is on hard coatings such as (Cr,Al)N deposited by the combination of the direct current magnetron sputtering (dcMS) and the high power pulsed magnetron sputtering (HPPMS) method. The elasto-plastic properties of the coating are identified indirectly by a nanoindentation test. The morphology of the coating layer is visualized by a scanning electron microscope (SEM). Further the deformation profile after nanoindentation is measured using confocal laser scanning microscopy (CLSM). The numerical results on the measured surface profile and the required force in the nanoindentation test are compared. The coating fracture behavior is investigated by studying the influence of different parameters such as elasto-plastic properties of the coating system and the cohesive zone parameters.

Damage and fracture can be induced not only by mechanical loading but also due to chemical interactions within a solid. On one hand, species concentration may embrittle or toughen the material and on the other hand, the mechanical state adds additional driving force for diffusion. We propose a chemo-mechanically coupled cohesive fracture model with several novel features. It distinguishes the mode-dependent damage progression and its influence on lithium transport. Coupled with mode-dependent cohesive zone damage, the model recaptures both the normal and tangential transport behavior of lithium at the interface. Moreover, it tackles concentration-dependent crack initiation, various softening behavior, as well as the cyclic damage accumulation. The thermodynamic consistency of the proposed model with the mentioned features is demonstrated. The model is numerically implemented with the finite element method. Numerical results, along with comparison with related experimental data, demonstrate that the model can be applied to study diffusion-induced fracture in general solid ionic conductors in Lithium-ion batteries. In particular, illustrative numerical results are presented for both the intergranular fracture inside active material or solid electrolyte and the interface fracture between active material and solid electrolyte. Furthermore, it is discussed how the solid electrolyte influences the dominant crack patterns. The current contribution is applicable to address similar problems on hydrogen-induced cracking and moister-dependent fracture.

The unique mechanical properties and transport features of grain boundaries (GBs) in polycrystalline materials have been widely investigated. However, studies which focus on the unique chemo-mechanics phenomena resulting from GBs’ are exceedingly sparse. In this work, a thermodynamically consistent framework has been developed to explore the multi-physics coupling between mechanics and species diffusion. Constitutive laws for the bulk and the across-GB interaction laws have been derived for large deformations from the system free energies. A chemo-mechanically coupled cohesive zone model is developed which takes into account mode-dependent fracture properties in the presence of GBs. Polycrystalline LiNi x Mn y Co z O 2 (NMC) particles and Li x V 2 O 5 nanowires haveueen selected to demonstrate the impact of GBs on the modeled and observed chemo-mechanics. The model has been implemented in the open-source finite element (FE) package MOOSE. Simulation results indicate that the chemical process and the mechanical degradation go hand-in–hand, where enhanced intergranular chemical inhomogeneities weaken the mechanical strength of the GBs, while damage to the GBs affects or even block transport across the GB. Furthermore, experimentally observed characteristics of chemo-mechanical degradation, e.g., chemical “hot-spots” and surface layer delamination can be accurately predicted by the model.

For reasons of wear resistance and better performance, manufacturing tools are frequently applied with coatings. Such coatings should be made of materials with satisfactory lifetime and degradation resistance. By using the cohesive zone (CZ) element technique embedded at grain boundaries as well as in the interface between the coating and the substrate, a numerical model is developed which serves to predict the damage behavior and crack propagation in the coating system. Different numerical treatments are tested to improve the convergence of the CZ modeling. The present numerical studies reveal the most effective parameters in order to produce stronger coatings. These parameters can be summarized as: grain morphology, elastic and plastic properties, residual stresses and the form of the traction-separation law for the cohesive zone model.

We propose a new phase-field damage formulation which takes into account anisotropic damage evolution in solids. Such anisotropy projects itself in fracture energy values which depend on the direction of the crack surface. Therefore, instead of one constant scalar parameter for the fracture energy value, we use a direction-dependent fracture energy function. By incorporating a direction-dependent fracture energy function, only a single damage variable as well as a first order damage gradient need to be used within the standard phase-field damage model. This is in contrast to other available anisotropic phase-field models which typically use multiple variables or higher order gradient terms. To obtain values for the fracture energy function, atomistic calculations are performed. Here, molecular static simulations are utilized to calculate the energy of free surfaces within an Aluminum crystal. As a result, we report the fracture energy value as a function of the surface orientation. The obtained fracture energy function is passed directly to the phase-field damage formulation to investigate transgranular fracture within a single crystalline. Moreover, the grain boundary is represented via a cohesive zone model to take into account intergranular fracture in a bi-crystalline structure. The predicted crack path is in good agreement with obtained results from molecular dynamics simulations. Finally, by calibrating the length scale parameter in the phase-field damage model, it is possible to compare the reaction forces from finite element calculations with atomistic ones.

Understanding and modeling the interface behavior is an important task for predicting materials response in various applications. To formulate the behavior of an arbitrary interface, one needs to construct the relation between acting tractions and displacement jumps at the interface. In addition to capturing the correct physics of the interface, the so-called traction–separation relation must also be thermodynamically consistent and satisfy the basic balance laws. Apart from many attempts in the literature to address these issues, a new and simple method to capture the complex mechanical behavior at an arbitrary interface is proposed. The new formulation is based on introducing a new quantity called “traction density”. As a result, the traction–separation relation for any arbitrary interface is automatically computed by integrating the traction density over the interface. The traction density can be formulated based on understandings and observations from lower scales. As will be shown, the mathematical representation of the traction density is relatively simple and therefore its consistency can be verified easily. When it comes to the grain boundary (GB) behavior, the proposed methodology is able to represent not only intergranular fracture but also grain boundary sliding . For calibration and verification of the model, molecular dynamics (MD) simulations for aluminum Σ 5 GB are utilized. Interestingly, the calculations from current MD simulations show size-dependent behavior for the GB. By introducing a healing parameter in the new interface model, it is now possible to explain and predict possible GB size-dependent behavior.

The elaborated method of eigenfunction expansion in elliptic coordinates is employed to obtain an exact time-domain series solution, involving products of angular and radial Mathieu functions, for the forced flexural vibrations of a thin elastic plate of elliptical planform. The plate is supported by a constant moduli two-parameter foundation, while elastically restrained against translation and rotation at its edge, and subjected to the combined action of uniform in-plane static edge forces and general arbitrary time-dependent transverse loads with arbitrary initial conditions. Numerical calculations are carried out for the displacement response of clamped or simply supported elliptical plates of selected aspect ratios in various practical loading configurations (i.e., an impulsive point load, a point force in circular motion, a uniformly distributed harmonic load, and a blast load), with or without an elastic foundation, while taking the effects of initial tension or compression below the buckling load into consideration. Limiting cases are considered and good agreements with available results as well as with the computations made by using a commercial finite element package are obtained.

A general fully coupled three-dimensional vibro-acoustic model is developed to investigate the forced non-stationary acousto-structural response of a thin elastic plate of elliptical planform which is backed by a reverberant, rigid, and finite (closed) elliptic cylindrical acoustic enclosure, while under the action of general external transverse loads of arbitrary temporal and spatial variations. The Laplace transform with respect to the time coordinate is invoked, and the classical method of separation of variables in elliptic coordinates is used to obtain the transformed solutions as a linear combination of even and odd modes in terms of products of radial and angular Mathieu functions. A linear system of coupled algebraic equations is ultimately obtained, which is truncated and then solved numerically by implementing Durbin’s numerical Laplace transform inversion scheme. Detailed numerical simulations are conducted for the temporal histories of plate center-point displacement and on-axis cavity acoustic pressure for air-coupled elliptic aluminum plates of selected aspect ratios when subjected to external loadings of practical interest (i.e., an impulsive point load, a uniformly distributed pulse load, and a blast load). Also, acoustic radiation into the backing enclosure is examined by using appropriate 2D images of the internal sound field for selected cavity depths and plate eccentricities. The presented results confirm that the acousto-elastic characteristics of the coupled plate-cavity system are significantly influenced by the plate aspect ratio, cavity depth and the transverse loading configuration. Validity of the work is established through the computations made by using a commercial finite element package.

A two-dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions, based on the Navier displacement equation of motion for a state of plane stress. The model has been simplified by invoking the Helmholtz decomposition theorem, and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of (even and odd) angular and radial Mathieu functions. Extensive numerical results are presented in an orderly fashion for the first three anti-symmetric/symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical (clamped and free) and flexible boundary conditions. Also, the occurrences of “frequency veering” between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed. Moreover, selected 2D deformed mode shapes are presented in vivid graphical form. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature. The set of data reported herein is believed to be the first rigorous attempt to obtain the in-plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities.

The elaborated method of eigenfunction expansion in terms of transcendental Mathieu and modified Mathieu functions is employed to present the first known exact time-domain series solution for transverse vibrations of a uniform elastic membrane of elliptical shape under arbitrary loading and initial conditions. Numerical calculations for displacement time response of an elliptical membrane of typical aspect ratio excited by various common types of transient loads are presented and discussed. Limiting cases are considered and good agreements with available results are obtained.

International audience; This paper presents a time-discretization scheme for the simulation of nonsmooth mechanical systems. These consist of rigid and flexible bodies, joints as well as contacts and impacts with dry friction. The benefit of the proposed formalism is both the consistent treatment of velocity jumps, e.g. due to impacts, and the automatic local order elevation in non-impulsive intervals at the same time. For an appropriate treatment of constraints in impulsive and non-impulsive intervals, constraints are implicitly formulated on velocity level in terms of an augmented Lagrangian technique [1]. They are satisfied exactly without any penetration. For efficiency reasons, all other evaluations are explicit which yields a half-explicit method [2–8]. The numerical scheme is an extended timestepping scheme for nonsmooth dynamics according to Moreau [9]. It is based on time-discontinuous Galerkin methods to carry over higher order trial functions of event-driven integration schemes to consistent timestepping schemes for nonsmooth dynamical systems with friction and impacts. Splitting separates the portion of impulsive contact forces from the portion of non-impulsive contact forces. Impacts are included within the discontinuity of the piecewise continuous trial functions, i.e., with first-order accuracy. Non-impulsive contact forces are integrated with respect to the local order of the trial functions. In order to satisfy the constraints, a set of nonsmooth equations has to be solved in each time step depending on the number of stages; the solution of the velocity jump together with the corresponding impulse yields another nonsmooth equation. All nonsmooth equations are treated separately by semi-smooth Newton methods. The integration scheme on acceleration level was first introduced in [10] labeled "forecasting trapezoidal rule". It was analyzed and applied to a decoupled bouncing ball example concerning principal suitability without taking friction into account. In this work, the approach is algorithmically specified, improved and applied to nonlinear multi-contact examples with friction. It is compared to other numerical schemes and it is shown that the newly proposed integration scheme yields a unified behavior for the description of contact mechanical problems.