Inelastic Material Behavior Research Articles

Polyconvex inelastic constitutive artificial neural networks

AbstractThe characterization of the material behavior of inelastic materials requires a high degree of expert knowledge to identify and constitutively describe the material response. In addition, specific models are usually pre‐selected in the course of characterization and only the best parameters for these specific models are determined, but therefore not necessarily the best models. Unfortunately, a more general description of these pre‐selected models results in an increased effort during characterization, which is barely practicable by hand. This is where machine learning algorithms may help us. To get the best of both worlds, powerful machine learning and sound thermodynamic considerations, inelastic Constitutive Artificial Neural Networks (iCANNs) discover generic formulations of the Helmholtz free energy and pseudo potential. Constitutive relations guide us towards thermodynamically consistent descriptions of stresses and inelastic strains; a concept that is applicable to a wide range of inelastic phenomena from viscoelasticity, elastoplasticity, phase transformations to growth and remodeling of living tissues. Here, we equip the original iCANN framework to guarantee polyconvexity a priori, which ensures at least one minimizing deformation. We investigate the ability of our iCANN to identify the material behavior of a viscoelastic polymer at different stretch levels and strain rates. We made our source code, data, and example accessible to the public at https://doi.org/10.5281/zenodo.11084354.

Utilization of Short Span Web-Tapered Beams Using Flexible Nodal Bracing

For many years member with varying cross-sectional dimensions along the length have been widely used in steel structures. The use of nodal braces to enhance the load-carrying capacity and stability of these elements is prevalent. These supports ensure structural stability by regulating the distribution of loads within the structural system while also enhancing the structure's resistance to lateral loads. Presently, the design of these nodal braces are carried out according to requirements outlined in Design Guide 25 and AISC Specification (2022). However, these standards offer a general framework for elements with fixed brace conditions and prismatic cross-section. This study aims to be a pioneering investigation into the variation of brace force values in short-span and compact web-tapered beams under different loading conditions. The article seeks to comprehensively examine the requirements and limits of brace force in short-span web-tapered compact beams. To achieve this goal, parametric finite element analyses is utilized to explore how brace force changes concerning beam geometry, material properties, and loading conditions. The beam used was considered as doubly symmetric and divided into 100 nodes and supported by a nodal brace at the middle node. The beam is 50 inches long, tapering from 42 inches to 36.40 inches over its span of 100 nodes. In terms of finite element analysis, the software utilized significantly influences the accuracy and reliability of results, particularly in scenarios involving inelastic nonlinear analysis. In this study, the Abaqus program was employed specifically to conduct parametric finite element analyses, considering the complexities of inelastic material behavior. The findings of this research are intended to contribute to the development of a new design method for determining the requirements and limits of brace force in short-span and variable cross-sectional dimension compact beams. Consequently, the aim is to enable the safe and economical design of such beams, providing engineers with ideas to consider and data to utilize in their designs.

Numerical examination on seismic response behavior of a piping system considering the plastic deformation of supports

A previous study proposed a passive safety structure to mitigate severe damage to crucial components under the beyond design basis event (BDBE). To examine the applicability of this concept to the piping system, the effect of the elastic-plastic behavior of the piping support structure on the seismic response of the piping system was investigated through elastic-plastic time history response analyses on a piping system model with multiple support structures. The numerical analyses considered the elastic-plastic behavior of supports and/or the inelastic characteristics of pipe material. The analysis results confirmed that the response of the piping system can be suppressed by adequately introducing the inelastic behavior of support structures. The results show the possibility of realizing of passive safety structure in piping systems and addressing some issues, such as changing vibration modes due to the elastic-plastic deformation of the support or the order of generation of inelastic behavior of pipe material and supports.

Efficient and accurate uncertainty quantification in engineering simulations using time-separated stochastic mechanics

A robust method for uncertainty quantification is undeniably leading to a greater certainty in simulation results and more sustainable designs. The inherent uncertainties of the world around us render everything stochastic, from material parameters, over geometries, up to forces. Consequently, the results of engineering simulations should reflect this randomness. Many methods have been developed for uncertainty quantification for linear elastic material behavior. However, real-life structure often exhibit inelastic material behavior such as visco-plasticity. Inelastic material behavior is described by additional internal variables with accompanying differential equations. This increases the complexity for the computation of stochastic quantities, e.g., expectation and standard deviation, drastically. The time-separated stochastic mechanics is a novel method for the uncertainty quantification of inelastic materials. It is based on a separation of all fields into a sum of products of time-dependent but deterministic and stochastic but time-independent terms. Only a low number of deterministic finite element simulations are then required to track the effect of (in)homogeneous material fluctuations on stress and internal variables. Despite the low computational effort the results are often indistinguishable from reference Monte Carlo simulations for a variety of boundary conditions and loading scenarios.

A finite element implementation of phase-field approach of fracture for nonlinear solid shells including inelastic material behavior

The parametrization of shell structures using the so-called solid shell concept has been widely exploited in the last decades. This trend is mainly attributed to the relatively simple kinematic treatment of solid shells in the corresponding finite element formulation in conjunction with the use of unmodified three-dimensional material laws, among other aspects. In the present investigation, we provide a comprehensive finite element implementation of solid shells incorporating: (i) the use of Enhanced Assumed Strain (EAS) and the Assumed Natural Strain (ANS) methods to prevent locking issues, (ii) the phase-field approach for triggering fracture events, and (iii) some representative inelastic material models. The current modular implementation has been integrated into the FE package ABAQUS via the user-defined routine UEL. Several representative examples demonstrate the applicability of the present formulation.

Inelastic mechanical behaviour of an additively manufactured titanium alloy: a statistical continuum mechanics theory perspective

Statistical continuum mechanics theory was used to simulate the inelastic stress of polycrystalline materials using two-point statistics. For the experimental part, the Electron beam melting (EBM) technique (Arcam EBM Q10 additive machine) was used to fabricate cylindrical rods of Ti-6Al-4V both in horizontal and vertical directions. Electron backscatter diffraction (EBSD) technique was employed to achieve statistically reliable orientation maps of vertically and horizontally printed samples. In this study, high strain rate compression tests at six different strain rates were performed, and the stress–strain curves were generated. This work is amongst the first attempts to model the microstructure of additively manufactured hexagonal alloys under compressive loadings using the statistical continuum mechanics theory. The model is capable of simulating reasonably large microstructures (statistically representative) with a practical computational cost and accuracy, unlike numerical models that require a high computational cost. It should be noted that in additive manufacturing, due to large grains and high anisotropy, microstructures used in the simulations should be large enough to include sufficient information from the material’s structure. Therefore, using finite element models would be very challenging here. On the other hand, the statistical continuum mechanics theory uses the statistical representation of the material’s characteristics for solving the governing equations with Green’s function that enables this methodology to use more microstructure characteristic information without having a noticeable change to the computational cost. The proposed model in this study uses different microstructure characteristics such as crystal grain orientation, total slip systems, active slip systems, gain morphology, and chemical phases that are obtained from EBSD images for simulating the inelastic mechanical behavior of polycrystalline materials. Although this model simulates polycrystalline materials by considering various crystal and grain information, unlike numerical methods, it doesn’t simulate the grain interactions well and we cannot study local deformation and crack nucleation sites. This model works very well for simulating the overall behavior of material instead of each individual grain and failure analysis. This model has shown a good combination of computational cost and accuracy in which the error between the simulated and experimental strength for vertical and horizontal samples was 6.21% and 8.07%, respectively.

High fidelity nonlinear finite element tire modeling for dynamic analysis: Total Lagrangian formulation in rolling contact

Tire design with respect to vibrational behavior and noise emission is a challenging task due to the large number of parameters and their coupling. Therefore, accurate high fidelity nonlinear Finite Element (FE) models are needed to aid in the design process. The widely used Arbitrary Lagrangian Eulerian (ALE) models, despite their computational efficiency, are limited to rib-slicked tires and require significant effort to consider inelastic material behavior and frictional phenomena. To overcome these limitations a novel nonlinear FE tire model is presented based on the Total Lagrangian (TL) formulation. The TL model is compared to the reference ALE model for the case of stationary rolling on a smooth and a rough road surface. The observed match of the TL model with respect to the hub and contact forces proves the feasibility of using the TL formulation in the context of rolling contact tire simulations. Therefore, the TL model can be seen as a valid alternative to the frequently used ALE models.

A constitutive model for describing decoupled material behavior in thickness and in‐plane directions

AbstractAn increasing demand for paper and paperboard in engineering applications has emerged throughout the years. This leads to a growing interest in simulation tools to predict the material behavior, which is challenging due to the anisotropic structure of the material. It can be divided into in‐plane and out‐of‐plane behavior, which is mostly independent of each other resulting from the microstructure. Thus, a constitutive material model that captures in‐plane and out‐of‐plane behavior in a consistent manner is desired to simulate complex deformation states. The decoupling will be incorporated by a decomposition of the kinematics and the energy formulation into in‐plane and out‐of‐plane terms by use of structural tensors. This approach is flexible and can easily be extended to inelastic material behavior. In this work, the introduction of the decoupling into a constitutive material model for inelastic behavior is described. Visco‐elasticity is included through inelastic potentials. Three different potential formulations are given where focus is on the third, novel formulation, which defines inelastic strain evolution only in out‐of‐plane directions. The capabilities of this formulation are compared to the remaining potentials by simulations of a pinched cylinder that is loaded in the out‐of‐plane direction.

A displacement‐driven approach to frictional contact mechanics

SummaryThe contribution at hand aims at a numerical method to represent contact between two or more different solid bodies. The proposed computational contact algorithm is based on a displacement‐driven approach to describe the geometrical constraint resulting from the impenetrability of the solid bodies in contact. A solution method of contact between one deformable and one rigid body is assumed for the initial computationally fast implementation of the underlying equations. Elastic or inelastic material behavior can be considered for the deformable body. Furthermore, the proposed approach is capable of finite deformations. Moreover, a node‐to‐segment (NTS) formulation is utilized for contact element discretization. Various numerical examples are presented to demonstrate the numerical method's applicability for different contact mechanics tasks. First, the presented examples verify the displacement‐driven approach by investigating its convergence behavior and consistency. Then, comparisons of solutions by the proposed formulation to the analytical solution and the penalty method are provided to validate its accuracy. Moreover, the stick and slip condition of large sliding motion in a frictional contact problem is successfully depicted.

A data-driven approach for plasticity using history surrogates: Theory and application in the context of truss structures

Data-driven methods and algorithms show immense potential for future advancements in the modeling and simulation of complex mechanical systems. However, in order to actually exploit this potential, the methods must be extended to consider inelastic and path-dependent material behavior—a step that appears more complex than in conventional material modeling, where this can be achieved with the help of additional, mostly internal, history variables. The effect of such history variables must thereby be transferred to the data-driven framework. This is achieved in the present paper by defining an appropriate history surrogate as well as a so-called propagator. The history surrogate contains tangible quantities that represent the backward path–uniquely in the ideal case–and enrich the conventional database entries of matching pairs of stresses and strains. The propagator defines the update of the history surrogate from one discrete time step to another. As a major advantage, the newly developed method retains the structures of the data-driven algorithm for elastic material behavior and therefore allows a rather straightforward extension of preexisting program codes. Thus, for instance, heterogeneous problems in which purely elastic and elastoplastic materials are present can be solved without further significant coding effort. Furthermore, several material classes covering different inelastic phenomena such as plasticity with isotropic and kinematic hardening as well as phase transformations in shape memory alloys can be considered in our framework.

Dynamics of creeping landslides controlled by inelastic hydro-mechanical couplings

Slow-moving landslides affect proximal infrastructures and communities, often causing extensive economic loss. While many of these landslides exhibit slow and episodic sliding for decades or more, they sometimes accelerate rapidly and fail catastrophically. Although it is known that the landslide dynamics are controlled by hydro-mechanical processes, few analytical models enable a versatile incorporation of the inelastic behavior of the shear zone materials, thus hindering an accurate quantification of how their properties modulate the magnitude and rate of coupled fluid flow and landslide motion. To address this problem, we develop a simulation framework incorporating rainfall-induced, deformation-mediated pore-water pressure transients at the base of active landslides. The framework involves the computation of two sequential diffusion processes, one within an upper rigid-porous landslide block, and another within the inelastic shear zone. Although the framework can be linked to any elastoplastic constitutive laws, here we model landslide motion through an elastic-perfectly plastic frictional model, which enables us to account for standard properties of earthen materials such as elastic moduli, friction angle, dilation angle, and hydraulic conductivity. Numerical case studies relevant to slow-moving landslides in the California Coast Ranges show that the proposed formulation captures different temporal patterns of movement induced by precipitation. In each of the case, we achieved a relatively accurate match between data and simulations by incorporating positive dilation coefficients, which leads to spontaneous generation of negative excess pore-water pressure and self-regulating motion. Conversely, simulations with no dilation (hence, reflecting the approach of critical state) produce sharp acceleration, typical of catastrophic runaway acceleration. Our findings encourage the use of the proposed framework in conjunction with constitutive laws tailored to site-specific geomaterial properties and data availability, thus favoring a versatile representation of the variety of creeping landslide trends observed in nature.

Polymorphic Uncertain Structural Analysis: Challenges in Data‐Driven Inelasticity

AbstractThis contribution addresses polymorphic uncertainty quantification within structural analysis of reinforced concrete structures composed of heterogeneous concrete and reinforcement (e.g. steel bars or carbon fibres). The macroscopic material behaviour of concrete is strongly dependent on the mesoscopic heterogeneities, which are considered by multiscale modelling. The heterogeneous mesoscopic material behaviour is characterized by representative volume elements (RVE) and the transition of scales is carried out by utilizing numerical homogenization methods.The concept of data‐driven computational mechanics enables material model free finite element analyses directly based on material data sets, overcoming the necessity of assumptions in material modelling. This approach mainly consists in assigning a stress‐strain state, which leads to a minimum of an objective function and fulfils equilibrium as well as compatibility constraints of every integration point. Inelastic material behaviour is taken into account through the definition of local data sets containing only data set states which are consistent with respect to the past local history . The realistic modelling of structures requires the consideration of data uncertainty. Generalized polymorphic uncertainty models are utilized in order to take variability, imprecision, inaccuracy and incompleteness of material data into account by combining aleatoric and epistemic uncertainty models.In this contribution, a computationally efficient approach for the consideration of data sets containing uncertain stress‐strain states within data‐driven analysis based on information reduction measurements is presented. Due to generalization, the approach is applicable to various aleatoric, epistemic and polymorphic uncertainty models. The identification of admissible local data sets for taking inelastic material behaviour into account within data‐driven analysis is realized by an energy based history parametrization which is extended to uncertain data. An approach for the efficient selection of these local data sets is presented and challenges in data‐driven inelasticity, particularly in the use case of polymorphic uncertain analyses of concrete structures, are pointed out.

On the dynamic simulation of viscoelastic structures with random material properties using time‐separated stochastic mechanics

AbstractSimulating stochastic structures with inelastic material behavior is often done with Monte Carlo simulations. The method is robust but needs a huge computational effort. We propose a new approach for the dynamic simulation of viscoelastic stochastic structures with a computational effort in the magnitude of one deterministic simulation. It is based on a separation of the viscous evolution equation in time‐dependent but deterministic and stochastic but time‐independent terms. We present the calculation of the effective quantities as stresses and reaction forces with the time‐separated stochastic mechanics. Numerical experiments showcase the high approximation quality compared to reference Monte Carlo simulations.

The use of full-scale tensometry for studying the stress state of new power equipment

Creation of the new generation of power equipment (nuclear reactors, gas turbine plants, special power plants) with increased operational parameters and meeting high safety requirements, entails the goal of creating novel domestic methods for determining stresses and strains that occur in the most critical elements during operation of such installations. The new approaches, including the restoration and development of domestic competencies in the creation of means for experimental control of deformations in elements of the equipment at high temperatures are proposed. The results of developing indirect methods providing determination of the deformations at the most critical points of the structure based on solutions of the inverse problems of experimental mechanics are presented. Improved algorithms for processing experimental information and determining strains from measured stresses are considered for the case of inelastic behavior of the structural material in the zones of tensometric measurements. The improved hermetic strain gauges resistant to lead and sodium coolant are proposed for experimental determination of the deformations that occur on the internal surfaces of the liquid metal coolant circulation circuit of new nuclear reactors of BN and BREST types. The results of tests of the developed measuring instruments in liquid sodium medium at a temperature of 540°C during bench tests are presented. In conditions of increased requirements for assessing the effect of the strain gauge creep on the measurement results obtained at high temperatures, a stand design has been developed that makes it possible to determine the boundaries of a possible error at temperatures up to 600°C under dynamic loading of the structure. To determine the stresses arising at dangerous points of the heat exchange equipment of BN type reactors located in inaccessible areas of the inner surface of the facility, an iterative algorithm for solving the inverse problem of thermoelasticity is proposed, using the measured values of stresses and temperatures on the outer surface of the structure. A set of improvements to the traditional method of full-scale tensometric study is proposed proceeding from of the analysis and implementation of the data obtained.

Experimental-numerical study of wrinkling in rotary-draw bending of Tight Fit Pipes

The paper focuses on tight fit pipe (TFP) wrinkling in rotary-draw bending (RDB), where TFP is a double-walled pipe. To this end, a corrosion-resistant alloy liner is fitted inside an outer carbon steel pipe through a thermal-hydraulic manufacturing process. A 3D elastic–plastic finite element (FE) model (the dynamic explicit FEM code ABAQUS/Explicit) is developed in this study. This model simulates the manufacturing process in the first step of the analysis and proceeds in the rotary-draw bending analysis of the lined pipe. Then, in the third step, the springback of the two-layer pipe is analyzed. This integrated three-stage process considers geometric nonlinearities, geometric and thickness imperfections, inelastic material behavior, and contact between the two pipes. Contact pressure during the process in the presence of various imperfections is monitored by the histogram to predict wrinkling. Furthermore, the best bending angle, with minimum possibility of wrinkling in the inner pipe, is obtained by these histograms. ABAQUS scripting via the Python programming language is used to study the effect of various imperfections on RDB. Numerical results on imperfection sensitivity demonstrate the significant influence of imperfections’ amplitude on liner wrinkling. The ovality and thinning of the lined pipe via bending angle were investigated. By comparing the experimental results with those of the simulation, an appropriate agreement is observed.

Inelastic material formulations based on a co-rotated intermediate configuration—Application to bioengineered tissues

In the field of material modeling, there is a general trend to include more and more complex phenomena in the modeling, making the models’ theoretical derivation and numerical implementation extremely difficult. In particular, modeling inelastic material behavior under finite deformations in a continuum mechanical manner, e.g. to simulate soft biological tissues, remains one of the most challenging tasks. Unfortunately, the multiplicative decomposition of the deformation gradient usually utilized in this context suffers from an inherent rotational non-uniqueness, making it not straightforward to combine this approach with algorithmic differentiation (AD)—a very helpful tool in modern computational mechanics. To address this issue in the growth and remodeling model proposed herein, a novel co-rotated intermediate configuration is introduced. This configuration shares essential characteristics with the intermediate one, but is uniquely defined and applicable to a wide range of inelastic materials. In this regard, the concept of structural tensors, hardening effects, and a thermodynamically consistent derivation are discussed as well. Since the stress-driven growth model presented is based on the approach of homeostatic surfaces by Lamm et al. (2022), a large number of derivatives of potentials and energies are required, which can be elegantly implemented using AD due to the co-rotated formulation. Moreover, fiber remodeling of collagen fibers is taken into account in a stress-driven manner using AD. Finally, qualitative comparisons are made with recently published experiments by Eichinger et al. (2020) in uniaxial and multiaxial settings, revealing the efficient combination of the proposed framework and the material model.

A finite difference-based approach for strain demand prediction of inelastic pipes subjected to permanent ground displacements

Permanent ground displacement triggered by geohazards constitutes a major threat to the integrity of long-distance pipelines. In this study, a novel and simple method is proposed to evaluate inelastic pipes’ behavior under ground movements considering soil-pipe interaction based on the finite difference method. The existing finite difference-based method previously proposed by the authors for strain analysis of buried pipelines subjected to ground movements excludes material nonlinearity in pipes, which limits its applicability in engineering practice. To remedy this situation, the method presented herein maintains the well-established concepts of the existing finite-difference-based method but also introduces the scheme to consider the inelastic material behavior of steel pipes. More specifically, the expressions of internal axial force and bending moment, required in the finite difference equations, are explicitly derived based on the actual stress distribution on the pipe cross-section. The proposed method is validated against the finite element method (one-dimensional beam model) in terms of the strain and deformation demands using two indicative case studies, referred to as symmetric and non-symmetric soil resistance conditions, respectively. The comparison between finite element analysis and the proposed method indicates good agreement. Additionally, the proposed method is compared with four existing analytical methods for pipes subjected to strike-slip fault displacement. The proposed method is a simple but general technique to analyze pipes’ response under a wide range of geohazards, and thus can be potentially used as an alternative method for preliminary design, safety prescreening, and reliability-based assessment of pipes against geohazards.

Effect of damping model and inelastic deformation on the prediction of vertical seismic acceleration demand on steel frames

This study addresses the modeling of different energy dissipation mechanisms for numerical prediction of the vertical acceleration demand in regular moment-resisting steel frame structures. One of the issues discussed is the consideration of viscous damping in the structural model. It is shown that well-established Rayleigh-damping may highly overestimate the damping of the vertical modes, resulting in much too low vertical acceleration response predictions. A study with different damping models provides an appropriate damping modeling strategy that leads to reasonable predictions of both horizontal and vertical frame acceleration demands. Another open question is the effect of inelastic material behavior on the vertical acceleration demand on the considered regular structures. The results of a shell model of a frame structure exposed to high intensity ground motion excitation demonstrate that inelastic material behavior has virtually no impact on the vertical acceleration demand, while structural inelasticity leaves the horizontal acceleration response significantly smaller compared to the elastic demand. This leads to the conclusion that common frame models that capture the inelastic horizontal response but behave elastic in the vertical direction are suitable for the computation of both the horizontal and vertical acceleration demand.

A Review of Simplified Numerical Beam-like Models of Multi-Storey Framed Buildings

Modern computational techniques have greatly influenced the numerical analyses of structures, not only in terms of calculation speed, but also in terms of procedural approach. In particular, great importance has been given to structural modelling, that is, the process by which a structure and the actions to which it is subjected are reduced to a simplified scheme. The use of a simplified calculation scheme is necessary since the structures are, in general, considerably complex physical systems whose behaviour is influenced by a large number of variables. The definition of a structural scheme that is at the same time simple enough to be easily computable as well as sufficiently reliable in reproducing the main characteristics of the behaviour of the analysed structure is, therefore, a crucial task. In particular, with reference to multi-storey framed buildings, the extensive use of three-dimensional finite element models (FEM) has been made in recent decades by researchers and structural engineers. However, an interesting and alternative research field concerns the possibility of studying multi-storey buildings through the use of equivalent beam-like models in which the number of degrees of freedom and the required computational effort are reduced with respect to more demanding FEM models. Several researchers have proposed single or coupled continuous beams to simulate either the static or dynamic response of multi-storey buildings assuming elastic or inelastic behaviour of the constitutive material. In this paper, a review of several scientific papers proposing elastic or inelastic beam-like models for the structural analyses of framed multi-storey buildings is presented. Considerations about limits and potentialities of these models are also included.

Mixed isogeometric collocation for geometrically exact 3D beams with elasto-visco-plastic material behavior and softening effects

A geometrically nonlinear, shear-deformable 3D beam formulation with inelastic material behavior and its numerical discretization by a mixed isogeometric collocation method are presented. In particular, the constitutive model captures elasto-visco-plasticity with damage/softening from Mullin’s effect, which applies to the modeling of metallic and polymeric materials, e.g., in additive manufacturing applications and metamaterials. The inelastic material behavior is formulated in terms of thermodynamically consistent internal variables for viscoelastic and plastic strains and isotropic and kinematic hardening variables, as well as accompanying evolution equations. A mixed isogeometric collocation method is applied for the discretization of the strong form of the quasi-static nonlinear differential equations. Thus, the displacements of the centerline curve, the cross-section orientations, and the stress resultants (forces and moments) are discretized as B-spline or NURBS curves. The internal variables are defined only locally at the collocation points, and an implicit return-mapping algorithm is employed for their time discretization. The method is verified in comparison to 1D examples as well as reference results for 3D beams. Furthermore, its applicability to the simulation of beam lattice structures subject to large deformations and instabilities is demonstrated.