Inelastic Material Response Research Articles

Flexibility, toughness, and load bearing of 3D-printed chiral kerf composite structures

Chiral kerf structures are formed by arranging chiral and coiled unit cells which allows for multi-dimensional and multi-scale shape configurations under mechanical loadings. In this study, we investigate how the mechanical properties of materials and microstructural topologies interact to control the flexibility, toughness, and load bearing of 3D-printed chiral kerf structures. We explore chiral kerf structures with two different kerf patterns, i.e., square and hexagon, and three coiling densities. We consider three materials, namely brittle Polylactic Acid (PLA), compliant thermoplastic polyurethane (TPU), and a ductile composite made by alternating PLA and TPU, referred to as a programmable composite. The chiral kerf structures undergo two deformation mechanisms when subjected to mechanical loadings. The first one results from reconfigurations of kerf cells such as uncoiling, rotation of cells, and cell packing, and the second mechanism arises from nonlinear and inelastic material responses. The use of brittle material limits cell reconfigurations before material failure, reducing the overall flexibility and toughness of kerf structures. While the compliant material enables full cell reconfigurations, it results in low load bearing. The use of PLA:TPU composite allows for cell reconfigurations and inelastic material response, enhancing flexibility and toughness while maintaining a relatively high load bearing. We demonstrate that stress distribution in kerf structures can be controlled by using multiple materials or coil densities. This strategy can delay failure and improve the toughness and load-bearing capabilities of kerf structures.

The spectrum of fault slip in elastoplastic fault zones

Natural faults are typically surrounded by damage zones that exhibit inelastic material response. This study investigates the role of fault zone strength in modulating the spectrum of fault slip across different spatio-temporal scales. We carry out long-term simulations of seismic and aseismic slip for an elastoplastic spring slider model with rate-and-state friction as well as a continuum model of a 2D anti-plane rate-and-state fault embedded in an elastoplastic bulk. Results of the elastoplastic spring slider model show the emergence of a new stability boundary, depending on the bulk yield strength relative to fault frictional strength, that limits the rupture size regardless of the fault length. Continuum simulations generate a spectrum of slip analogous to the spring slider model including localized or migrating events of slow and fast slip. A fault may remain locked for yield strength sufficiently low and close to fault reference strength even if it is intrinsically rate weakening and larger than the nucleation length scale predicted by the elastic analysis. These findings shed new light on the nature of fault frictional stability and suggest the critical role of the fault zone rheological properties in modulating the spectrum of fault slip.

Viscoelastic Response of Elastohydrodynamically Lubricated Compliant Contacts below Glass-Transition Temperature.

The widespread use of polymers in the high-performance engineering applications brings challenges in the field of liquid lubrication in order to separate the rubbing surfaces by the coherent fluid-film thickness relative to not only the inelastic material response of the polymers. The determination of the mechanical properties by the nanoindentation and the dynamic mechanical analysis represents the key methodology to identify the viscoelastic behavior with respect to the intense frequency and temperature dependance exhibited by polymers. The fluid-film thickness was examined by the optical chromatic interferometry on the rotational tribometer in the ball-on-disc configuration. Based on the experiments performed, first, the complex modulus and the damping factor for the PMMA polymer describing the frequency and temperature dependence were obtained. Afterwards, the central as well as minimum fluid-film thickness were investigated. The results revealed the operation of the compliant circular contact in the transition region very close to the boundary between the Piezoviscous-elastic and Isoviscous-elastic modes of the elastohydrodynamic lubrication regime, and a significant deviation of the fluid-film thickness from the prediction models for both modes in dependence on the inlet temperature.

Modeling of Ultra-high Performance Fiber Reinforced Concrete Filled Steel Tube Columns under Eccentric Loading

Recently, ultra-high performance fiber reinforced concrete (UHPFRC) has been commonly used as a structural material. In this study, finite element (FE) model is constructed to investigate the behavior of ultra-high performance fiber reinforced concrete filled steel tube columns (UHPFRCFSTs) under axial or eccentric loading. The analysis included three-dimensional FE model using solid elements. The novelty of the suggested FE model is the consideration of the confinement of the UHPFRC in-filled material. Furthermore addition, the numerical model includes initial local and overall geometric imperfections, as well as the inelastic response of both UHPFRC and steel materials. The interaction between the steel tube and UHPFRC in-filled is modelled using surface to surface contact. Experimental results from the literature are used to validate the FE model. It is proved that the FE model can predict the ultimate capacities, failure modes, and post-cracking behavior accurately for both short and long UHPFRCFSTs. The FE interaction diagram agreed very well with the experimental results. Using the verified FE model, a parametric study on UHPFRCFSTs is carried out. Several parameters, including concrete strength material, steel yield strength, and aspect ratio of the columns (column diameter/tube thickness), are investigated. Eventually, comparisons are conducted between the results obtained from FE simulation and the existing design codes for predicting load-moment interaction diagram of UHPFRCFSTs. It has been found that the Eurocode 4 predictions in most analyzed cases are conservative for UHPFRCFSTs.

An elastic‐inelastic model and embedded bounce‐back control for layered printing with cementitious materials

AbstractThis paper presents a finite‐deformation model for extrusion‐based layered printing with cementitious materials. The evolution of mechanical properties as the printed material cures and stiffens results in nonphysical reduction in the magnitude of elastic strains when standard constitutive models are employed. This elastic recovery of the printing induced deformation contradicts the experimentally observed behavior of the printed cementitious materials that harden at a nearly‐frozen deformed state. A thermodynamically motivated constraint on the evolution of elastic strains is imposed on the constitutive model to remedy the nonphysical bounce‐back effect. An algorithm that is based on a strain‐projection technique for the elastic part of deformation is developed that complements the inelastic response given by the Drucker–Prager model. It is then embedded in a finite strain finite element framework for the modeling and simulation of cure hardening and inelastic response of the early age cementitious materials. A ghost mesh method is proposed for continuous layer‐wise printing of the material without the need for intermittent mesh generation technique or adaptive remeshing methods. The model is validated via comparison with experimental data and representative test cases are presented that investigate the mathematical and computational attributes of the proposed model.

Formulation and numerical implementation of a variationally consistent multi-scale model based on second-order computational homogenisation at finite strains for quasi-static problems

A multi-scale model which couples a macro-scale strain-gradient theory with a classical continuum at representative volume element (RVE) level is formulated in this contribution for quasi-static problems. It enables modelling the inelastic response of heterogeneous materials at finite strains incorporating their microstructural size and accounting for non-uniform macroscopic deformation fields within the RVE. The development of the multi-scale model is based on the Method of Multi-Scale Virtual Power from which equilibrium equations and homogenisation relations are derived, ensuring a variationally consistent scale transition. The kinematical constraints at the RVE level are enforced with the Lagrange multiplier method and particularised for direct, periodic and minimal conditions. It is shown that the homogenised stresses can be obtained exclusively from the Lagrange multipliers. The finite element solution of the micro and macro equilibrium problems is presented in detail together with the numerical implementation. An efficient strategy is devised to compute the consistent macroscopic tangents required for the FE2 framework. The Newton–Raphson scheme is employed for the iterative solution of the corresponding non-linear system of equations at both scales. Numerical results show the effectiveness of the numerical strategy. A systematic analysis of the influence of the RVE length, the number and size of the micro-constituents, with different constitutive laws at the micro-scale and subjected to several macroscopic deformation states, is performed. The determination of an RVE that is both representative and descriptive of the macroscopic characteristic length is discussed. The accuracy of the numerical results is assessed against a reference direct numerical simulation.

Nonlinear material modelling for fibre-based progressive collapse analysis of RC framed buildings

The choice of the most suitable constitutive models for fibre-based progressive collapse analysis of reinforced concrete (RC) structures is still an open issue, mainly because nonlinear material modelling has never been treated as a variable involved in the assessment problem so far. To support analysts in selecting which models could be more representative of the actual inelastic material response for progressive collapse analysis of RC framed buildings, this paper presents a numerical investigation where fibre modelling was integrated with different combinations of stress–strain relationships for concrete and reinforcing steel. A series of pushdown simulations of a two-bay perimeter frame mock-up were then carried out to assess the gravity load capacity under downward displacement. Analysis results are compared with available experimental test data for performance quantification of numerical models, which is based upon an experimental-to-numerical load capacity ratio and overall statistical parameters. Sensitivity to the material modelling approach, load eccentricity and boundary conditions is evaluated, involving strain indicators that can be used in performance-based robustness assessment of RC framed buildings. Finally, a number of parametric analyses are presented to show how the load capacity is influenced by capacity model properties, such as material strengths, beam span length, and span length ratio of asymmetric frames.

Localization of deformation in fluid-saturated sandstone

A laboratory study of inelastic deformation and localization was conducted to evaluate microseismicity and fracture in fluid-saturated rock. Plane strain compression experiments were performed on oil- and water-saturated Berea sandstone under dry, drained, undrained, and quasi-unjacketed conditions. Parameters associated with the elastic and inelastic material response were accurately measured and used in a constitutive model to predict localization of deformation under the drained and undrained conditions. Acoustic emission (AE) activity was recorded and the onset of inelastic response in fluid-saturated rock, similar to dry rock, coincided with an increase in AE rate. Fracture initiation and propagation were successfully monitored by AE locations. Clustering of AE events and the start of inhomogeneous deformation inferred from displacement measurements are shown to be well correlated with the model prediction of the onset of localized deformation under various rock-fluid interaction conditions.

Computational modeling of viscoplastic polymeric material response during micro-indentation tests

The computational modeling of instrumented indentation tests used to characterize material properties is challenging. It is mainly due to the computational techniques demanded to couple the complex physical mechanisms involved, such as, for example, the time-dependent inelastic material response to loads during contact. Therefore, this work aims to simulate the mechanical response of the poly vinylidene fluoride (PVDF) during a micro-indentation test considering a viscoplastic material model, and a prescribed load approach, using the finite element method. Further, model validation is performed based on experimental data measured during the contact between the indenter and the PVDF. Numerical analyses were performed using COMSOL Multiphysics finite element software considering the loading scheme of the experimental tests of 800 mN/min rate during loading and unloading, and a 400 mN constant load, held by 30 s. Finally, a viscoplastic Chaboche constitutive model is presented considering two cases: (1) a perfectly plastic behavior, and (2) a nonlinear isotropic hardening behavior based on Voce and Hockett–Sherby exponential laws. While the latter models exhibit some discrepancy in capturing the experimental behavior, the former one has shown excellent agreement with the load-depth curves obtained experimentally, achieving the best fitting for the set of Chaboche parameters: $$A=1\,{\mathrm{{s}}}^{-1}, {n}=4.62$$ and $$\sigma _\mathrm{{ref}}=132$$ MPa. Moreover, several phenomenological features of viscoplastic behavior such as rate dependence, plastic flow (or creep) and stress relaxation were accurately provided by the Chaboche model when describing the behavior of the PVDF material.

Modeling anisotropic inelastic effects in sheet metal forming using microstructural vectors—Part I: Theory

This paper presents a constitutive equation for anisotropic inelastic deformation rate applied to sheet metal forming based on recently developed invariants for nonlinear orthotropic elastic–inelastic material response. These invariants are based on Eulerian evolution equations for microstructural vectors that characterize directions of anisotropy and elastic deformations which cause stress. These microstructural vectors are used here to model directions of orthotropy for sheet metal forming. This Part I of a two-part paper modifies an anisotropic yield function for sheet metal analysis based on stress components determined by elastic deformations of microstructural vectors. Specific expressions are proposed for the direction of inelastic distortional deformation rate, which automatically satisfies the rate of material dissipation inequality, and for the direction of inelastic spin. The material parameters are calibrated for the aluminum alloy AA6022-T43, steel MP980 sheets and cold rolled low carbon steel sheets. The results of stress anisotropy and R-value distributions are shown to be in good agreement with the experimental data. Part II describes the implementation of this model into a finite element program and presents a comparison of Eulerian and Lagrangian formulations.

The generalized viscoelastic spring

A spring/rod model is presented that describes one-dimensional behaviour of solids susceptible to large or small viscoelastic deformation. Derivation of its constitutive equation is underpinned by the fact that the internal energy, which the elastic part of deformation stores in the spring, changes in time with the observed strain as well as with some, unknown part of the strain-rate. The latter emerges through the action of a viscous flow potential and is the source of inelastic deformation. Thus, unlike its conventional viscoelasticity counterparts, the model does not postulate a priori a rule that relates strain with viscous flow formation. Instead, it considers that such a rule, as well as other important features of combined elastic and inelastic material response, should become known a posteriori through the solution of a relevant, well-posed boundary value problem. This paper begins with considerations compatible with large viscoelastic deformations and gradually progresses through simpler modelling situations. The latter also include the case of small viscoelastic strain that underpins formulation of classical, spring-dashpot viscoelastic models. In an example application, a relevant closed-form solution is obtained for a spring undergoing small viscoelastic deformation under the influence of a viscous flow potential which is quadratic in the stress.

Prediction of plastic deformation and wear in railway crossings – Comparing the performance of two rail steel grades

Railway crossings are subjected to a severe load environment leading to damage and degradation of rail profiles. The damage is the result of the high magnitudes of contact pressure and slip generated in the wheel–rail contacts during each wheel transition between wing rail and crossing nose. In this paper, numerical predictions of the long-term accumulation of plastic deformation and wear are presented and compared for two rail grades used in crossings. The comparison is performed using a framework that includes simulations of dynamic vehicle–track interaction, wheel–rail contact and rail damage. A load sequence generated by means of Latin hypercube sampling, taking into account variations in worn wheel profile, vehicle speed and wheel–rail friction coefficient, is considered. A Hertzian-based metamodel for wheel–rail normal contact accounting for inelastic material response is used for the simulation of wheel–rail contact. The methodology is demonstrated by calculating the plastic deformation for one cross-section of the crossing nose using an Ohno-Wang cyclic plasticity model, while wear is predicted by means of the Archard model for sliding wear. For the studied conditions and after the simulation of 41400 load cycles, representing an accumulated traffic load of 0.8 MGT, it is concluded that the fine-pearlitic rail grade R350HT experiences half of the ratchetting strain compared to the austenitic hot-rolled manganese steel Mn13. Based on a wear model calibrated for Mn13, the maximum wear of the studied cross-section is about 2% of the maximum plastic deformation.

Geometrically nonlinear inelastic analysis of steel–concrete composite beams with partial interaction using a higher-order beam theory

A comprehensive finite element model based on a higher-order beam theory (HBT) is developed for an accurate prediction of the response of steel–concrete composite beams with partial shear interaction. The formulation of the proposed one dimensional finite element model incorporated nonlinearities due to large deformations of the beam as well as inelastic material behaviour of its constituent components. The higher-order beam model is achieved by taking a third order variation of the longitudinal displacement over the beam depth for the steel and concrete layers separately. The deformable shear studs used for connecting the concrete slab with the steel girder are modelled as distributed shear springs along the interface between these two material layers. The Green–Lagrange strain vector is used to capture the effect of geometric nonlinearity due to large deflections. The von Mises plasticity theory with an isotropic hardening rule and a damage mechanics model are incorporated within the proposed finite element model for simulating the inelastic response of the beam materials. The nonlinear governing equations are solved by an incremental-iterative technique following the Newton–Raphson method. A dissipation based arc-length method is employed to capture the post peak response of these beams successfully. The capability of the proposed model is assessed through its validation and verification using existing experimental results and numerical results produced by detailed finite element modelling of these beams.

Non-coherent energetic interfaces accounting for degradation

Within the continuum mechanics framework, there are two main approaches to model interfaces: classical cohesive zone modeling (CZM) and interface elasticity theory. The classical CZM deals with geometrically non-coherent interfaces for which the constitutive relation is expressed in terms of traction---separation laws. However, CZM lacks any response related to the stretch of the mid-plane of the interface. This issue becomes problematic particularly at small scales with increasing interface area to bulk volume ratios, where interface elasticity is no longer negligible. The interface elasticity theory, in contrast to CZM, deals with coherent interfaces that are endowed with their own energetic structures, and thus is capable of capturing elastic resistance to tangential stretch. Nonetheless, the interface elasticity theory suffers from the lack of inelastic material response, regardless of the strain level. The objective of this contribution therefore is to introduce a generalized mechanical interface model that couples both the elastic response along the interface and the cohesive response across the interface whereby interface degradation is taken into account. The material degradation of the interface mid-plane is captured by a non-local damage model of integral-type. The out-of-plane decohesion is described by a classical cohesive zone model. These models are then coupled through their corresponding damage variables. The non-linear governing equations and the weak forms thereof are derived. The numerical implementation is carried out using the finite element method and consistent tangents are derived. Finally, a series of numerical examples is studied to provide further insight into the problem and to carefully elucidate key features of the proposed theory.

Implementation and validation of a three-dimensional plasticity-based deformation model for orthotropic composites

A new orthotropic elasto-plastic constitutive model has been developed to predict the inelastic response of composite materials under high velocity impact conditions. The model is driven by experimental stress-strain curve data stored as tabular input allowing for a very general material description. The theoretical details of the elasto-plastic deformation part of the material model are briefly summarized. This summary is then followed by details of the numerical implementation of the model as MAT213 (suitable for use with solid elements) into the commercial transient dynamic finite element code, LS-DYNA. The theoretical basis and the numerical implementation of the constitutive model are validated by using two sets of validation tests involving a widely used unidirectional composite, T800/F3900 - composite laminates used in coupon level tests and a low velocity impact test on a flat panel. Results show that the implementation is efficient, robust and accurate.

Hybrid finite element for analysis of functionally graded beams

ABSTRACTA hybrid finite element model is presented, where stiffness and mass distributions over a beam with functionally graded material (FGM) are accurately modeled for both elastic and inelastic material responses. Von Mises and Drucker-Prager plasticity models are implemented for metallic and ceramic parts of FGM, respectively. Three-dimensional stress-strain relations are solved by a general closest point projection algorithm, and then condensed to the dimensions of the beam element. Numerical examples and verification studies on a proposed element demonstrate accuracy and robustness under inelastic material response as well as capturing fundamental, higher, and mix modes of vibration frequencies and shapes.

An internal-state-variable based viscoelastic-plastic model for polymers

In this study, a new viscoelastic-plastic constitutive model which has been formulated by utilizing the formalism of stress-like internal state variables is introduced. The developed constitutive equation allows for a good description of the inelastic material response of polymeric materials over a wide range of strain rates. An algorithm for numerical integration of the model equations has been derived. The FE implementation of the constitutive equation is widely discussed and the results of solving several exemplary problems are presented.

Inelastic response of solids described by implicit constitutive relations with nonlinear small strain elastic response

We develop a model to describe the inelastic response of bodies that exhibit non-linear response even in the small strain regime. We introduce a new approach to modeling the inelastic response of materials by gainfully exploiting the discontinuity of the functions that appear in the constitutive relations to describe a plethora of inelastic responses that have been observed. The model that has been developed and its generalizations can be used to describe the response of geomaterials such as clay and rocks, traditional materials such as Aluminum and several polymeric solids, as well as modern intermetallic alloys.

Elastic-viscoplastic modeling of soft biological tissues using a mixed finite element formulation based on the relative deformation gradient.

The characteristic highly nonlinear, time-dependent, and often inelastic material response of soft biological tissues can be expressed in a set of elastic-viscoplastic constitutive equations. The specific elastic-viscoplastic model for soft tissues proposed by Rubin and Bodner (2002) is generalized with respect to the constitutive equations for the scalar quantity of the rate of inelasticity and the hardening parameter in order to represent a general framework for elastic-viscoplastic models. A strongly objective integration scheme and a new mixed finite element formulation were developed based on the introduction of the relative deformation gradient-the deformation mapping between the last converged and current configurations. The numerical implementation of both the generalized framework and the specific Rubin and Bodner model is presented. As an example of a challenging application of the new model equations, the mechanical response of facial skin tissue is characterized through an experimental campaign based on the suction method. The measurement data are used for the identification of a suitable set of model parameters that well represents the experimentally observed tissue behavior. Two different measurement protocols were defined to address specific tissue properties with respect to the instantaneous tissue response, inelasticity, and tissue recovery.

On Micromechanically‐Based Approaches to Failure in Polymers

AbstractRubbery polymers comprise of three‐dimensional network structures, formed by flexible and mobile chains. They exhibit highly non‐linear ground state elastic combined with rate‐dependent inelastic material response. The key challenge in the micromechanically‐based modeling of polymer materials is the construction of micro‐macro transition that provides a bridge between microscopic kinematic variables of a single chain and macroscopic deformation measures of a continuum. The fracture toughness of rubbery polymers shows distinct rate dependency which can be identified through tearing tests. In this contribution, an overview on the micromechanical‐motivated modeling of polymeric materials will be given and different micromechanically‐based approaches to the modeling of rubbery polymers will be discussed. The novel aspect of the contribution is the incorporation of micromechanically‐motivated crack phase field model for the description of rate‐dependent fracture process in rubbery polymers. To do so, the viscoelastic spectrum of rubber network is idealized by a number of Maxwell branches superimposed to a ground‐state elastic spring representing the crosslinked network. The core of the ansatz is the incorporation of rupture of the superimposed network of dangling chains to the fracture process at different loading rates. We demonstrate the modeling capability of the proposed ansatz by means of representative numerical examples. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)