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Dieser Artikel stellt eine bahnbrechende Methode zur Erkennung des Shakedown-Verhaltens von Materialien vor, die zyklischer Belastung ausgesetzt sind. Der Schwerpunkt liegt auf der Verwendung minimaler Ergebnisse der digitalen Bildkorrelation (DIC) in Kombination mit einem elastischen Finite-Elemente-Modell, was den erforderlichen Rechen- und Versuchsaufwand deutlich reduziert. Die Studie demonstriert diesen Ansatz auf AA6061 auxetic sheets, die für ihre einzigartigen mechanischen Eigenschaften bekannt sind. Die Forschungsergebnisse unterstreichen die Bedeutung eines Shakedown im Konstruktionsbereich, insbesondere in Sektoren wie Luft- und Raumfahrt, Automobilindustrie und Energie, wo traditionelle ertragsbeschränkte Designansätze möglicherweise unzureichend sind. Der Artikel gibt einen detaillierten Überblick über den Versuchsaufbau, einschließlich des Einsatzes einer servohydraulischen Prüfmaschine und DIC zur Messung von Oberflächenverformungen. Außerdem werden die Simulationen Finite-Elemente diskutiert, die das experimentelle Design geleitet haben, und die Kriterien, anhand derer Shakedown identifiziert werden konnte. Die Ergebnisse zeigen, dass die vorgeschlagene Methode Shakedown-Zustände effektiv erkennen kann und ein robustes und flexibles Werkzeug für technische Anwendungen bietet. Die Schlussfolgerung betont das Potenzial dieses Ansatzes für experimentelles Shakedown-Screening mit hohem Durchsatz und seine Anwendbarkeit auf komplexe architektonische Strukturen.
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Diese Zusammenfassung des Fachinhalts wurde mit Hilfe von KI generiert.
Abstract
Background
Shakedown occurs in elastoplastic bodies subject to cyclic loading when limited plastic deformation in the early stages of cycling gives rise to residual stresses that arrest the plastic response. As a result, purely elastic behavior is recovered upon further load cycling.
Objective
This paper outlines a new procedure for identifying macroscopic shakedown responses using minimal kinematic measurements from digital image correlation and a purely elastic finite element model.
Method
In contrast to traditional shakedown detection methods that track the evolution of total or equivalent plastic strains upon cycling, the proposed approach obviates assumptions about the inelastic nature of the constituent material of interest altogether. The proposed approach is inspired by Melan’s static lower bound shakedown theorem and it offers advantages in terms of computational and experimental effort, providing robustness against material uncertainties, and flexibility for a variety of applications.
Results
The approach is demonstrated on aluminum metamaterial sheets that are auxetic due to their arrangement of periodic perforations. In the demonstration, a set of uniaxial experiments at room temperature are analyzed where a range of cyclic stress amplitudes are imposed at non-zero mean stresses.
Conclusions
The proposed shakedown determination protocol was verified by comparing with traditional methods that track the incremental evolution of equivalent plastic strain. The new procedure provides a rapid and robust approach that is suitable for high-throughput experimentation.
Graphical abstract
On Detecting Elastoplastic Shakedown Using Minimal Digital Image Correlation Results and an Elastic Model: Demonstration for AA6061 Auxetic Sheets
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Highlights
A new analysis method for detecting the shakedown behavior in cyclic tests is proposed
Auxetic perforated sheets are tested with cyclic stress amplitudes and non-zero mean stresses
Digital Image Correlation (DIC) is used only to measure boundary condition displacements
Comparing experiment and auxiliary elastic finite element analyses with DIC-measured boundary conditions identifies shakedown
Introduction
When a solid is subject to a cyclic force it may experience shakedown (Fig. 1) – a mechanical behavior wherein limited plastic deformation is experienced in the early stages of cycling, giving rise to internal, residual stresses that arrest the plastic response [1‐3]. A purely elastic behavior results during any further loading cycles. Traditional (i.e., ultra-conservative) design approaches allow for no plastic deformation, called “yield-limited" design. They do this as a way to safeguard against potential failures, such as low cycle fatigue from alternating plasticity or (incremental) plastic collapse from ratchetting, Fig. 1. This yield-limited approach, however, neglects the underutilized shakedown phenomenon that (i) can also be used to safeguard against failure, and (ii) more importantly, has proven to be the only feasible design approach in some sectors [4, 5]. For example, nuclear energy and roadway engineers have long recognized the security and benefits of shakedown, and shakedown behavior has been widely adopted in these sectors as it allows for designs that would otherwise be inaccessible using traditional design approaches [1, 4, 6‐12].
Fig. 1
Schematic cyclic elastoplastic behaviors with an elastic-perfectly-plastic assumption
While shakedown concepts, limit theorems, and numerical methods, have been developed since the 1920s and 1930s [9, 13‐16], their widespread acceptance and application in engineering design communities remains limited [9, 17, 18]. A major obstacle in the wider application and acceptance of shakedown analyses and design is the lack of systematic full-field experimental assessment of shakedown.
Historically, experimentally detecting shakedown has followed norms established by the tools first and most widely available (i.e., displacement transducers, strain gauges, extensometers) [4, 19‐23]. For example, some of the earliest published shakedown experiments [20, 24‐26] relied on strain gauges to detect cyclic behaviors of the structures. Ng et al. [20] extended the strain gauge measurement ability in shakedown experiments from room temperature to \(400^\circ \)C by introducing a hot wheel test rig for 316 stainless steel. In the European Brite-EuRam project LISA [27], the main results were reported by Heitzer et al. [4], where an INSTRON 1343 test rig subjected hollow cylindrical ferritic steel samples commonly used in the nuclear industry to cyclic tension with nonzero mean stress and constant torque under ambient conditions. An INSTRON extensometer was used to monitor strains and the torsional angle was recorded. Several experiments with various combinations of the axial and torsional loads were tested to identify combinations that resulted in shakedown (manifested by the stabilization of the torsional angle with the cyclic alternating axial force) or inadmissible ratchetting behavior (the torsional angle increases in an unbounded manner despite the constant moment loading). Methods involving hole-drilling and x-ray diffraction have also been proposed to track residual stress fields responsible for shakedown [28‐31]. At the mesoscale of polycrystals, Charkaluk et al. [32‐37] have quantified plastic dissipation related to shakedown through combined thermal and kinematic measurements (via Digital Image Correlation, DIC) and have supplemented these data with crystal plasticity modeling efforts. Recently, state-of-the-art synchrotron studies of fatigue have also reported shakedown states by monitoring crack initiation and growth, though shakedown was not the focus of the studies but an unintended byproduct of the loadings and materials considered [38]. This article will only focus on macroscopic experimental shakedown detection strategies that are based on DIC and will leave the discussion of mesoscale and sub-mesoscale approaches like diffraction and synchrotron analysis methods for future studies.
Increasingly, experimental studies have identified shakedown using full-field DIC as a kind of replacement for strain gauges and extensometers. This history and trend has translated into a majority of DIC-based macroscopic experimental shakedown detection procedures that incrementally and step-by-step track the change of the total strain (\(\varepsilon \)) or, with post-processing using the additive decomposition of strain, the equivalent plastic strain (\(\varepsilon ^p\)) with cycles (N) and when \(\text {d}\varepsilon ^p/\text {d}N = 0\), determine that shakedown has been achieved. In practice, this clean mathematical condition is relaxed to \(\text {d}\varepsilon ^p/\text {d}N \approx 0\) and historically accepted thresholds have been adopted, such as \(\text {d}\varepsilon ^p/\text {d}N \le 2\times 10^{-5}\) [39‐41]. Related procedures have tracked the evolution of the width of cyclic stress-strain hysteresis loops in order to detect their collapse and return to a linear elastic behavior. Again in practice, thresholds for this critical (near-zero) width where shakedown is detected have been adopted [40‐42].
While straightforward, these macroscopic DIC-based experimental shakedown detection strategies rely on full-field computational processing of images that are increasingly rich in resolution and data size. Under cyclic loading conditions, the incremental step-by-step analysis approach then requires the full processing of larger and larger image and data sets to identify shakedown states. This paper outlines a new procedure to detect shakedown using minimal kinematic measurements from DIC and an auxiliary purely elastic finite element model. In contrast to traditional macroscopic DIC-based shakedown detection methods, the proposed approach offers advantages in terms of computational and experimental effort, provides robustness against material uncertainties, and flexibility for a variety of applications. The approach is demonstrated on aluminum metamaterial sheets that are auxetic due to their arrangement of periodic perforations.
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To the best of the authors’ knowledge, there have only been two prior studies on the elastoplastic shakedown behavior of auxetics [43, 44]. Wang et al. [43] predicted the maximum load bearing capacities of various auxetic tubular lattice structures using a new direct numerical shakedown method. The study mainly focused on developing an effective numerical tool for their shakedown analysis and it was entirely computational. Wang et al. [44] examined the shakedown performance of an auxetic perforated sheet structure subjected to cyclic uniaxial loading. From experimental, numerical, and analytical perspectives, a non-monotonic relationship between the shakedown multiplier and auxeticity was identified. Thus the main contributions of the present study lie in (i) proposing a new methodology for analyzing cyclic experiments to identify whether or not elastoplastic shakedown has occurred, and (ii) presenting a combined experimental and numerical investigation with the proposed approach exploring the conditions for achieving shakedown in AA6061-T6 auxetic sheets through a Bree-like load interaction diagram. Unlike the classical Bree problem considering mechanical and thermal loads, the loading in this study is cyclic uniaxial asymmetric tension, which is defined by a mean force \(F_m\) and a force amplitude \(F_{amp}\), forming a (\(F_m\), \(F_{amp}\)) Bree-like load space [45].
The article is organized as follows. First, the experimental case study of the auxetic sheets is presented. The next section describes the finite element model used to guide the experimental design, and that is later modified for experimental post-processing. The shakedown test set-up and experiment details are then given. The new approach for experimental shakedown determination is presented as well as the results from the proposed methodology. The auxetic structure behaviors under asymmetric cyclic uniaxial loadings are discussed using a Bree-like load interaction diagram. Last, a map of the spatial extent of shakedown across the structure is presented.
Auxetic Sheet Case-Study: Structure & Material
Architected, cellular, and/or mechanical metamaterial structures offer the ability to tailor effective properties arising not only from the base constituent material but also from the command of “spatial heterogeneity.” That is, combinations of multiple materials or of material(s) and space are arranged in configurations and with connectivities that offer enhanced performance. Many such structures have been shown to exhibit auxetic behaviors (effective negative Poisson’s ratio) – such as re-entrant unit cells, chiral unit cells, perforated plates/cylinders, missing-rib unit cells, and more [46]. Unlike conventional non-auxetic components, when auxetic components are axially loaded, they expand in the direction perpendicular to the applied force (rather than the expected contraction) [47‐49]. This unconventional behavior may improve energy absorption and failure resistance [50‐53], and is thus also of interest for a variety of transportation, energy, and aerospace applications [54‐56]. Furthermore, low porosity auxetics have recently been shown to improve high cycle fatigue lifetimes [57, 58]. In this way, it is promising to now also examine auxetics in the relatively low-cycle elastoplastic shakedown domain (i.e., around 100 cycles).
The present work will demonstrate a new methodology for analyzing cyclic experiments to identify the shakedown response of a simple auxetic metamaterial sheet. The auxetic structure is very similar to the design proposed by Taylor et al. [59] and used by Wang et al. [44]; it is a perforated metallic sheet with identical mutually orthogonal slotted holes, as shown in Fig. 2. The differences in the present design compared with the structure studied by Taylor et al. [59] is that the present aspect ratio is fixed (8.7) and lower than Taylor et al. (who studied ratios between \(1-40\)) while the porosity \(16\%\) is higher than those (\(2\%-5\%\)) studied in Taylor et al. [59]. Compared with the perforated AA-5083 sheet examined by Wang et al. [44], the structure in this study was made of AA-6061 and the width (50 mm), thickness (1.016 mm), and porosity (\(16\%\)) were lower (Fig. 2) than those of Wang et al. [44] (63.5 mm, 3 mm, and \(16.4\%\), respectively). As two-dimensional Digital Image Correlation (2D-DIC) was chosen as the method to measure surface deformations and calculate strains, only the outer surface deformation would be observed. Thus, a specimen thickness that allowed for conditions close to plane stress (with negligible through-thickness variations in maximum stress and strain) was desired. An iterative simulation process (not included herein), combined with sheet supplier constraints, was used to identify 1.016 mm as appropriate. This thickness resulted in less than \(1.2\%\) difference in maximum allowable forces between surface and plane stress condition critical element locations (see below).
Fig. 2
Specimen (a) design and (b) manufactured with dimensions in millimeters. The thickness is 1.016 mm
Typical material properties for AA6061-T6 sheet (0.254mm - 6.35mm) at room temperature [61]
Young’s Modulus
Poisson’s ratio
Yield strength
Ultimate strength
Ultimate strain
E
\(\nu \)
\(\sigma _{y}^{0.2\%}\)
\(\sigma _{ult}\)
\(\varepsilon _{ult}\)
68 GPa
0.33
245 MPa
310 MPa
12%
Following the study of Taylor et al. [59] that characterized the auxetic nature of perforated sheets with elliptical holes of various aspect ratios, the common aluminum alloy AA6061-T6 was used. This type of aluminum alloy combines relatively high strength, good workability, high resistance to corrosion, and is widely used across aerospace, marine, transportation, and energy-harvesting industries (Table 1). In particular, all specimens were machined from \(1020 \times 1300\) mm size sheets of 1.016 mm thickness [60].
The choice to use test specimens with three vertically stacked pseudo unit cells (Fig. 2) was also based on the prior numerical and experimental work of Taylor et al. [59] on the auxetic nature of this structure. Taylor et al. [59] determined this configuration as ideal to avoid grip boundary condition effects under monotonic uniaxial loading. The authors also found that additional horizontal unit cells were unnecessary (results from a planar array of \(3 \times 3\) unit cells were negligibly different from a \(3 \times 1\) stack and a range of aspect ratios and void volume fractions).
Simulations to Guide Initial Experiments
In the spirit of the proposed approach, which strives to offer a new methodology to analyze experiments and detect shakedown behavior without having to resolve the actual distribution and evolution of equivalent plastic strain in the structure of interest and without having to make any assumptions about the actual inelastic behavior of the constituent material, finite element shakedown simulations with only simple elastic-perfectly-plastic (EPP) assumptions were conducted to guide experiments (Young’s modulus, Poisson’s ratio, and yield strength are listed in Table 1). These simulations are used strictly to help identify an approximate target experimental loading space where shakedown might occur. AA6061-T6 has been reported in the literature to exhibit limited combined or kinematic hardening [61‐63]. Future studies will explore the use of DIC-based Finite Element Method Updating (FEMU) approaches to determine appropriate hardening model parameters and obtain more accurate shakedown simulations for materials with greater uncertainty associated with their constitutive behaviors (e.g. graded material structures or additively manufactured structures). One of the major contributions of the proposed experimental analysis procedure is that this material characterization information about hardening models is not at all needed to identify whether or not elastoplastic shakedown has occurred in an experiment.
Three-dimensional Model
The cyclic elastoplastic behavior of the auxetic structure of interest was first investigated numerically to help guide the choice of experimental loading conditions (mean forces and cyclic force amplitudes). Finite element simulations (Fig. 3) were conducted in the commercial finite element software ABAQUS/Standard [64] to determine the shakedown response, following the classical incremental step-by-step strategy [10, 65] detailed below.
Fig. 3
(a) Three-dimensional finite element model, (b) one-eighth model with boundary conditions, (c) one-eighth model mesh with an enlargement of the central slotted hole highlighting the critical location at the central stress concentration where the onset of shakedown is expected, and (d) schematic loading
A three-dimensional model was established based on the manufactured dimensions (Fig. 2). Note that the model did not include the solid gripping extensions at the sample extremities used for the MTS machine fixtures. Taking advantage of symmetries, a one-eighth model (Fig. 3(b)) was used in the simulations for computational efficiency. The model was discretized with 8-node linear brick elements with reduced integration (C3D8R). The slotted hole root region (boxed in red in Fig. 3(c)) was identified as the critical location where the highest stress concentrations and stress gradients were expected, and a finer mesh was used in such regions. The critical element shown in Fig. 3(c) is where first-yield occurs. The critical element has dimensions of 0.2 mm \(\times \) 0.15 mm \(\times \) 0.2 mm, which was determined by an iterative mesh convergence study (not shown). The total number of elements in the model was 4032.
Boundary Conditions
Two loading steps were used in the analysis. First, a force ramp-up step and then a cyclic force step. In the first step, the axial force was increased from zero up to the prescribed mean force level \({F}_{m}\) by an applied traction at the top surface. Then, in the next step, the force was cycled with an amplitude of \({F}_{amp}\) for 150 cycles. The choice of 150 cycles was made based on established shakedown investigation practices [20, 27, 44, 65‐68]. In order to establish a one-eighth simulation model, the symmetry boundary conditions (Fig. 3 (b)) along the three directions were active throughout the whole simulation (steps 1 and 2).
Confirming Auxeticity
The auxetic nature of the structure (Figs. 2 and 3(a)) is confirmed by observing the lateral expansion when subjected to the vertical tensile force (Fig. 3(b,c)). Figure 4 shows the simulation results under load case T25 (Table 3). The light blue showing the deformed configuration without scaling is superimposed over the unloaded configuration (gray color) in Fig. 4(a). This comparison illustrates the structure’s lateral expansion, therefore confirming its auxetic behavior. Figure 4(b) presents the von Mises stress contours with gray areas indicating levels greater than the yield stress (Table 1). As expected, these highly stressed areas are concentrated at the roots of the slotted holes. As reported in the literature [59, 69], this observation suggests that the deformation mechanism responsible for the auxeticity is driven by the rotational effect in these critical regions.
Fig. 4
Example of FEA output at the onset of yielding for load case T25 (Table 3). (a) Deformed configuration in light blue superimposed over the unloaded configuration, demonstrating the auxetic nature of the structure. (b) von Mises stress contours with levels above the nominal yield stress in gray
Numerical load interaction diagram with black circular markers superimposed indicating load cases that were experimentally tested (Table 3). The loading combinations resulting in compressive applied loads were not considered and are grayed out
Classical Incremental-Approach to Numerical Shakedown Determination
Using an incremental step-by-step approach to shakedown detection, the von Mises equivalent stress and equivalent plastic strain (PEEQ) were numerically tracked through cycling at the critical element located at the root of the slotted hole where the stress concentrations were most severe and representative of the structure critical response (Fig. 3(c)). If \(\Delta \varepsilon ^{p}/\Delta N < 2\times 10^{-5}\) was satisfied over the last 50 cycles [10, 39‐41, 65, 68] (where \(\varepsilon ^{p}\) is the equivalent plastic strain and N the loading cycle), the element was deemed in a shakedown state. The blue solid line in Fig. 5 shows a typical non-shakedown response at the critical element where \(\Delta \varepsilon ^{p}/\Delta N<2\times 10^{-5}\) is violated over the last 50 cycles. In contrast, the black dashed line shows a typical shakedown response at the critical element where the equivalent plastic strain does not increase with cycling.
Numerical Bree-like Load Interaction Diagram
The structural loading design space is illustrated by a Bree-like load interaction diagram [1]. Figure 6 shows the numerical load interaction diagram for the perforated structure (Fig. 2) subjected to a cyclic tensile force with a mean level of \(F_m\) and an amplitude of \(F_{amp}\) (Fig. 3(c)). The elastic limit line (red solid line) represents the design limit using traditional yield-based design criteria – any loading combination below the line ensures the elastic response of the structure. In contrast, the loading combinations between the shakedown limit (red dashed line) and elastic limits allow plastic deformation to occur in the structure in early stages of force cycling. The induced residual stresses prohibited further plastic strain increases in the subsequent cycles. The inadmissible region refers to undesirable behaviors such as alternating plasticity, ratchetting, and collapse [18], which are not further delineated in this study. To give an approximation of the plastic collapse limit (gray dashed-dotted line), and further guide experiments to focus on shakedown (somewhere below the collapse limit), n (optional) FE simulations were conducted. A bilinear isotropic hardening model under monotonic loading (\(F_m + F_{amp}/2\)) was considered using available ultimate stress and strain values from the literature [61], Table 1. Plastic collapse was estimated by checking for load cases where the FE simulations terminated due to excessive plastic deformation and/or the increments required to proceed being less than 0.05 N. The loading combinations resulting in compressive applied loads were not considered (to avoid buckling) and are grayed out in the diagram.
Experimental Study
In the present section, the experimental protocol is introduced as well as different load cases chosen for the tests.
Test Set-Up and Specimen Preparation
The tests were conducted using a servohydraulic 810 MTS testing machine with an axial load capacity of 100 kN. Hydraulic grips (flat edge) were used to clamp the specimens (40-bar gripping pressure). To facilitate gripping, four 48-50 mm square AA6061 tabs of the same thickness as the specimens (1.016 mm) were attached to the front and back gripping sections of the samples (see specimen drawing in Fig. 2). These gripping aides were prepared by first lightly sanding (Struers, SiC P#120 grit) all of the attachment surfaces, followed by cleaning with Acetone, and the application of toughened structural adhesive (Araldite, 420 A/B UN2735). Following 24 h of adhesive curing at room temperature, a black marker (BIC Velleda) was used to trace and paint over the interior hole cut-out surfaces (through-thickness surfaces) to avoid camera glare off of the base material.
The samples were then prepared for DIC imaging via speckling. Masking tape was applied on the gripping sections during the painting steps and removed before mechanical testing. Matte white aerosol spray paint (Ront Production, 400 mL RAL-9010) was applied to the specimen surface until the metal was completely covered. After drying for approximately 5-10 min, a speckle pattern was applied by hand by lightly spraying matte black aerosol paint from a distance of approximately 60 cm (Ront Production, 400 mL RAL-9005). The quality of the speckle pattern was preliminarily judged through evaluation of the image gray levels with the histogram revealing a broad span with minimal saturation, and the achievement of black spots that are approximately 5 pixels wide, in accordance with IDICS recommendations [70]. For a majority of tests (denoted with an asterisk in Table 3), a Manta G145B camera and telecentric lens (Edmund Optics, \(\times 0.08\) lens) was mounted on a lab rail platform placed approximately 25 cm away from the specimen gripped in the testing machine (Fig. 7). The remaining tests were conducted without the telecentric lens and with a 150 cm stand-off distance. In both cases, the camera was positioned to capture a majority of the slotted holes, while remaining perpendicular to the specimen surface.
Table 2
DIC hardware parameters
Camera
Manta G145B
Definition
\(1390\times 1038\) pixels
Gray Levels amplitude
8 bits
Lens
Edmund Optics telecentric lens 0.08X or AF Micro Nikkon
An LED bank light was angled between the specimen and camera (out of the field of view) to provide additional lighting. Table 2 gathers the DIC hardware parameters.
In order to quantify measurement uncertainties, a series of 100 images of the specimen surface were acquired in the gripped configuration, but before any mechanical loading was applied. A piece of cardboard painted using the same matte black spray paint (Ront Production, 400 mL RAL-9005) was temporarily inserted behind the specimen to facilitate clear edge detection in the images for backtracking purposes (Fig. 8).
Fig. 8
Close-up view of the gripped and speckled specimen
Using the testing devices described above, a series of shakedown tests at different loadings (Table 3) were carried out to create a Bree-like load interaction diagram. The specimens (Fig. 2(b)) were firstly gripped and then the testing machine was zeroed and set to load control. The dwell time for the specimen under zero force and zero torque was set to be longer than 100 s to measure the standard force uncertainty \(\gamma _f\) (i.e., the standard deviation of the corresponding force acquisitions). Then, the axial force started to ramp to \(F_m\) in \(t_m\) as shown in Fig. 9. The force loading rate was kept between 50 N/s to 100 N/s [71] to eliminate the strain rate effect (Table 3). After a short dwell time d/2 (half of column 6 in Table 3), the axial force started to cycle with an amplitude of \(F_{amp}\) for 150 cycles. 150 cycles were chosen to allow for the shakedown state to be reached. To ensure image acquisition at the highest and lowest force levels during cycling, a constant dwell time d was used at \(F_m \pm F_{amp}/2\). Because the loading rates are kept within a certain range, the time (ramping time and dwell time) for each load case was different. The imaging frequency was set to be constant (see last column of Table 3).
Fig. 10
Example load case T21\(^{*}\) with 100 initial images for uncertainty quantification purposes and reduced data set used for analysis during cycling
Schematic illustration of proposed methodology. (a) DIC measured displacement field, and (b) top and bottom (Dirichlet) nodes where measured boundary conditions are applied (depicted in yellow)
To detect shakedown with the new analysis method (next section), and following the numerical load interaction diagram (Fig. 6), 14 tests corresponding to 11 different loading combinations were chosen (depicted in Fig. 6 with black markers) and conducted with the previously described setup. The variety of test cases included a significant portion away from the approximate shakedown limit (T16, T22, T15, T19, T21, and T27). This choice was made to identify a threshold force amplitude offset level from which to detect shakedown or non-shakedown status. Four test cases were selected close to (\(0.1~\text {kN}/0.9~\text {kN}=11\%\)), the approximate expected shakedown limit (T17, T18, T2 (T24 is a repeat of T2), and T3). These cases were chosen with a conservative margin to account for testing and prediction uncertainties.
Table 3 summarizes the 14 tests conducted for the 11 different loading combinations, in which three of them are repeat tests, namely, T19 and T21\(^{*}\) are repeat tests for T15 and T24\(^{*}\) is a repeat test for T2. The load cases with an asterisk indicate that a telecentric lens was used in the tests to make the setup more compact.
Figure 10 displays the experimentally measured force response from test T21\(^{*}\) (\(F_{m}=0.75\) kN, \(F_{amp}=0.5\) kN) recorded with the same frequency as DIC images were acquired. This figure includes the initial 100 images for the loading-free dwell period to perform uncertainty quantification. In the following, only the force levels at \(F_{max}\) and \(F_{min}\) are shown.
Representative comparison between reaction forces from elastic FEAs with DIC-measured boundary conditions and measured forces from load-controlled (a) T15 and (b) T24 experiments showing a shakedown case (a) and non-shakedown case (b), respectively
The shakedown identification protocol proposed in this study is inspired by Direct Methods in shakedown analysis [9] like Melan’s lower bound theorem [15, 45, 72], that utilizes linear elastic stress solutions to predict cyclic elastoplastic shakedown states without the need for computationally costly incremental elastoplastic FEA. In particular, minimal kinematic measurements from DIC are combined with a purely elastic finite element model to rapidly identify shakedown states. The key is to focus on the fact that if shakedown is achieved, purely elastic response should be recovered. One simple way to check if the end of a cyclic experiment has reverted to elastic behavior is to compare it with a purely elastic FEA simulation. The actual load amplitudes recorded during the last 50 cycles of the experiments are compared with the corresponding reaction force amplitudes from an auxiliary linear elastic FEA model that takes as input only the DIC-measured cyclic displacements at the structural boundaries. Taking measurement errors into account, a shakedown state is identified if the differences between these force amplitudes are negligible; conversely, a non-shakedown state is identified if the differences between these force amplitudes are significant. For the load-control cyclic tests conducted in the present study, the procedure is as follows. (i) First, the displacement fields of the specimen are captured via DIC analysis. Only the cyclic displacements at the specimen boundaries noted in Fig. 11 are needed (top/bottom node sets). (ii) Then those cyclic displacement fields are applied as boundary conditions in an elastic-only FEA on the undeformed specimen mesh. (iii) The corresponding predicted reaction force amplitudes from the elastic FEA are computed. (iv) Finally, the shakedown/non-shakedown state is identified by comparing the experimentally measured force amplitudes with the elastically predicted ones. The shakedown detection approach is general and not restricted to the load-control scenario investigated herein. It is worth noting that the proposed shakedown detection approach does not apply for the cyclic behaviors in the inadmissible regime (i.e., potentially alternating plasticity, ratchetting, and/or collapse).
Fig. 13
Load interaction diagram with markers indicating experimental results: black dots for shakedown cases, red dots for non-shakedown cases, and red squares for fracture cases. The value in parentheses below (gray) is the average force amplitude offset for each test
Figure 11 illustrates the proposed methodology that is demonstrated on the auxetic structure tests. The acquired images were processed using the Correli 3.0 framework [73] (Table 4), which utilizes meshes made of three-noded (T3) triangles. The DIC mesh was constructed once the surface mesh used in FE simulations (i.e., made of 4-noded quadrilaterals) was backtracked and transformed into a T3 mesh by splitting one Q4 element into 2 T3 elements so that the same nodes are considered in both meshes.
A representative displacement field measured by DIC is illustrated in Fig. 11(a). The displacements at the top and bottom node sets, as illustrated in Fig. 11(b), are applied to elastic FEAs as boundary conditions. Thus, the only material parameters needed in these analyses are the Young’s modulus and Poisson’s ratio. 2D meshes were considered and the FEAs were run under plane stress assumption.
The variable of interest from this auxiliary FEA simulation is the reaction force at the node sets with prescribed displacement boundary conditions (either top node set or bottom node set). Figure 12 compares the reaction force amplitude from the elastic FEA predictions with DIC-measured boundary conditions (red hollow circles) and the measured force for two load-controlled experiments (black stars). The force amplitude spans from the start of force cycling to the 150th cycle. Figure 12 shows the results for load cases T15 and T24, respectively, representing a shakedown case and a non-shakedown case (detailed descriptions are discussed hereafter).
Load Interaction Diagram with New Methodology Results
Using the proposed methodology described above, the comparison between measured and predicted forces (Fig. 12) has been made for all 14 tests. They are divided into 4 groups, as shown in Appendix A. Table 5 gathers the results of load case T15 and two repeat tests T19 and T21. The comparison plot for each test is provided in the second column, along with the standard force uncertainty \(\gamma _f\). To quantify the comparison, the simulated force amplitudes (red hollow markers) and the experimentally measured ones (black star markers) were averaged over the last 50 cycles to calculate the offset that is reported in the last column. For these three tests with the same loading, the standard force cell uncertainties \(\gamma _f\) were all less than 1.3 N, and the maximum averaged force amplitude offset was 12.3 N.
Fig. 14
Spatial distribution of elements satisfying \(\Delta \sigma _{VM}/(2\sigma _y)>1\) in the region of interest (a,b,d,e) and histogram plots (c,f) for load cases \(\#15\) and \(\#24\)
The shakedown state is determined based on the offset of the averaged force amplitude (\(\text {F}_{amp}\)). If the offset is greater than 10 times \(\gamma _f\), it indicates that the structure is in a non-shakedown state, otherwise it is deemed as shakedown. The load case with these 3 tests (Table 5) has a maximum averaged force amplitude offset of 12.3 N, which is 9.5 times \(\gamma _f\). Therefore, load case T15 (with T19 and T21 repeats) is recognized as shakedown. Similar results are obtained for load cases T16, T18, T22, and T26 with higher \(F_{max}\) than T15 (Table 6). The standard force uncertainties were less than 1.3 N for these 4 tests. In particular, test T26 had an averaged force amplitude offset of 48.8 N while 3 tests led to offsets less than 8.7 N. This observation indicates that load case T26 is the only shakedown case among these 4 tests. Likewise, Table 7 summarizes the results of load case T2 and its repeat tests T24, T17, and T27, with a force amplitude offset of 94.6 N, 67.8 N, 22.7 N, and 6.1 N, respectively. The maximum force uncertainty level was 1.5N for this group of tests. Thus experiment T27 can be found to be the only non-shakedown case among all 4 tests in this table. Last, the results of load cases T25, T3, and T23 are shown in Table 8. Note that fracture was observed for all 3 tests; the force cycling stage was not finished. Therefore, the predicted reaction force and measured force comparison are shown by measured force versus image number. The predicted force levels (via elastic FEAs) increased rapidly up to fracture. The maximum force uncertainty was 1.3 N, and the averaged force amplitude offset ranged from 169.3 N to 217.6 N. These fracture cases are regarded as non-shakedown cases.
Finally, from Tables 5 to 8, the maximum force \(F_{max}\) of the tests in each group generally increases (second column of Table 3). However, the average force amplitude offsets of shakedown cases in Tables 5 to 8 generally remain within 15 N (except for case T17 which is close to the shakedown limit (Fig. 13)). This consistency indicates that the force offset is independent of loading magnitude, making it a robust index for shakedown identification.
With the shakedown states of 14 tests determined using the proposed approach, the load interaction diagram has been updated with experimental results as shown in Fig. 13. The load case markers have been revised, namely, the black dots represent shakedown states, the red dots indicate non-shakedown cases, and the red squares stand for fracture cases. Each test is annotated with a value in parentheses (gray) below it, showing the averaged force amplitude offset (Tables 5–8). In addition, the fracture line (red dash-dotted line) has been estimated to separate the fracture cases from other non-fracture cases. With these results, the original inadmissible design space (denoted by red dashed line) is indicated by the pink shading. Based on this revised load interaction diagram, the maximum allowable force considering shakedown is about 3.6 (1.9/0.53) times that allowed by the elastic limit from a traditional first-yield criterion.
Discussion & Conclusions
In addition to the predicted reaction forces and experimentally measured force comparison for load cases T15 (shakedown) and T24 (no shakedown) shown in Fig. 12, element-based distribution maps and histograms of the region of interest (Fig. 14(a,d)) have also been created for each case. The histograms show the number of elements with respect to the von Mises stress amplitude under cyclic loading normalized by twice the yield stress (\(\Delta \sigma _{VM}/(2\sigma _y)\)). This quantity is significant because a value of 1 indicates that cyclic plastic mechanisms could be activated. The maps in Fig. 14(a,b,d,e) show the distribution of this normalized quantity. The structure is identified as non-shakedown if there is at least one element with \(\Delta \sigma _{VM}/(2\sigma _y)>1\). Both Fig. 14(c) and (f) show that most of the elements in the region of interest have a value of \(\Delta \sigma _{VM}/(2\sigma _y)\) less than 0.5. However, no element in load case T15 (shakedown) has a value exceeding 1, while in load case T24 (not shakedown) there are 8 elements with values greater than 1. In this way the structure under load case T25 is deemed in a macroscopic non-shakedown state and a closer inspection of the distribution of elements with \(\Delta \sigma _{VM}/(2\sigma _y) > 1 \) gives an indication of the extent of local shakedown/non-shakedown. The local non-shakedown region is limited to the 8 elements that are at the root of the slotted hole where stress concentrations are expected.
Traditional experimental shakedown detection of architected structures relies on strain observations, whose accuracy may be highly affected by the surface preparation, imaging accessibility, and architecture length scale (e.g., slotted hole size in this study). The proposed approach utilizes measured displacements and forces away from the region of interest, which are more easily accessible. Therefore, the length scale and architecture are separated from shakedown detection. This feature provides an opportunity for future shakedown experimental studies involving smaller length scales and complex multidimensional architected structures.
In conclusion, a new protocol to determine the macroscopic shakedown response of cyclically loaded structures has been proposed and demonstrated on a perforated auxetic structure. Specifically, this approach identifies shakedown based on minimal digital image correlation results combined with pure elastic FEA predictions instead of classical incremental plastic strain analysis. The proposed approach is especially suitable for high-throughput experimental shakedown screening where automated and/or real-time DIC analysis could drive the experimental program.
Declarations
Conflicts of Interest
The authors have no conflict of interest to declare that is relevant to the content of this article.
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Comparisons between experimentally prescribed force levels and the numerical shakedown detection analysis for load cases resulting in fracture upon reaching load maximum (Table 3)
Note that no load cycles were performed, but the force level (F) between consecutive images (\(\Delta \)Image) is reported
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