Viscoelastic Response of Silicone Additively Manufactured Direct Ink Write (DIW) Foams under Repetitive Compression

Viscoelastic Response of Silicone Additively Manufactured Direct Ink Write (DIW) Foams under Repetitive Compression | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Viscoelastic Response of Silicone Additively Manufactured Direct Ink Write (DIW) Foams under Repetitive Compression Moira M. Foster, Daisy Philtron, Mark D. Herynk, Ziad Ammar, Siddharthan Selvasekar, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6497954/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Additively manufactured (AM) foam was compressively strained into the plateau region through a reduced design of experiments. Its dynamic stiffness, or complex modulus, declined across 10,000 cycles of small deformation in the plateau region. This rate of change of stiffness, when fit to a simple power law, is correlated with the degree of nonlinearity of the material’s deformation. Nonlinearity is defined as the nonlinear viscoelastic parameter, calculated as total harmonic distortion. Experiments with low nonlinearity tend to maintain their complex modulus across cycles, while materials undergoing highly nonlinear deformations decreased their modulus. The nonlinearity of the experiment is dependent upon the stress that the material experiences as well as its strain rate. Strain rate correlates with the odd resonance nonlinearity while the complex modulus relaxation rate is correlated to the even resonances. Two different material structures were tested, face centered tetragonal (FCT) and simple cubic (SC). As SC has more overlapping strand areas, and thus a shorter load path length through the sample, the initial stress is higher at the same relative porosities. The SC material therefore experiences greater nonlinearity, 5% total harmonic distortion, than the FCT material with 2% total harmonic distortion. Thus, the SC structure shows much lower stiffness retention than FCT. These findings indicate that when selecting material for repetitive cyclic compressions, a more stable low stress material will maintain its dynamic performance more uniformly than an instable high stress material. Dynamic modulus complex modulus nonlinear polymer Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1. Introduction Understanding the correlations between different material properties and how they transform over time has many industrial and aerospace applications (Mills 2007 ). In a combined pre-loaded compression and vibration environment, material properties are exhibited at both low-rate stiffness (static) and dynamic stiffness. The rate at which the static stiffness declines over time due to relaxation has been characterized through relaxation and creep testing (Schapery 1969 ). The dynamic modulus and how it changes over time, however, has been less explored. Static stiffness, or Young’s modulus, can be calculated via the stress divided by the strain (Gibson and Ashby 1997 ). The dynamic modulus or complex modulus is kinematically defined as the steady-state solution due to an imposed sinusoidal excitation. The dynamic modulus can be calculated using Fourier transform rheology (FTR) (Wilhelm et al. 1998 ). The Fourier transforms of the stress, \(\:\mathcal{F}\left(\sigma\:\right)\) [MPa], and strain, \(\:\mathcal{F}\left(\epsilon\:\right)\) [mm mm-1], are divided in the frequency domain to calculate the linear dynamic stiffness, E* [MPa], at the excitation frequency (Wilhelm 2002 ) $$\:\left|{E}^{*}\right|=\left|\frac{\mathcal{F}\left(\sigma\:\right)}{\mathcal{F}\left({\epsilon\:}\right)}\right|.$$ 1 The storage modulus, E’ [MPa], and loss modulus, E” [MPa], can be defined as the imaginary and real components of the complex modulus, $$\:{E}^{*}={E}^{{\prime\:}}+i\:E".$$ 2 Through the kinematic solution of a simple Voigt or Maxwell model, loss modulus is shown to be a linear function of the excitation frequency, while storage modulus is stable in frequencies well under the natural resonance of the system. Subtracting the phases of the stress and strain at the excitation resonance produces the damping factor, or phase lag, ẟ , $$\:\delta\:=\mathcal{F}\left({\sigma\:}\right)-\mathcal{F}\left({\epsilon\:}\right).$$ 3 When the amplitude or frequency of the sinusoidal excitation is increased, the dynamic stiffness of foamed materials will decrease then increase, exhibiting a second order relationship with strain rate (Gibson and Ashby 1997 ; White et al. 2000 ). The bulk material stiffens as the material is excited faster, however, at higher strain rates inertial effects -such as the flow of air in and out of the cellular structure - begin to take effect as well, causing nonlinearity in the stiffness response (Mills 2007 ). Nonlinear energy can occur in the dynamic modulus’s response when the material undergoes nonuniform or unstable deformation (Hyun et al. 2011 ). Nonlinear energy can be calculated via the nonlinear viscoelastic parameter (NVP) (Wilhelm 2002 ). NVP is defined as the ratio of magnitudes of a higher order resonance normalized by the magnitude of the excitation resonance (Hirschberg et al. 2020 , 2021 ). In this work, the definition from the total harmonic distortion was used as the root mean square approach akin to total harmonic distortion, $$\:NV{P}_{n,\:m}=\frac{\sqrt{\left|{\mathcal{F}}_{n}^{2}\left(\sigma\:\right)\right|+\left|{\mathcal{F}}_{m}^{2}\left(\sigma\:\right)\right|}}{\left|{\mathcal{F}}_{1}^{2}\left(\sigma\:\right)\right|}$$ 4 where NVP n,m is the nonlinear viscoelastic parameter at resonance n and m . Mills suggests that nonlinearity could be a function of the frictional interfaces of the cell walls rubbing together in the material (Mills 2007 ). Thus, nonlinearity is higher with material undergoing larger pre compressions, or lower relative porosity (White et al. 2000 ; Wang et al. 2020 ). Odd resonances appear with increased frequency of excitation, or the strain rate, of the material (Hyun et al. 2011 ). Even resonances appear when there are nonuniform strain distributions in the material. Nonuniform compression in cellular and honeycomb materials have been attributed to instabilities within the lattice structure (Kyriakides 1993 ; Rajput et al. 2017 ; Luan et al. 2022 ). In AM material, for example, overlapping strands in a lattice are areas of stress concentrations (Van Meerbeek et al. 2022 ; Zhu et al. 2023 ). These stress concentrations are relieved via buckling in the surrounding strands, which cause sudden load releases and nonuniform deformation in the sample (Hyun et al. 2011 ). Wilhelm proposed in his 1998 paper that even resonances may also be influenced by “time-dependent memory effects or nonlinear elastic contributions in the system” (Wilhelm et al. 1998 ). The appearance of odd and even resonances could be utilized as an in-situ technique for determining failure of the material system, similar to structural health analysis. Studies on brittle polystyrene and PMMA, for example, have mapped an increase in the nonlinear second and third resonances to small and large cracking (respectively) in brittle material (Hirschberg et al. 2017 , 2018 ). Flexible, non-brittle, materials like polyurethane also exhibit changes over time, however, these changes are not discrete such as fracture (Mills 2007 ). This work aims to connect how the nonlinearity in dynamic system attributes to the decay of the complex modulus across cycles. Viscoelastic materials exhibit decay of their stiffness as they are held under a fixed strain (Menard 1999 ). The rate of stress relaxation, or relaxation modulus, is governed by both topology and bulk material properties in a sample (Gibson and Ashby 1997 ; Zhu et al. 2023 ). The rate of this decay is commonly modeled via an exponential or power law decay and is dependent on the applied strain during relaxation (Schapery 1969 ). AM silicone lattices showed that face centered tetragonal (FCT) structures had a slower relaxation rate than the simple cubic (SC) structures (Zhu et al. 2023 ) (Fig. 1 ). This was hypothesized to be due to the lower stress levels in the FCT structure as the offset strands were able to collapse nicely into the next layer. Limited studies have fit the rate of decay of the complex modulus under a constant cyclic deformation. In this study, long cycle and large deformation DMA is conducted on FCT and SC materials. As the material is pre-compressed to the mean excitation strain, the material begins to undergo a stress relaxation test. This relaxation, however, is overlaid with sinusoidal excitations that themselves have a calculated dynamic stiffness element. Thus the material has a relaxation modulus associated with this test as well as a decaying dynamic stiffness as calculated by FTR. Previous studies have shown that an increased strain rate increases the decay of the dynamic stiffness in this testing, however, no correlation has been conducted to the nonlinear properties of the excitation and the structural effects of introduced topological differences. 2. Methodology Foam samples were additively manufactured using direct ink write printing (Maiti 2021 ). Simple cubic (SC) and face centered tetragonal (FCT) lattices were printed using Llama 50 as the base material with a strand diameter of 500 µm on a Teflon coated aluminum plate (Small et al. 2019 ; Lenhardt 2022 ). Subsequently each part was thermally cured in an oven using the same temperature profile of 2 hours ramping to 150°C then holding at 150°C for 16 hours. A total of six unique lattice structures, three strand pitches of 1.19 mm, 1.35 mm, and 1.56 mm were used to create three initial nominal porosity values of 55%, 60%, and 65% (Fig. 2 ). The initial rectangular print geometry was 30 layers, approximately 12 mm thick and 127 mm wide. The samples were trimmed to a width of 65 mm to remove printing edge effects while maintaining a 5:1 ratio to the thickness. During printing and curing, structure sag and shrinkage occurred to various degrees in all samples, producing sample thicknesses ranging from 11.4 to 12.3 mm. A calculated porosity based on the initial measured thickness and intended porosity were used in for the design of experiments. Three replicates were printed for each unique lattice structure and were each tested to a unique strain and frequency environment. The samples were placed between two eye-leveled six-inch compression platens on a MTS hydraulic load frame outfitted with a FLIR blackfly camera. The samples were strained to an initial strain value, ε_0 [mm mm − 1 ]. After a 2 minute static stress relaxation period, the samples were sinusoidally excited with an amplitude, ε 1 [mm mm − 1 ], of 0.3 mm (0.3% strain) at an excitation frequency, ω [rad s − 1 ], $$\:\epsilon\:\left(t\right)\:=\:{\epsilon\:}_{0}\:+\:{\epsilon\:}_{1}\:sin\left(\omega\:t\right)$$ 5 Each sample was cyclically excited for 10,000 cycles. Displacement and force information was recorded via the loadcell and crosshead readouts at a rate of 1000 Hz. The data was recorded for 100 cycle increments logarithmically spaced throughout the 10,000 cycles. The displacement information was then viscoelastically compliance corrected using linear interpolation to shift the data in time and displacement. The 50 kN load cell was calibrated annually in house using ASTM E4-16 without NIST traceability. The time history of displacement and force from the load frame were post processed using DMA Fourier Transform Rheology (FTR) (Wilhelm et al. 1998 ). After filtering using a Butterworth filter, the Fourier transform of the stress and strain time histories were calculated, to produce the magnitude and damping of the complex modulus (Eqs. 1 and 2 ). For this work, NVP 2 and NVP 3 were analyzed via Eq. 3 . The total NVP 2 − 5 was also assessed as the summation of resonances 2 through 5. To supplement the information provided via the time history of displacement and force, a FLIR Blackfly camera with a 105 mm lens captured images at 100 frames per second. Post processing of the images in MatchID used spatial reference updating, and affine shape function. The SC samples exhibited large out of plane deformation which prevented usable DIC data. FCT samples underwent DIC analysis for qualitative description of the strain distribution within the material layers using virtual extensometers. Since the samples were dynamically strained within the plateau region, the dynamic strain changes were mainly isolated to the lattice deformation rather than compression within strands. A large subset size allowed tracking of filament displacement relative to other filaments, instead of material strain within filaments. In the design of experiments, the samples were compressed to five different initial strains, ε 0 , ranging from 15 to 33%. The combination of initial porosities, p i , and strains, ε 0 , produced three relative porosities, p r , calculated as $$\:{p}_{r}=\frac{{p}_{i}-{\epsilon\:}_{0}}{1-{\epsilon\:}_{0}}.$$ 6 This calculation assumes that the compression remains in the plateau strain region, such that all the displacement is taken by the deformation of the lattice structure and collapse of air pores, rather than compression of the actual base material. This assumption was verified using DIC strain field results (Morrison et al. 2024 ). Using the strain and porosities together with the initial porosity calculation, a Latin Square Design of Experiments was used to determine the combination of strains and porosities of interest such that there were repeating values of strain and relative porosities between the initial print parameter porosities (Table 1 ). Based on previously published work from 2024 on polyurethane materials under similar test conditions, an assumption was made that there was no interaction between the relative porosity and the frequency (Foster et al.). Thus, different frequencies were able to be overlaid on the Latin Square for another dimension in correlations. This design led to a comparison of both the strain rate among the diagonal samples that had the same relative porosity, and then comparison of strain rate and relative porosity combined effects using the other eight conditions. This Latin Square Design was duplicated with both the FCT and SC print structures for a total of 18 different structure, frequency, strain, porosity combinations. A single set of replicates for one condition were included in the design of experiments and were shown to have good repeatability. Table 1 Latin Square Design showing frequency and relative porosity information overlaid. strain Porosity 15-19% 25% 33% 55% 47 % @ 5 Hz 40 % @ 10 Hz 33 % @ 0.5 Hz 60% 53 % @ 5 Hz 47 % @ 10 Hz 40 % @ 0.5 Hz 65% 57 % @ 5 Hz 54 % @ 10 Hz 47 % @ 0.5 Hz Including the control parameters, there were 15 different variables considered for causation and correlation in this experiment (Table 2 ). The output variables relaxed and changed over the course of the 10,000 cycles of excitation, thus each output variable also includes a time-dependent variable. To investigate relationships between variables, p-values from the Pearson correlation coefficient (PCC) and Akaike information criterion (AIC) were used to quantify significance. Table 2 Input and output experimental variables. Variable Input (Control) Input (Calculated) Output (Response) Porosity X Strain X Relative Porosity X Frequency X Amplitude X Strain Rate X Stress X Static Modulus X Complex (Dynamic) Modulus X Tan delta (Damping) X Nonlinear Viscoelastic Parameter(s) X Static Modulus Relaxation Rate X Dynamic Modulus Decay Rate X Damping Decay Rate X NVP Decay Rate X 3. Results and Discussion Nine samples of SC and FCT DIW silicone material were compressively strained to within the stress plateau region, then cyclically strained about the initial compression point at a given excitation frequency as indicated by the Latin square design of experiments. The stress response was recorded across cycles and used in the calculation for static and dynamic stiffnesses. First the initial viscoelastic response was analyzed, then across time. 3.1 Initial DMA Results The average stress during the first 100 cycles of excitation was linearly related to the relative porosity and offset by lattice print structure (Fig. 3a). Static stiffness was calculated as the average stress divided by the initial strain. Unlike stress, initial static stiffness could not be simplified to a linear function of relative porosity and structure, instead best linearly related to stress (Fig. 3b). While relative porosity effectively normalizes the stress strain curve in regard to stress, the stiffness does not scale linearly with the normalization. The dynamic stiffness, or complex modulus, of the material during the initial 100 cycles of compression demonstrated some damping in the material, with mainly elastic components. Initial damping values were correlated to strain rate and the damping trends across time were correlated to the initial damping values. Graphically, the hysteresis loops became steep (increased in stiffness), widened (increased in damping), and became more nonlinear (shown through coloring) with increased excitation frequency and smaller relative porosity (Fig. 4 ). The initial complex modulus was linearly correlated to the static modulus by a factor of 1.5. When the samples compressed to the initial strain, ε 0 , the strain was nonuniformly distributed across the vertical length of the samples. DIC full field strain maps were used to assess strain localization, although quantitative values were not reliable due to the large displacement for DIC calculation (Morrison et al. 2024 ). In most FCT samples, the strain localized near the upper and lower boundaries at the platen-sample interface, similar to previously published observations of foam under compression (Sriram et al. 2023 ). The side of the sample that experienced sagging due to manufacturing showed the most strain, likely inciting earlier collapse to the unstable structure (Kyriakides 1993 ). Like the initial strain distribution, the dynamic strain distribution was also nonuniform (Fig. 5 ). The static strain is the strain due to the initial compression. The dynamic strain is the strain that is produced about the initial compression due to the dynamic sinusoidal excitation amplitude. The relative porosity and the excitation frequency both influenced the dynamic strain distribution. For example, uniform strain occurred in a 5 Hz sample while nonuniform dynamic strain occurred in a 0.5 Hz sample (Fig. 5 ). In general, however, as excitation frequency increased the strain distribution further isolated within the material. The samples’ strain distribution remained steady across cycle counts with no significant changes detected across the 10,000 cycles of compression in any sample. Higher excitation frequency and non-uniform strain distribution within the material invoked larger NVP values. Like strain distribution, NVP 2 was correlated to both the strain rate and relative porosity, however, relative porosity had limited effect on the statistical fit and was omitted. All three NVP definitions (resonance 2, resonance 3, and total) were found to be nonlinearly related to the excitation frequency with an offset for lattice print structure. Based on prior work demonstrating the importance of strain rate in addition to excitation frequency, the analysis used strain rate, \(\:\dot{\epsilon\:}\) [s − 1 ], as the dependent term in the quadratic fit as shown with NVP 3 (Foster et al.) (Fig. 6 ). The transition of nonlinearity at 5 Hz could be due to the transition between the quasistatic and intermediate rate regimes. NVP 2 and NVP 2 − 5 were also nonlinearly related to the excitation frequency but inverted from NVP 3 . Increased excitation frequency causing increased nonlinearity has been observed in previous studies (Grieninger et al. 2019 ). As NVP is a dimensionless quantity, the strain rate in the statistical fit is a normalized strain rate to a reference strain rate. 3.2 Stiffness Decay over Time As the material was held at the initial pre-compression with the overlaid oscillation, the average stress across cycles decayed through stress relaxation. Through this viscoelastic relaxation, static stiffness, calculated from the average stress, also decayed according to a nonlinear power law function as described by Schapery (Schapery 1969 ), $$\:E\left(t\right)={E}_{\infty\:}+\:{E}_{0}h{t}^{-m},$$ 10 where m is the decay rate, E ∞ and E 0 are constants across same porosity materials, and h is a parameterization function for different loading conditions. For this study, the variability in the initial porosity, strain, and print structure of the samples made determining a parameterization function, h , unpractical. For this reason, E 0 h was simplified to a fully empirically derived value, A . Furthermore, the infinite modulus, E ∞ , was taken to be zero as the static stiffness was never observed to reach a steady state condition. These simplifications created a linear regression model on a log-log scale, $$\:\text{log}\left(E\left(t\right)\right)=\text{log}\left(A\right)-m\:\text{l}\text{o}\text{g}\left(t\right).$$ 11 Using a least squares regression approach, each sample’s static stiffness across time was fitted with resulting R 2 values of 0.75 to 0.99 (Fig. 7 ). The same linear power law was fitted to the dynamic stiffness data (Fig. 8 ). To maintain the simple linear power law relationship with the dynamic stiffness decay, the first excitation data point was omitted. A large change in stiffness occurred within the first 300 cycles of excitation. This change was not aligned with cycles 250 onward that demonstrated a highly linear rate of change on a log-log scale. The initial change in dynamic modulus could be due to Mullin’s effect, or other chemical and physical relations (Cantournet et al. 2009 ). The resulting R 2 of the power law fit for dynamic stiffness ranged from 0.25 to 0.99 per sample. The overall R 2 across all samples was close to 1. Samples that experienced minimal change in dynamic stiffness had lower R 2 values than samples experiencing either an increase or decrease in stiffness over time. The optimized solution for the static stiffness decay differed from that of the dynamic stiffness decay. As Menard mentions in their textbook on DMA, the decay rates can differ between slow rate tests like creep and dynamic excitation like DMA (Menard 1999 ). The dynamic and static stiffness rates of change were overlaid to demonstrate the difference between their decay rates (Fig. 9 ). The dynamic modulus includes both the decay of the storage modulus and the decay of the loss modulus. The storage modulus and complex modulus decayed at approximately the same rate. The loss modulus, however, had a slower decay rate. The slower rate of the loss modulus is likely due to the preservation of damping in the sample, even when the elastic stiffness decays. Plotting NVP 2 versus the resulting m values, a linear relationship was uncovered (Fig. 10 ). Therefore, NVP 2 can be mapped to the decay rate of the dynamic modulus using a simple linear fit with a slope of 2.34 with an R 2 value of 0.78. Unlike NVP 2 , NVP 3 was not found to have a relationship with the decay rate. The relationship between the complex modulus’s decay rate and the initial second resonance NVP, NVP 2 , exposes the possibility that the stability of the dynamically excited foam system is dependent on its initial linearity. As NVP 2 has been qualitatively observed to increase with localized strain distribution both in this study and others, this implies that the decay of the complex modulus is dependent on the stability of the structure under deformation. Interestingly, the change in nonlinearity was not correlated to the change in complex modulus, but instead its initial value. Previously, Hirschberg correlated changes in NVP 2 and NVP 3 with cracking and macroscopic damage in material (Hirschberg et al. 2020 ). In this experiment, however, no macroscopic failure occurred and the initial condition, rather than change in NVP, was the indication of change in material performance. Based on long standing work from Kyriakides, instabilities within a material dictate the material’s deformation and collapse patterns (Kyriakides 1993 ). With a uniform material structure, these instabilities invoke nonuniformity in the material deformation (Luan et al. 2022 ). Nonuniformity in the strain field increases with frequency and thus increases the nonlinear energy in the material system. As shown in this work, the nonlinear energy is linearly correlated to the rate of decay of dynamic stiffness. Therefore, the more unstable the material system is due to structure, excitation frequency, or relative porosity, the more time dependent the dynamic stiffness response becomes. 4. Conclusion In this study, additively manufactured direct ink write silicone foams were placed in cyclic compression for 10,000 cycles. It was found that their static and dynamic stiffnesses both decayed over time but at different rates. A semi-empirical power law for static stiffness deformation was applied to the dynamic stiffness decay and was found to be correlated, but not well aligned with experimental data. A linear relationship with an R squared value of 0.86, however, was found between the dynamic stiffness decay rate and the second order resonance nonlinear viscoelastic parameter, NVP 2 . The nonlinear viscoelastic parameter was also found to be correlated to the nonuniformity in the dynamic strain distribution of the material which was influenced by the porosity and the frequency of the excitation. Thus, the dynamic response of the material over time is dependent on the initial instability and nonuniformity of the strain field within the material. Declarations The authors have no competing interests to declare that are relevant to the content of this article. Author Contribution Z.A. and S.S prepared and printed the samples. M.F. conducted the experimental characterization and authored the manuscript. D.S. performed statistical analysis. M.H. and L.L. advised the work. All authors reviewed the manuscript. Acknowledgement This material is based upon research partially supported by the U.S. Office of Naval Research under PANTHER award number N00014-21-1-2916 through Dr. Timothy Bentley. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-JRNL-2002451 Data Availability Data is available from the authors upon reasonable request. References Cantournet S, Desmorat R, Besson J (2009) Mullins effect and cyclic stress softening of filled elastomers by internal sliding and friction thermodynamics model. Int J Solids Struct 46:2255–2264. https://doi.org/10.1016/j.ijsolstr.2008.12.025 Foster MM, Morrison DC, Landauer AK, et al Assessment of frequency and amplitude dependence on the cyclic degradation of polyurethane foams. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6497954","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":447264181,"identity":"379eb381-fb6c-4a76-a551-a9a5f5c30f7f","order_by":0,"name":"Moira M. Foster","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyklEQVRIiWNgGAWjYBAC9gYw9U+On525geEBMVp4DjAwAnUdMJZsBtIJpGhJ3HCYaC3s7c8ffNxzh3HzYca2BwkMtXIGBwhp4Tlj2Djj2TNms8OM7QYJDMeNCWqxl8hhbOY5wMwG1NImkcBwLHFmAyFbJNIfNv85wMxj3Ey8lgTDZoYDhyUMmMFaahL7CegA+2Vmz4E0AwmwwwwOGPMT1MLe/uDDjwM29f3tzcckPlTUybER0oIGDA6TqAEI6kjXMgpGwSgYBcMeAACixEIm5psPrwAAAABJRU5ErkJggg==","orcid":"","institution":"Lawrence Livermore National Laboratory","correspondingAuthor":true,"prefix":"","firstName":"Moira","middleName":"M.","lastName":"Foster","suffix":""},{"id":447264182,"identity":"e8614a31-6320-4cae-ac08-bac6e850fd7d","order_by":1,"name":"Daisy Philtron","email":"","orcid":"","institution":"Colorado School of Mines","correspondingAuthor":false,"prefix":"","firstName":"Daisy","middleName":"","lastName":"Philtron","suffix":""},{"id":447264183,"identity":"2aab7cf1-cda9-4eea-90be-73e004044974","order_by":2,"name":"Mark D. Herynk","email":"","orcid":"","institution":"Lawrence Livermore National Laboratory","correspondingAuthor":false,"prefix":"","firstName":"Mark","middleName":"D.","lastName":"Herynk","suffix":""},{"id":447264185,"identity":"101645bb-84eb-412a-8b30-43a24c4d0c52","order_by":3,"name":"Ziad Ammar","email":"","orcid":"","institution":"Lawrence Livermore National Laboratory","correspondingAuthor":false,"prefix":"","firstName":"Ziad","middleName":"","lastName":"Ammar","suffix":""},{"id":447264186,"identity":"dcdac466-acf6-481e-ade6-3b83728662a9","order_by":4,"name":"Siddharthan Selvasekar","email":"","orcid":"","institution":"Lawrence Livermore National Laboratory","correspondingAuthor":false,"prefix":"","firstName":"Siddharthan","middleName":"","lastName":"Selvasekar","suffix":""},{"id":447264188,"identity":"50c38b18-8e10-4c0e-b3e6-5d72b16c5698","order_by":5,"name":"Leslie E. Lamberson","email":"","orcid":"","institution":"Colorado School of Mines","correspondingAuthor":false,"prefix":"","firstName":"Leslie","middleName":"E.","lastName":"Lamberson","suffix":""}],"badges":[],"createdAt":"2025-04-21 17:23:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6497954/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6497954/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":81374940,"identity":"47d28d30-cb24-49f2-a0a7-76afe9232504","added_by":"auto","created_at":"2025-04-25 11:23:15","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":62029,"visible":true,"origin":"","legend":"\u003cp\u003eFCT and SC structures\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/d4537e243d05355e2fb42801.jpg"},{"id":81373721,"identity":"10ef60a9-1d5f-4f82-a270-baba0b720866","added_by":"auto","created_at":"2025-04-25 11:07:15","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":57525,"visible":true,"origin":"","legend":"\u003cp\u003eKeyence imaging of a 60 % porous FCT sample.\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/8614cff6c44888ca5aefb39f.jpg"},{"id":81373735,"identity":"8fd40ef2-7a46-428c-a818-55d116ca0302","added_by":"auto","created_at":"2025-04-25 11:07:15","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":45575,"visible":true,"origin":"","legend":"\u003cp\u003ea) Stress versus relative porosity grouped by structure type. b) Static stiffness versus stress linear relationship grouped by structure type.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/ab30ac6ba980f3baf62d02f2.jpg"},{"id":81373727,"identity":"459b0906-a012-4468-9d52-4d10af7ab058","added_by":"auto","created_at":"2025-04-25 11:07:15","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":40421,"visible":true,"origin":"","legend":"\u003cp\u003eHysteresis loops of SC samples plotted on stress versus strain plot. The loops are colored by increasing NVP.\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/fce08c664307ada4d9465814.jpg"},{"id":81374507,"identity":"578c3714-0ae5-457e-810a-7674798b1ce6","added_by":"auto","created_at":"2025-04-25 11:15:15","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":86733,"visible":true,"origin":"","legend":"\u003cp\u003eDIC overlay of strain field for Left) 55 % initial porosity at 40 % relative porosity at 10 Hz with nonuniform strain distribution, Right) 65 % initial porosity at 57 % relative porosity at 5 Hz with uniform strain field.\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/b8f42be1c164f408a763eb3d.jpg"},{"id":81373733,"identity":"84b8f69a-0550-4791-8c19-ba4905ada4b2","added_by":"auto","created_at":"2025-04-25 11:07:15","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":40381,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e versus strain rate separated by structure type overlaid with quadratic fit.\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/11b11b07d2f414a69bffd527.jpg"},{"id":81373731,"identity":"4d1c56d7-a9be-47b3-8051-c7f38523e444","added_by":"auto","created_at":"2025-04-25 11:07:15","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":52518,"visible":true,"origin":"","legend":"\u003cp\u003eStatic stiffness decay versus time fit to power law function for FCT family of samples.\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/c2a74d089e77ca544c1d8d23.jpg"},{"id":81373738,"identity":"a0493b75-b4da-4481-8346-01a1825711cb","added_by":"auto","created_at":"2025-04-25 11:07:15","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":50935,"visible":true,"origin":"","legend":"\u003cp\u003eDynamic stiffness decay versus time fit to power law function for FCT family of samples.\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/ba8af22d49d4eb91ced696d3.jpg"},{"id":81374505,"identity":"e8b05e7d-f953-41c0-bc4e-36292db1897c","added_by":"auto","created_at":"2025-04-25 11:15:15","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":42437,"visible":true,"origin":"","legend":"\u003cp\u003eRate of decay for static stiffness versus dynamic stiffness for each sample.\u003c/p\u003e","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/993c9c3bae6eb296e663163a.jpg"},{"id":81373736,"identity":"3b84c396-b787-4297-9dfe-e2e2e4b7d081","added_by":"auto","created_at":"2025-04-25 11:07:15","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":43989,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e versus complex modulus decay rate, \u003cem\u003em\u003c/em\u003e.\u003c/p\u003e","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/ca0d454a83414601fb1d3205.jpg"},{"id":83286182,"identity":"6924300f-be8d-4990-996c-4f97b5397583","added_by":"auto","created_at":"2025-05-22 11:32:06","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1098638,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6497954/v1/e3e9e82c-4393-4351-bdd5-639fa80c7c38.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Viscoelastic Response of Silicone Additively Manufactured Direct Ink Write (DIW) Foams under Repetitive Compression","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eUnderstanding the correlations between different material properties and how they transform over time has many industrial and aerospace applications (Mills \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). In a combined pre-loaded compression and vibration environment, material properties are exhibited at both low-rate stiffness (static) and dynamic stiffness. The rate at which the static stiffness declines over time due to relaxation has been characterized through relaxation and creep testing (Schapery \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1969\u003c/span\u003e). The dynamic modulus and how it changes over time, however, has been less explored.\u003c/p\u003e \u003cp\u003eStatic stiffness, or Young\u0026rsquo;s modulus, can be calculated via the stress divided by the strain (Gibson and Ashby \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e1997\u003c/span\u003e). The dynamic modulus or complex modulus is kinematically defined as the steady-state solution due to an imposed sinusoidal excitation. The dynamic modulus can be calculated using Fourier transform rheology (FTR) (Wilhelm et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1998\u003c/span\u003e). The Fourier transforms of the stress, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathcal{F}\\left(\\sigma\\:\\right)\\)\u003c/span\u003e\u003c/span\u003e [MPa], and strain, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathcal{F}\\left(\\epsilon\\:\\right)\\)\u003c/span\u003e\u003c/span\u003e [mm mm-1], are divided in the frequency domain to calculate the linear dynamic stiffness, \u003cem\u003eE*\u003c/em\u003e [MPa], at the excitation frequency (Wilhelm \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2002\u003c/span\u003e)\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\left|{E}^{*}\\right|=\\left|\\frac{\\mathcal{F}\\left(\\sigma\\:\\right)}{\\mathcal{F}\\left({\\epsilon\\:}\\right)}\\right|.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe storage modulus, \u003cem\u003eE\u0026rsquo;\u003c/em\u003e [MPa], and loss modulus, \u003cem\u003eE\u0026rdquo;\u003c/em\u003e [MPa], can be defined as the imaginary and real components of the complex modulus,\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{E}^{*}={E}^{{\\prime\\:}}+i\\:E\u0026quot;.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThrough the kinematic solution of a simple Voigt or Maxwell model, loss modulus is shown to be a linear function of the excitation frequency, while storage modulus is stable in frequencies well under the natural resonance of the system. Subtracting the phases of the stress and strain at the excitation resonance produces the damping factor, or phase lag, \u003cem\u003eẟ\u003c/em\u003e,\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\delta\\:=\\mathcal{F}\\left({\\sigma\\:}\\right)-\\mathcal{F}\\left({\\epsilon\\:}\\right).$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhen the amplitude or frequency of the sinusoidal excitation is increased, the dynamic stiffness of foamed materials will decrease then increase, exhibiting a second order relationship with strain rate (Gibson and Ashby \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e1997\u003c/span\u003e; White et al. \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). The bulk material stiffens as the material is excited faster, however, at higher strain rates inertial effects -such as the flow of air in and out of the cellular structure - begin to take effect as well, causing nonlinearity in the stiffness response (Mills \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2007\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eNonlinear energy can occur in the dynamic modulus\u0026rsquo;s response when the material undergoes nonuniform or unstable deformation (Hyun et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Nonlinear energy can be calculated via the nonlinear viscoelastic parameter (NVP) (Wilhelm \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). NVP is defined as the ratio of magnitudes of a higher order resonance normalized by the magnitude of the excitation resonance (Hirschberg et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2020\u003c/span\u003e, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). In this work, the definition from the total harmonic distortion was used as the root mean square approach akin to total harmonic distortion,\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:NV{P}_{n,\\:m}=\\frac{\\sqrt{\\left|{\\mathcal{F}}_{n}^{2}\\left(\\sigma\\:\\right)\\right|+\\left|{\\mathcal{F}}_{m}^{2}\\left(\\sigma\\:\\right)\\right|}}{\\left|{\\mathcal{F}}_{1}^{2}\\left(\\sigma\\:\\right)\\right|}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003en,m\u003c/em\u003e\u003c/sub\u003e is the nonlinear viscoelastic parameter at resonance \u003cem\u003en\u003c/em\u003e and \u003cem\u003em\u003c/em\u003e. Mills suggests that nonlinearity could be a function of the frictional interfaces of the cell walls rubbing together in the material (Mills \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). Thus, nonlinearity is higher with material undergoing larger pre compressions, or lower relative porosity (White et al. \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Wang et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Odd resonances appear with increased frequency of excitation, or the strain rate, of the material (Hyun et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Even resonances appear when there are nonuniform strain distributions in the material. Nonuniform compression in cellular and honeycomb materials have been attributed to instabilities within the lattice structure (Kyriakides \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1993\u003c/span\u003e; Rajput et al. \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Luan et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). In AM material, for example, overlapping strands in a lattice are areas of stress concentrations (Van Meerbeek et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Zhu et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). These stress concentrations are relieved via buckling in the surrounding strands, which cause sudden load releases and nonuniform deformation in the sample (Hyun et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Wilhelm proposed in his 1998 paper that even resonances may also be influenced by \u0026ldquo;time-dependent memory effects or nonlinear elastic contributions in the system\u0026rdquo; (Wilhelm et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1998\u003c/span\u003e). The appearance of odd and even resonances could be utilized as an in-situ technique for determining failure of the material system, similar to structural health analysis. Studies on brittle polystyrene and PMMA, for example, have mapped an increase in the nonlinear second and third resonances to small and large cracking (respectively) in brittle material (Hirschberg et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2017\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Flexible, non-brittle, materials like polyurethane also exhibit changes over time, however, these changes are not discrete such as fracture (Mills \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). This work aims to connect how the nonlinearity in dynamic system attributes to the decay of the complex modulus across cycles.\u003c/p\u003e \u003cp\u003eViscoelastic materials exhibit decay of their stiffness as they are held under a fixed strain (Menard \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1999\u003c/span\u003e). The rate of stress relaxation, or relaxation modulus, is governed by both topology and bulk material properties in a sample (Gibson and Ashby \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e1997\u003c/span\u003e; Zhu et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). The rate of this decay is commonly modeled via an exponential or power law decay and is dependent on the applied strain during relaxation (Schapery \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1969\u003c/span\u003e). AM silicone lattices showed that face centered tetragonal (FCT) structures had a slower relaxation rate than the simple cubic (SC) structures (Zhu et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). This was hypothesized to be due to the lower stress levels in the FCT structure as the offset strands were able to collapse nicely into the next layer. Limited studies have fit the rate of decay of the complex modulus under a constant cyclic deformation.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn this study, long cycle and large deformation DMA is conducted on FCT and SC materials. As the material is pre-compressed to the mean excitation strain, the material begins to undergo a stress relaxation test. This relaxation, however, is overlaid with sinusoidal excitations that themselves have a calculated dynamic stiffness element. Thus the material has a relaxation modulus associated with this test as well as a decaying dynamic stiffness as calculated by FTR. Previous studies have shown that an increased strain rate increases the decay of the dynamic stiffness in this testing, however, no correlation has been conducted to the nonlinear properties of the excitation and the structural effects of introduced topological differences.\u003c/p\u003e"},{"header":"2. Methodology","content":"\u003cp\u003eFoam samples were additively manufactured using direct ink write printing (Maiti \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Simple cubic (SC) and face centered tetragonal (FCT) lattices were printed using Llama 50 as the base material with a strand diameter of 500 \u0026micro;m on a Teflon coated aluminum plate (Small et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Lenhardt \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Subsequently each part was thermally cured in an oven using the same temperature profile of 2 hours ramping to 150\u0026deg;C then holding at 150\u0026deg;C for 16 hours. A total of six unique lattice structures, three strand pitches of 1.19 mm, 1.35 mm, and 1.56 mm were used to create three initial nominal porosity values of 55%, 60%, and 65% (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The initial rectangular print geometry was 30 layers, approximately 12 mm thick and 127 mm wide. The samples were trimmed to a width of 65 mm to remove printing edge effects while maintaining a 5:1 ratio to the thickness. During printing and curing, structure sag and shrinkage occurred to various degrees in all samples, producing sample thicknesses ranging from 11.4 to 12.3 mm. A calculated porosity based on the initial measured thickness and intended porosity were used in for the design of experiments. Three replicates were printed for each unique lattice structure and were each tested to a unique strain and frequency environment.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe samples were placed between two eye-leveled six-inch compression platens on a MTS hydraulic load frame outfitted with a FLIR blackfly camera. The samples were strained to an initial strain value, ε_0 [mm mm\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e]. After a 2 minute static stress relaxation period, the samples were sinusoidally excited with an amplitude, ε\u003csub\u003e1\u003c/sub\u003e [mm mm\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e], of 0.3 mm (0.3% strain) at an excitation frequency, ω [rad s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e],\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:\\epsilon\\:\\left(t\\right)\\:=\\:{\\epsilon\\:}_{0}\\:+\\:{\\epsilon\\:}_{1}\\:sin\\left(\\omega\\:t\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eEach sample was cyclically excited for 10,000 cycles. Displacement and force information was recorded via the loadcell and crosshead readouts at a rate of 1000 Hz. The data was recorded for 100 cycle increments logarithmically spaced throughout the 10,000 cycles. The displacement information was then viscoelastically compliance corrected using linear interpolation to shift the data in time and displacement. The 50 kN load cell was calibrated annually in house using ASTM E4-16 without NIST traceability. The time history of displacement and force from the load frame were post processed using DMA Fourier Transform Rheology (FTR) (Wilhelm et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1998\u003c/span\u003e). After filtering using a Butterworth filter, the Fourier transform of the stress and strain time histories were calculated, to produce the magnitude and damping of the complex modulus (Eqs.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). For this work, \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e were analyzed via Eq.\u0026nbsp;\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The total \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u0026thinsp;\u0026minus;\u0026thinsp;5\u003c/em\u003e\u003c/sub\u003e was also assessed as the summation of resonances 2 through 5.\u003c/p\u003e \u003cp\u003eTo supplement the information provided via the time history of displacement and force, a FLIR Blackfly camera with a 105 mm lens captured images at 100 frames per second. Post processing of the images in MatchID used spatial reference updating, and affine shape function. The SC samples exhibited large out of plane deformation which prevented usable DIC data. FCT samples underwent DIC analysis for qualitative description of the strain distribution within the material layers using virtual extensometers. Since the samples were dynamically strained within the plateau region, the dynamic strain changes were mainly isolated to the lattice deformation rather than compression within strands. A large subset size allowed tracking of filament displacement relative to other filaments, instead of material strain within filaments.\u003c/p\u003e \u003cp\u003eIn the design of experiments, the samples were compressed to five different initial strains, \u003cem\u003eε\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e, ranging from 15 to 33%. The combination of initial porosities, \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e, and strains, \u003cem\u003eε\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e, produced three relative porosities, \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e, calculated as\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{p}_{r}=\\frac{{p}_{i}-{\\epsilon\\:}_{0}}{1-{\\epsilon\\:}_{0}}.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThis calculation assumes that the compression remains in the plateau strain region, such that all the displacement is taken by the deformation of the lattice structure and collapse of air pores, rather than compression of the actual base material. This assumption was verified using DIC strain field results (Morrison et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2024\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eUsing the strain and porosities together with the initial porosity calculation, a Latin Square Design of Experiments was used to determine the combination of strains and porosities of interest such that there were repeating values of strain and relative porosities between the initial print parameter porosities (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Based on previously published work from 2024 on polyurethane materials under similar test conditions, an assumption was made that there was no interaction between the relative porosity and the frequency (Foster et al.). Thus, different frequencies were able to be overlaid on the Latin Square for another dimension in correlations. This design led to a comparison of both the strain rate among the diagonal samples that had the same relative porosity, and then comparison of strain rate and relative porosity combined effects using the other eight conditions. This Latin Square Design was duplicated with both the FCT and SC print structures for a total of 18 different structure, frequency, strain, porosity combinations. A single set of replicates for one condition were included in the design of experiments and were shown to have good repeatability.\u003c/p\u003e\u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;font-size:15px;font-family:\"Times New Roman\",serif;'\u003eTable 1\u0026nbsp;Latin Square Design showing frequency and relative porosity information overlaid.\u003c/p\u003e\n\u003cdiv style='margin-top:0in;margin-right:0in;margin-bottom:6.0pt;margin-left:0in;line-height:110%;font-size:15px;font-family:\"Times New Roman\",serif;'\u003e\n \u003ctable style=\"border: none;width:244.55pt;border-collapse:collapse;\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width:22.85pt;padding:0in 5.4pt 0in 5.4pt;height:1.6pt;\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd style=\"width:32.65pt;padding:0in 5.4pt 0in 5.4pt;height:1.6pt;\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd colspan=\"3\" style=\"width:189.05pt;padding:0in 5.4pt 0in 5.4pt;height:1.6pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003estrain\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"4\" style=\"width:22.85pt;padding:0in 5.4pt 0in 5.4pt;height:11.85pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:5.65pt;margin-bottom:0in;margin-left:5.65pt;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003ePorosity\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:32.65pt;padding:0in 5.4pt 0in 5.4pt;height:11.85pt;\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd style=\"width:64.7pt;padding:0in 5.4pt 0in 5.4pt;height:11.85pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e15-19%\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:61.8pt;padding:0in 5.4pt 0in 5.4pt;height:11.85pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e25%\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:62.55pt;padding:0in 5.4pt 0in 5.4pt;height:11.85pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e33%\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width:32.65pt;padding:0in 5.4pt 0in 5.4pt;height:25.9pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e55%\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:64.7pt;border:solid windowtext 1.0pt;background:#D1D1D1;padding:0in 5.4pt 0in 5.4pt;height:25.9pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e47 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 5 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:61.8pt;border:solid windowtext 1.0pt;border-left:none;background: #ADADAD;padding:0in 5.4pt 0in 5.4pt;height:25.9pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e40 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 10 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:62.55pt;border:solid windowtext 1.0pt;border-left:none;background: #747474;padding:0in 5.4pt 0in 5.4pt;height:25.9pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e33 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 0.5 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width:32.65pt;padding:0in 5.4pt 0in 5.4pt;height:11.5pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e60%\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:64.7pt;border:solid windowtext 1.0pt;border-top:none;background: #E8E8E8;padding:0in 5.4pt 0in 5.4pt;height:11.5pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e53 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 5 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:61.8pt;border-top:none;border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;background:#D1D1D1;padding:0in 5.4pt 0in 5.4pt;height:11.5pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e47 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 10 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:62.55pt;border-top:none;border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;background:#ADADAD;padding:0in 5.4pt 0in 5.4pt;height:11.5pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e40 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 0.5 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width:32.65pt;padding:0in 5.4pt 0in 5.4pt;height:11.65pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e65%\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:64.7pt;border:solid windowtext 1.0pt;border-top:none;padding: 0in 5.4pt 0in 5.4pt;height:11.65pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e57 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";'\u003e@ 5 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:61.8pt;border-top:none;border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;background:#E8E8E8;padding:0in 5.4pt 0in 5.4pt;height:11.65pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e54 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 10 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width:62.55pt;border-top:none;border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt;background:#D1D1D1;padding:0in 5.4pt 0in 5.4pt;height:11.65pt;\"\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e47 %\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\n \u003cp style='margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:0in;line-height:normal;font-size:15px;font-family:\"Times New Roman\",serif;text-align:center;'\u003e\u003cspan style='font-family:\"Times New Roman\";color:black;'\u003e@ 0.5 Hz\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e \u003cp\u003eIncluding the control parameters, there were 15 different variables considered for causation and correlation in this experiment (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The output variables relaxed and changed over the course of the 10,000 cycles of excitation, thus each output variable also includes a time-dependent variable. To investigate relationships between variables, p-values from the Pearson correlation coefficient (PCC) and Akaike information criterion (AIC) were used to quantify significance.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eInput and output experimental variables.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVariable\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eInput (Control)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eInput (Calculated)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eOutput (Response)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePorosity\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStrain\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRelative Porosity\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFrequency\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAmplitude\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStrain Rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStress\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStatic Modulus\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eComplex (Dynamic) Modulus\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTan delta (Damping)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNonlinear Viscoelastic Parameter(s)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStatic Modulus Relaxation Rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDynamic Modulus Decay Rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDamping Decay Rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNVP Decay Rate\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eX\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"3. Results and Discussion","content":"\u003cp\u003eNine samples of SC and FCT DIW silicone material were compressively strained to within the stress plateau region, then cyclically strained about the initial compression point at a given excitation frequency as indicated by the Latin square design of experiments. The stress response was recorded across cycles and used in the calculation for static and dynamic stiffnesses. First the initial viscoelastic response was analyzed, then across time.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Initial DMA Results\u003c/h2\u003e \u003cp\u003eThe average stress during the first 100 cycles of excitation was linearly related to the relative porosity and offset by lattice print structure (Fig.\u0026nbsp;3a). Static stiffness was calculated as the average stress divided by the initial strain. Unlike stress, initial static stiffness could not be simplified to a linear function of relative porosity and structure, instead best linearly related to stress (Fig.\u0026nbsp;3b). While relative porosity effectively normalizes the stress strain curve in regard to stress, the stiffness does not scale linearly with the normalization.\u003c/p\u003e \u003cp\u003eThe dynamic stiffness, or complex modulus, of the material during the initial 100 cycles of compression demonstrated some damping in the material, with mainly elastic components. Initial damping values were correlated to strain rate and the damping trends across time were correlated to the initial damping values. Graphically, the hysteresis loops became steep (increased in stiffness), widened (increased in damping), and became more nonlinear (shown through coloring) with increased excitation frequency and smaller relative porosity (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003e). The initial complex modulus was linearly correlated to the static modulus by a factor of 1.5.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWhen the samples compressed to the initial strain, \u003cem\u003eε\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e, the strain was nonuniformly distributed across the vertical length of the samples. DIC full field strain maps were used to assess strain localization, although quantitative values were not reliable due to the large displacement for DIC calculation (Morrison et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). In most FCT samples, the strain localized near the upper and lower boundaries at the platen-sample interface, similar to previously published observations of foam under compression (Sriram et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). The side of the sample that experienced sagging due to manufacturing showed the most strain, likely inciting earlier collapse to the unstable structure (Kyriakides \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1993\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eLike the initial strain distribution, the dynamic strain distribution was also nonuniform (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003e). The static strain is the strain due to the initial compression. The dynamic strain is the strain that is produced about the initial compression due to the dynamic sinusoidal excitation amplitude. The relative porosity and the excitation frequency both influenced the dynamic strain distribution. For example, uniform strain occurred in a 5 Hz sample while nonuniform dynamic strain occurred in a 0.5 Hz sample (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003e). In general, however, as excitation frequency increased the strain distribution further isolated within the material. The samples\u0026rsquo; strain distribution remained steady across cycle counts with no significant changes detected across the 10,000 cycles of compression in any sample.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eHigher excitation frequency and non-uniform strain distribution within the material invoked larger NVP values. Like strain distribution, \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e was correlated to both the strain rate and relative porosity, however, relative porosity had limited effect on the statistical fit and was omitted.\u003c/p\u003e \u003cp\u003eAll three NVP definitions (resonance 2, resonance 3, and total) were found to be nonlinearly related to the excitation frequency with an offset for lattice print structure. Based on prior work demonstrating the importance of strain rate in addition to excitation frequency, the analysis used strain rate, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\dot{\\epsilon\\:}\\)\u003c/span\u003e\u003c/span\u003e [s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e], as the dependent term in the quadratic fit as shown with \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e (Foster et al.) (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e6\u003c/span\u003e). The transition of nonlinearity at 5 Hz could be due to the transition between the quasistatic and intermediate rate regimes. \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u0026thinsp;\u0026minus;\u0026thinsp;5\u003c/em\u003e\u003c/sub\u003e were also nonlinearly related to the excitation frequency but inverted from \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e. Increased excitation frequency causing increased nonlinearity has been observed in previous studies (Grieninger et al. \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). As NVP is a dimensionless quantity, the strain rate in the statistical fit is a normalized strain rate to a reference strain rate.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Stiffness Decay over Time\u003c/h2\u003e \u003cp\u003eAs the material was held at the initial pre-compression with the overlaid oscillation, the average stress across cycles decayed through stress relaxation. Through this viscoelastic relaxation, static stiffness, calculated from the average stress, also decayed according to a nonlinear power law function as described by Schapery (Schapery \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1969\u003c/span\u003e),\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:E\\left(t\\right)={E}_{\\infty\\:}+\\:{E}_{0}h{t}^{-m},$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003em\u003c/em\u003e is the decay rate, \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u0026infin;\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e are constants across same porosity materials, and \u003cem\u003eh\u003c/em\u003e is a parameterization function for different loading conditions. For this study, the variability in the initial porosity, strain, and print structure of the samples made determining a parameterization function, \u003cem\u003eh\u003c/em\u003e, unpractical. For this reason, \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003eh\u003c/em\u003e was simplified to a fully empirically derived value, \u003cem\u003eA\u003c/em\u003e. Furthermore, the infinite modulus, \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u0026infin;\u003c/em\u003e\u003c/sub\u003e, was taken to be zero as the static stiffness was never observed to reach a steady state condition. These simplifications created a linear regression model on a log-log scale,\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:\\text{log}\\left(E\\left(t\\right)\\right)=\\text{log}\\left(A\\right)-m\\:\\text{l}\\text{o}\\text{g}\\left(t\\right).$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eUsing a least squares regression approach, each sample\u0026rsquo;s static stiffness across time was fitted with resulting R\u003csup\u003e2\u003c/sup\u003e values of 0.75 to 0.99 (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe same linear power law was fitted to the dynamic stiffness data (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e8\u003c/span\u003e). To maintain the simple linear power law relationship with the dynamic stiffness decay, the first excitation data point was omitted. A large change in stiffness occurred within the first 300 cycles of excitation. This change was not aligned with cycles 250 onward that demonstrated a highly linear rate of change on a log-log scale. The initial change in dynamic modulus could be due to Mullin\u0026rsquo;s effect, or other chemical and physical relations (Cantournet et al. \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). The resulting R\u003csup\u003e2\u003c/sup\u003e of the power law fit for dynamic stiffness ranged from 0.25 to 0.99 per sample. The overall R\u003csup\u003e2\u003c/sup\u003e across all samples was close to 1. Samples that experienced minimal change in dynamic stiffness had lower R\u003csup\u003e2\u003c/sup\u003e values than samples experiencing either an increase or decrease in stiffness over time.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe optimized solution for the static stiffness decay differed from that of the dynamic stiffness decay. As Menard mentions in their textbook on DMA, the decay rates can differ between slow rate tests like creep and dynamic excitation like DMA (Menard \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1999\u003c/span\u003e). The dynamic and static stiffness rates of change were overlaid to demonstrate the difference between their decay rates (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e). The dynamic modulus includes both the decay of the storage modulus and the decay of the loss modulus. The storage modulus and complex modulus decayed at approximately the same rate. The loss modulus, however, had a slower decay rate. The slower rate of the loss modulus is likely due to the preservation of damping in the sample, even when the elastic stiffness decays.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003ePlotting \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e versus the resulting \u003cem\u003em\u003c/em\u003e values, a linear relationship was uncovered (Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e10\u003c/span\u003e). Therefore, \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e can be mapped to the decay rate of the dynamic modulus using a simple linear fit with a slope of 2.34 with an R\u003csup\u003e2\u003c/sup\u003e value of 0.78. Unlike \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e was not found to have a relationship with the decay rate.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe relationship between the complex modulus\u0026rsquo;s decay rate and the initial second resonance NVP, \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, exposes the possibility that the stability of the dynamically excited foam system is dependent on its initial linearity. As \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e has been qualitatively observed to increase with localized strain distribution both in this study and others, this implies that the decay of the complex modulus is dependent on the stability of the structure under deformation.\u003c/p\u003e \u003cp\u003eInterestingly, the change in nonlinearity was not correlated to the change in complex modulus, but instead its initial value. Previously, Hirschberg correlated changes in \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e with cracking and macroscopic damage in material (Hirschberg et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). In this experiment, however, no macroscopic failure occurred and the initial condition, rather than change in NVP, was the indication of change in material performance.\u003c/p\u003e \u003cp\u003eBased on long standing work from Kyriakides, instabilities within a material dictate the material\u0026rsquo;s deformation and collapse patterns (Kyriakides \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1993\u003c/span\u003e). With a uniform material structure, these instabilities invoke nonuniformity in the material deformation (Luan et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Nonuniformity in the strain field increases with frequency and thus increases the nonlinear energy in the material system. As shown in this work, the nonlinear energy is linearly correlated to the rate of decay of dynamic stiffness. Therefore, the more unstable the material system is due to structure, excitation frequency, or relative porosity, the more time dependent the dynamic stiffness response becomes.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eIn this study, additively manufactured direct ink write silicone foams were placed in cyclic compression for 10,000 cycles. It was found that their static and dynamic stiffnesses both decayed over time but at different rates. A semi-empirical power law for static stiffness deformation was applied to the dynamic stiffness decay and was found to be correlated, but not well aligned with experimental data. A linear relationship with an R squared value of 0.86, however, was found between the dynamic stiffness decay rate and the second order resonance nonlinear viscoelastic parameter, \u003cem\u003eNVP\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e. The nonlinear viscoelastic parameter was also found to be correlated to the nonuniformity in the dynamic strain distribution of the material which was influenced by the porosity and the frequency of the excitation. Thus, the dynamic response of the material over time is dependent on the initial instability and nonuniformity of the strain field within the material.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eThe authors have no competing interests to declare that are relevant to the content of this article.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eZ.A. and S.S prepared and printed the samples. M.F. conducted the experimental characterization and authored the manuscript. D.S. performed statistical analysis. M.H. and L.L. advised the work. All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThis material is based upon research partially supported by the U.S. Office of Naval Research under PANTHER award number N00014-21-1-2916 through Dr. Timothy Bentley. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-JRNL-2002451\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eData is available from the authors upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eCantournet S, Desmorat R, Besson J (2009) Mullins effect and cyclic stress softening of filled elastomers by internal sliding and friction thermodynamics model. 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Compos Commun 38:101475. https://doi.org/10.1016/j.coco.2022.101475\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Dynamic modulus, complex modulus, nonlinear, polymer","lastPublishedDoi":"10.21203/rs.3.rs-6497954/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6497954/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAdditively manufactured (AM) foam was compressively strained into the plateau region through a reduced design of experiments. Its dynamic stiffness, or complex modulus, declined across 10,000 cycles of small deformation in the plateau region. This rate of change of stiffness, when fit to a simple power law, is correlated with the degree of nonlinearity of the material\u0026rsquo;s deformation. Nonlinearity is defined as the nonlinear viscoelastic parameter, calculated as total harmonic distortion. Experiments with low nonlinearity tend to maintain their complex modulus across cycles, while materials undergoing highly nonlinear deformations decreased their modulus. The nonlinearity of the experiment is dependent upon the stress that the material experiences as well as its strain rate. Strain rate correlates with the odd resonance nonlinearity while the complex modulus relaxation rate is correlated to the even resonances. Two different material structures were tested, face centered tetragonal (FCT) and simple cubic (SC). As SC has more overlapping strand areas, and thus a shorter load path length through the sample, the initial stress is higher at the same relative porosities. The SC material therefore experiences greater nonlinearity, 5% total harmonic distortion, than the FCT material with 2% total harmonic distortion. Thus, the SC structure shows much lower stiffness retention than FCT. These findings indicate that when selecting material for repetitive cyclic compressions, a more stable low stress material will maintain its dynamic performance more uniformly than an instable high stress material.\u003c/p\u003e","manuscriptTitle":"Viscoelastic Response of Silicone Additively Manufactured Direct Ink Write (DIW) Foams under Repetitive Compression","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-25 11:07:10","doi":"10.21203/rs.3.rs-6497954/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"95bb7c9f-55c4-4fbf-b5c1-31edd9ac5e77","owner":[],"postedDate":"April 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-05-22T11:23:53+00:00","versionOfRecord":[],"versionCreatedAt":"2025-04-25 11:07:10","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6497954","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6497954","identity":"rs-6497954","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}