Depth-dependent Model Research Articles

Exploring methane cycling in an arctic lake in Kangerlussuaq Greenland using stable isotopes and 16S rRNA gene sequencing

Lakes are currently responsible for a significant amount of total natural methane emission. Microbial oxidation of methane plays a central role in Arctic carbon cycling, potentially reducing methane emissions from lakes, though little is known about methane cycling in the water column of Arctic lakes. We previously detected surprisingly large enrichments of heavy carbon and hydrogen isotopes of methane in three small lakes in Greenland suggesting unusually efficient methanotrophic communities in these Arctic lakes. Using stable isotope and 16S rRNA gene sequencing we determined carbon and hydrogen isotopes and microbial community composition down the water column of Teardrop lake, under open-water conditions. We found that isotopic values of methane in Teardrop lake were again highly enriched 13C and 2H at 4 m depth with −13.2‰ and −27.1‰ values for carbon and hydrogen isotopes, respectively. Methane concentrations slightly increased at the depth interval with isotope enrichment, not typical of classic methanotrophy. Consistent with isotopic enrichment of the heavy isotopes we detected the highest relative abundance of putative methanotrophs, in particular Methylovulum at 4 m. The highest relative abundance of putative methanogens was detected at 3 m as well as at 5 m. At the same depth interval, temperature and oxidation reduction potential also increase, supporting increased microbial activity within the water column. Based on geochemical and microbial observations, we suggest that the methane cycling in Teardrop lake is decoupled from a traditional depth dependent model where the dominant source of methane is in the anoxic sediments. Instead, methane in the water column is likely from a combination of anoxic sediment, littoral transport and oxic methanogenesis in the mid-water column, and recycling of carbon within the water column is leading to extreme isotope enrichments. Thus, understanding linkages between depth-dependent microbial dynamics and methane biogeochemistry are necessary to constrain the sensitivity of the methane cycle to future climate change.

Uncertainty in reliability of thick high strength pipelines with corrosion defects subjected to internal pressure

This paper assesses the uncertainties in the structural reliability levels of thick high strength pipelines with active corrosion defects subjected to internal pressure. A general burst strength model and other applicable to thick high strength pipes are adopted and their predictions uncertainty over a wide range of corrosion defect depths is assessed by Monte Carlo simulation. A 3-parameter lognormal probability model is adopted to describe the uncertainty in the burst pressure. Global and corrosion defect depth dependent model uncertainty factors are derived by comparing the predictions of the burst strength models with burst test data of corroded pipes. Burst failure limit states are formulated in terms of the burst strength models and corresponding model uncertainty factors. Linear and nonlinear corrosion models are adopted to describe the growth of corrosion defects. The uncertainty on the safety levels of pipelines due to the uncertainty on the corrosion defect, corrosion growth model and reliability algorithm is assessed. The results presented for the two strength models provide insights on the adequacy of a thick high strength burst model for probabilistic applications in comparison with a general burst strength model.

A dynamical model of the laminar BOLD response

High-resolution functional magnetic resonance imaging (fMRI) using blood oxygenation dependent level-dependent (BOLD) signal is an increasingly popular tool to non-invasively examine neuronal processes at the mesoscopic level. However, as the BOLD signal stems from hemodynamic changes, its temporal and spatial properties do not match those of the underlying neuronal activity. In particular, the laminar BOLD response (LBR), commonly measured with gradient-echo (GE) MRI sequence, is confounded by non-local changes in deoxygenated hemoglobin and cerebral blood volume propagated within intracortical ascending veins, leading to a unidirectional blurring of the neuronal activity distribution towards the cortical surface. Here, we present a new cortical depth-dependent model of the BOLD response based on the principle of mass conservation, which takes the effect of ascending (and pial) veins on the cortical BOLD responses explicitly into account. It can be used to dynamically model cortical depth profiles of the BOLD signal as a function of various baseline- and activity-related physiological parameters for any spatiotemporal distribution of neuronal changes. We demonstrate that the commonly observed spatial increase of LBR is mainly due to baseline blood volume increase towards the surface. In contrast, an occasionally observed local maximum in the LBR (i.e. the so-called “bump”) is mainly due to spatially inhomogeneous neuronal changes rather than locally higher baseline blood volume. In addition, we show that the GE-BOLD signal laminar point-spread functions, representing the signal leakage towards the surface, depend on several physiological parameters and on the level of neuronal activity. Furthermore, even in the case of simultaneous neuronal changes at each depth, inter-laminar delays of LBR transients are present due to the ascending vein. In summary, the model provides a conceptual framework for the biophysical interpretation of common experimental observations in high-resolution fMRI data. In the future, the model will allow for deconvolution of the spatiotemporal hemodynamic bias of the LBR and provide an estimate of the underlying laminar excitatory and inhibitory neuronal activity.

Quantitative Hydrogeological Framework Interpretations Using Heliborne Electromagnetic Surveys for the North Platte Valley, Western Nebraska Groundwater Model

The U.S. Geological Survey and its partners have collaborated to use airborne geophysical surveys for areas of the North Platte River valley in western Nebraska. The objective of the surveys was to map the aquifers and bedrock topography of the area to help improve the understanding of groundwater?surface-water relations to be used in water management decisions. Frequency-domain heliborne electromagnetic (HEM) surveys were completed, using a unique survey flight line design, to collect resistivity data that can be related to lithologic information for refinement of groundwater model inputs. To make the geophysical data useful to multidimensional groundwater models, numerical inversion is necessary to convert the measured data into a depth-dependent subsurface resistivity model. This inverted model, in conjunction with sensitivity analysis, geological ground truthing (boreholes), and geological interpretation, is used to characterize hydrogeologic features. The two- and three- dimensional interpretation provides the groundwater modeler with a high-resolution hydrogeologic framework and a quantitative estimate of framework uncertainty. This method of creating hydrogeologic frameworks improved the understanding of the actual flow path orientation by redefining the location of the paleochannels and associated bedrock highs. The improved models represent the actual hydrogeology at a level of accuracy not achievable using previous data sets.

Estimating the Modulus and Yield Strength of the Top-Layer Film on Multilayer BEOL Stacks

The manufacturing process-caused variation in the modulus and the yield strength of individual layers in passivated metal stacks critically impacts the reliability of back-end-of-line (BEOL) structures. However, experimentally characterizing the elastic modulus and yield strength of thin films as fabricated, with sufficient sensitivity to distinguish process-induced property variations, remains a significant challenge. Towards this end, we utilize nanoindentation experiments to estimate the elastic modulus and yield strength of top-layer films in multilayer stacks. To address the challenge of extracting individual layer modulus from the composite modulus, in the present paper, we propose a depth-dependent mathematical model (dominant regime theory) by which the modulus of the top-layer can be estimated accurately. Additionally, we employ optimization-based inverse finite element analysis (IFEA) to numerically estimate the modulus and the yield strength of the top-layer film. The uniqueness of the properties estimated by IFEA is investigated through a full-factorial statistically designed numerical experiment. The developed techniques are demonstrated by estimating the modulus as well as the yield strength of tetraethylorthosilicate (TEOS) film on a two-layer stack (TEOS, Silicon) and the same film deposited on a multilayer stack (TEOS, Aluminum, Silicon Nitride, Silicon).

Modelling effects of an asphalt road at a dike crest on dike cover erosion onset during wave overtopping

Structures integrated in a grass-covered dike may increase erosion development. Currently, safety assessment methods for flood defences are only applicable for a conventional grass-covered dike and the effects of structures on dike cover erosion are poorly understood. Since many dikes have a road on top, it is important to study the effect of such a road structure on erosion onset during wave overtopping. To investigate this effect, a coupled hydrodynamic–erosion model was developed. The erosion onset caused by overtopping waves was predicted by combining the time-varying bed shear stresses from the hydrodynamic model with a depth-dependent erosion model. The results show that roads on top of a dike increase the erosion of the neighbouring grass cover. This increase in erosion may have a negative impact on dike stability. Therefore, we recommend considering effects of constructions on top of dike profiles during safety assessments. Explicitly, consideration of the roughness transitions in the safety assessments of dikes is recommended.

Depth-Dependent Three-Layer Model for the Surface Second-Harmonic Generation Yield

We present a generalization of the three-layer model to calculate the surface second harmonic generation (SSGH) yield, that includes the depth dependence of the surface nonlinear second order susceptibility tensor $\boldsymbol{\chi}(-2\omega;\omega,\omega)$. This model considers that the surface is represented by three regions or layers. The first layer is a semi-infinite vacuum region with a dielectric function $\epsilon_{v}(\omega)=1$, from where the fundamental electric field impinges on the material. The second layer is a thin layer ($\ell$) of thickness $d$ characterized by a dielectric function $\epsilon_{\ell}(\omega)$, and it is in this layer where the SSHG takes place. We consider the position of $\boldsymbol{\chi}(-2\omega;\omega,\omega)$ within this surface layer. The third layer is the bulk region denoted by $b$ and characterized by $\epsilon_{b}(\omega)$. We include the effects caused by the multiple reflections of both the fundamental and the second-harmonic (SH) fields that take place within the thin layer $\ell$. {\color{red} As a test case, we calculate $\boldsymbol{\chi}(-2\omega;\omega,\omega)$ for the Si(111)(1$\times$1):H surface, and present a layer-by-layer study of the susceptibility to elucidate the depth dependence of the SHG spectrum. We then use the depth dependent three-layer model to calculate the SSHG yield, and contrast the calculated spectra with experimental data. We produce improved results over previous published work, as this treatment can reproduce key spectral features, is computationally viable for many systems, and most importantly remains completely \emph{ab initio}.

Interval inversion approach for an improved interpretation of well logs

The quality analysis of well-logging inversion results has always been an important part of formation evaluation. The precise calculation of hydrocarbon reserves requires the most accurate possible estimation of porosity, water saturation, and shale and rock-matrix volumes. The local inversion method conventionally used to predict the above model parameters depth by depth represents a marginally overdetermined inverse problem, which is rather sensitive to the uncertainty of observed data and limited in estimation accuracy. To reduce the harmful effect of data noise on the estimated model, we have suggested the interval inversion method, in which an increase of the overdetermination ratio allows a more accurate solution of the well-logging inverse problem. The interval inversion method inverts the data set of a longer depth interval to predict the vertical distributions of petrophysical parameters in a joint inversion procedure. In formulating the forward problem, we have extended the validity of probe response functions to a greater depth interval assuming the petrophysical parameters are depth dependent, and then we expanded the model parameters into a series using the Legendre polynomials as basis functions for modeling inhomogeneous formations. We solved the inverse problem for a much smaller number of expansion coefficients than data to derive the petrophysical parameters in a stable overdetermined inversion procedure. The added advantage of the interval inversion method is that the layer thicknesses and suitably chosen zone parameters can be estimated automatically by the inversion procedure to refine the results of inverse and forward modeling. We have defined depth-dependent model covariance and correlation matrices to compare the quality of the local and interval inversion results. A detailed study using well logs measured from a Hungarian gas-bearing unconsolidated formation revealed that the greatly overdetermined interval inversion procedure can be effectively used in reducing the estimation errors in shaly sand formations, which may refine significantly the results of reserve calculation.

Bayesian inversion of marine controlled source electromagnetic data offshore Vancouver Island, Canada

This paper applies nonlinear Bayesian inversion to marine controlled source electromagnetic (CSEM) data collected near two sites of the Integrated Ocean Drilling Program (IODP) Expedition 311 on the northern Cascadia Margin to investigate subseafloor resistivity structure related to gas hydrate deposits and cold vents. The Cascadia margin, off the west coast of Vancouver Island, Canada, has a large accretionary prism where sediments are under pressure due to convergent plate boundary tectonics. Gas hydrate deposits and cold vent structures have previously been investigated by various geophysical methods and seabed drilling. Here, we invert time-domain CSEM data collected at Sites U1328 and U1329 of IODP Expedition 311 using Bayesian methods to derive subsurface resistivity model parameters and uncertainties. The Bayesian information criterion is applied to determine the amount of structure (number of layers in a depth-dependent model) that can be resolved by the data. The parameter space is sampled with the Metropolis–Hastings algorithm in principal-component space, utilizing parallel tempering to ensure wider and efficient sampling and convergence. Nonlinear inversion allows analysis of uncertain acquisition parameters such as time delays between receiver and transmitter clocks as well as input electrical current amplitude. Marginalizing over these instrument parameters in the inversion accounts for their contribution to the geophysical model uncertainties. One-dimensional inversion of time-domain CSEM data collected at measurement sites along a survey line allows interpretation of the subsurface resistivity structure. The data sets can be generally explained by models with 1 to 3 layers. Inversion results at U1329, at the landward edge of the gas hydrate stability zone, indicate a sediment unconformity as well as potential cold vents which were previously unknown. The resistivities generally increase upslope due to sediment erosion along the slope. Inversion results at U1328 on the middle slope suggest several vent systems close to Bullseye vent in agreement with ongoing interdisciplinary observations.

Mapping CS in starburst galaxies: Disentangling and characterising dense gas

Aims. We observe the dense gas tracer CS in two nearby starburst galaxies to determine how the conditions of the dense gas varies across the circumnuclear regions in starburst galaxies. Methods. Using the IRAM-30m telescope, we mapped the distribution of the CS(2-1) and CS(3-2) lines in the circumnuclear regions of the nearby starburst galaxies NGC 3079 and NGC 6946. We also detected the formaldehyde (H2CO) and methanol (CH3OH) in both galaxies. We marginally detect the isotopologue C34S. Results. We calculate column densities under LTE conditions for CS and CH3OH. Using the detections accumulated here to guide our inputs, we link a time and depth dependent chemical model with a molecular line radiative transfer model; we reproduce the observations, showing how conditions where CS is present are likely to vary away from the galactic centres. Conclusions. Using the rotational diagram method for CH3OH, we obtain a lower limit temperature of 14 K. In addition to this, by comparing the chemical and radiative transfer models to observations, we determine the properties of the dense gas as traced by CS (and CH3OH). We also estimate the quantity of the dense gas. We find that, provided that there are a between 10^5 and 10^6 dense cores in our beam, for both target galaxies, emission of CS from warm (T = 100 - 400 K), dense (n(H2) = 10^5-6 cm-3) cores, possibly with a high cosmic ray ionisation rate (zeta = 100 zeta0) best describes conditions for our central pointing. In NGC 6946, conditions are generally cooler and/or less dense further from the centre, whereas in NGC 3079, conditions are more uniform. The inclusion of shocks allows for more efficient CS formation, leading to an order of magnitude less dense gas being required to replicate observations in some cases.

Mine-Water Flow between Contiguous Flooded Underground Coal Mines with Hydraulically Compromised Barriers

Groundwater flow entering closed contiguous underground coal mines may be strongly influenced by leakage across inter-mine barriers. This study examines a complex of multiple closed and flooded mines that developed into a nearly steady-state groundwater flow system within 10 to 50 years after closure. Field water-level observations, mine geometry, barrier hydraulic conductivity, recharge rates, and late-stage storage gains were parameterized to match known pumping rates and develop a fluid mass balance. Vertical infiltration (recharge and leakage) estimates were developed using a depth-dependent model based on the assumption that most vertical infiltration is focused in areas with <75 m of overburden. A MODFLOW simulation of the nearly steady-state flow conditions was calibrated to hydraulic heads in observation wells and to known pumping rates by varying barrier hydraulic conductivity. The calibrated model suggests significant head-driven leakage between adjacent mines, both horizontally through coal barriers and vertically through inter-burden into a shallower mine in an overlying seam. Calibrated barrier hydraulic conductivities were significantly greater than literature values for other mines at similar depths in the region. This suggests that some barriers may be hydraulically compromised by un-mapped entries, horizontal boreholes, or similar features that act as drains between mines. These model results suggest that post-mining inter-annual equilibrium conditions are amenable to quantitative description using mine maps, sparse observation-well data, accurately estimated pumping rates, and depth-dependent vertical infiltration estimates. Results are applicable to planning for post-flooding water-control schemes, although hydraulic testing may be required to verify model results.

Patterned corneal collagen crosslinking for astigmatism: Computational modeling study

To test the hypothesis that spatially selective corneal stromal stiffening can alter corneal astigmatism and assess the effects of treatment orientation, pattern, and material model complexity in computational models using patient-specific geometries. Cornea and Refractive Surgery Service, Academic Eye Institute, Cleveland, Ohio, USA. Computational modeling study. Three-dimensional corneal geometries from 10 patients with corneal astigmatism were exported from a clinical tomography system (Pentacam). Corneoscleral finite element models of each eye were generated. Four candidate treatment patterns were simulated, and the effects of treatment orientation and magnitude of stiffening on anterior curvature and aberrations were studied. The effect of material model complexity on simulated outcomes was also assessed. Pretreatment anterior corneal astigmatism ranged from 1.22 to 3.92 diopters (D) in a series that included regular and irregular astigmatic patterns. All simulated treatment patterns oriented on the flat axis resulted in mean reductions in corneal astigmatism and depended on the pattern geometry. The linear bow-tie pattern produced a greater mean reduction in astigmatism (1.08 D ± 0.13 [SD]; range 0.74 to 1.23 D) than other patterns tested under an assumed 2-times increase in corneal stiffness, and it had a nonlinear relationship to the degree of stiffening. The mean astigmatic effect did not change significantly with a fiber- or depth-dependent model, but it did affect the coupling ratio. In silico simulations based on patient-specific geometries suggest that clinically significant reductions in astigmatism are possible with patterned collagen crosslinking. Effect magnitude was dependent on patient-specific geometry, effective stiffening pattern, and treatment orientation. Proprietary or commercial disclosures are listed after the references.

Modeling scale-dependent runoff generation in a small semi-arid watershed accounting for rainfall intensity and water depth

Observed scale effects of runoff on hillslopes and small watersheds derive from complex interactions of time-varying rainfall rates with runoff, infiltration and macro- and microtopographic structures. A little studied aspect of scale effects is the concept of water depth-dependent infiltration. For semi-arid rangeland it has been demonstrated that mounds underneath shrubs have a high infiltrability and lower lying compacted or stony inter-shrub areas have a lower infiltrability. It is hypothesized that runoff accumulation further downslope leads to increased water depth, inundating high infiltrability areas, which increases the area-averaged infiltration rate. A model was developed that combines the concepts of water depth-dependent infiltration, partial contributing area under variable rainfall intensity, and the Green–Ampt theory for point-scale infiltration. The model was applied to rainfall simulation data and natural rainfall–runoff data from a small sub-watershed (0.4ha) of the Walnut Gulch Experimental Watershed in the semi-arid US Southwest. Its performance to reproduce observed hydrographs was compared to that of a conventional Green–Ampt model assuming complete inundation sheet flow, with runon infiltration, which is infiltration of runoff onto pervious downstream areas. Parameters were derived from rainfall simulations and from watershed-scale calibration directly from the rainfall–runoff events. The performance of the water depth-dependent model was better than that of the conventional model on the scale of a rainfall simulator plot, but on the scale of a small watershed the performance of both model types was similar. We believe that the proposed model contributes to a less scale-dependent way of modeling runoff and erosion on the hillslope-scale.

On the time and indentation depth dependence of hardness, dissipation and stiffness in polydimethylsiloxane

Nanoindentation tests were performed on polydimethylsiloxane to characterize its mechanical behavior at different indentation depths and loading times. Astonishing indentation size effects have been observed in these experiments where the universal hardness increases by about 15 times from indentation depths of 5000 down to 100 nm. The hardness was found to depend on the loading time at small indentations, while at larger indentation depths the hardness hardly changed with loading time. In an attempt to unveil the underlying deformation mechanisms, an in-depth experimental study is pursued in this article with detailed analysis of the experimental data. Applying different loading times, the indentation experiments were evaluated at indentation depths from 100 to 5000 nm with respect to (a) universal hardness, (b) ratio of remaining indentation depth after unloading to maximum indentation depth, (c) ratio between elastic and total indentation works, and (d) indentation stiffness at maximum applied force. All these characteristics are found to be significantly different compared to a reference material that does not exhibit indentation size effects. The corresponding experimental data has been analyzed with an existing indentation depth dependent hardness model for polymers that has been motivated by a Frank elasticity related theory incorporating rotation gradients.

Wave mixed, wind-generated near-surface shear observed over a tidal flat

Around high tide over a flooded tidal flat, vertical shear measured 0.13–0.4m beneath the sea surface by upward-looking Acoustic Doppler Profilers (ADPs) was usually aligned with the wind. However, shear did not increase with increasing windspeed, likely owing to an increase in wave heights and wave-generated mixing with increasing windspeed. The associated increase in water-side drag coefficient with windspeed was predicted, with substantial scatter, by a model for near-surface mixing by breaking waves. A depth-independent eddy-viscosity model yielded only slightly less skill than a fitted depth-dependent model. Farther (0.3–0.6m) from the surface, model skill was low, suggesting that shear at depth was often determined by factors other than wind. Wave heights tended to increase with increasing water depth. Consequently, the simulated wave-generated surface eddy viscosity tended to increase with increasing depth.

The Bathymetric Stripping Corrections to Gravity Field Quantities for a Depth-Dependent Model of Seawater Density

In geophysical studies investigating the lithosphere structure, topographic, bathymetric, and density contrasts stripping corrections are applied to gravity data. The ocean density contrast is typically calculated as the difference between the mean densities of crust and seawater. The approximation of the actual seawater density by its mean value yields relative errors up to 2%. To reduce these errors, we adopt a depth-dependent seawater density model to account for increasing density with pressure/depth. This approximation reduces errors to less than 0.1%. This density model is utilized in newly derived expressions for the bathymetric stripping corrections.

On the accuracy of the bathymetry-generated gravitational field quantities for a depth-dependent seawater density distribution

In geophysical studies investigating the lithosphere structure, the gravitational field generated by the ocean density contrast (i.e., bathymetry-generated gravitational field) represents a significant amount of the signal to be modelled and subsequently removed from the Earth’s gravity field. The ocean density contrast is typically calculated as the difference between the mean density values of the Earth’s crust and seawater. The approximation of the actual seawater density distribution by its mean value yields relative errors up to about 2% in computed quantities of the gravitational field. To reduce these errors, a more realistic model of the seawater density distribution is utilized based on the analysis of existing oceanographic data of salinity, temperature, and pressure (depth). We study the accuracy of the bathymetry-generated gravitational field quantities formulated for a depth-dependent model of the seawater density distribution. This density distribution approximates the seawater density variations due to an increasing pressure with depth, whereas smaller lateral density variations caused by salinity, temperature, and other oceanographic factors are not taken into consideration. The error analysis reveals that the approximation of the seawater density by the depth-dependent density model reduces the maximum errors to less than 0.6%. The corresponding depth-averaged errors are below 0.1%. The depth-dependent seawater density model is further facilitated in expressions for computing the bathymetry-generated gravitational field quantities by means of the spherical bathymetric (ocean bottom depth) functions. The numerical realization reveals large differences in the results obtained with and without consideration of the depth-dependent seawater density distribution. The maxima of absolute differences reach 201 m2/s2 and 16.5 mGal in computed values of the potential and attraction, respectively. The application of the depth-dependent seawater density model thus significantly improves the accuracy in the forward modelling of the bathymetric gravitational field quantities.

A depth-dependent model of the pericellular microenvironment of chondrocytes in articular cartilage

Experimental studies suggest that the magnitude of chondrocyte deformation is much smaller than expected based on the material properties of extracellular matrix (ECM) and cells, and that this result could be explained by a structural unit, the chondron, that is thought to protect chondrocytes from large deformations in situ. We extended an existing numerical model of chondrocyte, ECM and pericellular matrix (PCM) to include depth-dependent structural information. Our results suggest that superficial zone chondrocytes, which lack a pericellular capsule (PC), are relatively stiff, and therefore are protected from excessive deformations, whereas middle and deep zone chondrocytes are softer but are protected by the PC that limits cell deformations in these regions. We conclude that cell deformations sensitively depend on the immediate structural environment of the PCM in a depth-dependent manner, and that the functional stiffness of chondrocytes in situ is much larger than experiments on isolated cells would suggest.

Accurate Depth Dependent Lens Distortion Models: An Application to Planar View Scenarios

In order to calibrate cameras in an accurate manner, lens distortion models have to be included in the calibration procedure. Usually, the lens distortion models used in camera calibration depend on radial functions of image pixel coordinates. Such models are well-known, simple and can be estimated using just image information. However, these models do not take into account an important physical constraint of lens distortion phenomena, namely: the amount of lens distortion induced in an image point depends on the scene point depth with respect to the camera projection plane. In this paper we propose a new accurate depth dependent lens distortion model. To validate this approach, we apply the new lens distortion model to camera calibration in planar view scenarios (that is 3D scenarios where the objects of interest lie on a plane). We present promising experimental results on planar pattern images and on sport event scenarios. Nevertheless, although we emphasize the feasibility of the method for planar view scenarios, the proposed model is valid in general and can be used in any scenario where the point depth can be estimated.

Influence of the molecular structure on indentation size effect in polymers

Size dependent deformation of polymers has been observed by various researchers in various experimental settings including micro beam bending, foams and indentation testing. Here in this article the indentation size effect in polymers is examined which manifests itself in increased hardness at decreasing indentation depths. Based on previously suggested rationale of size dependent deformation and depth dependent hardness model the depth dependent hardness of various polymers are analyzed. It is found that polymers containing aromatic rings in their molecular structure exhibit depth dependent hardness above the micron length scale. For polymers not containing aromatic rings polymers the indentation size effect starts at smaller indentation depths if they are present at all.