Creep analysis of silicone for podiatry applications

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Available online at www.sciencedirect.com

www.elsevier.com/locate/jmbbm

Research Paper

Creep analysis of silicone for podiatry applications Julia Janeiro-Arocasa, Javier Tarrı´o-Saavedrab,n, Jorge Lo´pez-Beceiroc, Salvador Nayab, Adria´n Lo´pez-Canosac, Nicola´s Heredia-Garcı´ac, Ramo´n Artiagac a

Department of Health Science, Faculty of Nursing and Podiatry, Universidade da Coruña, Spain Department of Mathematics, Higher Polytechnic University College, Universidade da Coruña, Spain c Department of Industrial Engineering II, Higher Polytechnic University College, Universidade da Coruña, Spain b

ar t ic l e in f o

abs tra ct

Article history:

Purpose. This work shows an effective methodology to characterize the creep–recovery behavior

Received 26 March 2016

of silicones before their application in podiatry. The aim is to characterize, model and compare

Received in revised form

the creep–recovery properties of different types of silicone used in podiatry orthotics.

7 July 2016

Methods. Creep–recovery phenomena of silicones used in podiatry orthotics is characterized

Accepted 10 July 2016

by dynamic mechanical analysis (DMA). Silicones provided by Herbitas are compared by

Available online 18 July 2016

observing their viscoelastic properties by Functional Data Analysis (FDA) and nonlinear

Keywords:

regression. The relationship between strain and time is modeled by fixed and mixed effects

Silicones

nonlinear regression to compare easily and intuitively podiatry silicones.

Orthotics

Results. Functional ANOVA and Kohlrausch–Willians–Watts (KWW) model with fixed and

Podiatry

mixed effects allows us to compare different silicones observing the values of fitting parameters

Dynamic Mechanical Analysis

and their physical meaning. The differences between silicones are related to the variations of

Functional Data Analysis

breadth of creep–recovery time distribution and instantaneous deformation-permanent strain.

Statistical learning

Nevertheless, the mean creep-relaxation time is the same for all the studied silicones. Silicones used in palliative orthoses have higher instantaneous deformation-permanent strain and narrower creep–recovery distribution. Conclusions. The proposed methodology based on DMA, FDA and nonlinear regression is an useful tool to characterize and choose the proper silicone for each podiatry application according to their viscoelastic properties. & 2016 Elsevier Ltd. All rights reserved.

1.

Introduction

engineering. The manufacture of foot orthoses or foot orthotic intends to correct, palliate or to compensate deformities,

Podiatry orthotics is an interdisciplinary domain that

defects of locomotion and different pathologies by distribut-

combines anatomy, pathophysiology, biomechanics and

ing the pressures involved (Hawke and Burns, 2012; Delacroix

n

Corresponding author. E-mail addresses: [email protected] (J. Janeiro-Arocas), [email protected] (J. Tarrío-Saavedra), [email protected] (J. López-Beceiro), [email protected] (S. Naya), [email protected] (A. López-Canosa), [email protected] (N. Heredia-García), [email protected] (R. Artiaga). http://dx.doi.org/10.1016/j.jmbbm.2016.07.014 1751-6161/& 2016 Elsevier Ltd. All rights reserved.

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

457

Fig. 2 – Creep recovery diagram.

Fig. 1 – Example of interdigital foot orthoses. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.) et al., 2014). The aim is to reduce pain in the damaged area and to improve the usability of all the parts of the foot. Taking into account the application and the aims of these types of elements, in-depth studies of their mechanical properties should be performed, for instance creep–recovery analysis. Taking also into account the polymeric nature of the foot orthoses, the use of dynamic mechanical analysis (DMA) should be generalized. Broadly, there are two main types of orthoses applied in podiatry, accommodative orthoses and functional foot orthoses:


Accommodative foot orthoses are applied to muffle or relieve pressure from damaged areas on sole of the foot, such as callouses, ulcerations, and other related diseases (Guldemond et al., 2006; Kirby, 2016). Three dimensional models, obtained from a plaster model of the foot, by scanning the foot with optical and mechanical devices or using a box of compressible foam, are used to produce accommodative foot orthoses. They are made of materials such as foams, leather, rubber, cork, etc., thus they tend to be softer and more flexible than functional orthoses.


On the other hand, functional foot orthoses are usually applied to amend abnormal lower extremity function and broadly abnormal foot usability. They are fabricated from a three dimensional model of the foot and commonly made of flexible, rigid and semi-rigid polymeric materials, therefore they are characterized by an easy adaptation and fit to the foot inside the shoe. The treatments of the pain produced in the big toe joint of the foot, in the instep, arch, ankle and heel are the main applications of this type of orthoses (Guldemond et al., 2006; Kirby, 2016).

There is an important group of tailor-made orthoses based on silicone used in digital orthopodiatry. Such silicones are designed to treat forefoot pathologies, where the presence of digital deformations with deviations of the toes in three planes leads to the occurrence of injuries by friction from shoes or other bone structure. These injuries are many a times the cause of alterations in the gait pattern, discomfort and pain. The hardness is one of the properties that characterize silicones and that will affect largely their biomechanical performance on foot. So, as softer or harder, the orthoses will provide greater protection or less pressure, shock, cushioning, strength, elasticity or damping. Fig. 1 presents an application example performed in a podoscope by the authors, where the footprint of a patient is shown before and after applying a silicone orthosis in the toe. The contact surface can be observed in green due to the fluorescent light. A larger contact of the entire sole of the foot can be observed when applying silicone orthoses (look at the toes). Muscles, connective tissues (James, 2001), Achilles tendon (Maganaris et al., 2008) and other parts of the human body are subjected to creep phenomena. Closely linked to the efforts undergone by human body, podiatry orthotics also are subjected to creep. To study the creep behavior of orthoses used in podiatry is essential due to their operation conditions (constant stress due to body weight, recovery when this weight is suddenly released, the heat inside the shoes due to friction, etc.). In materials science, creep can be defined as the slow deformation increasing suffered by a viscoelastic material when subjected to constant stress. Creep is promoted by long-term exposure to stress (even below to the yield strength) and temperatures close to melting point (Chartoff et al., 2009). The test conducted was a creep– recovery test. This is a test that is carried out to study some of the viscoelastic properties of a material. Deformation response of a viscoelastic material at a given stress is a function of time. In a creep–recovery experiment, a constant stress is applied to the sample and the deformation is measured as a function of time. At a defined time, the stress is removed and the material tends to recover its initial dimensions. The creep recovery experiment is the most direct measurement of material elasticity. When the force being applied is released, the total strain will have a permanent part of plastic deformation and an elastic part that is recovered. In this test, the temperature is maintained constant for the effects of thermal expansion (Chartoff et al.,

458

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

2009; Instron, 2016). In order to illustrate the creep–recovery tests, Fig. 2 shows the characteristic output of this type of experimental analysis. The increasing use of silicones (viscoelastic material) in podiatry orthoses justifies performing deeper creep–recovery studies by analytical techniques. Static and dynamic viscoelastic properties of materials such as silicones can be studied by dynamic mechanical analysis (Chartoff et al., 2009). DMA is the most used thermal analysis and rheological analysis technique to study the viscoelastic properties of viscoelastic materials. Thus, the strain versus time DMA curves are obtained in creep–recovery tests to obtain reliable information about static viscoelastic properties of silicones used in podiatry applications. The viscoelastic characterization of silicones used as orthoses in podiatry by DMA could provide very useful information, complementary to other studies focused on the effect of orthoses in the mechanical efforts suffered by foot (Meardon et al., 2009). The aim of this work is to establish an analytical method that could model the creep behavior of silicones used in orthopodiatry when they are subjected to a constant stress that is suddenly released at room temperature. Many regression models are commonly used for modeling the viscoelastic behavior of materials. The simplest are the Maxwell (spring and dashpot in series) and Voigt (spring and dashpot in parallel) models; the former can be used for representing stress relaxation, the latter for creep and recovery conditions. A common factor in these models is that viscoelastic

deformation is considered to vary smoothly. Some authors object strongly to mechanical models, since the real materials are not made of springs and dashpots (Fancey, 2005), many times they do not obey as a simple behavior as statistical models define. Nevertheless, understanding the behavior of polymeric materials is greatly facilitated by adjusting statistical models, although this may be a simplification of a complexer reality. In the present work, an alternative approach proposed by some authors in which viscoelastic changes are suggested to occur through incremental jumps is considered. On a molecular level, the phenomenon could be envisaged as segments of molecules jumping between positions of relative stability (Fancey, 2005). The main models based on this approach are the following: The first model based on this approach consists of two mathematical formulations: For the creep part of the curve, a potential (or power law) model performed by Findley and Peterson (1958) is used. It is one of the most commonly used constitutive models to describe the creep behavior of viscoelastic material, used by Liu et al. (2015) for studying the creep behavior of poly(trimethylene terephthalate)/mesoporous silica composites. The equation is as follows: ϵðtÞ ¼ ϵ0 þ Atn ; where ϵðtÞ is the creep strain at time t, ϵ0 the instantaneous initial strain, directly obtained from the creep curves, A the amplitude of transient creep strain, and n the time exponent. Here A and n are used as the two adjusting parameters to apply power law equation to fit ϵðtÞ. For the recovery part of

Fig. 3 – Graphical output with the pressure distribution in feet measured in kPa and obtained from patient to whom a silicone orthosis in the second interdigital space of left foot has been placed.

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

the strain vs. time curve, a Weibull distribution equation can be used to describe the factors influencing the transient recovery strain during creep recovery (Fancey, 2005). On removing the loads, the instantaneous recovery strain ϵr ðtÞ is 

  t t0 ϵðtÞ ¼ ϵIV exp  βr þ ϵVP ; ηr where ηr is the characteristic life parameter, βr the shape parameter, ϵIV is the viscoelastic strain recovery determined by the parameters ηr and βr, t0 is the time of no stress, and ϵVP is the permanent strain caused by viscous flow effects. The second model is the Kohlrausch–Willians–Watts (KWW), introduced by the german physicist Rudolf Kohlrausch in 1854 to describe the discharge of a capacitor, and also known as stretched exponential function or complementary cumulative Weibull distribution (Matsuoka, 1992). In physics, this function is often used as a phenomenological description of relaxation in disordered systems and it was applied to model the storage modulus variation of viscoelastic materials (Matsuoka, 1992). The KWW function is defined as follows: "
βKWW !# t ; ϵðtÞ ¼ AKWW exp  τKWW where, ϵ is the strain (response variable), t the time, AKWW parameter is the initial strain when a constant stress is applied or permanent strain in relaxation test, the βKWW parameter is the distribution factor and it is related to the breadth of the distribution of creep times or relaxation times, and τKWW accounts for the mean creep time or relaxationrecovery time. In this study, KWW model is applied to explain the creep– recovery trends of silicones used in podiatry. Nonlinear regression and optimization processes were developed using R statistical software (R Development, 2016), the most flexible and comprehensive computational tool for statistical analysis. Previously to any regression analysis, the application of exploratory analysis tools are necessary in order to identify different populations, estimate the data mean and variability, etc. Taking into account the functional nature of the strain curves obtained in creep–recovery tests, functional data analysis (FDA) techniques are the most adequate to perform descriptive analysis of data. More information about this branch of statistics can be retrieved in the monographs of Ferraty and Vieu (2006) and Ramsay and Silverman (2005). In addition, there are some works focused on the application of FDA techniques to thermal analysis that can provide a brief introduction about the domain in a more applied way (Naya et al., 2014; Francisco-Fernández et al., 2012; Tarrío-Saavedra et al., 2013, 2011; Francisco-Fernández et al., 2014). FDA exploratory analysis, design of experiments tools, and the application of nonlinear parametric statistical models are useful to analyze and compare silicones in order to establish selection criteria. This work is organized as follows: In Section 2, the materials, sample preparation and testing methods are described. In Section 3, the applied statistical tools to describe and model the creep–recovery data are described. In Section 4 the results are presented and discussed. Finally, Section 5 corresponds to the conclusions.

2.

459

Experimental

Four samples of Master silicone, four samples of Podiabland silicone, and five samples of Blanda silicone were obtained and tested. The sample size was chosen in order to obtain a balance between experimental time to obtain a representative sample and the reliable estimation of the data variability.

2.1.

Materials

Three different types of silicones provided by Herbitas were used. 1. Blanda: very low density, ideal for palliative orthoses, easily malleable,does not smell, incorporates medicinal oils, can be mixed with other silicones, shore hardness from 6 to 8 Å. 2. Podiabland: elastic and low density, does not adhere to the skin, easy and quick catalysis, nice touch and appearance, can be used in all kinds of orthoses, shore hardness from 12 to 15 Å. 3. Master: medium-hard density, does not break easily, does not adhere to the skin, nice touch and appearance, can be mixed with other silicones, shore hardness from 20 to 24 Å, does not produce rejection, ideal for corrective orthoses.

2.2.

Sample preparation

One drop of Reaktol (liquid catalyst) for each gram of silicone is added according to the supplier (Herbitas) to obtain cylindrical samples defined by a diameter of about 17 mm and a thickness of 6.5 mm (mean values). The mean weights were 1.5 g for “Blanda”, 1.75 g for “Podiabland”, and 2.08 g for “Master”.

2.3.

Pressure distribution measurements

There are many situations where materials used in podiatric orthoses are subjected to static loading and suffer creep phenomena. For example, standing on the two feet with the weight distributed into both feet (or even into one of them), and when remaining seated (where the applied force on the feet is much lower). Thus, the implementation of creep analysis of these materials is justified to estimate their behavior under these conditions. Focusing on the stress applied in experimental test, peak plantar pressures to which feet are subjected during human walking are around of 100 kPa and greater, depending obviously on patient weight (Kato et al., 1996; Burnfield et al., 2004). However, there are also many cases where these materials are placed between the little toes where the pressure is lower than in other parts of the foot. Thus, in order to have an estimation of the pressure distribution in feet, under static mode, the authors have obtained new experimental results using a T-Plate pedobarographic platform of Medicapteurs (610  580  45 mm). Fig. 3 shows the output image with pressure distribution estimates obtained from an adult patient to whom a relatively hard silicone orthosis in the 2nd interdigital space of left foot has been placed. The number shown in each pixel corresponds to

460

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

the pressure measured by each sensor in kPa. Fig. 3 shows that the maximum pressure is detected in the left heel (91.6 kPa), but pressures in the area of interest ranging between 11 and 16 kPa. These pressures are obtained from the contact area between the orthosis and the platform. Moreover, the pressure on the interdigital space between two adjacent toes (the area where the silicone should really show their viscoelastic ability to achieve the desired therapeutic effect) can be even lower. Fig. 3 shows that pressures of about 20 kPa could be common in the area of foot where these types of orthoses are placed, therefore a constant stress of this magnitude has been the applied to perform the creep analysis.

2.4.

Creep recovery study

This work is focused on the creep behavior at room temperature so that the results provide an insight for the aforementioned situations where forces similar to those used in the work are continuously applied on the orthoses. Creep recovery tests were performed in a Rheometrics DMTA IV analyzer. Cylindrical tension/compression geometry was used and the test setup were as follows: Temperature equal to 15 1C. Displacement mode less than 0.1 mm. Time per measure 2 s. Stress 20 kPa. The stress was applied for 200 s and then removed.

3.

Statistical tools

In order to model the performance, static viscoelastic properties variation and differences between the different podiatry silicones, proper statistical learning techniques have been applied. Since implementing exploratory analysis tools is needed in the first step of statistical analysis, and taking into account the functional nature of the data, position statistics such as the functional mean with bootstrap confidence bands and functional analysis of variance (FANOVA) are applied. Creep recovery data modeling is also necessary to characterize the performance of silicones and to compare them in order to obtain application criteria. Otherwise, nonlinear regression techniques based on the Kohlrausch–Willians– Watts (KWW) function are implemented. Further analysis has provided the comparison between the different silicones adding random effects (related to the type of silicone) to nonlinear regression models. From the fact that the R software is the most popular, flexible and comprehensive statistical software (R Development, 2016), it has been used to apply the statistical learning methods above mentioned, e.g. fda.usc, nls, nlme and DEoptim libraries.

3.1.

FDA exploratory analysis and FANOVA

FDA is a new branch of statistics that analyzes the information involved to curves, surfaces or other structure varying in a continuous (Ferraty and Vieu, 2006): time, volume, etc. X is functional if it takes values in a functional normed or semi-normed space E and a functional dataset fX1 ; …; Xn g is the n observation of functional variables X1 ; …; Xn identical distributed such as X. FDA exploratory analysis techniques

should be implemented since the DMA curves obtained in creep–recovery analysis can be considered functional data, such as other type of curves obtained by thermal analysis (Naya et al., 2014; Francisco-Fernández et al., 2012; TarríoSaavedra et al., 2013, 2011; Francisco-Fernández et al., 2014). The aim of functional exploratory analysis is to obtain a descriptive idea about the location and variability of the DMA curves, i.e. an estimation of the overall functional mean and variance are sought. The application of FDA techniques prevents the data loss due to the use of multivariate and univariate approaches. In this way, the three different type of commercial silicones could be compared attending to DMA mean and variability. A resampling smoothed bootstrap procedure is implemented to estimate the functional mean of each type of silicones with its bootstrap confidence bands at 95% confidence level, as it was previously applied to study timber and epoxy based composites (Naya et al., 2014; TarríoSaavedra et al., 2011; Cuevas et al., 2006). The confidence bands provides an intuition about the area where the functional mean of the DMA curves could be placed. More information about bootstrap resampling procedure can be retrieved in Naya et al. (2014) and Cuevas et al. (2006). Furthermore, the implementation of techniques such as analysis of variance (ANOVA) (Tarrío-Saavedra et al., 2011; Cuesta-Albertos and Febrero-Bande, 2010) is absolutely necessary when statistical evidence of differences in a number of population is sought. The statistical ANOVA test provides information about whether the mean of a quantitative response variable is significantly different depending on the level of one or more qualitative variables (factors). The ANOVA analysis tests the influence of each factor on the quantitative response. In the current work, the application of a FDA version of ANOVA test (FANOVA) is necessary because the response is functional (DMA curves of strain vs. time), and the studied factor is the type of commercial silicone, qualitative variable composed of three levels: Blanda, Podiabland, and Master. The loss of important information is prevented using FDA techniques instead of dimension reduction and multivariate statistical methods. In FANOVA test the null hypothesis H0 : m1 ¼ m2 ¼ … ¼ mk is tested (where mi is the functional mean of the i-level of the factor and k is the number of levels) with respect to the alternative hypothesis H1 : at least one of the means is different from the others. In order to perform the FANOVA test, Random Projection procedure is applied. It consists on the test of k random one-dimensional projections (CuestaAlbertos and Febrero-Bande, 2010). Since k random projections are managed, the application of a method that prevents the effect of multiple testing is also needed. In this case, False Discovery Rate (FDR) (Cuesta-Albertos and Febrero-Bande, 2010) method is implemented, although there are other alternative methods such as Bonferroni. The FANOVA test can be performed using the anova.RPm function, available in the fda.usc R package. In Naya et al. (2014) a more complete description about the FANOVA test can be retrieved.

3.2.

Nonlinear regression models

The DMA creep–recovery curve fitting can be tackled as a problem of statistical regression analysis (Cao et al., 2004;

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Ríos-Fachal et al., 2014). There are two main statistical approaches to solve a regression problem, parametric and nonparametric regression. To obtain more information about nonparametric regression, the work of Wasserman can be consulted (Wasserman, 2006). Taking into account that the regression models that are applied in the present study are parametric, this section is focused on the introduction of these type of models. One of the goals of parametric regression models is to obtain a set of parameter with physical–chemical meaning that characterize the curve that is fitted and therefore the material that is tested. This is the case of the celebrated Kohlrausch–Williams–Watts (KWW) function. The parametric model expression can be shown in the following simple way: yi ¼ mðti Þ þ εi ;

i ¼ 1; 2; …; n;

with Eðεi Þ ¼ 0;

where m is the regression function of the response variable Y given the independent variable T, ϵi are 0 mean i.i.d. residuals and the fixed design satisfies 0 rt1 ot2 o⋯otn r1. When the   model m is parametric, M ¼ mθ ð
Þ=θ AΘ , where θ is the parameters vector and Θ a subset of Rk . In statistics, the parametric regression models can be divided into linear and nonlinear. In nonlinear regression analysis, the real data are modeled by a nonlinear function such as exponential, logistic, Weibull, Gaussian, trigonometric, logarithmic, and power functions. These are defined by a nonlinear combination of the parameters of regression model. These parameters are estimated by fitting procedures composed by successive approximations. In contrast to linear regression, there is yet no closed-form expression to obtain the optimal parameters for reaching the best curve fitting by nonlinear regression. In fact, there are many numerical optimization algorithms to determine the optimal fitting parameters but they do not prevent the possibility to reach many local minima of the curve fitting, producing biased parameter estimates. In order to prevent this problem, the parameters of nonlinear functions can be estimated together with global optimization algorithms applied to least squares objective functions. The statistical R software has been used to fit nonlinear regression models to the creep–recovery data obtained by DMA. Namely, the function nlsLM of the package minpack.

461

lm was applied to obtain the optimal parameters of the nonlinear function by nonlinear least squares minimization. This function provides nonlinear least squares estimates of the parameters by the application of a modification of the Levenberg–Marquardt algorithm. The Levenberg–Marquardt algorithm is capable to provide parameter estimates where other algorithms such as port or Gauss–Newton, port or plinear fail. Since the Levenberg–Marquardt algorithm needs an initial solution to begin the iterative process, the previous application of a global optimization algorithm to obtain a reliable initial solution is advisable. The global optimization by differential evolution (DE) algorithm has been applied in this work to obtain a reliable initial solution (Storn and Price, 1997; Tarrío-Saavedra et al., 2014). DE algorithm is a global optimization heuristic evolutionary method to find the global optimum of a function depending on the value of its parameters that works fixing an interval of possible values of parameters. Further information can be retrieved in RíosFachal et al. (2014). In this work, the strain vs. time curves corresponding to creep–recovery analysis are fitted using the Kohlrausch– Williams–Watts (KWW) nonlinear function described in the introduction section.

3.3.

Mixed effects nonlinear regression models

This work presents a problem of repeated measurements taking into account that four or five different creep–recovery curves retrieved by DMA curves are obtained for each type of commercial silicone. When repeated measurements are obtained, the addition of random effects to nonlinear regression fittings is very useful to model the dependence between variables. The addition of random effects provides more parsimonious models. For instance, more or less different DMA traces are expected if we take some replicates, thus a random effect of the replicate can be introduced, obtaining just one expression for the regression model instead one per replicate. If a nonlinear regression model includes fixed and random effects in the parameters, we call it mixed effect nonlinear regression model.

Fig. 4 – The 13 creep curves of the three types of silicone.

462

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Table 1 – Instantaneous, elastic and viscous strain measured in the mean ϵðtÞ curves for each type of silicone (in creep and recovery modes). ϵ comp.

BLANDA Silicone

Instant. Viscoel. Elas. recov. Permanent Total

PODIABLAND Silicone

MASTER Silicone

ϵ=%

% of ϵ

ϵ=%

% of ϵ

ϵ=%

% of ϵ

0.0784 0.0097 0.0569 0.0139 0.0881

89.01 10.99 64.61 15.72 100

0.0588 0.0105 0.0390 0.0131 0.0693

84.90 15.10 56.34 18.87 100

0.0395 0.0070 0.0264 0.0075 0.0465

84.98 15.02 56.80 16.17 100

taken into account. In fact, we assume that the data obtained is a sample oextracted from the population of the different types of commercial silicones. A more formal description is provided through the following expressions, taking into account the random effect related to the types of silicone, in the seek of simplicity: "
ϕ3i !# tij þ ϵij ; ϵðtÞ ¼ ϕ1i exp  ϕ2i

Fig. 5 – The 13 creep and recovery curves of the three types of silicone with functional means and 95% confidence bands. In the present case, the KWW function is proposed to model the dependence between strain and time in creep and recovery analysis that have been implemented to silicone used in podiatry by DMA. Fixed and random effects are scheduled. Taking into account that this is a repeated measurements problem, a random effect related to different replicates is added to the KWW parameters. The DMA curves variability due to the replicate effect will be defined by the random part of the KWW parameter estimation. In addition, a second random effect related to the type of silicone is also

where i is the type of polymer, ϕi is the vector of fitted parameters, ϕ1i ¼ AKWW is the initial strain when a constant stress is applied or the permanent strain in recovery test, ϕ2i ¼ τKWW accounts for the mean creep or recovery time, ϕ3i ¼ βKWW is the distribution factor and it is related to the breadth of the distribution of creep times or recovery times, and ϵij are the model errors. With respect to fixed and random effects parametrization, β accounts for the fixed effects, i.e. the average value of the individual parameters in the commercial silicones population, ψ the covariance matrix of random effects, s is the standard deviation, and bi deviations from the silicones population average. The random effects on the parameters are assumed to be normal distributed with zero mean. The mixed effects approach permits us to model the relationship between strain and time with a less quantity

Table 2 – Parameter estimates, and parameter significance test study (t-test) obtained from the fittings of creep data corresponding to each type of silicone and replicate. The first column corresponds to Blanda, the second to Podiabland, and the third to Master silicone samples. Parameters

Sample 1 Estimates

Sample 2

Sample 3

Sample 4

Sample 5

p-value

Estimates

p-value

Estimates

p-value

Estimates

p-value

Estimates

p-value

BLANDA Silicone 0.0709 AKWW 0.0008 τKWW βKWW  0.2219

0 2.31e-07 0

0.0872 0.0597  0.4741

0 0.0022 0

0.1019 1.6e-04 0.1686

0 0.001 0

0.1087 0.0005  0.2052

0 1.12e-09 0

0.1007 0.0005  0.2145

0 1.58e-07 0

PODIABLAND AKWW τKWW βKWW

0 0 0

0.0821 0.0002  0.0094

0 7.4e-05 0

0.0883 0.0021 0.2064

0 2.49e-08 0

0.00632 0.0006  0.1420

0 0 0

0 0 0

0.0464 0.0045  0.2023

0 0 0

0.0513 0.0020 0.1731

0 0 0

0.0550 0.0058  0.1734

0 0 0

Silicone 0.0915 0.0034  0.2094

MASTER Silicone AKWW 0.0576 0.0020 τKWW -0.2235 βKWW

463

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Table 3 – Parameter estimates, and parameter significance test study (t-test) obtained from the fittings of recovery data corresponding to each type of silicone and replicate. The first column corresponds to Blanda, the second to Podiabland, and the third to Master silicone samples. Parameters

Sample 1 Estimates

Sample 2

Sample 3

Sample 4

Sample 5

p-value

Estimates

p-value

Estimates

p-value

Estimates

p-value

Estimates

p-value

BLANDA Silicone 0.0231 AKWW 2566 τKWW 0.2004 βKWW

0 0 0

0.0263 911.2 0.1563

0 0.007 0

0.0342 1851 0.1398

0 0.003 0

0.0250 1007 0.1916

0 0 0

0.0273 794 0.1981

0 0 0

PODIABLAND Silicone 0.0212 AKWW 738.1 τKWW βKWW 0.3796

0 0 0

0.0227 1477 0.3850

0 0 0

0.0249 914.9 0.3806

0 0 0

0.0200 1748 0.3531

0 0 0

MASTER Silicone AKWW 0.0143 522.9 τKWW 0.4049 βKWW

0 0 0

0.0118 774.8 0.4049

0 0 0

0.0138 607.2 0.3916

0 0 0

0.0155 545.4 0.3657

0 0 0

of parameters (just three parameters for fixed effects and the covariance matrix parameters corresponding to random effects). More information about mixed effects models can be retrieved in Pinheiro et al. (2015) and Pinheiro and Bates (2000).

silicones show lower overall strain than Blanda, and Master silicones show lower strain that Podiabland (this trend is inverse to the hardness trend). Table 1 shows the components of the strain in creep– recovery tests (Fig. 2) measured in the mean ϵðtÞ curves for each type of silicone. If the absolute measurements are observed, we can infer that Blanda silicone undergoes the

4.

Results and discussion

This section presents the results obtained from the application of descriptive analysis, FANOVA, and KWW nonlinear and mixed nonlinear model to the strain vs. time curves obtained by DMA. The aim is to characterize and model the creep–recovery behavior of the three types of silicone for podiatry applications and to compare them to asses an application criteria. Characterization and modeling of creep– recovery phenomena is essential due to the operation condition of silicones in podiatry: constant stress is applied due to body weight, and recovery is experimented when this weight is suddenly released. Five DMA tests for Blanda, four for Podiabland, and four for Master silicone were performed under the same conditions, detailed in the experimental section. Fig. 4 shows the 13 DMA traces of the 3 different silicone classes. The material strain depending on time is measured when a constant stress of 20 kPa is applied (creep) and after relieving stress (recovery). On the right area of each panel in Fig. 4, the creep behavior is shown. The strain decay corresponds to the recovery phenomena. Some differences between replicates corresponding to each silicone are observed as expected. These are due to the inherent variability of the material and experimental procedure. In addition, there is another source of variability related to the type of silicone. DMA creep–recovery analysis provides important information about the different static viscoelastic behavior of the types of silicone. Comparing the curves in Fig. 4 with the typical diagram of a creep test (Fig. 2), the soft silicone (Blanda) has experimented the greatest deformation, including instantaneous strain and viscous creep strain. The Podiabland

largest total strain (followed by Podiabland and Master, respectively), although the higher viscoelastic strain is suffered by Podiabland silicone. When percentage with respect to the overall strain are observed, the Master and Podiabland silicones have a similar behavior for each strain component, one differs from the other in the scale of deformation (absolute values). It is important to stress that the permanent strain of Blanda and Podiabland, in absolute terms, is almost the same: they differ in the instantaneous elastic strain and in the elastic strain recovery (both higher in Blanda). In addition, the viscoelastic strain creep and recovery percentages are rather higher in Podiabland and Master silicones, revealing a viscoelastic behavior. In conclusion, DMA analysis can be useful to chose the proper silicone taking into account the application: we can choose Master, Blanda or Podiabland depending on the requirements of elasticity and plasticity. Taking into account that some overlap between the deviations of the type Blanda and Podiabland silicones has been observed (Fig. 4), the application of statistical tools to discern if the silicones are significantly different is necessary. Namely, since the DMA curves can be considered functional data, FDA exploratory analysis can be applied. Thus, Fig. 5 shows the functional or FDA means of ϵðtÞ with their confidence bands (95% confidence level) obtained by performing a smoothed bootstrap process with 150 resamples. In the creep step, the ϵðtÞ mean curves could be considered different from each other due to the 95% confidence bands are not overlapped. Although, the confidence bands of Blanda and Podiabland functional means for the recovery trends are completely overlapped. Thus, the strain recovery behavior of the two type of materials could be considered the same.

464

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Fig. 6 – Fittings of the KWW model to creep data.

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Fig. 7 – Fittings of the KWW model to recovery data.

465

466

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Table 4 – Asymptotic confidence intervals at 95% for the model parameters corresponding to creep and recovery trends. PODIABLAND Silicone

MASTER Silicone

Low. lim.

Low. lim.

Upp. lim.

Confidence intervals at 95% for the model parameters corresponding to creep trends 0.0753 0.1125 0.0619 0.1037 AKWW 0.0206 0.0452 0.0008 0.0039 τKWW 0.4098  0.1039 0.2513  0.0744 βKWW

0.0448 0.0006  0.3007

0.0603 .0066 0.0179

Confidence intervals at 95% for the model parameters corresponding to recovery trends 0.0220 0.0324 0.0188 0.0256 AKWW 480.14 2371.5 467.36 1971.9 τKWW 0.1431 0.2113 0.3515 0.3976 βKWW

0.0114 431.36 0.3585

0.0163 793.79 0.4120

Param.

BLANDA Silicone Low. lim.

Upp. lim.

Upp. limit

FANOVA test (Naya et al., 2014) based on random projec-

0 (2  10  16). Therefore the relationship between strain and

tions and false discovery rate (FDR) is implemented to verify if the populations (the three silicones) are really different

time in a creep–recovery test is different depending on the

with respect to their creep–recovery behavior. In short, the

performed considering each creep and recovery curves as one

null hypothesis of equality of functional ϵðtÞ means is rejected at a confidence level of 95%, in fact all the FDR corrected p-

curve or datum. Considering the results of the exploratory analysis, the

values corresponding to each pairwise comparison are almost

following step is to model the creep and relaxation behavior

type of silicone. It is important to note that the test has been

Fig. 8 – Application of mixed effects nonlinear KWW model to recovery data. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

of each silicone. The aim is to obtain a reliable model to analyze the creep–recovery behavior of each type of silicone to estimate, predict their performance. Thus, the silicone comparison will be possible considering the model parameters. The fitting model used in this work is the KWW model. Tables 2 and 3 show the KWW parameter estimates and their significance analysis corresponding to creep and recovery data, respectively. The procedure to obtain these fitting parameters was described in Section 3. First of all the DE evolutionary global optimization algorithm was applied indicating a wide range of possible initial solutions for each parameter. The parameters AKWW, βKWW, and τKWW are allowed to vary from 0 to 10,000 when recovery phenomena is fitted. When creep behavior is adjusted, AKWW, and τKWW vary from 0 to 10,000 while βKWW is between 10,000 and 0. These wide intervals are scheduled in order to prevent to reach a local minimum (in this case the computing runtime is not a critical parameter). Then, the fitting parameters of KWW model are obtained applying the Levenberg–Marquardt algorithm from the initial solution obtained by DE. Nonlinear regression significance analysis is also performed. Tables 2 and 3 show that all the parameters are significant different from zero (all p-values of t-tests are lesser than 0.05). In addition, Figs. 6 and 7 plot the real ϵðtÞ curves and the KWW fittings to creep and recovery trends, respectively. They can be considered a measure of the goodness of fit. All the fittings reliably reproduce all the real trends. In fact, the confidence interval for the conditioned mean (regression model estimates) and the confidence interval for prediction are very narrow and content the real trend (Figs. Figs. 6 and 7), mainly in the recovery case. In conclusion, KWW model is a proper model to describe the creep–recovery behavior of the studied silicones for podiatry applications. This model provides reliable estimates and could be useful to obtain strain predictions taking into account the very narrow prediction confidence intervals. The KWW model permits to compare the different silicones considering the values of the fitting parameters. Moreover, the KWW model provides silicone comparisons considering the physical meaning of its parameters. Table 4 shows the asymptotic confidence intervals for the parameters AKWW, βKWW, and τKWW corresponding to each type of silicone. The parameter τKWW, that accounts for the mean creep or recovery time, seems to be the same for all the silicones considering that the corresponding confidence intervals are overlapped. Thus, the creep and recovery time seems invariant with respect to the type of silicone. If the confidence intervals for AKWW and βKWW are observed (Table 4), the AKWW parameter of Master silicone is significantly lower than in the other silicones, both in creep and recovery analysis. Thus, Blanda silicone is the one with a higher instantaneous deformation and permanent strain, considering that the parameter AKWW is related to the initial strain of the material when a constant stress is applied and also with the permanent strain in the recovery test. βKWW accounts for the curvature of the theoretical curves and the breadth of the distribution of creep–recovery times. Blanda βKWW parameter is significantly lower than the βKWW parameters in the other silicones. The τKWW and βKWW

467

parameters of Blanda and Podiabland could be considered equal. This is according to the supplier specifications, in fact Master and Podiabland silicones can be used in corrective orthoses while Blanda can be only utilized in palliative orthoses, where a higher deformation and an elastic behavior is required. In order to complete the comparison study of silicones for podiatry application, mixed effects nonlinear regression is applied to creep data. The use of mixed effects models provides some advantages with respect to standard nonlinear models: (1) the same relationship is modeled with a less number of parameters, (2) the variability of the group silicones and replicates is estimated apart from residual variability, (3) the creep–recovery behavior of the three silicones can be compared easily at a glance, separating the effects of replicates and type of silicone. Fig. 8 shows the KWW fittings to creep data assuming mixed effects approach. The data are grouped following a nested structure: the replicate factor is nested in the type of silicone factor in order to see the effects of each factor in the ϵðtÞ curves. All the fittings were performed using the package nlme (Pinheiro and Bates, 2000). Random effects are added to βKWW and AKWW parameters assuming the results of fixed effects nonlinear regression: the differences of the three silicones are related to the variations of βKWW and AKWW parameters (τKWW seems invariant with respect to the type of silicone). The fitting parameters are obtained using a maximum likelihood procedure (Pinheiro et al., 2015). Fig. 8 also illustrates the fitting corresponding to the fixed parameters (in blue) that accounts for the average value of the individual parameters in the commercial silicones population. In pink we plot the fittings obtained for each type of silicone, while the fittings corresponding to each replicate are shown in black. We can evaluate the effect of the factor type of silicone considering the fitted mean trend as reference. Therefore, Podiabland silicone has the closest creep behavior to the creep behavior of the population, while Master silicone suffers lower strain than the mean and Blanda higher. The obtained mixed effects nonlinear model fits well the creep data adding random effects just to βKWW and AKWW parameters. Therefore, the differences between the silicones are related to the variation of this two parameters.

5.

Conclusions

The static viscoelastic properties of silicones applied in palliative and corrective orthoses related to podiatry applications were studied by DMA. Creep–recovery tests were performed. The creep–recovery behavior, ϵðtÞ, of Blanda, Podiabland and Master silicones provided by Herbitas were characterized, modeled and compared applying statistical FDA techniques, fixed and mixed nonlinear regression models based on the KWW function. This work shows how DMA analysis can be useful to chose the proper silicone for each different application according to their viscoelastic properties. FDA exploratory analysis and functional ANOVA statistical techniques have shown that the creep–recovery behavior of podiatry silicones, defined by ϵðtÞ, is significant different with

468

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

respect to each silicone. Nevertheless, if only the recovery strain is studied, the recovery behavior of Blanda and Podiabland could be considered the same (through application of bootstrap confidence bands for the functional ϵðtÞ mean). KWW model fits accurately creep–recovery behavior of the studied silicones for podiatry applications defined by the ϵðtÞ trends. KWW model can be used to provide reliable strain estimates and predictions taking into account the accuracy of the fits, their very narrow confidence intervals. In addition, KWW model has allowed us to compare the different silicones considering the values of the fitting parameters and their physical meaning. Thus Blanda silicone presents higher instantaneous deformation and permanent strain, AKWW. The τKWW parameter remains the same in the three types of silicones, considering the confidence intervals of this parameter. Thus, the mean creep–recovery time is the same for all the studied silicones. βKWW accounts for the curvature of the theoretical curves and the breadth of the distribution of creep–recovery times. Blanda βKWW parameter is significantly lower than the βKWW parameters in the other silicones, therefore it presents a different ϵðtÞ curvature and breadth of the creep–recovery times distribution. Moreover, Blanda and Podiabland show the same τKWW and AKWW parameters taking into account their confidence intervals. This fact is according to the supplier specifications: Master is used in corrective orthoses, Blanda is only applied in palliative orthoses while Podiabland silicones can be used in both corrective and palliative orthoses. The mixed effects regression models are an useful statistical tool to model and compare these type of silicones for podiatry applications. The ϵðtÞ trends are fitted with a less number of parameters and the creep–recovery behavior of the three silicones can be compared easier, observing the effects of replicates and type of silicone. The differences of the three silicones are related to the variations of βKWW and AKWW parameters, considering that random effects were added to βKWW and AKWW parameters and a reliable model was obtained. Nevertheless, τKWW, the mean time of creep, seems invariant with respect to the type of silicone.

Acknowledgements This research has been supported by the Spanish Ministry of Science and Innovation, Grant MTM2014-52876-R (ERDF included).

references

Burnfield, J.M., Few, C.D., Mohamed, O.S., Perry, J., 2004. The influence of walking speed and footwear on plantar pressures in older adults. Clin. Biomech. 19 (1), 78–84. Cao, R., Naya, S., Artiaga, R., Garcia, A., Varela, A., 2004. Logistic approach to polymer degradation in dynamic tga. Mater. Sci. Forum 85, 667–674. Chartoff, R., Menczel, J.D., Dillman, S.H., 2009. Dynamic mechanical analysis (DMA). In: Menczel, J.D., Prime, R.B. (Eds.), Thermal Analysis of Polymers Fundamentals and Applications. John Wiley & Sons: San Jose´.

Cuesta-Albertos, J., Febrero-Bande, M., 2010. A simple multiway anova for functional data. TEST 19, 537–557. Cuevas, A., Febrero, M., Fraiman, R., 2006. On the use of the bootstrap for estimating functions with functional data. Comput. Stat. Data Anal. 51 (2), 1063–1074. Delacroix, S., Lavigne, A., Nuytens, D., Che`ze, L., 2014. Effect of custom foot orthotics on three-dimensional kinematics and dynamics during walking. Comput. Methods Biomech. Biomed. Engin 17 (sup1), 82–83. Fancey, K.S., 2005. A mechanical model for creep, recovery and stress relaxation in polymeric materials. J. Mater. Sci. 40 (18), 4827–4831. Ferraty, F., Vieu, P., 2006. Nonparametric Functional Data Analysis: Theory and Practice. Springer-Verlag, New York. Findley, W., Peterson, D., 1958. Prediction of long-time creep with ten-year creep data on four plastic laminates. In: Proc. ASTM; 58:851. Francisco-Ferna´ndez, M., Tarrı´o-Saavedra, J., Mallik, A., Naya, S., 2012. A comprehensive classification of wood from thermogravimetric curves. Chemom. Intell. Lab. Syst. 118, 159–172. Francisco-Ferna´ndez, M., Tarrı´o-Saavedra, J., Naya, S., Lo´pezBeceiro, J., Artiaga, R., 2014. Classification of wood using differential thermogravimetric analysis. J. Therm. Anal. Calorim. 120 (1), 541–551. Guldemond, N., Leffers, P., Sanders, A., Emmen, H., Schaper, N., Walenkamp, G., 2006. Casting methods and plantar pressure. J. Am. Podiat. Med. Assoc. 96 (1), 9–18. Hawke, F., Burns, J., 2012. Brief report: custom foot orthoses for foot pain: what does the evidence say?. Foot Ankle Int. 33 (12), 1161–1163. Instron, Barcelona, Spain. 2016. Functional and Accommodative Foot Orthoses 〈http://www.instron.es/es-es? region ¼Spain. James, M. SE Farmer, 2001. Contractures in orthopaedic and neurological conditions: a review of causes and treatment. Disabil. Rehabil. 23 (13), 549–558. Kato, H., Takada, T., Kawamura, T., Hotta, N., Torii, S., 1996. The reduction and redistribution of plantar pressures using foot orthoses in diabetic patients. Diabetes Res. Clin. Pract. 31 (1), 115–118. Kirby, K.A., 2016. Functional and Accommodative Foot Orthoses. PodiatryNetwork 〈http://www.podiatrynetwork.com/docu ment_disorders.cfm?id ¼148.. Liu, H., Yin, L., Wu, D., Yao, Z., Zhang, M., Chen, C., Fu, P., Qin, J., 2015. Mechanical properties and creep behavior of poly(trimethylene terephthalate)/mesoporous silica composites. Polym. Compos 36 (8), 1386–1393. Maganaris, C.N., Narici, M.V., Maffulli, N., 2008. Biomechanics of the achilles tendon. Disabil. Rehabil. 30 (20–22), 1542–1547. Matsuoka, S., 1992. Relaxation Phenomena in Polymers. Hanser, Munich. Meardon, S.A., Edwards, W.B., Ward, E., Derrick, T.R., 2009. Effects of custom and semi-custom foot orthotics on second metatarsal bone strain during dynamic gait simulation. Foot Ankle Int. 30 (10), 998–1004. Naya, S., Tarro-Saavedra, J., Lpez-Beceiro, J., Francisco-Fernndez, M., Flores, M., Artiaga, R., 2014. Statistical functional approach for interlaboratory studies with thermal data. J. Therm. Anal. Calorim. 118 (2), 1229–1243. Pinheiro, J., Bates, D., 2000. Mixed-Effects Models in S and S-PLUS. Springer, New York. Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D., 2015. R Core Team. nlme: Linear and Nonlinear Mixed Effects Models. URL 〈http:// CRAN.R-project.org/package ¼ nlme〉, r package version 3.1– 122. R Development Core Team. 2016. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. 〈http://www.R-project.org〉.

journal of the mechanical behavior of biomedical materials 63 (2016) 456 –469

Rı´os-Fachal, M., Tarrı´o-Saavedra, J., Lo´pez-Beceiro, J., Naya, S., Artiaga, R., 2014. Optimizing fitting parameters in thermogravimetry. J. Therm. Anal. Calorim. 116 (3), 1141–1151. Ramsay, J.O., Silverman, B.W., 2005. Functional Data Analysis. Springer-Verlag, New York. Storn, R., Price, K., 1997. Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11 (4), 341–359. Tarrı´o-Saavedra, J., Naya, S., Francisco-Ferna´ndez, M., Artiaga, R., Lo´pez-Beceiro, J., 2011. Application of functional anova to the study of thermal stability of micro-nano silica epoxy composites. Chemom. Intell. Lab. Syst. 105, 114–124.

469

Tarrı´o-Saavedra, J., Francisco-Ferna´ndez, M., Naya, S., Lo´pezBeceiro, J., Artiaga, R., 2013. Wood identification using pressure dsc data. J. Chem. http://dx.doi.org/10.1002/cem.2561. Tarrı´o-Saavedra, J., Lo´pez-Beceiro, J., Naya, S., FranciscoFerna´ndez, M., Artiaga, R., 2014. Simulation study for generalized logistic function in thermal data modeling. J. Therm. Anal. Calorim. 118 (2), 1253–1268. Wasserman, L., 2006. All of Nonparametric Statistics (Springer Texts in Statistics). Springer-Verlag Inc., New York Secaucus, NJ, USA.