Hyperelasticity and the Failure of Averages

Transcription

1 Paper 204 Civil-Comp Press, 2015 Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing, J. Kruis, Y. Tsompanakis and B.H.V. Topping, (Editors), Civil-Comp Press, Stirlingshire, Scotland Hyperelasticity and the Failure of Averages D. Robertson and D. Cook New York University Abu Dhabi Abu Dhabi, United Arab Emirates Abstract Hyperelastic constitutive models are frequently used to characterize the stress-strain response of highly flexible materials. Literature surveys reveal most researchers report statistical quantities (e.g. mean and standard deviation) to describe and summarize hyperelastic coefficient values. However, statistical quantities are not appropriate for describing and summarizing coefficient values of hyperelastic constitutive models. Average coefficient values do not produce average behaviors and evaluating a hyperelastic constitutive model within one standard deviation of the mean coefficient values can produce extreme behavior that is at times completely unrealistic. The purpose of this paper is to address this issue, providing the theoretical background for this problem as well as examples and recommendations for avoiding this common pitfall. Keywords: average, coefficient, constitutive, error, fail, hyperelastic, material, model, nonlinear, properties, standard deviation, statistics. 1 Introduction Hyperelasticity theory [1] is commonly used to quantify the nonlinear, large-strain response of elastomers, rubbers, and other highly flexible materials. The theory provides numerous advantages including relatively easy implementation into finite element analyses [2]. Consequently, hyperelastic material models are widely available in engineering software packages and are frequently implemented by a broad range of users. However, some users may not be aware of certain, subtle, and non-obvious intricacies that can cause difficulty and errors when utilizing these material models. In particular, statistical quantities (e.g. mean and standard deviation) are typically not appropriate for describing and summarizing the hyperelastic constitutive coefficients of a group of experimental samples. The 1

2 purpose of this paper is to address this issue, providing the theoretical background for this problem as well as examples and recommendations for avoiding this common pitfall. Contrary to expectations, average hyperelastic constitutive coefficients generally do not produce a response that reflects the underlying data [3]. Furthermore, varying a hyperelastic constitutive coefficient within one standard deviation of its average does not necessarily produce behavior that is within one standard deviation of the average constitutive response. It is important for users of such models to be aware that averages and standard deviations of nonlinear hyperelastic constitutive coefficients generally produce nonaverage, nonphysical behaviors that can be completely unrelated to the actual response of the material in question [3]. This phenomenon (i.e. average coefficients failing to produce average behavior) is sometimes referred to as averaging failure [3-5], and is closely related to a number of logical fallacies (e.g. ecological fallacy [6]) that are widely recognized in other fields. In the field of probabilistic modeling Jensen s inequality states that when average measures are used as inputs to nonlinear models, average behavior is not produced [7]. Unfortunately, many users of hyperelastic material models are unaware of the consequences of pooling data over many trials or samples to compute statistical measures of hyperelastic material coefficients. Consequently, average constitutive coefficients are frequently reported in the literature even though they can produce erroneous material responses [3]. This is in part because of the non-intuitive nature of the problem. It is quite unnatural to think of average parameters producing nonaverage, unrealistic or even nonphysical behaviors. The problem is further complicated by the very strong tradition of reporting ecological or group data (and not data of individual samples) in scientific manuscripts as a means of data compression. For example, a recent literature survey found that 86% of the peer reviewed manuscripts investigating material properties of biological tissues reported average constitutive coefficients [3]. The purpose of this paper is to explain and demonstrate the theory behind averaging failure and to discuss the implications of averaging failure on hyperelastic constitutive models. Common hyperelastic material models will be investigated directly and guidelines are provided to allow the reader to determine if any given hyperelastic constitutive model will react badly to computation of statistical measures of its material coefficients. 2 Theory Errors associated with the computation of averages and standard deviations of hyperelastic material coefficients can arise from several sources. The following paragraphs highlight four potential sources of error and give examples of each. The definitions of several commonly used hyperelastic material models are then presented and the types of errors that can arise when using each model are reported. If constitutive coefficients possess non-normal or multimodal distributions or if coefficients are correlated in such a manner that they occupy a concave region in the parameter space, then the average of the measured coefficients may not lie within 2

3 the actual distribution of coefficient values. In these cases even the standard deviation of the coefficients may not substantially overlap with the actual measured coefficient values [4, 5]. For example, consider two normally distributed coefficients that are correlated in such a manner that the central values of each coefficient are associated with the outlying values of the other coefficient. If this were the case the average value of the two coefficients may lie entirely outside of their joint distribution. An example, of such a case is presented in Figure 1. The Figure shows that the mean values and one standard deviation ellipse of two normally distributed but correlated coefficients (X and Y) lie almost completely outside the joint distribution of X and Y. When such distributions occur in the parameter space of a constitutive model the set of average coefficient values as well as sets of coefficients that lie within one standard deviation of the mean coefficient values can produce stress-strain curves that are unrelated to the underlying sample. This occurs because the set of average coefficient values is not representative of the samples coefficient values. Figure 1: Both X and Y are normally distributed variables (mean = 0, standard deviation = 1). The variables are correlated such that the central values of each variable are associated with the tail or outing values of the other variable. As such the mean values of each variable and the one standard deviation ellipse shown in red lie almost entirely outside of the joint distribution of X and Y. A second source of error can arise from consistency or continuity conditions required by certain constitutive models. Many constitutive relations place certain bounds or relationships on constitutive coefficients to ensure that realistic material behavior is produced (e.g. coefficients must be positive; coefficient1 must be larger than coefficient 2, etc.). Even if every sample in an experimental test adheres to the given consistency conditions the set of average coefficients or coefficients that lie within one standard deviation of the average may not meet those same consistency conditions. This is because the averaging process accepts scalar values and produces a corresponding scalar: the mean value. Averaging ignores relationships between associated coefficients, thus effectually destroying these important relationships. 3

4 The destruction of relationships between coefficients is not obvious because such relationships are not frequently reported explicitly. Indeed, data obtained from experimental samples naturally satisfy these consistency conditions. In many cases, little attention is paid to these hidden relationships, but (as will be shown shortly), the loss of such relationships causes non-physical model behavior. One very obvious consistency condition is that of continuity. In the absence of fracture, load-deformation curves are continuous. Furthermore the derivative or slope of the curve is also continuous. However, numerous constitutive models can produce discontinuous curves when average coefficient values are employed. This type of error is illustrated using the commonly-employed exponential-linear model [8]: 1 (1) where is stress, a, b, c, d and are material coefficients and is strain. Figure 2 and Table 1 display stress-strain curves and constitutive coefficients of ten samples. Average coefficient values are also reported as is the stress-strain curve produced by the average coefficients. It can be seen in Figure 2 that the average coefficient curve is discontinuous. The slope of the curve is also discontinuous. This is because the average coefficients violate the consistency conditions of the model even though each individual sample adheres to the consistency conditions. It should be noted that constitutive models often possess more complex consistency conditions than that of continuity, and are sometimes used to ensure the model produces a purely elastic response. Figure 2: Stress-strain curves of ten samples subjected to large deformations are shown in grey. The curve produced by averaging the constitutive coefficients of the ten samples is shown in red. The average coefficient curve is discontinuous (indicated by dotted red line) because the process of averaging coefficients ignores the consistency conditions of the underlying constitutive model. 4

5 Sample a b c d ε toe Average Table 1: Constitutive Coefficients A third source of error can arise when a material s strain energy density definition contains nonlinear material coefficients. If a material coefficient appears in a denominator, exponential, or logarithm or is combined with a second material coefficient in any way other than through addition or subtraction serious errors can result from the process of averaging. This effect is due to Jensen s inequality which states that If f (Y) is a convex function, then E(f(Y)) > f(e(y)). If f (Y) is a concave function, then E(f(Y)) > f(e(y)). where E is the expected (or average) value and Y represents input variables to the function f. In other words Jensen s inequality states that applying average inputs (i.e. coefficients) to a nonlinear (constitutive) model will not produce average outputs. This fundamental mathematical law can be found in almost any standard probability theory textbook [7]. Jensen s inequality can be verified by analyzing the small deflection response of a cantilever beam. The deflected shape of a cantilever beam with a normal force at it tips can be computed as follows (2) where w is deflection of the beam, x is the length along the beam, F is force, L is length of the beam, E is modulus of elasticity, and I is the beam area moment of inertia. Suppose one wants to calculate the average deflected shape of ten beams of differing lengths. There are two possible approaches to this problem. The first approach involves measuring the physical characteristics of each beam and then using average physical characteristics as inputs to equation (2). Note this approach will not produce the average response because all three material coefficients (L,E, and I) undergo nonlinear operations (L is divided by both E and I). The second approach would be to enter the physical characteristics of each individual beam into 5

6 equation (2) to calculate their deflected shapes. Deflection data of the ten beams could then be averaged to produce the average deflected shape of the beams. This approach produces the true average response. Figure 3 depicts the average deformation shape of ten hypothetical beams using both of the approaches described above. Table 2 displays the characteristics of each beam. It can be seen that the beam with average physical properties does not produce the average deflected shape. 0 Normalized Length of Beam Delfection Figure 3: The average deflected shape of ten beams (see Table 2) is shown in grey and the deflected shape of a beam with average physical characteristics is shown as a dashed red line. The two deflected shapes are different because of Jensen s inequality. Sample I E L Average Table 2: Physical Characteristics of Beams Finally a fourth source of error can arise from inappropriately combining statistical measures (e.g. standard deviation). Evaluating a multi-coefficient constitutive model within one standard deviation of its material coefficients will most likely fail 6

7 to produce behavior that is within one standard deviation of the average material response. This is because some statistical measures such as standard deviation are nonlinear quantities. Therefore unless special conditions are met and the standard deviations are calculated in a very particular manner they cannot be simply combined. For example, using the constitutive model presented in equation (1) coefficient values of ten material curves were calculated. The standard deviation of each constitutive coefficient was then computed. Figure 4 displays the ten stressstrain curves as well as responses produced by evaluating the constitutive model one standard deviation above and below the mean constitutive coefficient values. This figure clearly demonstrates that evaluating the constitutive model within just one standard deviation of the mean coefficient values can generate very extreme behaviors that lie further from the mean behavior of the sample than any other single curve. Note that standard deviations are also susceptible to each of the previous sources of error mentioned in this manuscript. In this example, the standard deviation of the coefficients fails to meet the constitutive model s consistency conditions. Curves produced by the standard deviation are therefore discontinuous. Constitutive coefficients of all curves are shown in Table Stress Strain Figure 4: Ten stress-strain curves are shown in grey. Dashed red curves were produced by evaluating the constitutive model within one standard deviation of the average constitutive coefficient values. 7

8 Sample a b c d ε toe Average STDEV Table 3: Constitutive Coefficients 3 Failure of averaging in hyperelastic material models Each of the above paragraphs mentions potential errors that can occur when calculating statistical measures of hyperelastic material coefficients. However, some particular models may be immune to each of these errors while others may be subject each source of error. To determine which models are subject to averaging failure it is necessary to classify each model according to its strain energy density function. Models can be classified based on the number of material coefficients (single coefficient vs multiple coefficient models), and the manner in which such coefficients interact with one another in the mathematical definition of the materials strain energy density. Models in which a material coefficient in the strain energy density function is subjected to a nonlinear function (e.g. exponential, logarithm, etc.) or in which material coefficients are multiplied, divided or raised to the power of another material coefficient are classified as nonlinear. Models which contain multiple material coefficients that do not appear in a nonlinear term and are only added to or subtracted from other material coefficients are considered linear (note that a linear model in this sense can still generate a nonlinear stress-strain curve). The strain energy density functions of some of the most commonly employed hyperelastic material definitions and their accompanying classifications are given below where W = strain energy density function and I 1, I 2, and I 3 are the three invariants of the green deformation tensor given in terms of the principle stretch ratios λ 1, λ 2, λ 3 as follows: λ λ λ λ λ λ λ λ λ (3) λ λ λ A brief summary of each class of constitutive model is provided, and the types of errors to which that class are subjected is discussed. 8

9 3.1 Class I: single coefficient models Neo Hookean 3 (4) where C 1 is the only material coefficient. Most single coefficient models respond favorably to computation of statistical measures. The exception to this generality is if the single material coefficient appears in a nonlinear term. Barring this exception, the average and standard deviation of the material coefficient will produce the average and standard deviation of the measured material responses. However, depending on the distribution of measured responses the average and standard deviation may not be appropriate measures of central tendency in the data (e.g. multimodal or non-normal distributions). 3.2 Class II: multiple coefficient models with linear material coefficients St Venant Kirchoff Yeoh μ (5) where λ and µ are the Lamé Constants, E is the Lagrangian Green Strain and tr is the trace of the Lagrangian Green Strain tensor. 3 (6) where C i are material coefficients Mooney Rivlin 3 3 (7) where C 1 and C 2 are material coefficients. Polynomial or generalized Mooney Rivlin, 3 3 (8) where C ij are material coefficients and C 00 = 0. 9

10 Note that while none of the above strain energy density functions contains a material coefficient that appears in a nonlinear function, each definition does contain other nonlinearities. For example, the strain invariants may be raised to a power, multiplied by other strain invariants or even be multiplied by the material coefficients. Thus the defining feature of these models is that the actual material coefficients do not interact with other material coefficients except through addition or subtraction, and no material coefficients undergo nonlinear operations (e.g. being raised to power). As such, these types of constitutive models are immune to the third source of error mentioned in the theory section (e.g. nonlinearity), but are still subject to the other sources mentioned. 3.3 Class III: multiple coefficient models with nonlinear material coefficients Ogden, (9) λ λ λ 3 where α i and µ i are material coefficients. Arruda Boyce (10) Gent where C 1 and N are material coefficients. 1 (11) where µ and J m are material coefficients and I 1-3 < J m. Van der Waals or Kilian μ 1 where γ, µ, and J m are material coefficients. (12) These models are subject to all four sources of error outlined in the theory section and any attempts to compress constitutive data of the samples by reporting averages 10

11 or standard deviations of coefficients will likely produce errors. As such, it is best to completely avoid reporting or using any statistical measures to quantify such constitutive coefficients. In these cases individual sets of constitutive coefficients should be reported for each sample tested. 4 Discussion Average hyperelastic constitutive coefficients do not generally produce average material behaviors. Furthermore evaluating a multi-coefficient constitutive model within one standard deviation of the mean coefficient values can produce serious errors. The mathematical theory behind such non-intuitive behaviors has been summarized in section 2. The decision tree presented in Figure 4 can be used by future researchers to determine if any given constitutive model will react badly to averaging of coefficient values. Figure 4: Decision tree depicting when averaging hyperelastic coefficients will fail. Note that even if averaging the coefficients does not produce errors, evaluating any multi-coefficient model within one standard deviation of the mean coefficient values cannot be expected to produce behavior that lies within one standard deviation of the mean material behavior. 11

12 4.1 Implementing and reporting hyperelastic material model coefficients In some cases previous studies, have reported constitutive coefficients the best fit the average of the stress-strain data collected in the study. This approach avoids the previously mentioned errors. However, if the curves are highly variable averaging in this manner also presents several problems. For example, the average curve will tend to be more linear than any of the other curves. None the less, a quick review of the literature will reveal that this approach is not frequently employed [3]. Thus while it is apparent that some researchers recognize the inherent problems associated with application of average inputs to hyperelastic material models many researchers appear to be unaware of the consequences of applying averaged input parameters to nonlinear material definitions. This manuscript has attempted to raise awareness of and explain the theory behind averaging failure so that such errors can be avoided in the future. 4.2 Other factors for consideration This paper does not provide an exhaustive analysis of all possible hyperelastic models. Numerous other models have been introduced, including extensions and variations of the above material models. These variations can, in some cases, change the classification (e.g. nonlinear, linear, single coefficient) of the constitutive model. For example, the above constitutive relations are for purely incompressible materials. To account for material compressibility it may be necessary to add additional material coefficients that are subject to nonlinear operations thus changing the model classification to a multiple coefficient nonlinear model. To determine if reporting of average material coefficients is appropriate for a given constitutive model one can refer to the decision tree presented in Figure 4. Note that even if averaging the coefficients of a given model does not produce errors, evaluating any multi-coefficient model within one standard deviation of its mean coefficient values cannot be expected to produce behavior that lies within one standard deviation of the mean material behavior. 5 Conclusion The goal of this study was to examine the issue of coefficient averaging in hyperelastic material models and to raise awareness of the fact that common statistical measures (e.g. mean and standard deviation) of hypereslastic constitutive parameters can produce serious errors. Averaging of hyperelastic material coefficients is usually problematic, so caution and special attention should be given whenever such materials are used or reported. In terms of usage, we highly recommend the implementation of coefficient sets corresponding to actual physical samples. This practice is the most robust approach because it preserves all essential compatibility relationships. In doing so, the most average individual sample can be used to represent the typical case. 12

13 Researchers should refrain from reporting average coefficient sets for two reasons. First, they are often not representative of the data from which they were derived, as described in detail above. Second, subsequent researchers who may not be as well informed about these issues will naturally have the tendency to choose the average coefficient sets in future research efforts, including modeling, comparison, etc. Authors may also wish to explain why averages are not reported, to insure that the data is not averaged by subsequent researchers. Reviewers should encourage authors to follow these practices. It is our hope that the theory and examples in this paper may provide future researchers with a useful reference for correctly interpreting, implementing, and reporting hyperelastic material models. References [1] Y. Basar and D. Weichert, Nonlinear Continuum Mechanics of Solids: Fundamental Mathematical and Physical Concepts, Springer Berlin Heidelberg, [2] R. D. Cook, Finite element modeling for stress analysis, Wiley, [3] D. Robertson, D. Cook, Unrealistic statistics: How average constitutive coefficients can produce non-physical results, Journal of the mechanical behavior of biomedical materials, 40, , [4] M.S. Goldman, Failure of averaging, Encyclopedia of Computational Neuroscience, DOI / _15-1. [5] J. Golowasch, Failure of averaging in the construction of a conductancebased neuron model, Journal of Neurophysiology, 87(2), , [6] S. Piantadosi, D.P. Byar, S.B. Green, The ecological fallacy, American Journal of Epidemiology, 127(5), , [7] P. Westfall, K.S. Henning, Understanding advanced statistical methods, CRC Press, [8] E. Danso, Comparison of nonlinear mechanical properties of bovine articular cartilage and meniscus, Journal of biomechanics, 47(1), ,