Uncertainty quantification of elastic material responses: testing, stochastic calibration and Bayesian model selection
Abstract
Motivated by the need to quantify uncertainties in the mechanical behaviour of solid materials, we perform simple uniaxial tensile tests on a manufactured rubber-like material that provide critical information regarding the variability in the constitutive responses between different specimens. Based on the experimental data, we construct stochastic homogeneous hyperelastic models where the parameters are described by spatially independent probability density functions at a macroscopic level. As more than one parametrised model is capable of capturing the observed material behaviour, we apply Baye theorem to select the model that is most likely to reproduce the data. Our analysis is fully tractable mathematically and builds directly on knowledge from deterministic finite elasticity. The proposed stochastic calibration and Bayesian model selection are generally applicable to more complex tests and materials.
Keywords
Stochastic elasticity Finite strain analysis Hyperelastic material Bayes’ factor Experiments Probabilities“This task is made more difficult than it otherwise would be by the fact that some of the test-pieces used have to be moulded individually, and it is difficult to make two rubber specimens having identical properties even if nominally identical procedures are followed in preparing them.” - R.S. Rivlin and D.W. Saunders [55]
1 Introduction
The study of material elastic properties has traditionally used deterministic approaches, based on ensemble averages, to quantify constitutive parameters [36]. In practice, these parameters can meaningfully take on different values corresponding to possible outcomes of the experiments. The art and challenge of experimental setup is to get as close as possible to the ideal situations that can be analysed mathematically. From the mathematical modelling point of view, stochastic representations accounting for data dispersion are needed to improve assessment and predictions [9, 14, 18, 26, 45, 50, 61, 71].
Recently, stochastic models (described by a strain-energy density) were proposed for nonlinear elastic materials, where the parameters are characterised by probability distributions at a continuum level [37, 65, 66, 67, 68, 69]. These are advanced phenomenological models that rely on the finite elasticity theory [15, 48, 76] and on the maximum entropy principle to enable the propagation of uncertainties from input data to output quantities [62]. The principle of maximum entropy was introduced by Jaynes (1957) [19, 20] (see also [21]) and states that “The probability distribution which best represents the current state of knowledge is the one with largest entropy (or uncertainty) in the context of precisely stated prior data (or testable information).” The notion of entropy (or uncertainty) of a probability distribution was first defined by Shannon (1948) [57, 64] in the context of information theory. These models can further be incorporated into Bayesian approaches [3, 30] for model selection and updates [37, 45, 46, 56].
For non-deterministic material models, two important questions arise, namely “how do material constitutive laws influence possible equilibrium states and their stability?” and “what is the effect of the probabilistic parameters on the predicted elastic responses?” Presently, theoretical approaches have been able to successfully contend with cases of simple geometry such as cubes, spheres, shells and tubes. Specifically, within the stochastic framework, the classic problem of the Rivlin cube was considered in [38], the static and dynamic inflation of cylindrical and spherical shells was treated in [34, 35], the static and dynamic cavitation of a sphere was analysed in [33, 40], and the stretch and twist of anisotropic cylindrical tubes was examined in [39]. These problems were drawn from the analytical finite elasticity literature and incorporate random variables as basic concepts along with mechanical stresses and strains. Such problems are interesting in their own right and may also inspire further thinking about how stochastic extensions can be formulated before they can be applied to more complex systems.
To investigate the effect of probabilistic parameters in the case of more realistic geometries and loading conditions, computational approaches have been proposed in [68, 69]. However, for real materials, most available data consist of mean values from which deterministic models are usually derived, and there is a lack of experimental data reported in the literature that are directly suitable for stochastic modelling.
For rubber-like materials, the first experimental data for load–deformation responses of different material samples under large strains were provided by Rivlin and Saunders [55]. Following the phenomenological models for vulcanised rubber developed by Treloar (2005) [75] (read also [13]), many deterministic hyperelastic models calibrated to the mean data values were described (for example, in [7, 8, 17, 36, 49, 70, 77]). To capture the variability of Rivlin and Saunders’ data (see also [75, p. 224], or [76, p. 181]), probability distributions for the random shear modulus under relatively small strains were obtained in [34]. In recognition of the fact that a crucial part in assessing the elasticity of materials is to quantify the uncertainties in their mechanical responses, for rubber and soft tissues under large strain deformations, explicit stochastic hyperelastic models based on datasets consisting of mean values and standard deviations were developed in [37], while statistical models derived from numerically generated data were proposed in [6, 44]. In all these developments, the aim was to find the most appropriate general method of characterising the properties of nonlinear elastic materials at a macroscopic (non-molecular) level.
In this study, we first report on simple experimental results for different samples of silicone rubber material in finite uniaxial tension (Section 2). Full details of the manufactured material are presented in Section 2.1, while specific descriptions of the experimental setup and techniques used during testing are outlined in Sections 2.2 and 2.3. Employing the stochastic calibration method and Bayesian selection criterion proposed in [37], we then obtain stochastic homogeneous hyperelastic models, where the parameters are random variables that are constant in space, and are calibrated to our experimental measurements (Section 3). The assumptions and ideas that we adopt for the stochastic modelling are discussed in Section 3.1, while calibration results and details on how Bayes’ theorem can be employed explicitly to select the best performing model are given in Sections 3.3 and 3.4. In Section 4, we draw concluding remarks.
2 Experimental testing
In this section, we describe the experimental setup and techniques used to measure large elastic deformations of manufactured silicone rubber specimens. Our experimental results capture the inherent variability in the acquired data between different specimens during tensile tests.
2.1 Specimen manufacture
For Batch 1, tensile testing specimens were cast using Tech-Sil 25 Silicone (Technovent). This is a two part silicone, with a standard mixture ratio of 9:1 for Part A:Part B, respectively, as per the manufacturer’s recommendation, and is generally allowed to cure at room temperature for at least 24 h. The silicone was mixed and de-gassed prior to casting, ensuring an even mixture, and no air bubbles were present in the tensile specimens. Testing specimens of equal dimensions were made within a mould that is typically used for this purpose. The silicone was removed from the mould for testing after 4 weeks.
For Batch 2, the same make of silicone as in Batch 1 was used, with slight variations in the mixture components in order to simulate an error that would be within a realistic experimental range. Namely, the mixture ratio was 8.96:1. Tensile testing specimens with the specified geometry were created using this mixture. The silicone was also left to cure at room temperature, and taken out of the mould for testing after 2 weeks.
Full details of the silicone mixture for the two batches of tensile testing specimens
Batch number | 1 | 2 |
---|---|---|
Part A weight (g) | 180 | 92.3 |
Part B weight (g) | 20 | 10.3 |
Mixing ratio | 9:1 | 8.96:1 |
Curing period | 4 weeks | 2 weeks |
Number of specimens tested | 6 | 2 |
2.2 Experimental set-up
Specimen details and testing parameters
Specimen number | Silicone batch | Testing session | Testing speed (mm/min) | Grip length (mm) | Number of tests |
---|---|---|---|---|---|
1 | 1 | 1 | 30 | 30 | 3 |
2 | 1 | 1 | 30 | 30 | 3 |
3 | 1 | 1 | 30 | 30 | 3 |
4 | 1 | 1 | 30 | 20 | 2 |
5 | 1 | 2 | 30 | 30 | 3 |
6 | 1 | 2 | 30 | 20 | 3 |
7 | 2 | 2 | 30 | 30 | 3 |
8 | 2 | 2 | 30 | 20 | 3 |
2.3 Optical strain measurement
Post-processing parameters for the optical strain measurement system, with the gauge lengths as indicated in Fig. 3
Settings 1 | Settings 2 | Settings 3 | Settings 4 | |
---|---|---|---|---|
Tracking algorithm | Deform only | Stretch, rotate and deform | Deform only | Deform only |
Target 1 size | Random (based on user) | Random (based on user) | 45 × 45 pixels | 45 × 45 pixels |
Target 2 size | Random (based on user) | Random (based on user) | 45 × 45 pixels | 45 × 45 pixels |
Gauge length | A | A | A | B |
- The first Piola–Kirchhoff (PK) tensile stress, representing the force per unit area in the reference configuration,where F is the applied tensile force and A = 40 mm^{2} is the cross-sectional area.$$ P=\frac{F}{A}, $$(1)
3 Stochastic modelling
Next, we construct specific stochastic homogeneous hyperelastic models where the parameters are characterised by spatially independent probability distributions and optimised to the collected data.
3.1 Stochastic isotropic incompressible hyperelastic models
- (A1)
Material objectivity, stating that constitutive equations must be invariant under changes of frame of reference. This requires that the scalar strain-energy function, W = W(F), depending only on the deformation gradient F, with respect to the reference configuration, is unaffected by a superimposed rigid-body transformation (which involves a change of position) after deformation, i.e., \(W\left (\textbf {R}^{T}\textbf {F}\right )=W(\textbf {F})\), where R ∈ SO(3) is a proper orthogonal tensor (rotation). Material objectivity is guaranteed by defining strain-energy functions in terms of the scalar invariants.
- (A2)
Material isotropy, requiring that the strain-energy function is unaffected by a superimposed rigid-body transformation prior to deformation, i.e., W(FQ) = W(F), where Q ∈ SO(3). For isotropic materials, the strain-energy function is a symmetric function of the principal stretch ratios {λ_{i}}_{i= 1,2,3} of F, i.e., \(W(\textbf {F})=\mathcal {W}(\lambda _{1},\lambda _{2},\lambda _{3})\).
- (A3)Baker–Ericksen (BE) inequalities, which state that the greater principal (Cauchy) stress occurs in the direction of the greater principal stretch [2, 28],where {λ_{i}}_{i= 1,2,3} and {T_{i}}_{i= 1,2,3} denote the principal stretches and principal Cauchy stresses, respectively, such that$$ \left( T_{i}-T_{j}\right)\left( \lambda_{i}-\lambda_{j}\right)>0\quad \text{if}\quad \lambda_{i}\neq\lambda_{j},\quad i,j=1,2,3, $$(4)In Eq. 4, the strict inequality ‘>’ is replaced by “≥” if any two principal stretches are equal.$$ T_{i}=\lambda_{i}\frac{\partial\mathcal{W}}{\partial\lambda_{i}},\qquad i=1,2,3. $$(5)
- (A4)
For any given finite deformation, at any point in the material, the shear modulus, μ, and its inverse, 1/μ, are second-order random variables, i.e., they have finite mean value and finite variance [65, 66, 67, 68, 69].
Assumptions (A1)–(A3) are general principles in isotropic finite elasticity [15, 36, 48, 76]. In particular, for a homogeneous hyperelastic body under uniaxial tension, the deformation is a simple extension in the direction of tensile force if and only if the BE inequalities Eq. 4 hold [28]. Assumption (A3) implies that the shear modulus is always positive, i.e., μ > 0 [36], while (A4) places random variables at the foundation of hyperelastic models [12, 21, 37, 43, 62].
In practice, elastic moduli can take on different values, corresponding to possible outcomes of experimental tests. The maximum entropy principle allows us to explicitly construct the probability laws of the model parameters, given the available information. Approaches for the explicit derivation of probability distributions for the elastic parameters of stochastic homogeneous isotropic hyperelastic models calibrated to experimental data for rubber-like material and soft tissues were proposed in [37, 67].
3.2 Hypothesis testing
Parameters of the probability distributions derived from the data values for the random shear modulus at small strain
Probability density function (pdf) | \(\underline {\mu }\) | ∥μ∥ | ρ_{1} | ρ_{2} |
---|---|---|---|---|
Gamma pdf fitted to batch 1 data | 0.2837 | 0.0171 | 275.4403 | 0.0010 |
Normal pdf fitted to batch 1 data | 0.2837 | 0.0170 | - | - |
Gamma pdf fitted to batch 2 data | 0.2663 | 0.0083 | 1029.4047 | 0.0003 |
Normal pdf fitted to batch 2 data | 0.2663 | 0.0084 | - | - |
3.3 Stochastic calibration
Incompressible isotropic hyperelastic models, \(\mathcal {W}(\lambda _{1},\lambda _{2},\lambda _{3})\), their nonlinear shear modulus μ(a) at a given stretch a, and the shear modulus at infinitesimal deformation, \(\lim _{a\to 1}\mu (a)=\overline {\mu }\)
Material model | Strain-energy density | Shear moduli |
---|---|---|
\(\mathcal {W}(\lambda _{1},\lambda _{2},\lambda _{3})\) | ||
Mooney–Rivlin | \(\frac {C_{1}}{2}\left ({\lambda _{1}^{2}}+{\lambda _{2}^{2}}+{\lambda _{3}^{2}}-3\right )+\frac { C_{2}}{2}\left (\lambda _{1}^{-2}+\lambda _{2}^{-2}+\lambda _{3}^{-2}-3\right )\) | \(\mu (a)=C_{1}+\frac {C_{2}}{a}\) |
C_{1},C_{2} independent of deformation | \(\overline {\mu }=C_{1}+C_{2}\) | |
Gent–Gent | \(-\frac {C_{1}}{2\beta }\ln \left [1-\beta \left ({\lambda _{1}^{2}}+{\lambda _{2}^{2}}+{\lambda _{3}^{2}}-3\right )\right ]+\frac {3C_{2}}{2}\ln \frac {\lambda _{1}^{-2}+\lambda _{2}^{-2}+\lambda _{3}^{-2}}{3}\) | \(\mu (a)=\frac {C_{1}}{1-\beta \left (a^{2}+2/a-3\right )}+\frac {3C_{2}}{2a^{2}+1/a}\) |
C_{1},C_{2},β independent of deformation | \(\overline {\mu }=C_{1}+C_{2}\) | |
Ogden | \({\sum }_{p=1}^{3}\frac {C_{p}}{2{\alpha _{p}^{2}}}\left (\lambda _{1}^{2\alpha _{p}}+\lambda _{2}^{2\alpha _{p}}+\lambda _{3}^{2\alpha _{p}}-3\right )\) | \(\mu (a)={\sum }_{p=1}^{3}\frac {C_{p}}{\alpha _{p}}\frac {a^{1-\alpha _{p}}\left (1-a^{3\alpha _{p}}\right )}{1-a^{3}}\) |
[47] | C_{p} independent of deformation; α_{1} = 1, α_{2} = − 1, α_{3} = − 2, | \(\overline {\mu }={\sum }_{p=1}^{3}C_{p}\) |
Parameters of stochastic constitutive models given in Table 5 calibrated to the data, and the corresponding random nonlinear shear modulus μ = μ(a) at a = 1.15
Stochastic model | Calibrated parameters | Shear modulus | ||
---|---|---|---|---|
(mean value ± standard deviation) | (mean value ± standard deviation) | |||
Batch 1 | Batch 2 | Batch 1 | Batch 2 | |
Mooney–Rivlin | C_{1} = 0.0936 ± 0.0030 | C_{1} = 0.1029 ± 0.0001 | μ = 0.2411 ± 0.0130 | μ = 0.2277 ± 0.0056 |
C_{2} = 0.1696 ± 0.0115 | C_{2} = 0.1696 ± 0.0115 | |||
Gent–Gent | C_{1} = 0.0971 ± 0.0042 | C_{1} = 0.1007 ± 0.0011 | μ = 0.2532 ± 0.0136 | μ = 0.2397 ± 0.0056 |
C_{2} = 0.1826 ± 0.0110 | C_{2} = 0.1625 ± 0.0053 | |||
β = 0.0434 | β = 0.0421 | |||
Ogden | C_{1} = − 0.0645 ± 0.0143 | C_{1} = − 0.0437 ± 0.0111 | μ = 0.2719 ± 0.0720 | μ = 0.2563 ± 0.0272 |
α_{1} = 1, α_{2} = − 1, α_{3} = − 2 | C_{2} = − 0.0764 ± 0.0155 | C_{2} = − 0.0844 ± 0.0112 | ||
C_{3} = 0.4861 ± 0.0534 | C_{3} = 0.4505 ± 0.0345 |
3.4 Bayesian model selection
However, if the Bayes factor is equal to 1, then Occam’s razor [22, 23, 24, 73] would imply that a larger prior probability should be assigned to the simpler model than to the more complex one for reasons of parsimony.
4 Conclusion
We report on experimental tests on different samples of a manufactured rubber-like material under large tensile loading, and employ a stochastic strategy to derive constitutive models that take into account the variability in the collected data. Specifically, we construct isotropic incompressible hyperelastic models with model parameters defined as spatially independent random variables characterised by probability density functions at a continuum level. In addition, we provide a methodology where an explicit lower bound on the Bayes factor is used to compare different models, and then applied to select the model that is most likely to reproduce the data.
Ideally, these models are calibrated and validated on multiaxial test data [31, 32, 41, 63]. In this case also, our stochastic calibration and Bayesian model selection can be employed to obtain suitable models.
Our analysis is fully tractable mathematically and builds directly on knowledge from deterministic finite elasticity.
This study highlights the need for continuum models to consider the variability in the elastic behaviour of materials at large strains, and complements our previous theoretical investigations of how elastic solutions of fundamental problems in nonlinear elasticity can be extended to stochastic hyperelastic models [33, 34, 35, 38, 39, 40].
Notes
Acknowledgements
The authors are grateful to Professor Nikolai Leonenko of Cardiff School of Mathematics for discussions on probability theory, and to Richard Thomas of Cardiff School of Engineering for his valuable support with the experimental testing and setup.
Funding information
This study is supported by the Engineering and Physical Sciences Research Council of Great Britain under research grant EP/S028870/1 to L. Angela Mihai.
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