Abstract
For natural composite materials such as biological tissues, mechanically characterizing the individual constituents and elucidating their roles in supporting structural integrity has remained experimentally challenging since the constituents can often not be isolated without impairment and require non-standard testing devices. Adopting an inverse viewpoint, we examine, in this article, macroscopic samples whose constituent architecture is accessible and investigate whether it is possible to conclude on the stress–strain behaviour of the individual constituents based on experimental measurements from standard material tests. Focussing on isotropic hyperelastic composites, a direct discretization of the constituents’ strain energy densities in terms of global shape functions is explored. In order to assess the local deterministic identifiability of material parameters near an initial estimate, we adopt a sensitivity-based criterion and determine feasible combinations of candidate experiments without recourse to experimental measurements. Both the local identifiability and attainable reidentification accuracy are investigated in detail for a composite truss and a composite sheet whose force–strain responses in uniaxial or biaxial tension tests, respectively, are mainly determined by the constituents’ volume fractions.
Similar content being viewed by others
Notes
Strictly, homogenization techniques are frequently restricted to microstructured composites whose constituents are arranged on much smaller length scales than the sample extent.
For notational simplicity, we employ the same quadrature points and weights for all \({{\mathcal {E}}}_i\), here. However, in general (and, indeed, in Sect. 5.2), the quadrature approximation may be different for every force–strain curve i.
References
Böl M, Ehret AE, Leichsenring K, Weichert C, Kruse R (2014) On the anisotropy of skeletal muscle tissue under compression. Acta Biomater 10:3225–3234
Morales-Orcajo E, Siebert T, Böl M (2018) Location-dependent correlation between tissue structure and the mechanical behaviour of the urinary bladder. Acta Biomater 75:263–278
Meyer GA, Lieber RL (2011) Elucidation of extracellular matrix mechanics from muscle fibers and fiber bundles. J Biomech 44:771–773
Oskay C, Fish J (2008) On calibration and validation of eigendeformation-based multiscale models for failure analysis of heterogeneous systems. Comput Mech 42:181–195
Schmidt U, Mergheim J, Steinmann P (2012) Multiscale parameter identification. Int J Multiscale Comput Eng 10:327–342
Mazerolles L, Perriere L, Lartigue-Korinek S, Piquet N, Parlier M (2008) Microstructures, crystallography of interfaces, and creep behavior of melt-growth composites. J Eur Ceram Soc 28:2301–2308
Gillies AR, Lieber RL (2011) Structure and function of the skeletal muscle extracellular matrix. Muscle Nerve 44:318–331
Järvinen TAH, Józsa L, Kannus P, Järvinen TLN, Järvinen M (2002) Organization and distribution of intramuscular connective tissue in normal and immobilized skeletal muscles. J Muscle Res Cell Motil 23:245–254
Schüler T, Jänicke R, Steeb H (2016) Nonlinear modeling and computational homogenization of asphalt concrete on the basis of XRCT scans. Constr Build Mater 109:96–108
Vajda S, Rabitz H, Walter É, Lecourtier Y (1989) Qualitative and quantitative identifiability analysis of nonlinear chemical kinetic models. Chem Eng Commun 83:191–219
Reid JG (1977) Structural identifiability in linear time-invariant systems. IEEE Trans Autom Control 22:242–246
Gokhale NH, Barbone PE, Oberai AA (2008) Solution of the nonlinear elasticity imaging inverse problem: the compressible case. Inverse Probl 24:045010
Goenezen S, Barbone P, Oberai AA (2011) Solution of the nonlinear elasticity imaging inverse problem: The incompressible case. Comput Methods Appl Mech Eng 200:1406–1420
Schuster T, Wöstehoff A (2014) On the identifiability of the stored energy function of hyperelastic materials from sensor data at the boundary. Inverse Probl 30:105002
Seydel J, Schuster T (2017) Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method. Inverse Probl 33:124004
Klinge S (2012) Inverse analysis for multiphase nonlinear composites with random microstructure. Int J Multiscale Comput Eng 10:361–373
Klinge S (2012) Parameter identification for two-phase nonlinear composites. Comput Struct 108–109:118–124
Klinge S, Steinmann P (2015) Inverse analysis for heterogeneous materials and its application to viscoelastic curing polymers. Comput Mech 55:603–615
Schmidt U, Mergheim J, Steinmann P (2015) Identification of elastoplastic microscopic material parameters within a homogenization scheme. Int J Numer Methods Eng 104:391–407
Schmidt U, Steinmann P, Mergheim J (2016) Two-scale elastic parameter identification from noisy macroscopic data. Arch Appl Mech 86:303–320
Hartmann S, Gilbert RR (2018) Identifiability of material parameters in solid mechanics. Arch Appl Mech 88:3–26
Gurtin ME (1981) An introduction to continuum mechanics. Academic Press Inc, New York
Hartmann S (2001) Numerical studies on the identification of the material parameters of Rivlin’s hyperelasticity using tension-torsion tests. Acta Mech 148:129–155
Ogden RW (1972) Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc Lond Ser A Math Phys Sci 326:565–584
Arruda EM, Boyce MC (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41:389–412
Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. Prentice-Hall, Englewood Cliffs
Dahmen W, Reusken A (2008) Numerik für Ingenieure und Naturwissenschaftler, Springer-Lehrbuch, 2nd edn. Springer, Berlin
Rivlin RS (1956) Large elastic deformations. In: Eirich FR (ed) Rheology: Theory and applications, vol 1. Academic Press Inc, New York, pp 351–385
Yeoh OH (1993) Some forms of the strain energy function for rubber. Rubber Chem Technol 66:754–771
Hartmann S (2001) Parameter estimation of hyperelasticity relations of generalized polynomial-type with constraint conditions. Int J Solids Struct 38:7999–8018
Hartmann S, Neff P (2003) Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int J Solids Struct 40:2767–2791
Schröder J, Neff P (2003) Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int J Solids Struct 40:401–445
Ogden RW, Saccomandi G, Sgura I (2004) Fitting hyperelastic models to experimental data. Comput Mech 34:484–502
Walter É, Pronzato L (1997) Identification of parametric models from experimental data. Springer, Berlin
Bertsekas D (1996) Constrained optimization and Lagrange multiplier methods. Athena Scientific, Nashua
Björck Å (1996) Numerical methods for least squares problems. Society for Industrial and Applied Mathematics, Philadelphia
Rowley CW (2005) Model reduction for fluids, using balanced proper orthogonal decomposition. Int J Bifurc Chaos 15:997–1013
Holmes P, Lumley JL, Berkooz G, Rowley CW (2012) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge
Ljung L, Glad T (1994) On global identifiability for arbitrary model parametrizations. Automatica 30:265–276
Thomaseth K, Saccomani MP (2018) Local identifiability analysis of nonlinear ODE models: How to determine all candidate solutions. IFAC-PapersOnLine 51:529–534
Beck JV, Arnold KJ (1977) Parameter estimation in engineering and science. Wiley, New York
Trefethen LN, Bau D III (1997) Numerical linear algebra. Society for Industrial and Applied Mathematics, Philadelphia
Antoulas AC (2005) Approximation of large-scale dynamical systems, vol 6. Society for Industrial and Applied Mathematics, Philadelphia
Brun R, Reichert P, Künsch HR (2001) Practical identifiability analysis of large environmental simulation models. Water Resour Res 37:1015–1030
Tuncer N, Le TT (2018) Structural and practical identifiability analysis of outbreak models. Math Biosci 299:1–18
Hartmann S, Gilbert RR, Sguazzo C (2018) Basic studies in biaxial tensile tests. GAMM-Mitteilungen 41:e201800004
Simo JC, Pister KS (1984) Remarks on rate constitutive equations for finite deformation problems: Computational implications. Comput Methods Appl Mech Eng 46:201–215
Klisch SM, Van Dyke TJ, Hoger A (2001) A theory of volumetric growth for compressible elastic biological materials. Math Mech Solids 6:551–575
Belytschko T, Liu WK, Moran B, Elkhodary K (2008) Nonlinear finite elements for continua and structures. Wiley, New York
Acknowledgements
I gratefully acknowledge the support of Abimathi Siva Subramanian who assisted me in setting up the biaxial tension test case analyzed in Sect. 5.2 as part of her DAAD WISE internship at the Institute of Solid Mechanics in summer 2018.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Evaluating the sensitivities of predicted force resultants
Appendix: Evaluating the sensitivities of predicted force resultants
In this section, we address the semi-analytical evaluation of the sensitivities \(\partial F_i(\varepsilon ; {{\mathbf {w}}})/\partial {{\mathbf {w}}}\) for the ith force–strain curve \(F_i(\varepsilon ; {{\mathbf {w}}})\) and a particular imposed strain \(\varepsilon \in {{\mathcal {E}}}_i\) [5, 19]. Restricting the attention to a displacement-based finite element formulation, we first recall that the Galerkin approximation of the principle of virtual work leads to the following system of nonlinear equations
where \({{\mathbf {f}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}})\) denotes the vector of discrete internal forces, \({{\mathbf {f}}}^{(ext)}({{\mathbf {d}}})\) is the vector of discrete external forces and \({{\mathbf {d}}}\) summarizes all displacement degrees of freedom. In order to account for Dirichlet boundary conditions, \({{\mathbf {d}}}\) is commonly decomposed, after potential reordering, into degrees of freedom \({\hat{{{\mathbf {d}}}}}\) that are prescribed as \({\hat{{{\mathbf {d}}}}}_{\varepsilon }\) on the Dirichlet boundary,
and into remaining unknown degrees of freedom \({\tilde{{{\mathbf {d}}}}}\),
Here, the subscript \(\varepsilon \) indicates that the prescribed values of \({\hat{{{\mathbf {d}}}}}\) are set with reference to the externally imposed strain \(\varepsilon \). The order of displacement degrees of freedom in Eq. (62) is also reflected in the ordering of the entries in Eq. (60); for subsequent reference, we specifically record the block-partitioned representation of the discrete internal force vector
The Dirichlet boundary conditions in Eq. (61) are enforced by the discrete reaction forces \({\hat{{{\mathbf {a}}}}}\) in Eq. (60). For given constitutive parameters \({{\mathbf {w}}}\) and prescribed Dirichlet displacements \({\hat{{{\mathbf {d}}}}}_{\varepsilon }\), Eqs. (60) and (61) can be solved for the unknown displacement degrees of freedom \({\tilde{{{\mathbf {d}}}}}\) and the discrete reaction forces \({\hat{{{\mathbf {a}}}}}\) that establish equilibrium. The latter are particularly relevant as they determine the predictions of cumulative forces \(F_i(\varepsilon ; {{\mathbf {w}}})\) on the Dirichlet boundary. By consequence, the sensitivities \(\partial F_i(\varepsilon ; {{\mathbf {w}}})/\partial {{\mathbf {w}}}\) may be computed in terms of the sensitivities of the discrete reaction forces \(\partial {\hat{{{\mathbf {a}}}}}({{\mathbf {w}}})/\partial {{\mathbf {w}}}\).
If we consider \(\varepsilon \) and, hence, \({\hat{{{\mathbf {d}}}}}_{\varepsilon }\) as given, then the equilibrium solution \({{\mathbf {d}}}= {{\mathbf {d}}}^{\star }({{\mathbf {w}}})\) is smoothly parameterized by the constitutive parameters \({{\mathbf {w}}}\). Computing the gradient of Eq. (60) at \({{\mathbf {d}}}= {{\mathbf {d}}}^{\star }({{\mathbf {w}}})\) with respect to \({{\mathbf {w}}}\) and taking into account \(\partial {\hat{{{\mathbf {d}}}}}^\star ({{\mathbf {w}}})/\partial {{\mathbf {w}}}= {\mathbf {0}}\) by Eq. (61) yields the following two-step scheme for \(\partial {\hat{{{\mathbf {a}}}}} ({{\mathbf {w}}})/\partial {{\mathbf {w}}}\),
and
where \({{\mathbf {k}}}({{\mathbf {d}}}; {{\mathbf {w}}})\) represents the tangent stiffness matrix
Equations (64) and (65) were previously developed by Schmidt et al. [19] and show that, at an equilibrium solution \({{\mathbf {d}}}^\star ({{\mathbf {w}}})\), the evaluation of the sensitivities \(\partial {\hat{{{\mathbf {a}}}}} ({{\mathbf {w}}})/\partial {{\mathbf {w}}}\) requires the solution of n additional linear systems. Furthermore, the sensitivities of the discrete internal force vector \(\partial {{\mathbf {f}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}})/\partial {{\mathbf {w}}}\) appear here; we compute these analytically.
Rights and permissions
About this article
Cite this article
Sewerin, F. On the local identifiability of constituent stress–strain laws for hyperelastic composite materials. Comput Mech 65, 853–876 (2020). https://doi.org/10.1007/s00466-019-01798-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-019-01798-w