We present an innovative method for characterizing the distributive elastic properties in nonlinear membranes. The method hinges on an inverse elastostatic approach of stress analysis that can compute the wall stress in a deformed convex membrane structure using assumed elastic models without knowing the realistic material parameters. This approach of stress analysis enables us to obtain the wall stress data independently of the material in question. The stress and strain data collected during a finite inflation motion are used to delineate the elastic property distribution in selected regions. In this paper, we discuss the theoretical and computational underpinnings of the method and demonstrate its feasibility using numerical simulations involving a saclike structure of known material property.
1.
Rivlin
, R. S.
, and Saunders
, D. W.
, 1951, “Large Elastic Deformations of Isotropic Materials. VII. Experiments on the Deformation of Rubber
,” Philos. Trans. R. Soc. London, Ser. A
0962-8428, 243A
, pp. 251
–288
.2.
Wineman
, A.
, Wilson
, D.
, and Melvin
, J. W.
, 1979, “Material Identification of Soft Tissue Using Membrane Inflation
,” J. Biomech.
0021-9290, 12
, pp. 841
–850
.3.
Bylski
, D. I.
, Kriewall
, T. J.
, Akkas
, N.
, and Melvin
, J. W.
, 1986, “Mechanical Behavior of Fetal Dura Mater Under Large Deformation Biaxial Tension
,” J. Biomech.
0021-9290, 19
, pp. 19
–26
.4.
Kriewall
, T. J.
, Akkas
, N.
, Bylski
, D. I.
, and Melvin
, J. W.
, 1993, “Mechanical Behavior of Fetal Dura Mater Under Large Axisymmetric Inflation
,” ASME J. Biomech. Eng.
0148-0731, 105
, pp. 71
–76
.5.
Hsu
, F. P. K.
, Schwab
, C.
, Rigamonti
, D.
, and Humphrey
, J. D.
, 1994, “Identification of Response Functions for Nonlinear Membranes Via Axisymmetric Inflation Tests: Implications for Biomechanics
,” Int. J. Solids Struct.
0020-7683, 31
, pp. 3375
–3386
.6.
Hsu
, F. P. K.
, Liu
, A. M. C.
, Downs
, J.
, Rigamonti
, D.
, and Humphrey
, J. D.
, 1995, “A Triplane Video-Based Experimental System for Studying Axisymmetrically Inflated Biomembranes
,” IEEE Trans. Biomed. Eng.
0018-9294, 42
(5
), pp. 442
–450
.7.
Storakers
, B.
, 1979, “An Explicit Method to Determine Response Coefficients in Finite Elasticity
,” J. Elast.
0374-3535, 9
, pp. 207
–214
.8.
Sacks
, M. S.
, 2000, “Biaxial Mechanical Evaluation of Planar Biological Materials
,” J. Elast.
0374-3535, 61
, pp. 199
–246
.9.
Lu
, J.
, Zhou
, X.
, and Raghavan
, M. L.
, 2008, “Inverse Method of Stress Analysis for Cerebral Aneurysms
,” Biomech. Model. Mechanobiol.
1617-7959, 7
, pp. 477
–486
.10.
Govindjee
, S.
, and Mihalic
, P. A.
, 1996, “Computational Methods for Inverse Finite Elastostatics
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 136
, pp. 47
–57
.11.
Govindjee
, S.
, and Mihalic
, P. A.
, 1998, “Computational Methods for Inverse Deformations in Quasi-Incompressible Finite Elasticity
,” Int. J. Numer. Methods Eng.
0029-5981, 43
, pp. 821
–838
.12.
Yamada
, T.
, 1998, “Finite Element Procedure of Initial Shape Determination for Hyperelasticity
,” Struct. Eng. Mech.
1225-4568, 6
(2
), pp. 173
–183
.13.
Lu
, J.
, Zhou
, X.
, and Raghavan
, M. L.
, 2007, “Inverse Elastostatic Stress Analysis in Pre-Deformed Biological Structures: Demonstration Using Abdominal Aortic Aneurysm
,” J. Biomech.
0021-9290, 40
, pp. 693
–696
.14.
Lu
, J.
, Zhou
, X.
, and Raghavan
, M. L.
, 2007, “Computational Method of Inverse Elastostatics for Anisotropic Hyperelastic Solids
,” Int. J. Numer. Methods Eng.
0029-5981, 69
, pp. 1239
–1261
.15.
Zhou
, X.
, and Lu
, J.
, 2008, “Inverse Formulation for Geometrically Exact Stress Resultant Shells
,” Int. J. Numer. Methods Eng.
0029-5981, 74
, pp. 1278
–1302
.16.
Green
, A. E.
, and Rivlin
, R. S.
, 1964, “On Cauchy’s Equation of Motion
,” Z. Angew. Math. Phys.
0044-2275, 15
, pp. 290
–292
.17.
Spencer
, A. J. M.
, 1970, “The Static Theory of Finite Elasticity
,” IMA J. Appl. Math.
0272-4960, 6
(2
), pp. 164
–200
.18.
Truesdell
, C.
, and Noll
, W.
, 1965, “The Non-Linear Field Theories of Mechanics
,” Handbuch der Physik
, Vol. Bd. 3/3, S.
Flügge
, ed., Springer-Verlag
, Berlin
.19.
Noll
, W.
, 1972, “A New Mathematical Theory of Simple Material
,” Arch. Ration. Mech. Anal.
0003-9527, 48
, pp. 1
–50
.20.
Epstein
, M.
, and Maugin
, G. A.
, 1990, “The Energy-Momentum Tensor and Material Uniformity in Finite Elasticity
,” Acta Mech.
0001-5970, 83
(3–4
), pp. 127
–133
.21.
Epstein
, M.
, and Maugin
, G. A.
, 1996, “On the Geometrical Material Structure of Anelasticity
,” Acta Mech.
0001-5970, 115
, pp. 119
–131
.22.
Maugin
, G. A.
, and Epstein
, M.
, 1998, “Geometrical Material Structure of Elastoplasticity
,” Int. J. Plast.
0749-6419, 14
, pp. 109
–115
.23.
Miehe
, C.
, 1998, “A Constitutive Frame of Elastoplasticity at Large Strains Based on the Notion of a Plastic Metric
,” Int. J. Solids Struct.
0020-7683, 35
(30
), pp. 3859
–3897
.24.
Lu
, J.
, and Papadopoulos
, P.
, 2004, “A Covariant Formulation of Anisotropic Finite Plasticity: Theoretical Development
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 193
, pp. 5335
–5358
.25.
Lu
, J.
, 2006, “A Covariant Condition and Related Constitutive Results in Finite Plasticity
,” ZAMP
0044-2275, 57
, pp. 313
–323
.26.
Rodriguez
, E. K.
, Hoger
, A.
, and Mcculloch
, A. D.
, 1994, “Stress-Dependent Finite Growth in Soft Elastic Tissues
,” J. Biomech.
0021-9290, 27
, pp. 455
–467
.27.
Taber
, L. A.
, 1995, “Biomechanics of Growth, Remodeling, and Morphogenesis
,” Appl. Mech. Rev.
0003-6900, 48
, pp. 487
–545
.28.
Johnson
, B. E.
, and Hoger
, A.
, 1995, “The Use of a Virtual Configuration in Formulating Constitutive Equations for Residually Stressed Elastic Materials
,” J. Elast.
0374-3535, 41
, pp. 177
–215
.29.
Hoger
, A.
, 1997, “Virtual Configurations and Constitutive Equations for Residually Stressed Bodies With Material Symmetry
,” J. Elast.
0374-3535, 48
, pp. 125
–144
.30.
Olsson
, T.
, Stålhand
, J.
, and Klarbring
, A.
, 2006, “Modeling Initial Strain Distribution in Soft Tissues With Application to Arteries
,” Biomech. Model. Mechanobiol.
1617-7959, 5
, pp. 27
–38
.31.
Boehler
, J. P.
, 1979, “A Simple Derivation of Representations for Non-Polynomial Constitutive Equations in Some Case of Anisotropy
,” Z. Angew. Math. Mech.
0044-2267, 59
, pp. 157
–167
.32.
Lu
, J.
, and Papadopoulos
, P.
, 2000, “A Covariant Constitutive Description for Anisotropic Non-Linear Elasticity
,” ZAMP
, 51
, pp. 204
–214
. 0044-227533.
Rossettos
, J. N.
, 1966, “Nonlinear Membrane Solutions for Symmetrically Loaded Deep Membranes of Revolution
,” NASA
Technical Report No. NASA TN D-3297.34.
Wu
, C. H.
, and Peng
, D. Y. P.
, 1972, “On the Asymptotically Spherical Deformation of Arbitrary Membranes of Revolution Fixed Along an Edge and Inflated by Large Pressures—A Nonlinear Boundary Layer Phenomenon
,” SIAM J. Appl. Math.
0036-1399, 23
, pp. 133
–152
.35.
Taylor
, R. L.
, 2003, FEAP User Manual, v7.5, Technical report, Department of Civil and Environmental Engineering, University of California, Berkeley.36.
Gill
, P. E.
, Murray
, W.
, and Saunders
, M. A.
, 2005, “SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization
,” SIAM Rev.
0036-1445, 47
, pp. 99
–131
.37.
Lanir
, Y.
, and Fung
, Y. C.
, 1974, “Two-Dimensional Mechanical Properties of Rabbit Skin—I. Experimental System
,” J. Biomech.
0021-9290, 7
, pp. 29
–34
.38.
Lanir
, Y.
, and Fung
, Y. C.
, 1974, “Two-Dimensional Mechanical Properties of Rabbit Skin—II. Experimental Results
,” J. Biomech.
0021-9290, 7
, pp. 171
–182
.39.
Sun
, W.
, Sacks
, M. S.
, and Scott
, M. J.
, 2005, “Effects of Boundary Conditions on the Estimation of the Planar Biaxial Mechanical Properties of Soft Tissues
,” ASME J. Biomech. Eng.
0148-0731, 127
, pp. 709
–715
.40.
Kavanagh
, K. T.
, and Clough
, R. W.
, 1971, “Finite Element Applications in the Characterization of Elastic Solids
,” Int. J. Solids Struct.
0020-7683, 7
, pp. 11
–23
.41.
Iding
, R. H.
, Pister
, K. S.
, and Taylor
, R. L.
, 1974, “Identification of Nonlinear Elastic Solids by a Finite Element Method
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 4
, pp. 121
–142
.42.
Kyriacou
, S. K.
, Shah
, A. D.
, and Humphrey
, J. D.
, 1997, “Inverse Finite Element Characterization of Nonlinear Hyperelastic Membrane
,” ASME J. Appl. Mech.
0021-8936, 64
, pp. 257
–262
.43.
Seshaiyer
, P.
, and Humphrey
, J. D.
, 2003, “A Sub-Domain Inverse Finite Element Characterization of Hyperelastic Membranes Including Soft Tissues
,” ASME J. Biomech. Eng.
0148-0731, 125
, pp. 363
–371
.44.
Beatty
, M. F.
, 1987, “A Class of Universal Relations in Isotropic Elasticity
,” J. Elast.
0374-3535, 17
, pp. 113
–121
.45.
Waldman
, S. D.
, and Lee
, J. M.
, 2002, “Boundary Conditions During Biaxial Testing of Planar Connective Tissues. Part 1: Dynamic Behavior
,” J. Mater. Sci. Mater. Med.
, 13
, pp. 933
–938
. 0957-453046.
Waldman
, S. D.
, Sacks
, M. S.
, and Lee
, J. M.
, 2002, “Boundary Conditions During Biaxial Testing of Planar Connective Tissues. Part II: Fiber Orientation
,” J. Mater. Sci. Lett.
0261-8028, 21
, pp. 1215
–1221
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