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Salehabadi M, Nammari L, Luna A, Crutison J, Klatt D, Royston TJ. Quantifying uniaxial prestress and waveguide effects on dynamic elastography estimates for a cylindrical rod. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2023; 154:3580-3594. [PMID: 38038614 PMCID: PMC10693442 DOI: 10.1121/10.0022581] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2023] [Revised: 10/18/2023] [Accepted: 11/14/2023] [Indexed: 12/02/2023]
Abstract
Dynamic elastography attempts to reconstruct quantitative maps of the viscoelastic properties of materials by noninvasively measuring mechanical wave motion in them. The target motion is typically transversely-polarized relative to the wave propagation direction, such as bulk shear wave motion. In addition to neglecting waveguide effects caused by small lengths in one dimension or more, many reconstruction strategies also ignore nonzero, non-isotropic static preloads. Significant anisotropic prestress is inherent to the functional role of some biological materials of interest, which also are small in size relative to shear wavelengths in one or more dimensions. A cylindrically shaped polymer structure with isotropic material properties is statically elongated along its axis while its response to circumferentially-, axially-, and radially-polarized vibratory excitation is measured using optical or magnetic resonance elastography. Computational finite element simulations augment and aid in the interpretation of experimental measurements. We examine the interplay between uniaxial prestress and waveguide effects. A coordinate transformation approach previously used to simplify the reconstruction of un-prestressed transversely isotropic material properties based on elastography measurements is adapted with partial success to estimate material viscoelastic properties and prestress conditions without requiring advanced knowledge of either.
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Affiliation(s)
- Melika Salehabadi
- UIC Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA
| | - Lara Nammari
- UIC Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA
| | - Aime Luna
- UIC Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA
| | - Joseph Crutison
- UIC Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA
| | - Dieter Klatt
- UIC Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA
| | - Thomas J Royston
- UIC Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA
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2
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Pagé G, Bied M, Garteiser P, Van Beers B, Etaix N, Fraschini C, Bel-Brunon A, Gennisson JL. Comparison of ultrasound elastography, magnetic resonance elastography and finite element model to quantify nonlinear shear modulus. Phys Med Biol 2023; 68:205003. [PMID: 37703895 DOI: 10.1088/1361-6560/acf98c] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2023] [Accepted: 09/13/2023] [Indexed: 09/15/2023]
Abstract
Objective. The aim of this study is to validate the estimation of the nonlinear shear modulus (A) from the acoustoelasticity theory with two experimental methods, ultrasound (US) elastography and magnetic resonance elastography (MRE), and a finite element method.Approach. Experiments were performed on agar (2%)-gelatin (8%) phantom considered as homogeneous, elastic and isotropic. Two specific setups were built to ensure a uniaxial stress step by step on the phantom, one for US and a nonmagnetic version for MRE. The stress was controlled identically in both imaging techniques, with a water tank placed on the top of the phantom and filled with increasing masses of water during the experiment. In US, the supersonic shear wave elastography was implemented on an ultrafast US device, driving a 6 MHz linear array to measure shear wave speed. In MRE, a gradient-echo sequence was used in which the three spatial directions of a 40 Hz continuous wave displacement generated with an external driver were encoded successively. Numerically, a finite element method was developed to simulate the propagation of the shear wave in a uniaxially stressed soft medium.Main results. Similar shear moduli were estimated at zero stress using experimental methods,μ0US= 12.3 ± 0.3 kPa andμ0MRE= 11.5 ± 0.7 kPa. Numerical simulations were set with a shear modulus of 12 kPa and the resulting nonlinear shear modulus was found to be -58.1 ± 0.7 kPa. A very good agreement between the finite element model and the experimental models (AUS= -58.9 ± 9.9 kPa andAMRE= -52.8 ± 6.5 kPa) was obtained.Significance. These results show the validity of such nonlinear shear modulus measurement quantification in shear wave elastography. This work paves the way to develop nonlinear elastography technique to get a new biomarker for medical diagnosis.
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Affiliation(s)
- Gwenaël Pagé
- BioMaps, Laboratoire d'Imagerie Biomédicale Multimodale, Université Paris-Saclay, CEA, CNRS UMR 9011, Inserm UMR 1281, Service hospitalier Frédéric Joliot, F-91401 Orsay, France
| | - Marion Bied
- BioMaps, Laboratoire d'Imagerie Biomédicale Multimodale, Université Paris-Saclay, CEA, CNRS UMR 9011, Inserm UMR 1281, Service hospitalier Frédéric Joliot, F-91401 Orsay, France
| | - Philippe Garteiser
- Laboratory of imaging biomarkers, Center for Research on inflammation, UMR 1149, Université Paris-Cité, Inserm, F-75018 Paris, France
| | - Bernard Van Beers
- Laboratory of imaging biomarkers, Center for Research on inflammation, UMR 1149, Université Paris-Cité, Inserm, F-75018 Paris, France
- Department of Radiology, Beaujon university hospital Paris Nord, AP-HP, F-92110 Clichy, France
| | - Nicolas Etaix
- Hologic - Supersonic Imagine, F-13290 Aix en Provence, France
| | | | - Aline Bel-Brunon
- Univ Lyon, INSA Lyon, CNRS, LaMCoS, UMR5259, F-69621 Villeurbanne, France
| | - Jean-Luc Gennisson
- BioMaps, Laboratoire d'Imagerie Biomédicale Multimodale, Université Paris-Saclay, CEA, CNRS UMR 9011, Inserm UMR 1281, Service hospitalier Frédéric Joliot, F-91401 Orsay, France
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3
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Salehabadi M, Crutison J, Klatt D, Royston TJ. Decoupling Uniaxial Tensile Prestress and Waveguide Effects From Estimates of the Complex Shear Modulus in a Cylindrical Structure Using Transverse-Polarized Dynamic Elastography. JOURNAL OF ENGINEERING AND SCIENCE IN MEDICAL DIAGNOSTICS AND THERAPY 2023; 6:021003. [PMID: 36589925 PMCID: PMC9793439 DOI: 10.1115/1.4056411] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2022] [Revised: 11/26/2022] [Indexed: 12/12/2022]
Abstract
Dynamic elastography, whether based on magnetic resonance, ultrasound, or optical modalities, attempts to reconstruct quantitative maps of the viscoelastic properties of biological tissue, properties altered by disease and injury, by noninvasively measuring mechanical wave motion in the tissue. Most reconstruction strategies that have been developed neglect boundary conditions, including quasi-static tensile or compressive loading resulting in a nonzero prestress. Significant prestress is inherent to the functional role of some biological tissues currently being studied using elastography, such as skeletal and cardiac muscle, arterial walls, and the cornea. In the present article a configuration, inspired by muscle elastography but generalizable to other applications, is analytically and experimentally studied. A hyperelastic polymer phantom cylinder is statically elongated in the axial direction while its response to transverse-polarized vibratory excitation is measured. We examine the interplay between uniaxial prestress and waveguide effects in this muscle-like tissue phantom using computational finite element simulations and magnetic resonance elastography measurements. Finite deformations caused by prestress coupled with waveguide effects lead to results that are predicted by a coordinate transformation approach that has been previously used to simplify reconstruction of anisotropic properties using elastography. Here, the approach estimates material viscoelastic properties that are independent of the nonhomogeneous prestress conditions without requiring advanced knowledge of those stress conditions.
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Affiliation(s)
- Melika Salehabadi
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, IL 60607
| | - Joseph Crutison
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, IL 60607
| | - Dieter Klatt
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, IL 60607
| | - Thomas J. Royston
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, IL 60607,Corresponding author. e-mail:
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4
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Dore M, Luna A, Royston TJ. Biaxial Tensile Prestress and Waveguide Effects on Estimates of the Complex Shear Modulus Using Optical-Based Dynamic Elastography in Plate-Like Soft Tissue Phantoms. JOURNAL OF ENGINEERING AND SCIENCE IN MEDICAL DIAGNOSTICS AND THERAPY 2023; 6:011006. [PMID: 36590822 PMCID: PMC9793440 DOI: 10.1115/1.4056103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2022] [Revised: 10/23/2022] [Indexed: 11/24/2022]
Abstract
Dynamic elastography attempts to reconstruct quantitative maps of the viscoelastic properties of biological tissue, properties altered by disease and injury, by noninvasively measuring mechanical wave motion in the tissue. Most reconstruction strategies that have been developed neglect boundary conditions, including quasi-static tensile or compressive loading resulting in a nonzero prestress. Significant prestress is inherent to the functional role of some biological tissues, such as skeletal and cardiac muscle, arterial walls, and the cornea. In the present article a novel configuration, inspired by corneal elastography but generalizable to other applications, is studied. A polymer phantom layer is statically elongated via an in-plane biaxial normal stress while the phantom's response to transverse vibratory excitation is measured. We examine the interplay between biaxial prestress and waveguide effects in this plate-like tissue phantom. Finite static deformations caused by prestressing coupled with waveguide effects lead to results that are predicted by a novel coordinate transformation approach previously used to simplify reconstruction of anisotropic properties. Here, the approach estimates material viscoelastic properties independent of the nonzero prestress conditions without requiring advanced knowledge of those stress conditions.
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Affiliation(s)
- Marta Dore
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, IL 60607
| | - Aime Luna
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, IL 60607
| | - Thomas J. Royston
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, IL 60607,Corresponding author. e-mail:
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5
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Saccomandi G, Vergori L, Zanetti EM. Linear, weakly nonlinear and fully nonlinear models for soft tissues: which ones provide the most reliable estimations of the stiffness? PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20210321. [PMID: 36031840 DOI: 10.1098/rsta.2021.0321] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2022] [Accepted: 04/06/2022] [Indexed: 06/15/2023]
Abstract
Benign and malignant lesions in tissues or organs can be detected by elastographic investigations in which pathological regions are spotted from local alterations of the stiffness. As is known, the shear modulus provides a measure of the stiffness of an elastic material. Based on the classical theory of linear elasticity, an elastogram yields estimations of the linear shear modulus from measurements of the speed of small-amplitude transverse waves propagating in the medium tested. In this paper, we show that the estimation of the shear modulus can be improved significantly by employing the fourth-order weakly nonlinear theory of elasticity (FOE), and indicate how the stiffness can be assessed more precisely with the use of FOE. We discuss also why FOE provides more reliable results than the fully nonlinear theory of elasticity. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.
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Affiliation(s)
- G Saccomandi
- Dipartimento di Ingegneria, Università degli Studi di Perugia,06125 Perugia, Italy
| | - L Vergori
- Dipartimento di Ingegneria, Università degli Studi di Perugia,06125 Perugia, Italy
| | - E M Zanetti
- Dipartimento di Ingegneria, Università degli Studi di Perugia,06125 Perugia, Italy
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6
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Mukherjee S, Destrade M, Gower AL. Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0255] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022] Open
Abstract
Real-world solids, such as rocks, soft tissues and engineering materials, are often under some form of stress. Most real materials are also, to some degree, anisotropic due to their microstructure, a characteristic often called the ‘texture anisotropy’. This anisotropy can stem from preferential grain alignment in polycrystalline materials, aligned micro-cracks or structural reinforcement, such as collagen bundles in biological tissues, steel rods in pre-stressed concrete and reinforcing fibres in composites. Here, we establish a framework for initially stressed solids with transverse texture anisotropy. We consider that the strain energy per unit mass of the reference is an explicit function of the elastic deformation gradient, the initial stress tensor and the texture anisotropy. We determine the corresponding constitutive relations and develop examples of nonlinear strain energies that depend explicitly on the initial stress and direction of texture anisotropy. As an application, we then employ these models to analyse the stress distribution of an inflated initially stressed cylinder with texture anisotropy and the tension of a welded metal plate. We also deduce the elastic moduli needed to describe linear elasticity from stress reference with transverse texture anisotropy. As an example, we show how to measure the stress with small-amplitude shear waves.
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Affiliation(s)
- Soumya Mukherjee
- Department of Mechanical Engineering, National Institute of Technology Jamshedpur, Jamshedpur, Jharkhand 831014, India
| | - Michel Destrade
- School of Mathematical and Statistical Sciences, University of Galway, Galway, Ireland
- Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, People’s Republic of China
| | - Artur L. Gower
- Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK
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7
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Pivot shift and Lachman test simulation-based exploration in juvenile populations for accurately predicting anterior tibial translation. J Biomech 2022; 136:111069. [DOI: 10.1016/j.jbiomech.2022.111069] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2022] [Revised: 03/08/2022] [Accepted: 03/24/2022] [Indexed: 11/20/2022]
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8
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Crutison J, Sun M, Royston TJ. The combined importance of finite dimensions, anisotropy, and pre-stress in acoustoelastography. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2022; 151:2403. [PMID: 35461517 PMCID: PMC8993425 DOI: 10.1121/10.0010110] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/20/2021] [Revised: 02/21/2022] [Accepted: 03/18/2022] [Indexed: 06/14/2023]
Abstract
Dynamic elastography, whether based on magnetic resonance, ultrasound, or optical modalities, attempts to reconstruct quantitative maps of the viscoelastic properties of biological tissue, properties that are altered by disease and injury, by noninvasively measuring mechanical wave motion in the tissue. Most reconstruction strategies that have been developed neglect boundary conditions, including quasistatic tensile or compressive loading resulting in a nonzero prestress. Significant prestress is inherent to the functional role of some biological tissues currently being studied using elastography, such as skeletal and cardiac muscle, arterial walls, and the cornea. In the present article, we review how prestress alters both bulk mechanical wave motion and wave motion in one- and two-dimensional waveguides. Key findings are linked to studies on skeletal muscle and the human cornea, as one- and two-dimensional waveguide examples. This study highlights the underappreciated combined acoustoelastic and waveguide challenge to elastography. Can elastography truly determine viscoelastic properties of a material when what it is measuring is affected by both these material properties and unknown prestress and other boundary conditions?
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Affiliation(s)
- Joseph Crutison
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, Illinois 60607, USA
| | - Michael Sun
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, Illinois 60607, USA
| | - Thomas J Royston
- Richard and Loan Hill Department of Biomedical Engineering, University of Illinois Chicago, 851 South Morgan Street, MC 063, Chicago, Illinois 60607, USA
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9
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Sampath S, Sohn H. Cubic nonlinearity parameter measurement and material degradation detection using nonlinear ultrasonic three-wave mixing. ULTRASONICS 2022; 121:106670. [PMID: 35026609 DOI: 10.1016/j.ultras.2021.106670] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2021] [Revised: 12/20/2021] [Accepted: 12/21/2021] [Indexed: 06/14/2023]
Abstract
Early detection of accumulated damage or material degradation in structures is important to ensure their structural safety. Nonlinear ultrasonic techniques are widely used to measure the quadratic nonlinearity parameter that represents the third-order elastic constants of materials for material degradation detection. In addition, there are ongoing efforts to exploit both the third- and fourth-order elastic constants that describe the cubic nonlinearity parameter to detect material degradation. This study develops a nonlinear ultrasonic three-wave mixing technique to generate and measure third-order combined harmonics (TOCH) in plate-like structures and to measure cubic nonlinearity parameter. The proposed three-wave mixing technique generates three primary Lamb waves in the target structure and measures the TOCH produced by nonlinear cross-interaction of the primary Lamb waves. A theoretical model is developed to describe the generation of TOCH in a nonlinear elastic and homogeneous plate, and the effectiveness of the theoretical model is validated experimentally. Measurements of the TOCH are conducted for intact and degraded aluminum specimens with different degradation levels. Because inherent material nonlinearity and material degradation alter the third- and fourth-order elastic constants of a structure, the three-wave mixing technique for measuring the TOCH can be used to identify the inherent material nonlinearity and material degradation. In particular, the experimental results indicate that the proposed technique is more sensitive to early-stage material degradation than existing nonlinear ultrasonic techniques such as two-wave mixing, and third harmonic generation techniques.
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Affiliation(s)
- Santhakumar Sampath
- Department of Civil & Environmental Engineering, KAIST, 291 Daehakro, Yuseong-gu, Daejeon 34141, Republic of Korea
| | - Hoon Sohn
- Department of Civil & Environmental Engineering, KAIST, 291 Daehakro, Yuseong-gu, Daejeon 34141, Republic of Korea.
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10
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Kube CM, Roy A, Jensen DS, Branch DW. A unifying model of weakly nonlinear elastic waves; large on large theory. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2022; 151:1294. [PMID: 35232066 DOI: 10.1121/10.0009376] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2021] [Accepted: 01/09/2022] [Indexed: 06/14/2023]
Abstract
This article reconsiders traditional topics in nonlinear elastic waves and nonlinear ultrasonics. Herein, higher-order coupling between finite initial deformation and finite amplitude waves are considered. To allow for coupling, a large-on-large deformation model is developed and used to generate the equations of motion relative to the deformed and undeformed material configurations. Thus, the equations of motion provide a single setting to describe topics in nonlinear elastic waves such as acoustoelasticity, second harmonic generation, and coupling relations between these topics. The model is evaluated to recover the traditional linearized acoustoelastic relations and predicted second harmonic amplitudes. Then, the so-called large acoustoelasticity theory is developed for anisotropic materials with specific results given for isotropic materials. Last, the stress influence on second harmonic generation is presented.
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Affiliation(s)
- Christopher M Kube
- Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Anubhav Roy
- Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Daniel S Jensen
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - Darren W Branch
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
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11
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Bied M, Gennisson JL. Acoustoelasticity in transversely isotropic soft tissues: Quantification of muscle nonlinear elasticity. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2021; 150:4489. [PMID: 34972304 DOI: 10.1121/10.0008976] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2021] [Accepted: 11/19/2021] [Indexed: 06/14/2023]
Abstract
Recent developments in the field of elastography aim at developing the quantification of new mechanical properties of tissues, that are complementary to the shear modulus, which is characteristic of the linear elastic properties of a quasi-incompressible medium. In this context, measurement of the elastic nonlinearity of tissues was recently proposed based on acoustoelasticity. Up to now, most of the experimental applications of acoustoelasticity theory using Landau formalism in human tissues have assumed isotropy. However, this strong hypothesis does not hold in all human tissues, such as muscles that are generally considered as transversely isotropic (TI). In this work, after reviewing the constraints imposed by TI symmetry on the linear and nonlinear elastic properties of TI media, the acoustoelasticity theory in TI incompressible media is developed and implemented experimentally on a TI polyvinyl alcohol phantom and on ex vivo muscular tissues. Based on this theory and on the evolutions of the shear wave speed, with respect to uniaxial static stress, the nonlinear elastic parameter A is experimentally quantified. The estimations of A in ex vivo bovine and porcine muscles are on the order of hundreds of kPa. This work paves the way for more thorough muscle mechanical properties characterization as well as for the development of a potential new biomarker.
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Affiliation(s)
- Marion Bied
- BioMaps, Laboratoire d'Imagerie Biomédicale Multimodale à Paris-Saclay, Université Paris-Saclay, CEA, CNRS UMR 9011, INSERM UMR 1281, 4 Place du général Leclerc, 91401, Orsay, France
| | - Jean-Luc Gennisson
- BioMaps, Laboratoire d'Imagerie Biomédicale Multimodale à Paris-Saclay, Université Paris-Saclay, CEA, CNRS UMR 9011, INSERM UMR 1281, 4 Place du général Leclerc, 91401, Orsay, France
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12
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Callejas A, Melchor J, Faris IH, Rus G. Hyperelastic Ex Vivo Cervical Tissue Mechanical Characterization. SENSORS (BASEL, SWITZERLAND) 2020; 20:E4362. [PMID: 32764345 PMCID: PMC7472274 DOI: 10.3390/s20164362] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/20/2020] [Revised: 07/19/2020] [Accepted: 08/03/2020] [Indexed: 12/18/2022]
Abstract
This paper presents the results of the comparison between a proposed Fourth Order Elastic Constants (FOECs) nonlinear model defined in the sense of Landau's theory, and the two most contrasted hyperelastic models in the literature, Mooney-Rivlin, and Ogden models. A mechanical testing protocol is developed to investigate the large-strain response of ex vivo cervical tissue samples in uniaxial tension in its two principal anatomical locations, the epithelial and connective layers. The final aim of this work is to compare the reconstructed shear modulus of the epithelial and connective layers of cervical tissue. According to the obtained results, the nonlinear parameter A from the proposed FOEC model could be an important biomarker in cervical tissue diagnosis. In addition, the calculated shear modulus depended on the anatomical location of the cervical tissue (μepithelial = 1.29 ± 0.15 MPa, and μconnective = 3.60 ± 0.63 MPa).
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Affiliation(s)
- Antonio Callejas
- Department of Structural Mechanics, University of Granada, 18010 Granada, Spain; (I.H.F.); (G.R.)
- Instituto de Investigación Biosanitaria, ibs.GRANADA, 18012 Granada, Spain;
| | - Juan Melchor
- Instituto de Investigación Biosanitaria, ibs.GRANADA, 18012 Granada, Spain;
- Excellence Research Unit, “Modelling Nature” (MNat), University of Granada, 18010 Granada, Spain
- Department of Statistics and Operations Research, University of Granada, 18010 Granada, Spain
| | - Inas H. Faris
- Department of Structural Mechanics, University of Granada, 18010 Granada, Spain; (I.H.F.); (G.R.)
- Instituto de Investigación Biosanitaria, ibs.GRANADA, 18012 Granada, Spain;
| | - Guillermo Rus
- Department of Structural Mechanics, University of Granada, 18010 Granada, Spain; (I.H.F.); (G.R.)
- Instituto de Investigación Biosanitaria, ibs.GRANADA, 18012 Granada, Spain;
- Excellence Research Unit, “Modelling Nature” (MNat), University of Granada, 18010 Granada, Spain
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13
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Ng KM, Masurkar F, Tse PW, Yelve NP. Design of a new optical system to generate narrowband guided waves with an application for evaluating the health status of rail material. OPTICS LETTERS 2019; 44:5695-5698. [PMID: 31774756 DOI: 10.1364/ol.44.005695] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Accepted: 10/24/2019] [Indexed: 06/10/2023]
Abstract
The Letter presents a new design of Sagnac interferometer-based optical system (SIOS) that emits a line-arrayed pattern generating narrowband high-energy waves in a specimen. The SIOS is further used to excite Rayleigh waves in a pristine rail specimen to evaluate its intrinsic nonlinearity resulting from the lattice anharmonicity and dissolved impurities. Such a nonlinearity appears in the response in the form of a second harmonic that is sensed in this Letter using a scanning laser Doppler vibrometer. In addition to this noncontact measurement, a contact measurement of the nonlinearity of rail steel using wedge transducers is also carried out to compare the performance of the SIOS. Both experimentally evaluated nonlinearities are compared with those obtained using the nonlinear elasticity equations. The close agreement with the theoretically estimated nonlinearity and higher repeatability shows that the SIOS is effective in measuring the intrinsic nonlinearity of the rail steel and, thereby, predicting the health status of the rail specimens before fixing them on the track.
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14
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Destrade M, Pucci E, Saccomandi G. Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids. Proc Math Phys Eng Sci 2019; 475:20190061. [PMID: 31423090 DOI: 10.1098/rspa.2019.0061] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2019] [Accepted: 05/29/2019] [Indexed: 11/12/2022] Open
Abstract
We study elastic shear waves of small but finite amplitude, composed of an anti-plane shear motion and a general in-plane motion. We use a multiple scales expansion to derive an asymptotic system of coupled nonlinear equations describing their propagation in all isotropic incompressible nonlinear elastic solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear elasticity. We show that for a general isotropic incompressible solid, the coupling between anti-plane and in-plane motions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear two-dimensional shear waves. We then use the equations to study the evolution of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly polarized) shear beam source generates only odd harmonics, but that introducing a slight in-plane noise in the source signal leads to a second harmonic, of the same magnitude as the fifth harmonic, a phenomenon recently observed experimentally. Finally, we present examples of some special shear motions with linear polarization.
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Affiliation(s)
- Michel Destrade
- School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road Galway, Republic of Ireland
| | - Edvige Pucci
- Dipartimento di Ingegneria, Università degli studi di Perugia, 06125 Perugia, Italy
| | - Giuseppe Saccomandi
- School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road Galway, Republic of Ireland.,Dipartimento di Ingegneria, Università degli studi di Perugia, 06125 Perugia, Italy
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15
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Mohabuth M, Kotousov A, Ng CT. Large acoustoelastic effect for Lamb waves propagating in an incompressible elastic plate. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2019; 145:1221. [PMID: 31067922 DOI: 10.1121/1.5092604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2018] [Accepted: 02/11/2019] [Indexed: 06/09/2023]
Abstract
In this paper, the effect of a large pre-stress on the propagation of small amplitude Lamb waves in an incompressible elastic plate is investigated. Using the theory of incremental elasticity, the dispersion equations, which give the phase velocity of the symmetric and anti-symmetric wave modes as a function of the wavenumber, plate thickness, and pre-stress state, are derived for a general strain energy function. By considering the fourth-order strain energy function of incompressible isotropic elasticity, the correction to the phase velocity due to the pre-stress is obtained implicitly to the second order in the pre-strain/stress, and depends on the second, third, and fourth-order elastic constants. Numerical results are presented to show the dependence of the phase velocity of the Lamb wave modes upon the applied stress. These are compared to the first-order correction, and agree well with the limiting and asymptotic values obtained previously. It is envisaged that the present results may well find important practical applications in various guided wave based ultrasonic techniques utilising gels and rubber-like materials.
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Affiliation(s)
- Munawwar Mohabuth
- School of Mechanical Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
| | - Andrei Kotousov
- School of Mechanical Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
| | - Ching-Tai Ng
- School of Civil, Environmental and Mining Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
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16
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Radecki R, Su Z, Cheng L, Packo P, Staszewski WJ. Modelling nonlinearity of guided ultrasonic waves in fatigued materials using a nonlinear local interaction simulation approach and a spring model. ULTRASONICS 2018; 84:272-289. [PMID: 29179158 DOI: 10.1016/j.ultras.2017.11.008] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2017] [Revised: 09/29/2017] [Accepted: 11/13/2017] [Indexed: 05/27/2023]
Abstract
Modelling and numerical simulation - based on the framework of the Local Interaction Simulation Approach - was developed to have more insight into nonlinear attributes of guided ultrasonic waves propagating in fatigued metallic materials. Various sources of nonlinearity were considered in this modelling work, with particular emphases on higher-order harmonic generation and accumulation of nonlinearity along wave propagation. The material hyper-elasticity was considered in the model using an energy density approach based on the Landau-Lifshitz formulation; and the "breathing" motion pattern of a fatigue crack in the material was interrogated using a spring model. Upon the successful validation with the model prepared in the commercial software based on the Finite Element Methods, two scenarios were comparatively investigated, i.e. the lower and higher frequency regime. In the first case propagation of a basic symmetric mode pair was simulated using the model to observe a cumulative characteristic of the second harmonic mode with nonlinear hyper-elastic material definition upon appropriate selection of excitation frequency. In the second case, the higher-order symmetric mode pair was excited according to the "internal resonance" conditions, revealing a strong dependence of manifested nonlinearity on numerical parameters. Moreover, it was shown that with the use of the wave from the low frequency regime it was easier to differentiate later stages of the crack development, being in contrary to waves in the high frequency regime, which allowed to clearly observe early stages of the crack expansion. Such outcome lays the foundation to develop the damage detection and monitoring scheme in the field of Structural Health Monitoring based on utilising the nonlinear features of guided ultrasonic waves.
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Affiliation(s)
- Rafal Radecki
- Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Special Administrative Region; Department of Robotics and Mechatronics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland.
| | - Zhongqing Su
- Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Special Administrative Region
| | - Li Cheng
- Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Special Administrative Region
| | - Pawel Packo
- Department of Robotics and Mechatronics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
| | - Wieslaw J Staszewski
- Department of Robotics and Mechatronics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
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17
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Ye W, Bel-Brunon A, Catheline S, Combescure A, Rochette M. Simulation of nonlinear transient elastography: finite element model for the propagation of shear waves in homogeneous soft tissues. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2018; 34:e2901. [PMID: 28548237 DOI: 10.1002/cnm.2901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2016] [Revised: 05/09/2017] [Accepted: 05/22/2017] [Indexed: 06/07/2023]
Abstract
In this study, visco-hyperelastic Landau's model, which is widely used in acoustical physic field, is introduced into a finite element formulation. It is designed to model the nonlinear behaviour of finite amplitude shear waves in soft solids, typically, in biological tissues. This law is used in finite element models based on elastography, experiments reported in Jacob et al, the simulations results show a good agreement with the experimental study: It is observed in both that a plane shear wave generates only odd harmonics and a nonplane wave generates both odd and even harmonics in the spectral domain. In the second part, a parametric study is performed to analyse the influence of different factors on the generation of odd harmonics of plane wave. A quantitative relation is fitted between the odd harmonic amplitudes and the non-linear elastic parameter of Landau's model, which provides a practical guideline to identify the non-linearity of homogeneous tissues using elastography experiment.
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Affiliation(s)
- W Ye
- LaMCoS, INSA-Lyon, Université de Lyon, CNRS UMR 5259, Lyon, France
- ANSYS France, Montigny-le-Bretonneux, France
| | - A Bel-Brunon
- LaMCoS, INSA-Lyon, Université de Lyon, CNRS UMR 5259, Lyon, France
| | - S Catheline
- INSERM LabTAU Unit 1032, Université de Lyon, Lyon, France
| | - A Combescure
- LaMCoS, INSA-Lyon, Université de Lyon, CNRS UMR 5259, Lyon, France
| | - M Rochette
- ANSYS France, Montigny-le-Bretonneux, France
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18
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Li GY, Cao Y. Mechanics of ultrasound elastography. Proc Math Phys Eng Sci 2017; 473:20160841. [PMID: 28413350 PMCID: PMC5378248 DOI: 10.1098/rspa.2016.0841] [Citation(s) in RCA: 37] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2016] [Accepted: 01/23/2017] [Indexed: 12/19/2022] Open
Abstract
Ultrasound elastography enables in vivo measurement of the mechanical properties of living soft tissues in a non-destructive and non-invasive manner and has attracted considerable interest for clinical use in recent years. Continuum mechanics plays an essential role in understanding and improving ultrasound-based elastography methods and is the main focus of this review. In particular, the mechanics theories involved in both static and dynamic elastography methods are surveyed. They may help understand the challenges in and opportunities for the practical applications of various ultrasound elastography methods to characterize the linear elastic, viscoelastic, anisotropic elastic and hyperelastic properties of both bulk and thin-walled soft materials, especially the in vivo characterization of biological soft tissues.
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Affiliation(s)
- Guo-Yang Li
- Department of Engineering Mechanics, Institute of Biomechanics and Medical Engineering, AML, Tsinghua University, Beijing 100084, People's Republic of China
| | - Yanping Cao
- Department of Engineering Mechanics, Institute of Biomechanics and Medical Engineering, AML, Tsinghua University, Beijing 100084, People's Republic of China
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19
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Destrade M, Saccomandi G, Sgura I. Methodical fitting for mathematical models of rubber-like materials. Proc Math Phys Eng Sci 2017. [DOI: 10.1098/rspa.2016.0811] [Citation(s) in RCA: 68] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.
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Affiliation(s)
- Michel Destrade
- School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road, Galway, Ireland
| | - Giuseppe Saccomandi
- School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road, Galway, Ireland
- Dipartimento di Ingegneria, Università degli Studi di Perugia, Via G. Duranti, Perugia 06125, Italy
| | - Ivonne Sgura
- Dipartimento di Matematica e Fisica ‘E. De Giorgi’, Università del Salento, Via per Arnesano, 73100 Lecce, Italy
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20
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Giammarinaro B, Coulouvrat F, Pinton G. Numerical Simulation of Focused Shock Shear Waves in Soft Solids and a Two-Dimensional Nonlinear Homogeneous Model of the Brain. J Biomech Eng 2016; 138:041003. [PMID: 26833489 DOI: 10.1115/1.4032643] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2015] [Indexed: 12/22/2022]
Abstract
Shear waves that propagate in soft solids, such as the brain, are strongly nonlinear and can develop into shock waves in less than one wavelength. We hypothesize that these shear shock waves could be responsible for certain types of traumatic brain injuries (TBI) and that the spherical geometry of the skull bone could focus shear waves deep in the brain, generating diffuse axonal injuries. Theoretical models and numerical methods that describe nonlinear polarized shear waves in soft solids such as the brain are presented. They include the cubic nonlinearities that are characteristic of soft solids and the specific types of nonclassical attenuation and dispersion observed in soft tissues and the brain. The numerical methods are validated with analytical solutions, where possible, and with self-similar scaling laws where no known solutions exist. Initial conditions based on a human head X-ray microtomography (CT) were used to simulate focused shear shock waves in the brain. Three regimes are investigated with shock wave formation distances of 2.54 m, 0.018 m, and 0.0064 m. We demonstrate that under realistic loading scenarios, with nonlinear properties consistent with measurements in the brain, and when the shock wave propagation distance and focal distance coincide, nonlinear propagation can easily overcome attenuation to generate shear shocks deep inside the brain. Due to these effects, the accelerations in the focal are larger by a factor of 15 compared to acceleration at the skull surface. These results suggest that shock wave focusing could be responsible for diffuse axonal injuries.
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21
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Packo P, Uhl T, Staszewski WJ, Leamy MJ. Amplitude-dependent Lamb wave dispersion in nonlinear plates. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2016; 140:1319. [PMID: 27586758 DOI: 10.1121/1.4961489] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
The paper presents a perturbation approach for calculating amplitude-dependent Lamb wave dispersion in nonlinear plates. Nonlinear dispersion relationships are derived in closed form using a hyperelastic stress-strain constitutive relationship, the Green-Lagrange strain measure, and the partial wave technique integrated with a Lindstedt-Poincaré perturbation approach. Solvability conditions are derived using an operator formalism with inner product projections applied against solutions to the adjoint problem. When applied to the first- and second-order problems, these solvability conditions lead to amplitude-dependent, nonlinear dispersion corrections for frequency as a function of wavenumber. Numerical simulations verify the predicted dispersion shifts for an example nonlinear plate. The analysis and identification of amplitude-dependent, nonlinear Lamb wave dispersion complements recent research focusing on higher harmonic generation and internally resonant waves, which require precise dispersion relationships for frequency-wavenumber matching.
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Affiliation(s)
- Pawel Packo
- Department of Robotics and Mechatronics, AGH - University of Science and Technology, Al. A. Mickiewicza 30, 30-059 Krakow, Poland
| | - Tadeusz Uhl
- Department of Robotics and Mechatronics, AGH - University of Science and Technology, Al. A. Mickiewicza 30, 30-059 Krakow, Poland
| | - Wieslaw J Staszewski
- Department of Robotics and Mechatronics, AGH - University of Science and Technology, Al. A. Mickiewicza 30, 30-059 Krakow, Poland
| | - Michael J Leamy
- School of Mechanical Engineering, Georgia Institute of Technology, 771 Ferst Drive NW, Atlanta, Georgia, 30332-0405, USA
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22
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Mihai LA, Chin L, Janmey PA, Goriely A. A comparison of hyperelastic constitutive models applicable to brain and fat tissues. J R Soc Interface 2015; 12:0486. [PMID: 26354826 PMCID: PMC4614457 DOI: 10.1098/rsif.2015.0486] [Citation(s) in RCA: 107] [Impact Index Per Article: 11.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2015] [Accepted: 08/20/2015] [Indexed: 12/26/2022] Open
Abstract
In some soft biological structures such as brain and fat tissues, strong experimental evidence suggests that the shear modulus increases significantly under increasing compressive strain, but not under tensile strain, whereas the apparent Young's elastic modulus increases or remains almost constant when compressive strain increases. These tissues also exhibit a predominantly isotropic, incompressible behaviour. Our aim is to capture these seemingly contradictory mechanical behaviours, both qualitatively and quantitatively, within the framework of finite elasticity, by modelling a soft tissue as a homogeneous, isotropic, incompressible, hyperelastic material and comparing our results with available experimental data. Our analysis reveals that the Fung and Gent models, which are typically used to model soft tissues, are inadequate for the modelling of brain or fat under combined stretch and shear, and so are the classical neo-Hookean and Mooney-Rivlin models used for elastomers. However, a subclass of Ogden hyperelastic models are found to be in excellent agreement with the experiments. Our findings provide explicit models suitable for integration in large-scale finite-element computations.
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Affiliation(s)
- L Angela Mihai
- School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK
| | - LiKang Chin
- Institute for Medicine and Engineering, University of Pennsylvania, 3340 Smith Walk, Philadelphia, PA, USA
| | - Paul A Janmey
- Departments of Physiology and Physics and Astronomy, Institute for Medicine and Engineering, University of Pennsylvania, 3340 Smith Walk, Philadelphia, PA, USA
| | - Alain Goriely
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
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23
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Jiang Y, Li G, Qian LX, Liang S, Destrade M, Cao Y. Measuring the linear and nonlinear elastic properties of brain tissue with shear waves and inverse analysis. Biomech Model Mechanobiol 2015; 14:1119-28. [PMID: 25697960 DOI: 10.1007/s10237-015-0658-0] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2014] [Accepted: 02/06/2015] [Indexed: 10/24/2022]
Abstract
We use supersonic shear wave imaging (SSI) technique to measure not only the linear but also the nonlinear elastic properties of brain matter. Here, we tested six porcine brains ex vivo and measured the velocities of the plane shear waves induced by acoustic radiation force at different states of pre-deformation when the ultrasonic probe is pushed into the soft tissue. We relied on an inverse method based on the theory governing the propagation of small-amplitude acoustic waves in deformed solids to interpret the experimental data. We found that, depending on the subjects, the resulting initial shear modulus [Formula: see text] varies from 1.8 to 3.2 kPa, the stiffening parameter [Formula: see text] of the hyperelastic Demiray-Fung model from 0.13 to 0.73, and the third- [Formula: see text] and fourth-order [Formula: see text] constants of weakly nonlinear elasticity from [Formula: see text]1.3 to [Formula: see text]20.6 kPa and from 3.1 to 8.7 kPa, respectively. Paired [Formula: see text] test performed on the experimental results of the left and right lobes of the brain shows no significant difference. These values are in line with those reported in the literature on brain tissue, indicating that the SSI method, combined to the inverse analysis, is an efficient and powerful tool for the mechanical characterization of brain tissue, which is of great importance for computer simulation of traumatic brain injury and virtual neurosurgery.
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Affiliation(s)
- Yi Jiang
- Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, People's Republic of China
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24
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Whitford C, Studer H, Boote C, Meek KM, Elsheikh A. Biomechanical model of the human cornea: Considering shear stiffness and regional variation of collagen anisotropy and density. J Mech Behav Biomed Mater 2015; 42:76-87. [DOI: 10.1016/j.jmbbm.2014.11.006] [Citation(s) in RCA: 57] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2014] [Revised: 10/30/2014] [Accepted: 11/03/2014] [Indexed: 12/13/2022]
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25
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Chillara VK, Lissenden CJ. Nonlinear guided waves in plates: a numerical perspective. ULTRASONICS 2014; 54:1553-8. [PMID: 24792684 DOI: 10.1016/j.ultras.2014.04.009] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2014] [Accepted: 04/06/2014] [Indexed: 05/27/2023]
Abstract
Harmonic generation from non-cumulative fundamental symmetric (S0) and antisymmetric (A0) modes in plate is studied from a numerical standpoint. The contribution to harmonic generation from material nonlinearity is shown to be larger than that from geometric nonlinearity. Also, increasing the magnitude of the higher order elastic constants increases the amplitude of second harmonics. Second harmonic generation from non-phase-matched modes illustrates that group velocity matching is not a necessary condition for harmonic generation. Additionally, harmonic generation from primary mode is continuous and once generated, higher harmonics propagate independently. Lastly, the phenomenon of mode-interaction to generate sum and difference frequencies is demonstrated.
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Affiliation(s)
- Vamshi Krishna Chillara
- Department of Engineering Science and Mechanics, The Pennsylvania State University, 16802 PA, United States.
| | - Cliff J Lissenden
- Department of Engineering Science and Mechanics, The Pennsylvania State University, 16802 PA, United States
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26
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Destrade M, Ogden RW, Sgura I, Vergori L. Straightening: existence, uniqueness and stability. Proc Math Phys Eng Sci 2014; 470:20130709. [PMID: 24711723 PMCID: PMC3928959 DOI: 10.1098/rspa.2013.0709] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2013] [Accepted: 01/03/2014] [Indexed: 11/12/2022] Open
Abstract
One of the least studied universal deformations of incompressible nonlinear elasticity, namely the straightening of a sector of a circular cylinder into a rectangular block, is revisited here and, in particular, issues of existence and stability are addressed. Particular attention is paid to the system of forces required to sustain the large static deformation, including by the application of end couples. The influence of geometric parameters and constitutive models on the appearance of wrinkles on the compressed face of the block is also studied. Different numerical methods for solving the incremental stability problem are compared and it is found that the impedance matrix method, based on the resolution of a matrix Riccati differential equation, is the more precise.
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Affiliation(s)
- M. Destrade
- School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Republic of Ireland
- School of Mechanical and Materials Engineering, University College Dublin, Belfield, Dublin 4, Republic of Ireland
| | - R. W. Ogden
- School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK
| | - I. Sgura
- Dipartimento di Matematica e Fisica ‘Ennio De Giorgi’, Università del Salento, Lecce, Italy
| | - L. Vergori
- School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Republic of Ireland
- School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK
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27
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Destrade M, Saccomandi G, Vianello M. Proper formulation of viscous dissipation for nonlinear waves in solids. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2013; 133:1255-1259. [PMID: 23463998 DOI: 10.1121/1.4776178] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
To model nonlinear viscous dissipative motions in solids, acoustical physicists usually add terms linear in Ė, the material time derivative of the Lagrangian strain tensor E, to the elastic stress tensor σ derived from the expansion to the third (sometimes fourth) order of the strain energy density E=E(tr E,tr E(2),tr E(3)). Here it is shown that this practice, which has been widely used in the past three decades or so, is physically wrong for at least two reasons and that it should be corrected. One reason is that the elastic stress tensor σ is not symmetric while Ė is symmetric, so that motions for which σ+σ(T)≠0 will give rise to elastic stresses that have no viscous pendant. Another reason is that Ė is frame-invariant, while σ is not, so that an observer transformation would alter the elastic part of the total stress differently than it would alter the dissipative part, thereby violating the fundamental principle of material frame indifference. These problems can have serious consequences for nonlinear shear wave propagation in soft solids as seen here with an example of a kink in almost incompressible soft solids.
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Affiliation(s)
- Michel Destrade
- School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland.
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28
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Duenwald-Kuehl S, Lakes R, Vanderby R. Strain-induced damage reduces echo intensity changes in tendon during loading. J Biomech 2012; 45:1607-11. [PMID: 22542220 DOI: 10.1016/j.jbiomech.2012.04.004] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2011] [Revised: 03/30/2012] [Accepted: 04/04/2012] [Indexed: 12/21/2022]
Abstract
Tendon functionality is related to its mechanical properties. Tendon damage leads to a reduction in mechanical strength and altered biomechanical behavior, and therefore leads to compromised ability to carry out normal functions such as joint movement and stabilization. Damage can also accumulate in the tissue and lead to failure. A noninvasive method with which to measure such damage potentially could quantify structural compromise from tendon injury and track improvement over time. In this study, tendon mechanics are measured before and after damage is induced by "overstretch" (strain exceeding the elastic limit of the tissue) using a traditional mechanical test system while ultrasonic echo intensity (average gray scale brightness in a B-mode image) is recorded using clinical ultrasound. The diffuse damage caused by overstretch lowered the stress at a given strain in the tissue and decreased viscoelastic response. Overstretch also lowered echo intensity changes during stress relaxation and cyclic testing. As the input strain during overstretch increased, stress levels and echo intensity changes decreased. Also, viscoelastic parameters and time-dependent echo intensity changes were reduced.
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Affiliation(s)
- Sarah Duenwald-Kuehl
- Department of Orthopedics, University of Wisconsin-Madison, Madison, WI 53705, USA
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29
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Rotemberg V, Palmeri M, Nightingale R, Rouze N, Nightingale K. The impact of hepatic pressurization on liver shear wave speed estimates in constrained versus unconstrained conditions. Phys Med Biol 2011; 57:329-41. [PMID: 22170769 DOI: 10.1088/0031-9155/57/2/329] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
Increased hepatic venous pressure can be observed in patients with advanced liver disease and congestive heart failure. This elevated portal pressure also leads to variation in acoustic radiation-force-derived shear wave-based liver stiffness estimates. These changes in stiffness metrics with hepatic interstitial pressure may confound stiffness-based predictions of liver fibrosis stage. The underlying mechanism for this observed stiffening behavior with pressurization is not well understood and is not explained with commonly used linear elastic mechanical models. An experiment was designed to determine whether the stiffness increase exhibited with hepatic pressurization results from a strain-dependent hyperelastic behavior. Six excised canine livers were subjected to variations in interstitial pressure through cannulation of the portal vein and closure of the hepatic artery and hepatic vein under constrained conditions (in which the liver was not free to expand) and unconstrained conditions. Radiation-force-derived shear wave speed estimates were obtained and correlated with pressure. Estimates of hepatic shear stiffness increased with changes in interstitial pressure over a physiologically relevant range of pressures (0-35 mmHg) from 1.5 to 3.5 m s(-1). These increases were observed only under conditions in which the liver was free to expand while pressurized. This behavior is consistent with hyperelastic nonlinear material models that could be used in the future to explore methods for estimating hepatic interstitial pressure noninvasively.
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