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Polarized Image Correlation for Large Deformation Fiber Kinematics
Abstract and Figures
The underlying fiber architecture of soft tissues, like bat wing skin, plays an important role in the material’s overall mechanical behavior. The mesoscopic birefringent fiber architecture of the bat wing skin can be visualized directly under polarized light. This inherent property is the key to the new experimental technique developed in this work: polarized image correlation (PIC). PIC is a technique for determining full field material strains and fiber kinematics of mesoscopically resolved fibrous tissues during biaxial mechanical testing. Not only is the material birefringence used to determine fiber kinematics under finite deformations, but it is also used for image correlation and strain field computation. Pure translation tests performed with PIC verify the accuracy of the technique. A segmental image processing method was developed to solve an experimental issue of changing birefringent properties to construct accurate continuous deformation profiles. By integrating PIC with traditional digital image correlation, both surface and subsurface data give additional insight into through thickness tissue behavior. The PIC technique is applicable to semi-transparent tissues with birefringent mesosopic structures; incorporation of microscopy would resolve smaller fiber structures. Future work includes extending the techniques to three dimensions to analyze curved surfaces.
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... It possesses embedded mesoscopic fiber architecture consisting of two families of approximately perpendicular long, straight fibers, with one family oriented predominantly in the chordwise direction, and the other predominantly spanwise [1]. Based on optical properties, the spanwise fibers are primarily composed of elastin, and have diameters of approximately 50-70 μm and spacing of 1000-2000 μm [13,14]. The chordwise fibers are muscles known as plagiopatagiales, with diameters and spacing of approximately 70-200 μm and 300-900 μm, respectively [13,14]; the structures observed in the wing comprise striated skeletal muscle fibers and collagenous tendon. ...
... Based on optical properties, the spanwise fibers are primarily composed of elastin, and have diameters of approximately 50-70 μm and spacing of 1000-2000 μm [13,14]. The chordwise fibers are muscles known as plagiopatagiales, with diameters and spacing of approximately 70-200 μm and 300-900 μm, respectively [13,14]; the structures observed in the wing comprise striated skeletal muscle fibers and collagenous tendon. Like most soft tissue, the skin is considered incompressible due to its high water content [15,16]. ...
... Although multi-axial testing is required to fully characterize anisotropic tissues from a theoretical point of view (the strain energy depends on at least three independent measures of deformation, for example invariants I 1 , I 4 , and I 6 of the right Cauchy-Green tensor), biaxial testing is typically employed when testing thin tissues [18]. In this work, finite deformation biaxial testing was performed with in situ digital image correlation (DIC) of the tissue surface and polarized image correlation (PIC), a hybrid technique recently developed by the authors that incorporates polarized filters to visualize and compute strains in the underlying mesoscopic fiber architecture during biaxial deformation [13]. ...
The highly flexible and stretchable wing skin of bats, together with the skeletal structure and musculature, enables large changes in wing shape during flight. Such compliance distinguishes bat wings from those of all other flying animals. Although several studies have investigated the aerodynamics and kinematics of bats, few have examined the complex histology and mechanical response of the wing skin. This work presents the first biaxial characterization of the local deformation, mechanical properties, and fiber kinematics of bat wing skin. Analysis of these data has provided insight into the relationships among the structural morphology, mechanical properties, and functionality of wing skin. Large spatial variations in tissue deformation and non-negligible fiber strains in the cross-fiber direction for both chordwise and spanwise fibers indicate fibers should be modeled as two-dimensional elements. The macroscopic constitutive behavior was anisotropic and nonlinear, with very low spanwise and chordwise stiffness (hundreds of kilopascals) in the toe region of the stress–strain curve. The structural arrangement of the fibers and matrix facilitates a low energy mechanism for wing deployment and extension, and we fabricate examples of skins capturing this mechanism. We propose a comprehensive deformation map for the entire loading regime. The results of this work underscore the importance of biaxial field approaches for soft heterogeneous tissue, and provide a foundation for development of bio-inspired skins to probe the effects of the wing skin properties on aerodynamic performance.
... 2,29−31 Recent studies have employed polarized light to enhance image contrast and measure strains via DIC in sensitive systems such as cellulose nanocrystal films 16 and a bat wing skin. 32 The remarkable roles played by phase contrast on cell biology and polarized light in the study of liquid crystals and minerals led us to explore their potentials as practical contrast enhancing methods to generate acceptable trackable patterns from polymer films to perform measurements via DIC. ...
... PL reveals textural features related to domain orientations, and thickness gradients present in the samples as a function of the angle between polarizers ( Figure S2 in Supporting Information for PL patterns obtained for ultrahigh molecular weight polyethylene, and cellulose nanocrystals). Hence, PL can be highly beneficial for regions exhibiting different refractive indices 16,32 or even tracking crystalline areas embedded within amorphous regions. For example, within the explored films, polarized light highlighted distinctive textures in PEI films, which were homogeneously distributed over the entire film (Figure 2d), as opposite to the plain responses obtained for PI or PEN films ( Figure S1). ...
... Such a characteristic becomes very important in materials exhibiting structural anisotropy ( Figure S2 in Supporting Information), as polarized light can contrast different regions based on orientations to estimate displacements. 16,32 ROIs in PI, PEI, and PEN films were cross-correlated using different subsets sizes (Tables S1−S3) with respect to the first image (i.e., image taken at 30°C). Subsets of 60 pixels provided balance between accuracy and computation times as subsets of 30 pixels decreased correlation times and accuracy of the results, and subsets of 90 pixels increased correlation times above 10 min per run with minimal improvement of image correlations. ...
Thermal expansion represents a vital indicator of the processing history and dimensional stability of materials. Solvent-sensitive, thin, and compliant samples are particularly challenging to test. Here we describe how textures highlighted by contrast enhanced optical microscopy modes (i.e., polarized light (PL), phase contrast (PC)) and bright field (BF) can be used to determine the thermal expansion of polymer films in a contact-free way using digital image correlation (DIC). Three different films were explored polyetherimide (PEI), polyimide (PI) and polyethylene naphthalate (PEN). Image textural features (e.g., intensity, size, speckle pattern characteristics) obtained by BF, PC and PL were analyzed by two-dimensional Fourier transform and autocorrelations. The measured in-plane CTEs of PEI, PI and PEN films, 51.8, 20.5, and 10.2 ppm/K, respectively, closely approached those previously reported using DIC with artificially applied speckle patterns.
In-situ microscopic methods can help researchers to analyse microstructural changes of materials structures under different conditions (e.g., temperature and pressure) at various length scales. Digital Image Correlation (DIC) combines image registration and tracking to enable accurate measurements of changes in materials in 2D and 3D. This review focuses on combining microscopy and DIC to study the properties of materials (including natural/synthetic biomaterials, biological samples and their composites) in academic, public and industry settings, including exciting examples of bioimaging.
Hydrogels are a class of soft, highly deformable materials formed by swelling a network of polymer chains in water. With mechanical properties that mimic biological materials, hydrogels are often proposed for load bearing biomedical or other applications in which their deformation and failure properties will be important. To study the failure of such materials a means for the measurement of deformation fields beyond simple uniaxial tension tests is required. As a non-contact, full-field deformation measurement method, Digital Image Correlation (DIC) is a good candidate for such studies. The application of DIC to hydrogels is studied here with the goal of establishing the accuracy of DIC when applied to hydrogels in the presence of large strains and large strain gradients. Experimental details such as how to form a durable speckle pattern on a material that is 90% water are discussed. DIC is used to measure the strain field in tension loaded samples containing a central hole, a circular edge notch and a sharp crack. Using a nonlinear, large deformation constitutive model, these experiments are modeled using the finite element method (FEM). Excellent agreement between FEM and DIC results for all three geometries shows that the DIC measurements are accurate up to strains of over 10, even in the presence of very high strain gradients near a crack tip. The method is then applied to verify a theoretical prediction that the deformation field in a cracked sample under relaxation loading, i.e. constant applied boundary displacement, is stationary in time even as the stress relaxes by a factor of three.
Unique among animal flyers, bats have highly flexible and stretchable thin wing membranes. The connection between the structural constituents of bat wing skin, its material behavior, and flight abilities is not yet known. In this work we propose a structurally motivated constitutive model for the wing skin. Within a continuum mechanics framework, the proposed strain energy function for the wing skin is the sum of contributions due to the matrix and two mesoscopic fiber families, one oriented primarily spanwise consisting of elastin fiber bundles and the other family oriented chordwise consisting of muscle fibers. While the fibers are flat and straight when the wing is somewhat open, the matrix exhibits corrugations due to compressive loading from the pre-stretched spanwise fibers. This mismatch in the natural configurations of components is accounted for in the model by a decomposition of the deformation gradient of the spanwise fibers. The material parameters are fit with a procedure motivated by the underlying deformation mechanisms of the tissue corresponding to the regions of the j-shaped constitutive curves. The proposed model is fit to the first set of biaxial experimental stress-strain data for bat wing skin and captures the general features of the tissue response well.
The highly flexible and extensible wing skin of bats enables various wing shapes and flight modes, which distinguishes it from all other natural flyers making bats an ideal model for micro-aerial vehicles. We propose that an understanding of the relationship between the structure, properties and function of the wing tissue is essential to replicate and utilize the bat’s natural capabilities. In this work, we present the first biaxial mechanical characterization of bat wing skin, identify key mechanisms in its deformation, and employ these concepts to fabricate biomimetic skins. Ten Glossophaga soricina bat specimens were available for experiment obtained from Prof. Swartz or Brown University. Of the 20 excised wing skin samples, 11 were used for establishing testing protocols, 3 tore during preparation, and 6 were tested for the characterization presented in this work. The tissue was shown to be nonlinear, heterogeneous, anisotropic, and viscoelastic. The wrinkled tissue structure and substantial anisotropy promote great spanwise deployment and deformation increasing wing area and aspect ratio enabling greater lift generation. Comparison of the material structural organization with strain field responses demonstrated that the underlying fiber architecture corresponds to observed local strain variations and that the tissue represents a departure from traditional fiber reinforced materials since the mesoscopic elastin fiber architecture appears to be the soft component while the matrix provides the stiffening role. Fabricated skins capture the inherent mismatch in natural configurations of the spanwise elastin fibers and the matrix and exhibit the characteristic wrinkle pattern observed in the in vivo bat wing skin. Future work will include static mechanical testing of the synthetic skins as well as aerodynamic testing to investigate the link between tissue structure, properties and functional flight capabilities.
The mechanical testing of anisotropic nonlinearly elastic solids is a topic of considerable and increasing interest. The results of such testing are important, in particular, for the characterization of the material properties and the development of constitutive laws that can be used for predictive purposes. However, the literature on this topic in the context of soft tissue biomechanics, in particular, includes some papers that are misleading since they contain errors and false statements. Claims that planar biaxial testing can fully characterize the three-dimensional anisotropic elastic properties of soft tissues are incorrect. There is therefore a need to clarify the extent to which biaxial testing can be used for determining the elastic properties of these materials. In this paper this is explained on the basis of the equations of finite deformation transversely isotropic elasticity, and general planar anisotropic elasticity. It is shown that it is theoretically impossible to fully characterize the properties of anisotropic elastic materials using such tests unless some assumption is made that enables a suitable subclass of models to be preselected. Moreover, it is shown that certain assumptions underlying the analysis of planar biaxial tests are inconsistent with the classical linear theory of orthotropic elasticity. Possible sets of independent tests required for full material characterization are then enumerated.
The skin of the bat wing in functionally unique among mammals: it serves as a major locomotor organ in addition to its protective and regulatory functions. We used tensile testing to investigate the mechanical capabilities of wing membrane skin, and compared stiffness, strength, load at failure, and energy absorption among specific wing regions and among a variety of bat taxa. We related these characteristics to the highly architectural fibrous supporting network of the wing membrane. We found that all material properties showed a strong anisotropy. In particular, wing membrane skin shows maximum stiffness and stregth parallel to the wing skeleton, and greatest extensibility parallel to the wing's trailing edge. We also found significant variation among wing regions. The uropatagium (tail membrane) supported the greatest load at failure, and the plagiopatagium (proximal wing membrane between laterl body wall and hand skeleton) is the weakest and most extensible part of the wing. We believe that the increased load bearing ability of the uropatagium relats to its key role in capture of insect prey, and that the great extensibility of the plagiopatagium promotes development of camber near the wing's centre of lift. In interspecific comparisons, energy absorpion and load to failure were greatest in Artibeus jamaicensis, the largest bat in our sample and the species with the highest wing loading, suggesting that wing loading may play a role in dictating the fuctional design of wing membranes.
Digital image correlation techniques are commonly used to measure specimen displacements by finding correspondences between an image of the specimen in an undeformed or reference configuration and a second image under load. To establish correspondences between the two images, numerical techniques are used to locate an initially square image subset in a reference image within an image taken under load. During this process, shape functions of varying order can be applied to the initially square subset. Zero order shape functions permit the subset to translate rigidly, while first-order shape functions represent an affine transform of the subset that permits a combination of translation, rotation, shear and normal strains. In this article, the systematic errors that arise from the use of undermatched shape function, i.e., shape functions of lower order than the actual displacement field, are analyzed. It is shown that, under certain conditions, the shape functions used can be approximated by a Savitzky-Golay low-pass filter applied to the displacement functions, permitting a convenient error analysis. Furthermore, this analysis is not limited to the displacements, but naturally extends to the higher-order terms included in the shape functions. This permits a direct analysis of the systematic strain errors associated with an undermatched shape function. Detailed numerical studies are presented for the case of a second-order displacement field and first- and second-order shape functions. Finally, the relation of this work to previously published studies is discussed.
Selected aspects of the morphology of bats of the family Molossidae are described and the functional significance of these features are discussed. The structure and proportions of the ears and the wings are considered to reflect primarily the rapid enduring flight typical of molossids. Comparisons of some characteristics of the wings of three molossids and of four bats of the family Vespertilionidae were made, and several aerodynamic relationships were applied to a consideration of the styles and speeds of flight of these bats. Molossid bats in general seem adapted to fast flight in open areas, whereas the vespertilionids studied are apparently suited to slower flight fairly low to the ground, near vegetation and other obstacles.
Constitutive modeling of biological tissues plays an important role in the understanding of tissue behavior and the development of synthetic materials for medical and bio-inspired applications. A structural continuum model that incorporates principal structural features of the tissue can potentially provide the link between microstructure and the macroscopic mechanical response of biological tissues. For most soft biological tissues, including arterial walls and skin tissue, the main load-carrying constituent is presumed to be the distributed collagen fibers embedded in a base matrix. It is believed that the organization of the collagen fibers gives rise to the anisotropy of the material. In this paper, a semi-structural constitutive model is proposed to account for planar fiber distributions with more than one distributed planar fiber property. Motivated by histology information of the wing membrane of the bat, a statistical treatment is formulated in this paper to capture the overall effect of the distribution of fiber cross-sectional area and the distribution of the number of fibers. This formulation is suitable for general cases when more than one fiber property varies spatially. Furthermore, this model is a two-dimensional specialization within the framework of a three-dimensional theory, which is different the formulation based on a fundamentally two-dimensional theory.
Recently, digital image correlation has a tool for surface deformation measurements has found widespread use and acceptance in the field of experimental mechanics. The method is known to reconstruct displacements with subpixel accuracy that depends on various factors such as image quality, noise, and the correlation algorithm chosen. However, the systematic errors of the method have not been studied in detail. We address the systematic errors of the iterative spatial domain cross-correlation algorithm caused by gray-value interpolation. We investigate the position- dependent bias in a numerical study and show that it can lead to apparent strains of the order of 40% of the actual strain level. Furthermore, we present methods to reduce this bias to acceptable levels.
Digital image correlation (DIC) is an optical–numerical full-field displacement measuring technique, which is nowadays widely used in the domain of experimental mechanics. The technique is based on a comparison between pictures taken during loading of an object. For an optimal use of the method, the object of interest has to be covered with painted speckles. In the present paper, a comparison is made between three different speckle patterns originated by the same reference speckle pattern. A method is presented for the determination of the speckle size distribution of the speckle patterns, using image morphology. The images of the speckle patterns are numerically deformed based on a finite element simulation. Subsequently, the displacements are measured with DIC-software and compared to the imposed ones. It is shown that the size of the speckles combined with the size of the used pixel subset clearly influences the accuracy of the measured displacements.
The physical properties of single, 5–8-μm diameter, water-swollen elastin fibers have been investigated on a microtest apparatus attached to a polarizing microscope. Analysis of the mechanical and optical properties at extensions below 100% indicate that the elastic modulus (G) has a value of 4.1 × 105 N m−2, the average molecular weight of chains between crosslinks is in the range of 6000–7100, and the stress optical coefficient (C′) is 1 × 10−9 m2 N−1 at 24°C. Analysis of the temperature dependence of the stress optical coefficient indicates that the polarizability of the random link decreases with increasing temperature, with an apparent activation energy for this process of the order of 1.6 kcal/mol. Analysis of the non-Gaussian mechanical and optical properties at extensions above about 100% suggest that the chains between crosslinks contain approximately 10 “effective” random links, with each link consisting of 7–8 amino acid residues. These parameters for the random chains in the elastin network have been used to predict the dimensions of other random proteins. The close correlation of these predictions with published values for the dimensions of a series of proteins in solution in 6M guanidinium hydrochloride provides an independent test of the appropriateness of our analysis.
Digital image correlation techniques are commonly used to measure specimen displacements by finding correspondences between
an image of the specimen in an undeformed or reference configuration and a second image under load. To establish correspondences
between the two images, numerical techniques are used to locate an initially square image subset in a reference image within
an image taken under load. During this process, shape functions of varying order can be applied to the initially square subset.
Zero order shape functions permit the subset to translate rigidly, while first-order shape functions represent an affine transform
of the subset that permits a combination of translation, rotation, shear and normal strains.
In this article, the systematic errors that arise from the use of undermatched shape function, i.e., shape functions of lower
order than the actual displacement field, are analyzed. It is shown that, under certain conditions, the shape functions used
can be approximated by a Savitzky-Golay low-pass filter applied to the displacement functions, permitting a convenient error
analysis. Furthermore, this analysis is not limited to the displacements, but naturally extends to the higher-order terms
included in the shape functions. This permits a direct analysis of the systematic strain errors associated with an undermatched
shape function. Detailed numerical studies are presented for the case of a second-order displacement field and first- and
second-order shape functions. Finally, the relation of this work to previously published studies is discussed.