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...oad-controlled compression of the gelscylinders at a ramp speed of 1 mm/s. For all samples (at leaststhree of each concentration), the shear modulus was determinedsfrom the rubber elasticity equation =-=[35,36]-=-su = G(A − A−2) (7)swhere u is the normal component of the engineering, or nominal,sstress (force per undeformed area); G is the shear modulus; and Asis the deformation ratio (=L /L0, where L and L0 a...

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...iterated in the concluding remarks.sheorysContact Mechanics Models. The classical problem of an elas ic half-space indented by a rigid object was first treated by Hertz n 1881 and Boussinesq in 1885 =-=[18]-=-. Numerous others have con ributed to the theoretical framework, which is widely used in nterpreting the results of indentation tests at all length scales. xact solutions in the form of force-indenta...

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... are given in Table 1 for ndenters that are spherical (from the original Hertz theory of ontact between two ellipsoidal bodies and oftentimes referred to s the Hertz equation or model), sharp conical =-=[5,25]-=-, or sharpipped pyramidal [26] in geometry. For typical tapered indenters ith a blunt tip (Fig. 1), the relationship is [25,27,28]s3R tan ¢ 3Rs( ) [ ]{F = ao − m − arcsin1 −sv2 tan ¢ 2 a a3 b a2 − b2s...

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... automated, contact point indepen ent technique to generate relative elasticity maps by showing that he relative work of indentation at some force is proportional to he relative measure of stiffness =-=[31]-=-. However, an absolute mea ure of elastic properties requires fitting of force-indentation data o contact mechanics models. Since both F and o are dependent n the contact point, objective approaches ...

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... are given in Table 1 for ndenters that are spherical (from the original Hertz theory of ontact between two ellipsoidal bodies and oftentimes referred to s the Hertz equation or model), sharp conical =-=[5,25]-=-, or sharpipped pyramidal [26] in geometry. For typical tapered indenters ith a blunt tip (Fig. 1), the relationship is [25,27,28]s3R tan ¢ 3Rs( ) [ ]{F = ao − m − arcsin1 −sv2 tan ¢ 2 a a3 b a2 − b2s...

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...e compliant canti lever (0.06 N/m nominal spring constant) than the other gelss(0.58 N/m nominal spring constant). The spring constant of eachscantilever was determined using the thermal tune method =-=[38,39]-=-.sThe algorithm outlined previously was used to extract Young’ssmodulus from each dataset.sMultiple force curves for each PVA film were collected usingsthe “force-volume” mode of the AFM. In this auto...

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...ly applied force. When signifi cant adhesive forces exist between the contacting solids, theysmust be accounted for. Adhesive contact mechanics models willsbe discussed in a separate paper (Part II) =-=[29]-=-.sBasic Approaches to Extracting Elastic Properties FromsAFM Data. Indentation is performed by moving the cantilever insthe direction perpendicular to the sample surface, z, between asstarting positio...

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... to be successful in eliminating the need for subjective user input (e.g., specifying an estimate of the contact point based on visual inspection [19] or specifying the range of the noncontact region =-=[11]-=-) or capable of accurately processing diverse populations of curves. We have compiled a collection of complementary strate gies that addresses the known issues in the analysis of AFM force curves, an...

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...ata. In lieu of the flawed approach of pproximating the tip as either a sphere or a sharp cone, some esearchers generate calibration curves by combining nanoinden ation with macroscopic test methods =-=[15]-=-. The alternative ap roach introduced in Appendix B makes use of sharp tip models or dealing with blunt indenters and complements the strategies in ur algorithm. Applied in reverse order, the techniq...

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... and oftentimes referred to s the Hertz equation or model), sharp conical [5,25], or sharpipped pyramidal [26] in geometry. For typical tapered indenters ith a blunt tip (Fig. 1), the relationship is =-=[25,27,28]-=-s3R tan ¢ 3Rs( ) [ ]{F = ao − m − arcsin1 −sv2 tan ¢ 2 a a3 b a2 − b2s2 − b2)1/2− + (a m + (2a)[ ]} here 2¢ is the tip angle, b is the radius at which the tapered ides transition into a spherical tip ...

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...the cantilever tip is used as a nanoindenter, the AFM issell suited for probing the local elasticity of small, very soft, andsnhomogeneous samples. It has been used to study the elasticity ofsacteria =-=[8]-=-, cells [9–12], endothelia [13], articular cartilages14–16], and bony tissue [17]. However, despite its advantagessver more conventional testing methodologies, data processingsssues hinder the consist...

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...a nanoindenter, the AFM issell suited for probing the local elasticity of small, very soft, andsnhomogeneous samples. It has been used to study the elasticity ofsacteria [8], cells [9–12], endothelia =-=[13]-=-, articular cartilages14–16], and bony tissue [17]. However, despite its advantagessver more conventional testing methodologies, data processingsssues hinder the consistency of elasticity measurements...

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...rs that are spherical (from the original Hertz theory of ontact between two ellipsoidal bodies and oftentimes referred to s the Hertz equation or model), sharp conical [5,25], or sharpipped pyramidal =-=[26]-=- in geometry. For typical tapered indenters ith a blunt tip (Fig. 1), the relationship is [25,27,28]s3R tan ¢ 3Rs( ) [ ]{F = ao − m − arcsin1 −sv2 tan ¢ 2 a a3 b a2 − b2s2 − b2)1/2− + (a m + (2a)[ ]} ...

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... and oftentimes referred to s the Hertz equation or model), sharp conical [5,25], or sharpipped pyramidal [26] in geometry. For typical tapered indenters ith a blunt tip (Fig. 1), the relationship is =-=[25,27,28]-=-s3R tan ¢ 3Rs( ) [ ]{F = ao − m − arcsin1 −sv2 tan ¢ 2 a a3 b a2 − b2s2 − b2)1/2− + (a m + (2a)[ ]} here 2¢ is the tip angle, b is the radius at which the tapered ides transition into a spherical tip ...

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...ulus E as the fitting parameters. This approachscan be considered the semi-constrained counterpart to the visualsinspection method and is likewise prone to error when noise levelssare high.sKolambkar =-=[19]-=- formulated a more rigorous unconstrained ap proach by evaluating the error associated with assuming an incor rect contact point. He expanded the resulting force-indentationsrelation in a power seri...

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...ion [18]. The pointwise calculations of E will tend toward a constant value that issdependent on the chosen contact point. Typically, Poisson’s ratiosis assumed to be 0.5 for soft, gel-like materials =-=[5,20,30]-=-. To avoidsuncertainties about the compressibility of the sample, some re searchers define an elastic constant incorporating the terms E ands(1−v2), thereby combining the linear elastic and compressi...

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...H was adjusted by HCl. To createssamples of increasing stiffness, the PVA concentration was in creased while maintaining a constant crosslink density (one unitsof GDA per 100 units of vinyl alcohol) =-=[34]-=-. Gel cylinders andsthick layers (2 mmsto>1 cm) with final PVA concentrations ofs3%, 6%, 9%, and 12% (w/w) were cast in molds (1 cm diam ands1 cm high) and 35 mm Petri dishes, respectively. All sample...

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... In the interest of achieving consistency and accuracy in bsolute measurements of elastic properties, it is necessary to uti ize the calibration methods available for determining cantilever tiffness =-=[38,39,48,49]-=- and tip geometry [50–53].sAlthough the strategies discussed here have been applied to ontact mechanics models dealing with the indentation of an infi ite half-space, they are readily applicable to m...

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...the local elasticity of small, very soft, andsnhomogeneous samples. It has been used to study the elasticity ofsacteria [8], cells [9–12], endothelia [13], articular cartilages14–16], and bony tissue =-=[17]-=-. However, despite its advantagessver more conventional testing methodologies, data processingsssues hinder the consistency of elasticity measurements on softsiological tissues and preclude the use of...

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...ion [18]. The pointwise calculations of E will tend toward a constant value that issdependent on the chosen contact point. Typically, Poisson’s ratiosis assumed to be 0.5 for soft, gel-like materials =-=[5,20,30]-=-. To avoidsuncertainties about the compressibility of the sample, some re searchers define an elastic constant incorporating the terms E ands(1−v2), thereby combining the linear elastic and compressi...

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...oad-controlled compression of the gelscylinders at a ramp speed of 1 mm/s. For all samples (at leaststhree of each concentration), the shear modulus was determinedsfrom the rubber elasticity equation =-=[35,36]-=-su = G(A − A−2) (7)swhere u is the normal component of the engineering, or nominal,sstress (force per undeformed area); G is the shear modulus; and Asis the deformation ratio (=L /L0, where L and L0 a...

Citation Context

...e compliant canti lever (0.06 N/m nominal spring constant) than the other gelss(0.58 N/m nominal spring constant). The spring constant of eachscantilever was determined using the thermal tune method =-=[38,39]-=-.sThe algorithm outlined previously was used to extract Young’ssmodulus from each dataset.sMultiple force curves for each PVA film were collected usingsthe “force-volume” mode of the AFM. In this auto...

Citation Context

... In the interest of achieving consistency and accuracy in bsolute measurements of elastic properties, it is necessary to uti ize the calibration methods available for determining cantilever tiffness =-=[38,39,48,49]-=- and tip geometry [50–53].sAlthough the strategies discussed here have been applied to ontact mechanics models dealing with the indentation of an infi ite half-space, they are readily applicable to m...