Accuracy of beam theory for estimating bone tissue modulus and yield stress from 3-point bending tests on rat femora

Abstract The ability to measure bone tissue material properties plays a major role in diagnosis of diseases and material modeling. Bone’s response to loading is complex and shows a viscous contribution to stiffness, yield and failure. It is also ductile and damaging and exhibits plastic hardening until failure. When performing mechanical tests on bone tissue, these constitutive effects are difficult to quantify, as only their combination is visible in resulting stress–strain data. In this study, a methodology for the identification of stiffness, damping, yield stress and hardening coefficients of bone from a single cyclic tensile test is proposed. The method is based on a two-layer elasto-visco-plastic rheological model that is capable of reproducing the specimens’ pre- and postyield response. The model’s structure enables for capturing the viscously induced increase in stiffness, yield, and ultimate stress and for a direct computation of the loss tangent. Material parameters are obtained in an inverse approach by optimizing the model response to fit the experimental data. The proposed approach is demonstrated by identifying material properties of individual bone trabeculae that were tested under wet conditions. The mechanical tests were conducted according to an already published methodology for tensile experiments on single trabeculae. As a result, long-term and instantaneous Young’s moduli were obtained, which were on average 3.64 GPa and 5.61 GPa, respectively. The found yield stress of 16.89 MPa was lower than previous studies suggest, while the loss tangent of 0.04 is in good agreement. In general, the two-layer model was able to reproduce the cyclic mechanical test data of single trabeculae with an root-mean-square error of 2.91 ± 1.77 MPa. The results show that inverse rheological modeling can be of great advantage when multiple constitutive contributions shall be quantified based on a single mechanical measurement.

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Ultimate Load Analysis of RC Beam under Combined Action of Axial, Bending and Shear Loading

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.163-167.1702 ◽

2010 ◽

Vol 163-167 ◽

pp. 1702-1707

Author(s):

Bo Diao ◽

Yun Peng Zhang ◽

Ying Hua Ye

Keyword(s):

Ultimate Load ◽

Beam Theory ◽

Shear Loading ◽

Rc Beam ◽

Pure Bending ◽

Bending Tests ◽

Shear Loads ◽

Section Model ◽

Combined Action ◽

Load Analysis

Based on Timoshenko beam theory, a section model for the ultimate load analysis of RC beam under combined action of axial force, bending and shear force is presented in this paper. Especially, the transverse deformation is considered with the appearance of diagonal cracks. The validity of the model is established by comparing its results with several tests. These simulations include ultimate bending moments of pure bending tests and ultimate shear loads with different shear-span ratios. The analytical results show excellent agreement with experimental data.

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A Finite Element Model for Direction-Dependent Mechanical Response to Nanoindentation of Cortical Bone Allowing for Anisotropic Post-Yield Behavior of the Tissue

Journal of Biomechanical Engineering ◽

10.1115/1.4001358 ◽

2010 ◽

Vol 132 (8) ◽

Cited By ~ 31

Author(s):

D. Carnelli ◽

D. Gastaldi ◽

V. Sassi ◽

R. Contro ◽

C. Ortiz ◽

...

Keyword(s):

Finite Element ◽

Finite Element Model ◽

Cortical Bone ◽

Yield Stress ◽

Bone Tissue ◽

Mechanical Response ◽

Experimental Studies ◽

Element Model ◽

Yield Behavior ◽

Constitutive Parameters

A finite element model was developed for numerical simulations of nanoindentation tests on cortical bone. The model allows for anisotropic elastic and post-yield behavior of the tissue. The material model for the post-yield behavior was obtained through a suitable linear transformation of the stress tensor components to define the properties of the real anisotropic material in terms of a fictitious isotropic solid. A tension-compression yield stress mismatch and a direction-dependent yield stress are allowed for. The constitutive parameters are determined on the basis of literature experimental data. Indentation experiments along the axial (the longitudinal direction of long bones) and transverse directions have been simulated with the purpose to calculate the indentation moduli and the tissue hardness in both the indentation directions. The results have shown that the transverse to axial mismatch of indentation moduli was correctly simulated regardless of the constitutive parameters used to describe the post-yield behavior. The axial to transverse hardness mismatch observed in experimental studies (see, for example, Rho et al. [1999, “Elastic Properties of Microstructural Components of Human Bone Tissue as Measured by Nanoindentation,” J. Biomed. Mater. Res., 45, pp. 48–54] for results on human tibial cortical bone) can be correctly simulated through an anisotropic yield constitutive model. Furthermore, previous experimental results have shown that cortical bone tissue subject to nanoindentation does not exhibit piling-up. The numerical model presented in this paper shows that the probe tip-tissue friction and the post-yield deformation modes play a relevant role in this respect; in particular, a small dilatation angle, ruling the volumetric inelastic strain, is required to approach the experimental findings.

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Geometry and thickness dependant anomalous mechanical behavior of fabricated SU-8 thin film micro-cantilevers

Journal of Micromanufacturing ◽

10.1177/2516598420930988 ◽

2020 ◽

Vol 3 (2) ◽

pp. 113-120

Author(s):

Aviru Kumar Basu ◽

Anup Basak ◽

Shantanu Bhattacharya

Keyword(s):

Beam Theory ◽

Contradictory Result ◽

Cantilever Beams ◽

Bending Tests ◽

Cantilever Arrays ◽

Scale Thickness ◽

Peak Loads ◽

Intrinsic Length ◽

Micro Cantilever ◽

Similar Peak

SU-8 micro-cantilever arrays consisting of V- and M-shaped structures fabricated using a simplified single hard mask step. Bending tests were performed under similar peak loads (ranging 2–10 µN), with thickness ranging between micron (3.5 µm) and sub-micron (0.2 µm) scales. Various mechanical properties such as stiffness and hysteresis are determined from the load versus deflection curves. When the thickness of the V-shaped beam is decreased from 2 µm to 0.2 µm, the stiffness increases by a factor of 2.7, which is in contradiction with the classical beam theory according to which the stiffness for 0.2 µm beam should be three orders of magnitude less than that of 2 µm beam. Micropolar elasticity theory with a variable-intrinsic length scale (thickness dependant) is used to explain such an anomalous response. Experimentally obtained stiffness of two M-shaped beams of thickness 2 µm and 0.2 µm are almost identical. Reason behind this contradictory result is that the thicker beam has a residual strain with a large plastic deformation which usually increases the cross-linking network density, leading to increase in elastic modulus, hardness and thus stiffness of polymers. But the thinner beam has undergone an elastic deformation. The size effect of V- and M-shaped cantilever beams is discussed.

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Study of Formability of Sandwich Shells with Metal Foam Cores

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.554-557.2252 ◽

2013 ◽

Vol 554-557 ◽

pp. 2252-2255 ◽

Cited By ~ 1

Author(s):

H. Mata ◽

R. Natal Jorge ◽

A.D. Santos ◽

Marco P. L. Parente ◽

Robertt A.F. Valente ◽

...

Keyword(s):

Mechanical Behavior ◽

Yield Stress ◽

Composite Structure ◽

Metal Foam ◽

Numerical Models ◽

Solid Material ◽

Experimental Information ◽

Bending Tests ◽

Aluminum Sheets ◽

The Difference

The main goal of this work is made a first approach to the study of formability of sandwich structures, composed by two aluminum sheets (skins), separated by an aluminum metal foam core, using bending tests. The bending tests performed on this work, can be described as a cylindrical punch, which apply a force at the middle span of the specimens, which were have three different lengths, 114 mm, 167.5 mm and 230 mm, in a specific die. The bending tests were used to evaluate the type of yielding of the global composite structure, by measuring the force / displacement values at a middle span. The type of yielding of the global structure has extremely importance in the study of formability of this type of composite, because it was composed by two different materials; each one presents a different mechanical behavior separately. The difference in mechanical behavior can be explained by the axisymmetric compressive stress states test and by the influence of the hydrostatic pressure in the yield stress of the porous material (aluminum foam) and has no influence on the yield stress of the homogeneous solid material (aluminum sheet). With this work, experimental information on the yielding of this composite structure was obtained, to be used in numerical models to study its formability.

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Relationship between Bending Modulus and Test Sizes of Laminated Glass/Polyester Composites

Materials Science Forum ◽

10.4028/www.scientific.net/msf.537-538.71 ◽

2007 ◽

Vol 537-538 ◽

pp. 71-79 ◽

Cited By ~ 1

Author(s):

Z.L. Simon ◽

László Mihály Vas

Keyword(s):

Beam Theory ◽

Measured Data ◽

Bending Test ◽

Laminated Glass ◽

Bending Tests ◽

Bending Modulus ◽

Glass Fiber Reinforced ◽

Best Fitting ◽

Polyester Composites ◽

Woven Glass Fiber

The effect of span-to-thickness ratio (L/h) on the bending modulus was investigated in warp, 45°, 67.5° and weft directions in woven glass fiber reinforced polyester laminates. Using classical beam theory the results of the bending tests carried out for L/h = 5,..,50 were extended to the total range of definition (0≤L/h<∞) by applying rational fractional and exponential fractional functions. The extensions take the asymptotic behavior of the bending test for L/h = 0 and L/h = ∞ in account while providing the best fitting to the measured data. It has been shown that the exponential fractional estimations give better results regarding the form of functions that is independent from the directions.

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Failure mechanisms in three and four point short beam bending tests of unidirectional glass/epoxy

The Journal of Strain Analysis for Engineering Design ◽

10.1243/03093247v274235 ◽

1992 ◽

Vol 27 (4) ◽

pp. 235-243 ◽

Cited By ~ 45

Author(s):

W C Cui ◽

M R Wisnom ◽

M Jones

Keyword(s):

Shear Stress ◽

Shear Strength ◽

Shear Failure ◽

Beam Theory ◽

Interlaminar Shear Strength ◽

Bending Test ◽

Bending Tests ◽

Four Point Bending ◽

Short Beam ◽

Interlaminar Shear

Three and four point bending tests are compared both analytically and experimentally. In all the three point bending tests, damage was observed under the loading roller in addition to the interlaminar shear failure, while in the four point bending tests, only interlaminar shear failure was observed. Therefore, this four point bending test is valid for measuring the interlaminar shear strength. From the finite element analysis, it is found that the roller diameter is a critical parameter in determining the stress concentrations in short beam tests. In order to avoid damage under the roller and thus to make the short beam test a valid means for measuring the interlaminar shear strength, the appropriate roller diameters should be chosen. The damage under the loading roller in the three point bending test basically reduces the effective specimen thickness and thus this test underestimates the interlaminar shear strength. The interlaminar shear cracks in the short beam tests were found to be randomly distributed in a region between 30 percent and 70 percent through the thickness from the top surface. This is due to the non-linear shear response which means that the shear stress distribution is more uniform near the middle of the section. Also the maximum value of the shear stress is lower than the maximum value given by beam theory. A non-linear shear correction factor is suggested to account for this effect and for the glass/epoxy composite tested here, the actual interlaminar shear strength is only about 83 percent of the apparent value from classical beam theory. The interlaminar shear crack does not occur at the location of maximum shear stress. This may be because there is insufficient energy to propagate a crack at this location.

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