Lin David C, Shreiber David I, Dimitriadis Emilios K, Horkay Ferenc
Section on Tissue Biophysics and Biomimetics, NICHD, National Institutes of Health, Bethesda, MD 20892, USA.
Biomech Model Mechanobiol. 2009 Oct;8(5):345-58. doi: 10.1007/s10237-008-0139-9. Epub 2008 Nov 2.
The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.
缺乏实用的非线性弹性接触模型常常迫使人们在分析压痕数据时不适当地使用赫兹模型，这可能导致与生物原子力显微镜测量结果相关的不一致性。我们借助有限元方法，基于多种超弹性应变能函数推导并验证了力-压痕关系。这些模型被应用于现有的压痕数据，这些数据是使用微球作为压头对合成橡胶状凝胶、天然小鼠软骨组织和工程软骨进行压痕实验得到的。对于生物组织，冯氏模型和单参数奥格登模型对数据的拟合效果最佳，而所有测试的超弹性模型对合成凝胶都有很好的拟合效果。赫兹模型在小变形情况下（对于测试的样品，应变<0.05）对合成凝胶是可以接受的，但对生物组织则不然。尽管这一发现支持了普遍接受的观点，即许多软材料在小变形时可被假定为线弹性，但非线性模型有助于分析本质上非线性的组织和大应变压痕行为。
