Mak A F, Lai W M, Mow V C
Department of Bioengineering and Orthopaedic Surgery Research, University of Pennsylvania, Philadelphia 19104.
J Biomech. 1987;20(7):703-14. doi: 10.1016/0021-9290(87)90036-4.
A mathematical solution has been obtained for the indentation creep and stress-relaxation behavior of articular cartilage where the tissue is modeled as a layer of linear KLM biphasic material of thickness h bonded to an impervious, rigid bony substrate. The circular (radius = a), plane-ended indenter is assumed to be rigid, porous, free-draining, and frictionless. Double Laplace and Hankel transform techniques were used to solve the partial differential equations. The transformed equations and boundary conditions yielded an integral equation of the Fredholm type which was analyzed asymptotically and solved numerically. Our asymptotic analyses showed that the linear KLM biphasic material behaves like an incompressible (v = 0.5) single-phase elastic solid at t = 0+; the instantaneous response of the material is governed by the shear modulus (mu s) of the solid matrix. The linear KLM biphasic material behaves like a compressible elastic solid with material properties defined by those of the solid matrix, i.e. (lambda s, mu s) or (mu s, v s) as t----infinity. The transient viscoelastic creep and stress-relaxation behavior, 0 less than t less than infinity, of this material is controlled by the frictional drag (which is inversely proportional to the permeability k) associated with the flow of the interstitial fluid through the porous-permeable solid matrix. For given values of the Poisson's ratio of the solid matrix v s and the aspect ratio a/h, where a is the radius of the indenter and h is the thickness of the layer, the creep behavior with respect to the dimensionless time H Akt/a2 is completely controlled by the load parameter P/2 mu sa2 and the stress relaxation behavior is completely controlled by the rate of compression parameter R0 = kH A/V0h where H A = lambda s + 2 mu s and the equilibrium strain u0/h. This mathematical solution may now be used to describe an indentation experiment on articular cartilage to determine the intrinsic material properties of the tissue, i.e. permeability k, and the elastic coefficients of the solid phase (lambda s, mu s) or (mu s, v s).
已获得关节软骨压痕蠕变和应力松弛行为的数学解，其中将该组织建模为厚度为h的线性KLM双相材料层，该材料层粘结到不透水的刚性骨基质上。假定圆形（半径 = a）、平面端部压头是刚性、多孔、自由排水且无摩擦的。采用双拉普拉斯变换和汉克尔变换技术求解偏微分方程。变换后的方程和边界条件得到一个弗雷德霍姆型积分方程，对其进行了渐近分析并数值求解。我们的渐近分析表明，线性KLM双相材料在t = 0+时的行为类似于不可压缩（ν = 0.5）的单相弹性固体；材料的瞬时响应由固体基质的剪切模量（μs）控制。当t趋于无穷大时，线性KLM双相材料的行为类似于具有由固体基质特性定义的材料特性的可压缩弹性固体，即（λs，μs）或（μs，νs）。这种材料在0小于t小于无穷大时的瞬态粘弹性蠕变和应力松弛行为由与间隙流体通过多孔渗透固体基质流动相关的摩擦阻力（与渗透率k成反比）控制。对于固体基质泊松比νs和长宽比a/h的给定值（其中a是压头半径，h是层的厚度），相对于无量纲时间H Akt/a2的蠕变行为完全由载荷参数P/2μsa2控制，应力松弛行为完全由压缩速率参数R0 = kH A/V0h控制，其中H A = λs + 2μs以及平衡应变u0/h。现在这个数学解可用于描述关节软骨的压痕实验，以确定该组织的固有材料特性，即渗透率k以及固相的弹性系数（λs，μs）或（μs，νs）。
