超弹性橡胶材料的改进Rivlin模型

讨论了不可压缩橡胶材料的超弹性唯象本构模型。针对典型实验，给出选择应变能函数的原则。从物理机理上，分析了Neo-Hookean模型、Mooney模型、三阶Rivlin模型及Ogden模型的优缺点。在此基础上，将Rivlin模型改进成为 ，这种新形式具有三个优点：①若取前三项（N=1），则其结果与不可压缩线弹性的应变能相等，能够近似满足剪切的线性关系，但拉伸及压缩的线性关系是精确满足的。②当N≥2时，简单剪切中的应变能及剪应力τxy在小应变情况下是以剪应变γxy为等比的多项式展开；而Rivlin模型只能保证简单剪切实验中的应变能及剪应力τxy是以(γxy)2为等比的级数展开的形式，当取前两项的情况下，Rivlin模型只能精确保证常剪切，拉伸及压缩的线性关系无法得到保证。针对典型实验数据，若取同阶次多项式，本文模型的同类实验数据预测及不同类实验数据间相互预测的精度都比Rivlin模型的高。

AN IMPROVED RIVLIN MODEL OF HYPERELASTIC RUBBER MATERIALS

The phenomenological constitutive model of incompressible rubber materials is investigated in this paper. The principal of choosing strain-energy function is discussed. The advantages and disadvantages of several models, including the Neo-Hookean formulation, the Mooney formulation, the three-order Rivlin formulation and the Ogden formulation, is mechanically analyzed. Subsequently, an improved form of Rivlin formulation is proposed as , which has three significant advantages: ① in the case of N=1 and small strain, the new function is the same as the strain-energy of the linear elastic incompressible material, and gives a approximately linear shear deformation and a exact linear tension and compression deformation. ②when N≥2, in the condition of small strain, using the new formulation, the strain-energy and shear stress in simple shear experiment are polynomial with γxy as geometric proportion. While using Rivlin formulation, the strain-energy and shear stress τxy under small deformation are polynomial with (γxy)2 as geometric proportion. Besides, when the deformation is small, Rivlin formulation with N=1 provides constant shear module and non-linear relationship in tension and compression. The improved form, compared with the same order Rivlin formulation, shows significant advantages in prediction of test data in single experiment, and has a good performance in the mutual prediction between different tests.