The Research On The Bifurcation Problem Of The Rubberlike Material

Rubberlike material is a kind of the unique macromolecular material and plays the indispensable role in the industry because of its high elasticity. At present the rubber products are full of our daily life and manufacture, such as precision instrument, aviation, navigation, national defense, textile industry. In the world, the kinds of the practical rubber products add up to eighty thousand-ten thousand. So there are important theoretical meanings and vast engineering backgrounds to study the bifurcation problems of rubberlike materials. Just based on the above purpose, a series of researches are finished. This paper makes the following jobs.1. A kind of strain energy function proposed by Gao Y C was systematically studied, and the limit conditions as well as physical meanings of the constitutive parameters included in the strain energy function were also obtained. The results indicated: for the incompressible strain energy function, when the constitutive parameters met a > 0, n > 0, all kinds of limit conditions can be satisfied and the deformation was also stable; while for the compressible strain energy function, the constitutive parameters should meet a > 0, n>0, b> na3^n-2/2. The constitutive parameter n can be seen as the dimensionless strengthening parameter, while the constitutive parameters a, b were the quantities which had the same dimension as the elastic modulus. The results of the examples showed that these two kinds of strain energy function can reflect the finite deformation characters of rubberlike materials well.2. The Poisson function of the compressible membrane inflatable tube under plane strain was given, and with the help of Poisson function method, the explicit solutions to this question were firstly obtained. When the Poisson ratioγ₀ = 0.5, this solutions can degenerate into the incompressible case.3. The bifurcation problem of the compressible spherical membrane under a more universal disturbance displacement was built, and the bifurcation criterion same as the test was also given. If the compressible strain energy function proposed by Gao Y C was adopted, the detail bifurcation criterion can be gained. The results indicated: the controlling differential equations of the compressible situation were very similar with the incompressible case. They all had three absolute elastic coefficients, but the definitions of these coefficients were different. It was proved that the bifurcation would happen after the internal pressure reached the maximum from the theoretical analytical view. In the expanding process of the spherical membrane, its shape was no longer spherical after the internal pressure reached the maximum. That is to say the spherical membrane bifurcated, which was the same as the conclusion of the experiment.4. The questions about the cavity formation and bifurcation of the incompressible homogeneous solid sphere and circular cylinder were studied, and the analytical solutions were also acquired. If the bifurcation happened, the criterion on which bifurcation took place was also given. The results indicated: only if 0 < n < 1.5, the homogeneous solid sphere had bifurcation solution; while for the circular cylinder, the constitutive parameter n should meet 0 < n < 1. The constitutive parameter n had important influences on the bifurcation solutions. For the homogeneous circular cylinder, the influences of the axial principal stretch were also considered. If the right bifurcation took place, when the load p₀ was slightly bigger than the critical load p_cr in the stage of unload, the stresses had the phenomenon of boundary layer at the cavity wall. For the homogeneous solid sphere, the bifurcation criterion was relative to the critical load p_cr/a and the constitutive parameter n, while for the circular cylinder, it still did something with the axial principal stretchλ_z. The temperature field had strong effects on the bifurcation solution for the homogeneous solid circular cylinder. The bifurcation problem can be viewed as providing an idealized model for describing the sudden growth of a pre-existing micro-void.5. The questions about the cavity formation and bifurcation of the incompressible composite solid sphere and circular cylinder were also studied, and the analytical solutions were acquired as well. If the bifurcation happened, the criterion on which bifurcation took place was still given. The results indicated: only if 0 < n < 1.5, the solid sphere had the bifurcation solution; while for the circular cylinder, the constitutive parameter n should meet 0 < n < 1, which were the same as the homogeneous cases. The critical loads were only relative to the internal materials of the composite structures. The constitutive parameter n and combined coefficient m had important influences on the bifurcation solutions. For the composite circular cylinder, the influences of the axial principal stretch were also considered. If the right bifurcation took place, the stresses still had the phenomenon of boundary layer at the cavity wall under some special conditions which were similar with the homogeneous cases. For the composite solid sphere, the bifurcation criterion was relative to the critical load p_cr//a⁽¹⁾, constitutive parameter n, volume fraction f and combined coefficient m . while for the composite circular cylinder, it still did something with the axial principal stretchλ_z. It was also proved that the bifurcation problem can be viewed as providing an idealized model for describing the sudden growth of a pre-existing micro-void.6. With the help of the strain energy function proposed by Gao Y C and the theory of finite deformation dynamics, the questions about the cavity dynamical formation and bifurcation of the incompressible homogeneous solid sphere and circular cylinder under a suddenly applied uniform tensile dead-load were studied, and the analytical solutions were given too. The results indicated: the dynamical model can degenerate into the static model conveniently. If 0 < n < 1.5, the solid sphere had the dynamical bifurcation solutions; while for circular cylinder, the constitutive parameter n should meet 0 < n < 1, which were the same as the static cases. The constitutive parameter n had important influences on the dynamical bifurcation solutions. For the circular cylinder, the influences of the axial principal stretch were also considered. When the load was larger than the critical load, a cavity suddenly appeared in the centers of the structures and can grow quickly. The cavity displayed a periodic nonlinear oscillation. At the same time, the oscillation phase diagram of the cavity radius and the approximative oscillation period were also given.7. Based on the strain energy function which was proposed by Gao Y C, a rubber rectangular including a void under uniaxial compression was analyzed with the potential energy principle. The results indicated that this strain energy function can reflect the finite deformation character of the rubber material correctly, and that the rubber rectangular under uniaxial compression had a different mechanical features compared with the case under uniaxial extension.The analytical results in present paper may be helpful as a theoretical reference in the design, production and application of the rubberlike materials.