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The Korea Academia-Industrial cooperation Society (한국산학기술학회)
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General inflation and bifurcation analysis of rubber balloons

고무풍선의 일반화 팽창 및 분기 해석

  • Park, Moon Shik (Department of Mechanical Engineering, Hannam University)
  • 박문식 (한남대학교 기계공학과)
  • Received : 2018.08.31
  • Accepted : 2018.12.07
  • Published : 2018.12.31
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Abstract

Several typical hyper-elastic constitutive models that encompass both conventional and advanced ones were investigated for the application of instability problems, including the biaxial tension of a rubber patch and inflation of spherical or cylindrical balloons. The material models included the neo-Hookean model, Mooney-Rivlin model, Gent model, Arruda-Boyce model, Fung model, and Pucci-Saccomandi model. Analyses can be done using membrane equations with particular strain energy density functions. Among the typical strain energy density functions, Kearsley's bifurcation for the Treloar's patch occurs only with the Mooney-Rivlin model. The inflation equation is so generalized that a spherical balloon and tube balloons can be taken into account. From the analyses, the critical material parameters and limit points were identified for material models in terms of the non-dimensional pressure and inflation volume ratio. The bifurcation was then identified and found for each material model of a balloon. When the finite element method was used for the structural instability problems of rubber-like materials, some careful treatments required could be suggested. Overall, care must be taken not only with the analysis technique, but also in selecting constitutive models, particularly the instabilities.

몇 가지 전형적인 기존 및 진보된 초탄성 구성모델들의 고무패치 이축인장 및 구형 또는 원통형 풍선 팽창에서의 불안정성에 대해서 밝힌다. 적용할 구성모델은 neo-Hookean 모델, Mooney-Rivlin 모델, Gent 모델, Arruda-Boyce 모델, Fung 모델, Pucci-Saccomandi 모델 등이다. 팽창 및 분기 해석은 이들 변형에너지 함수들의 막 방정식을 이용하여 수행할 수 있다. 해석에는 사각패치에 대한 Kearsley의 분기현상, 고무풍선의 일반화 한 팽창현상, 고무풍선의 분기현상을 다룬다. 이들 변형에너지 함수들 중에서도 오직 Mooney-Rivlin 모델에서만 Kearsley의 분기현상이 일어남을 확인하였다. 팽창 방정식은 구형풍선과 원통형 풍선을 함께 다룰 수 있도록 일반화 시켰다. 팽창해석에 의하여 극한점과 임계 물성치들을 무차원 압력 및 팽창 부피의 항들로 구하였다. 그렇게 구해진 결과들로부터 분기현상을 구할 수 있었다. 또한 유한요소법을 사용하여 고무류의 구조적 불안정 문제들을 다룰 때 필요한 특별한 조처에 대해서 제안하였다. 결론적으로 고무류의 불안정성을 포함하는 문제를 다룰 때는 해석기법은 물론 구성모델의 선택에 따라 결과가 달라질 수 있으므로 신중한 처리가 요구된다.

Keywords

SHGSCZ_2018_v19n12_14_f0001.png 이미지

Fig. 1. Square rubber patch. (a) uniformly distributed biaxial edge loading (b) 10x10 finite element model with distributed loading control (c) 10x10 finite element model with constrained edge loading

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Fig. 2. Analysis of square rubber patch (α = 0.906 ) in biaxial loading. (a) symmetric and asymmetric solutions for the symmetric loading (b) bifurcation point with respect to the material parameter (c) solutions for the non-symmetric loading (d) load-stretch curves by analytic solutions and finite element methods

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Fig. 3. Rubber balloons inflations. (a) two balloons inflating by the same inflation pressure (b) spherical(ball) balloon inflation (c) cylindrical (tube) balloon inflation

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Fig. 4. Inflation curves. (a) ball balloon (b) tube balloon (c) inflation paths (d) critical material parameter. NH: neo-Hookean, MR: Mooney-Rivlin, GE: Gent, GE2: Gent or Pucci and Saccomandi, FU: Fung, AB: Arruda-Boyce

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Fig. 5. Tube balloon inflation and bifurcation. (a) ax-isymmetric FEM model with no restraint for inflation (b) axisymmetric FEM model with restraint for infla-tion (c) analytic results and FEM results (d) inflation and bifurcation instances of tube balloon.

Table 1. FEM calculation of bifurcation point using base state and eigenvalue analysis with perturbation

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Table 2. Inflation characteristics for spherical balloon

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Table 3. Inflation characteristics for cylindrical balloon

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