• 대한수학회논문집
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• 제20권3호
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• pp.589-602
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• 2005

We present the numerical algorithm for the model for high-strain rate deformation in hyperelastic-viscoplastic materials based on a fully conservative Eulerian formulation by Plohr and Sharp. We use a hyperelastic equation of state and the modified Steinberg and Lund's rate dependent plasticity model for plasticity. A two-dimensional approximate Riemann solver is constructed in an unsplit manner to resolve the complex wave structure and combined with the second order TVD flux. Numerical results are also presented.

• 한국자동차공학회논문집
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• 제7권5호
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• pp.141-153
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• 1999

A finite element procedure for the analysis of rubber-like hyperelastic material is developed. The volumetric incompressiblity conditions of the rubber deformation is included in the formulation by using penalty method. In this paper, the behavior of the rubber deformation is represented by hyperelastic constitutive relations based on a generalized Mooney-Rivlin model. The principle of virtual work is used to derive nonlinear finite element equation for the large displacement problem and presented in total-Lagrangian description. The finite element procedure using analytic differentiation resulted in very close solution to the result of the well known commercial packages NISAII AND ABAQUS. Numerical tests show that the results from the numerical differentiation method coincide very well with those from the analytic method and the well known commercial packages in static analysis. The convergency of rubber usingν iteration method is also discussed.

• Elastomers and Composites
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• 제56권2호
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• pp.79-84
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• 2021

A strain energy density function is used to characterize the hyperelasticity of rubber-like materials. Conventional models, such as the Neo-Hookean, Mooney-Rivlin, and Ogden models, are widely used in automotive industries, in which the strain potential is derived from strain invariants or principal stretch ratios. A fitting procedure for experimental data is required to determine material constants for each model. However, due to the complexities of the mathematical expression, these models can only produce an accurate curve fitting in a specified strain range of the material. In this study, a hyperelastic model for Neodymium Butadiene rubber is developed by using the Artificial Neural Network. Comparing the analytical results to those obtained by conventional models revealed that the proposed model shows better agreement for both uniaxial and equibiaxial test data of the rubber.

• Geomechanics and Engineering
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• 제7권4호
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• pp.335-353
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• 2014

The softening hyperelastic model based on the strain energy limitation is of clear concepts and simple forms to describe the failure of materials. In this study, a linear and a nonlinear softening hyperelastic model are proposed to characterize the deformation and the failure in granular materials by introducing a softening function into the shear part of the strain energy. A method to determine material parameters introduced in the models is suggested. Based on the proposed models the numerical examples focus on bearing capacity and strain localization of granular materials. Compared with Volokh softening hyperelasticity and classical Mohr-Coulomb plasticity, our proposed models are able to capture the typical characters of granular materials such as the strain softening and the critical state. In addition, the issue of mesh dependency of the proposed models is investigated.

• Current Optics and Photonics
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• 제5권4호
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• pp.391-401
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• 2021

In the present study, a hyperelastic constitutive model is built by complying with a simplified hyperelastic strain energy function, which yields the numerical solution for a deformed polydimethylsiloxane (PDMS) membrane in the case of axisymmetric hydraulic pressure. Moreover, a nonlinear equilibrium model is deduced to accurately express the deformation of the membrane, laying a basis for precise analysis of the optical transfer function. Comparison to experimental and simulated data suggests that the model is capable of accurately characterizing the deformation behavior of the membrane. Furthermore, the stretch ratio derived from the model applies to the geometrical optimization of the deformed membrane.

• Structural Engineering and Mechanics
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• 제35권6호
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• pp.677-697
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• 2010

This paper focuses on geometrically non-linear static analysis of a simply supported beam made of hyperelastic material subjected to a non-follower transversal uniformly distributed load. As it is known, the line of action of follower forces is affected by the deformation of the elastic system on which they act and therefore such forces are non-conservative. The material of the beam is assumed as isotropic and hyperelastic. Two types of simply supported beams are considered which have the following boundary conditions: 1) There is a pin at left end and a roller at right end of the beam (pinned-rolled beam). 2) Both ends of the beam are supported by pins (pinned-pinned beam). In this study, finite element model of the beam is constructed by using total Lagrangian finite element model of two dimensional continuum for a twelve-node quadratic element. The considered highly non-linear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. In order to use the solution procedures of Newton-Raphson type, there is need to linearized equilibrium equations, which can be achieved through the linearization of the principle of virtual work in its continuum form. In the study, the effect of the large deflections and rotations on the displacements and the normal stress and the shear stress distributions through the thickness of the beam is investigated in detail. It is known that in the failure analysis, the most important quantities are the principal normal stresses and the maximum shear stress. Therefore these stresses are investigated in detail. The convergence studies are performed for various numbers of finite elements. The effects of the geometric non-linearity and pinned-pinned and pinned-rolled support conditions on the displacements and on the stresses are investigated. By using a twelve-node quadratic element, the free boundary conditions are satisfied and very good stress diagrams are obtained. Also, some of the results of the total Lagrangian finite element model of two dimensional continuum for a twelve-node quadratic element are compared with the results of SAP2000 packet program. Numerical results show that geometrical nonlinearity plays very important role in the static responses of the beam.

• 대한기계학회논문집
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• 제16권7호
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• pp.1341-1350
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• 1992

본 연구에서는 시험 재료의 특성을 가장 잘 나타낼 수 있으며 시험 자료밖의 범위에서도 물리적 타당성을 갖는 변형 에너지 함수를 찾기 위한 방법의 일환으로 기 존의 여러 방법들을 이용하여 동일한 시험 자료에 적용함으로써, Rivlin 형태와 Ogden 형태에 대한 재 조사를 실시하였다. 그리고 비선형 대변형 재료의 실제 사용 예로서 앞에서 언급한 타이어용 고무를 들 수 있는데 이들에 대한 변형 에너지 함수를 결정하 기 위한 시험자료는 실제적 문제때문에 단순 인장과 같은 비교적 간단한 경우에 대하 여 한정되어 있으므로, 이러한 자료로부터 여러 종류의 하중 상태를 예측할 수 있는 변형 에너지 함수를 찾고자 하였다.

• Smart Structures and Systems
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• 제5권1호
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• pp.23-42
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• 2009

In order to reduce vibration or to control shape of structures made of metal or composites, piezoelectric materials have been extensively used since their discovery in 1880's. A recent trend is also seen to apply piezoelectric materials to flexible structures made of rubber-like materials. In this paper a non-linear finite element model using updated Lagrangian (UL) approach has been developed for static analysis of rubber-elastic material with surface-bonded piezoelectric patches. A compressible stain energy function has been used for modeling the rubber as hyperelastic material. For formulation of the nonlinear finite element model a twenty-node brick element is used. Four degrees of freedom u, v and w and electrical potential ${\varphi}$ per node are considered as the field variables. PVDF (polyvinylidene fluoride) patches are applied as sensors/actuators or sensors and actuators. The present model has been applied to bimorph PVDF cantilever beam to validate the formulation. It is then applied to study the smart rubber components under different boundary and loading conditions. The results predicted by the present formulation are compared with the analytical solutions as well as the available published results. Some results are given as new ones as no published solutions available in the literatures to the best of the authors' knowledge.

• Structural Engineering and Mechanics
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• 제57권6호
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• pp.1107-1124
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• 2016

Consistent finite strain Plate constitutive relations are derived based on a hyperelastic formulation for an isotropic material. Plate equilibrium equations under finite strain are derived following a static kinematic approach. Three Euler angles and four shear angles, based on Timoshenko beam theory, represent the kinematics of the deformations in the plate cross section. The Green deformation tensor has been expressed in term of a deformation tensor associated with the deformation and stretches of an embedded plate element. Buckling formulation includes the in-plane axial deformation prior to buckling and transverse as well as in-plane shear deformations. Numerical results for a simply supported thick plate under uni-axial compression force are presented.

• Elastomers and Composites
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• 제51권4호
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• pp.308-316
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• 2016

The rubber materials are widely used in automobile industry due to their capability of a large amount of elastic deformation under a force. Current trend of design process requires prediction of functional properties of parts at early stage. The behavior of rubber material can be modeled using strain energy density function. In this study, five different strain energy density functions - Neo-Hookean model, Reduced Polynomial $2^{nd}$ model, Ogden $3^{rd}$ model, Arruda Boyce model and Van der Waals model - were used to estimate the behavior of carbon black added natural rubber under uniaxial load. Two kinds of tests - uniaxial tension test and biaxial tension test - were performed and used to correlate the coefficients of the strain energy density function. Numerical simulations were carried out using finite element analysis and compared with experimental results. Simulation revealed that Ogden $3^{rd}$ model predicted the behavior of carbon added natural rubber under uniaxial load regardless of experimental data selection for coefficient correlation. However, Reduced Polynomial $2^{nd}$, Ogden $3^{rd}$, and Van der Waals with uniaxial tension test and biaxial tension test data selected for coefficient correlation showed close estimation of behavior of biaxial tension test. Reduced Polynomial $2^{nd}$ model predicted the behavior of biaxial tension test most closely.