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  • 1995-1999  (7)
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Material
Years
Year
  • 1
    Electronic Resource
    Electronic Resource
    Tangent modulus matrix for finite element analysis of hyperelastic materials (1995)
    Springer
    Acta mechanica 112 (1995), S. 187-201 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary In this study, using the VEC operator [1], compact expressions are formulated for the tangent modulus matrix of hyperelastic materials, in particular elastomers, using Lagrangian coordinates. Compressible, incompressible, and near-compressible materials are considered. Expressions are obtained for the corresponding finite element tangent stiffness matrices. It is observed that the incremental stress-strain relations should be considered anisotropic. Numerical procedures based on Newton iteration are sketched. The limiting case of small strain is developed. Finally, the tangent modulus matrix is presented for the Mooney-Rivlin material, with application to the rubber rod element.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01177488
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Incremental finite element equations for thermomechanical response of elastomers: Effect of boundary conditions including contact (1998)
    Springer
    Acta mechanica 128 (1998), S. 81-104 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary The present investigation concerns the solution of nonlinear finite element equations by Newton iteration, for which the Jacobian matrix plays a central role. In earlier investigations [1], [2], a compact expression for the Jacobian matrix was derived for incremental finite element equations governing coupled thermomechanical response of near-incompressible elastomers. A fully Lagrangian formulation was adopted, with three important restrictions: (a) the traction and heat flux vectors were referred to theundeformed coordinates; (b) Fourier's law for heat conduction was expressed in terms of theundeformed coordinates; and (c) variable contact was not considered. In contrast, in the current investigation, the boundary conditions and Fourier's law of heat conduction are referred to thedeformed coordinates, and variablethermomechanical contact is modeled. A thermohyperelastic constitutive equation introduced by the authors [3] is used and is specialized to provide a thermomechanical, near-incompressible counterpart of the two-term Mooney-Rivlin model. The Jacobian matrix is now augmented with several terms which are derived in compact form using Kronecker product notation. Calculations are presented on a confined rubber O-ring seal submitted to force and heat.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01463161
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Finite element method for thermomechanical response of near-incompressible elastomers (1997)
    Springer
    Acta mechanica 124 (1997), S. 181-198 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary The present study addresses finite element analysis of the coupled thermomechanical response of near-incompressible elastomers such as natural rubber. Of interest are applications such as seals, which often involve large deformations, nonlinear material behavior, confinement, and thermal gradients. Most published finite element analyses of elastomeric components have been limited to isothermal conditions. A basic quantity appearing in the finite element equation for elastomers is thetangent stiffness matrix. A compact expression for theisothermal tangent stiffness matrix has recently been reported by the first author, including compressible, incompressible, and near-incompressible elastomers. In the present study a compact expression is reported for the tangent stiffness matrix under coupled thermal and mechanical behavior, including pressure interpolation to accommodate near-incompressibility. The matrix is seen to have a computationally convenient structure and to serve as a Jacobian matrix in a Newton iteration scheme. The formulation makes use of a thermoelastic constitutive model recently introduced by the authors for near-incompressible elastomers. The resulting relations are illustrated using a near-incompressible thermohyperelastic counterpart of the conventional Mooney-Rivlin model. As an application, an element is formulated to model the response of a rubber rod subjected to force and heat.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01213024
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    On the tangent modulus tensor in hyperelasticity (1998)
    Springer
    Acta mechanica 131 (1998), S. 121-132 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary The tangent modulus tensor, denoted asD, plays a central role in nonlinear finite element simulation of elastomeric components such as seals. It is derived from the strain energy functionw for isotropic elastomers. Using Kronecker product notation, a compact expression forD has been derived in Nicholson [1] and Nicholson and Lin [2] for invariant-based strain energy functions such as the Mooney-Rivlin model. In the current investigation, a corresponding expression is derived for stretch ratio-based strain energy functions such as the Ogden model. Compressible, incompressible and near-incompressible elastomers are addressed. The derived expressions are considerably more elaborate than their counterparts for invariant based models. As illustration,D is evaluated and presented for the torsion of a natural rubber shaft described by a three term Ogden model, using coefficients reported by Treloar.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01178249
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Extensions of Kronecker product algebra with applications in continuum and computational mechanics (1999)
    Springer
    Acta mechanica 136 (1999), S. 223-241 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary Kronecker product algebra is widely applied in control theory. However, it does not appear to have been commonly applied to continuum and computational mechanics (CCM). In broad terms the goal of the current investigation is to extend Kronecker product algebra so that it can be broadly applied to CCM. Many CCM quantities, such as the tangent compliance tensor in finite strain plasticity, are very elaborate or difficult to derive when expressed in terms of tensor indicial or conventional matrix notation. However, as shown in the current article, with some extensions Kronecker product algebra can be used to derive compact expressions for such quantities. In the following, Kronecker product algebra is reviewed and there are given several extensions, and applications of the extensions are presented in continuum mechanics, computational mechanics and dynamics. In particular, Kronecker counterparts of quadratic products and of tensor outer products are presented. Kronecker operations on block matrices are introduced. Kronecker product algebra is extended to third and fourth order tensors. The tensorial nature of Kronecker products of tensors is established. A compact expression is given for the differential of an isotropic function of a second-order tensor. The extensions are used to derive compact expressions in continuum mechanics, for example the transformation relating the tangent compliance tensor in finite strain plasticity in undeformed to that in deformed coordinates. A compact expression is obtained in the nonlinear finite element method for the tangent stiffness matrix in undeformed coordinates, including the effect of boundary conditions prescribed in the current configuration. The aforementioned differential is used to derive the tangent modulus tensor in hyperelastic materials whose strain energy density is a function of stretch ratios. Finally, block operations are used to derive a simple asymptotic stability criterion for a damped linear mechanical system in which the constituent matrices appear explicitly.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01179259
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Theory of thermohyperelasticity for near-incompressible elastomers (1996)
    Springer
    Acta mechanica 116 (1996), S. 15-28 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary Elastomers are often used in hot and confining environments in which thermomechanical properties are important. It appears that published constitutive models for elastomers are mostly limited to isothermal conditions. In this study, athermohyperelastic constitutive model for near-incompressible elastomers is formulated in terms of the Helmholtz free energy density ϕ. Shear and volume aspects of the deformation are decoupled. Thermomechanical coupling occurs mostly as thermal expansion. Criteria for thermodynamic stability are derived in compact form. As illustration, a particular expression for ϕ is presented which represents the thermomechanical counterpart of the conventional two-term incompressible Mooney-Rivlin model. It is used to analyze several adiabatic problems in a rubber rod.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01171417
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Tangent modulus tensor in plasticity under finite strain (1999)
    Springer
    Acta mechanica 134 (1999), S. 199-215 
    Details
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary The tangent modulus tensor, denoted as $$\mathfrak{D}$$ , plays a central role in finite element simulation of nonlinear applications such as metalforming. Using Kronecker product notation, compact expressions for $$\mathfrak{D}$$ have been derived in Refs. [1]–[3] for hyperelastic materials with reference to the Lagrangian configuration. In the current investigation, the corresponding expression is derived for materials experiencing finite strain due to plastic flow, starting from yield and flow relations referred to the current configuration. Issues posed by the decomposition into elastic and plastic strains and by the objective stress flux are addressed. Associated and non-associated models are accommodated, as is “plastic incompressibility”. A constitutive inequality with uniqueness implications is formulated which extends the condition for “stability in the small” to finite strain. Modifications of $$\mathfrak{D}$$ are presented which accommodate kinematic hardening. As an illustration, $$\mathfrak{D}$$ is presented for finite torsion of a shaft, comprised of a steel described by a von Mises yield function with isotropic hardening.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01312655
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    Paper (German National Licenses)
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