Finite element approximations for near-incompressible and near-inextensible transversely isotropic bodies

This work comprises a detailed theoretical and computational study of the boundary value problem for transversely isotropic linear elastic bodies. General conditions for well-posedness are derived in terms of the material parameters. The discrete form of the displacement problem is formulated for conforming finite element approximations. The error estimate reveals that anisotropy can play a role in minimising or even eliminating locking behaviour for moderate values of the ratio of Young's moduli in the fibre and transverse directions. In addition to the standard conforming approximation, an alternative formulation, involving under-integration of the volumetric and extensional terms in the weak formulation, is considered. The latter is equivalent to either a mixed or a perturbed Lagrangian formulation, analogously to the well-known situation for the volumetric term. A set of numerical examples confirms the locking-free behaviour in the near-incompressible limit of the standard formulation with moderate anisotropy, with locking behaviour being clearly evident in the case of near-inextensibility. On the other hand, under-integration of the extensional term leads to extensional locking-free behaviour, with convergence at superlinear rates.     

Theory of thermohyperelasticity for near-incompressible elastomers

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 . 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 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.List of symbols A _i matrices appearing inD - c _e, _e, c _e specific heat at constant strain - C Cauchy strain tensor - C _R reduced Cauchy strain tensor - C ₁,C ₂ coefficients of Mooney-Rivlin model - c vectorial counterpart ofC: VEC (C) - c ₂ vectorial counterpart ofC ²: VEC (C ²) - D isothermal tangent stiffness matrix - e vectorial counterpart of : VEC () - deviatoric Lagrangian strain tensor - e _R reduced deviatoric Lagrangian strain tensor - e volume strain - e _T reduced volume strain - f thermal expansion function,=[1+(T–T ₀)/3]^–1 - F, F _T deformation gradient tensor - F _R reduced deformation gradient tensor - H Hessian matrix for the Gibbs free energy density - H related toH - I ₁,I ₂,I ₃ invariants ofC - I _1R,I _2R,I _3R invariants ofC _R - I, I ₉ identity matrix - i vectorial counterpart ofI: VEC (I) - J determinant ofF - J _R determinant ofF _R - J _T determinant ofF _T - J ₁,J ₂ invariants of /J _{R 2} ³ - J _1R,J _2R invariants of _R /R _J ^2/3 - k thermal conductivity coefficient - K ₁,K ₂,K ₃ invariants of /J ^2/3 - K _1R,K _2R,K _3R invariants ofe _R/J^2/3 - p hydrostatic pressure - s vectorial counterpart of stress : VEC () - s isotropic stress - T (absolute) temperature - T ₀ reference temperature - conventional (isothermal) strain energy density (per unit volume) - volumetric thermal expansion coefficient - thermal expansion vector - strain, Lagrangian strain - entropy density - isothermal bulk modulus - Lagrange multiplier - _i extension ratio - shear modulus - stability coefficient - mass density - stress, 2nd Piola-Kirchhoff stress - _i principal stress - Cauchy stress - _d deviatoric Cauchy stress - , _{M C T 0} Helmholtz free energy density - _i /I _i - _ij ²/I _i I _j - Gibbs free energy density - (.) variational operator - VEC (.) vectorization operator - operator for Kronecker product     

Incremental finite element equations for thermomechanical response of elastomers: Effect of boundary conditions including contact

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.List of symbols A contact area - A _i - A _MM coefficient matrix for foundation model - a _TM coefficient vector for foundation model - a _MT coefficient vector for foundation model - a _TT coefficient scalar for foundation model - B _t,B _q matrices related to boundary terms due to large deformations - B _{c t} ,B _{c q} matrices related to boundary terms due to variable thermomechanical contact - B _{f t} ,B _{f q} matrices related to boundary terms due to nonlinear foundation model - B _{c MM} submatrix inB _{c t} - B _{c TM} submatrix inB _{c q} - B _{c TT} submatrix inB _{c q} - B _{f MM} submatrix inB _{f t} - B _{f TM} submatrix inB _{c q} - B _{f TT} submatrix inB _{f q} - B _{t MM} submatrix inB _t - B _{q TM} submatrix inB _q - B _{q TT} submatrix inB _q - C Cauchy-Green strain tensor - C ₁,C ₂ constants in strain energy density functions for the elastomer - c vec (C) - c ₂ vec (C ²) - c _e specific heat at constant strain - _e - c _hi parameters in contact heat conductance model - D _nl stiffness matrix due to geometric nonlinearity - D _T isothermal tangent modulus matrix - D _{T e} modulus matrix at constantT ande - D _T tangent modulus matrix at constantT and - d g _i nodal vectors related to prescribed traction and heat flux - d g _{c q} nodal vector related to thermal contact - d g _{c t} nodal vector related to contact traction - d g _{f q} nodal vector related to heat flux - d g _{f q 0} nodal vector related to heat flux - d g _{f q 0} nodal vector related to heat flux - d g _{f t} nodal vector related to traction - d g _{f t 0} nodal vector related to prescribed traction - d g _{f q 0} nodal vector related to prescribed traction - d g _{q 0} nodal vector related to heat flux - d g _{q 0} nodal vector related to heat flux - d g _{t 0} nodal vector related to traction - d g _{t 0} nodal vector related to traction - d(n ^TT} q) prescribed heat flux increment - d tt} prescribed traction increment - e vec() - e _d vec( _d ) - e _r shift parameter in elastic foundation model - f, f(T) thermal expansion function, = - F deformation gradient tensor - f _c nodal vector from the contact traction - g gap function - g _i nodal vectors related to mechanical and thermal loads - g _M nodal vector related to mechanical load - g _{i T} nodal vectors related to thermal terms - h time step - h _n - I _i invariants ofC - I ₉ 9×9 identity tensor - I 3×3 identity tensor - i vectorial counterpart ofI: vec(I) - J Jacobian matrix for Newton iteration - J determinant ofF - k thermal conductivity - k _H high stiffness in elastic foundation model - k _L low stiffness in elastic foundation model - K(g) stiffness function for elastic foundation model - K tangent stiffness matrix - K _MM tangent stiffness submatrix - K _MT tangent stiffness submatrix - K _MP tangent stiffness submatrix - K _PP tangent stiffness submatrix - K _PT tangent stiffness submatrix - K _TT tangent stiffness submatrix - M ₁ strain-displacement matrix - M ₂ strain-displacement matrix - m unit vector normal to target surface - N interpolation matrix - n vector normal to current surface - n ₀ vector normal to undeformed surface - n _i - p (true) pressure - Q heat rate across contact surface - q heat flux vector referred to the deformed configuration - q ₀ heat flux vector referred to the undeformed configuration - qq} prescribed heat flux - r residual vector in combined equilibrium equation - r _M residual vector from mechanical equilibrium - r _T residual vector from thermal equilibrium - r residual vector from near-incompressibility constraint - R matrix of heat conduction in domain - S surface in current configuration - s vec () - S ₀ surface in undeformed configuration - S _c candidate contact surface in current configuration - S _{c 0} candidate contact surface in undeformed configuration - S _{f M} surface corresponding to nonlinear foundation in current configuration - S _{f M 0} surface corresponding to nonlinear foundation in undeformed configuration - S _{f T} surface corresponding to nonlinear foundation in current configuration - S _{f M 0} surface corresponding to nonlinear foundation in undeformed configuration - S _T prescribed temperature boundary surface in current configuration - S _{T 0} prescribed temperature boundary surface in undeformed configuration - S _t prescribed traction boundary surface in current configuration - S _{t 0} prescribed traction boundary surface in undeformed configuration - S _q prescribed heat flux boundary surface in current configuration - S _{q 0} prescribed heat flux boundary surface in undeformed configuration - S _u prescribed displacement boundary surface in current configuration - S _{u 0} prescribed displacement boundary surface in undeformed configuration - T current temperature - T ₀ reference temperature - T _r temperature of rigid foundation - t time - t _n contact traction normal to contact surface - t _{n, n} solution value oft _n at thenth load step - t _ti components of tangential contact traction vector - t traction referred to current configuration - tt} prescribed traction - u displacement vector - v combined vector of nodal parameters - V ₀ volume in undeformed configuration - V volume in deformed configuration - w - x position vector in deformed configuration - X position vector in undeformed configuration - y possible contact point in the target surface - volumetric thermal expansion coefficient - _i parameters in metal-metal thermal contact models - _hi coefficients in thermal contact model - _k coefficient in elastic foundation model - interpolation matrix for strain field - _T interpolation matrix for thermal gradient: ₀ T - - vector of nodal displacements - Lagrangian strain tensor - _d deviatoric portion of Lagrangian strain tensor - interpolation function for - entropy density - vector of nodal temperatures - þ isothermal bulk modulus - surface area factor - interpolation function forT - temperature-adjusted pressure, - mass density in the deformed configuration - ₀ mass density in the undeformed configuration - 2nd Piola-Kirchhoff stress tensor - Helmholtz free energy density function - _M Helmholtz free energy density function - ₀ Helmholtz free energy density function - _i - _i - _ij - _ij - nodal vector for pressure field - near-incompressibility constraint function - the target surface equation: (y)=0 - (·) variational operator - vec(·) vectorization operator - symbol for Kronecker product of two tensors - tr(·) trace of a tensor - det(·) determinant of a tensor - divergence operator with respect to current configuration - ₀ divergence operator with respect to undeformed configuration - · the norm of vector     

Finite element method for piezoelectric vibration

A finite element formulation which includes the piezoelectric or electroelastic effect is given. A strong analogy is exhibited between electric and elastic variables, and a ‘stiffness’ finite element method is deduced. The dynamical matrix equation of electroelasticity is formulated and found to be reducible in form to the well-known equation of structural dynamics, A tetrahedral finite element is presented, implementing the theorem for application to problems of three-dimensional electroelasticity.     

Finite element method for orthotropic micropolar elasticity

The total potential energy for a body composed of an anisotropic micropolar linear elastic material is developed and used to formulate a displacement type finite element method of analysis. As an example of this formulation triangular plane stress (and plane couple stress) elements are used to analyze several problems. The program is verified by computing the stress concentration around a hole in an isotropic micropolar material for which an exact analytical solution exists. Several anisotropic material cases are presented which demonstrate the dependence of the stress concentration factor on the micropolar material parameters.     

Finite element method for spectral modelling of tides

A spectral method for modelling of tides is proposed and applied to the calculation of the M₂ component of the tide in the English Channel. The classical non-linear hyperbolic problem of long wave propagation in shallow waters is transformed into a sequence of elliptic problems by looking at a multiperiodic solution the frequencies of which are previously known. The method is based upon a perturbation technique, but the principal difficulties arise from the non-analytical form of the quadratic friction term: the main conclusions of the corresponding study only are given here, because the purpose of this paper is to present a practical application of the method to the calculation of a tidal component by the finite element method. The variational formulation of the problem is presented, and the finite element package used is described. Some results are given for the M₂ tide in the English Channel: cotidal maps and current fields.     

Finite element method for analysis of frozen earth structures

A finite element model employing the “viscoplastic flow rule” and the von Mises yield function was developed for frozen soil in the multiaxial stress state. A weighting procedure, which evaluates “effective creep parameters”, was proposed to account for the substantial differences in creep parameters in tension and compression. For reinforced frozen earth structures, bond behavior between the reinforcement and frozen soil was modeled by bond interface elements with nonlinear properties inferred from experiments. Numerical examples include the creep behavior of an open excavation supported by plain and by reinforced frozen walls.     

A multiscale finite element method for modeling fully coupled thermomechanical problems in solids

This article proposes a two‐scale formulation of fully coupled continuum thermomechanics using the finite element method at both scales. A monolithic approach is adopted in the solution of the momentum and energy equations. An efficient implementation of the resulting algorithm is derived that is suitable for multicore processing. The proposed method is applied with success to a strongly coupled problem involving shape‐memory alloys.Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element formulation for a digital image correlation method

A finite element formulation for a digital image correlation method is presented that will determine directly the complete, two-dimensional displacement field during the image correlation process on digital images. The entire interested image area is discretized into finite elements that are involved in the common image correlation process by use of our algorithms. This image correlation method with finite element formulation has an advantage over subset-based image correlation methods because it satisfies the requirements of displacement continuity and derivative continuity among elements on images. Numerical studies and a real experiment are used to verify the proposed formulation. Results have shown that the image correlation with the finite element formulation is computationally efficient, accurate, and robust.     

Finite element method for non-linear forced vibrations of circular plates

Geometric non-linearities for large amplitude free and forced vibrations of circular plates are investigated. In-plane displacement and in-plane inertia are included in the formulation. The finite element method is used. An harmonic force matrix for non-linear forced vibration analysis is introduced and derived. Various out-of-plane and in-plane boundary conditions are considered. The relations of amplitude and frequency ratio for different boundary conditions and various load conditions are presented.     

A thermomechanical interface element formulation for finite deformations

Interfacial layers are thermally and mechanically described in the presented approach. The combination of temperature evolution and mechanical loading influences significantly the deformation and thermal behavior. A consistent framework is derived from principle thermodynamical laws and balance equations. The approach is incorporated in the finite element analysis framework, wherein the unknown temperature- and displacement fields are obtained, e.g. by a Newton-type solution scheme. The derived finite element equations are linearized and a fully coupled interface element formulation is given with respect to thermomechanical residuals and stiffnesses. Bonds between the opening crack flanks are the main mechanisms of the delamination process. These bonds can be of different nature, depending on the bulk material. They are constitutively described in the presented approach in terms of transmission of tractions and heat. Numerical examples are shown in order to demonstrate the predictive capabilities of the thermomechanical interface element.     

Finite element analysis of non-linear static and dynamic response

The paper presents the theoretical and computational procedures which have been applied in the design of a general purpose computer code for static and dynamic response analysis of non-linear structures. A general formulation of the incremental equations of motion for structures undergoing large displacement finite strain deformation is first presented. These equations are based on the Lagrangian frame of reference, in which constitutive models of a variety of types may be introduced. The incremental equations are linearized for computational purposes, and the linearized equations are discretized using isoparametric finite element formulation. Computational techniques, including step-by-step and iterative procedures, for the solution of non-linear equations are discussed, and an acceleration scheme for improving convergence in constant stiffness iteration is reviewed. The equations of motion are integrated using Newmark's generalized operator, and an algorithm with optional iteration is described. A solution strategy defined in terms of a number of solution parameters is implemented in the computer program so that several solution schemes can be obtained by assigning appropriate values to the parameters. The results of analysis of a few non-linear structures are briefly discussed.     

Finite element approximations for quasi-Newtonian flows employing a multi-field GLS method

This article concerns stabilized finite element approximations for flow-type sensitive fluid flows. A quasi-Newtonian model, based on a kinematic parameter of flow classification and shear and extensional viscosities, is used to represent the fluid behavior from pure shear up to pure extension. The flow governing equations are approximated by a multi-field Galerkin least-squares (GLS) method, in terms of strain rate, pressure and velocity (D-p-u). This method, which may be viewed as an extension of the formulation for constant viscosity fluids introduced by Behr et al. (Comput Methods Appl Mech 104:31–48, 1993), allows the use of combinations of simple Lagrangian finite element interpolations. Mild Weissenberg flows of quasi-Newtonian fluids—using Carreau viscosities with power-law indexes varying from 0.2 to 2.5—are carried out through a four-to-one planar contraction. The performed physical analysis reveals that the GLS method provides a suitable approximation for the problem and the results are in accordance with the related literature.     

Finite element method for solving mhd flow in a rectangular duct

A finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady MHD (magnetohydrodynamic) flow through a rectangular pipe having arbitrarily conducting walls. Linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and Hartmann number M (M ≤ 100). A special solution procedure has been used for the resulting block tridiagonal system of equations. Computations have been carried out for several values of Hartmann number (5 ≤ M ≤ 100) and wall conductivity. It is also found that, if the wall conductivity increases, the flux decreases. The same is the effect of increasing the Hartmann number. Selected graphs are given showing the behaviour of the velocity field and induced magnetic field.     

Finite heterogeneous element method using sliced microstructures for linear elastic analysis

A new finite heterogeneous element consisting of sliced microstructures (FHES) is applied in a multi?scale technique. The FHES represents a heterogeneous material with microscopic constituents without homogenization or microscopic finite element analysis. A representative volume element extracted from a heterogeneous structure is thinly sliced. Each slice is modeled as a combined spring to calculate properties of the FHES. Each FHES has the same number of nodes as an ordinary finite element, and the macroscopic analysis cost is the same as that for ordinary finite element analysis. However, the FHES retains information about the microscopic material layout (i.e., the distribution of a material's property) in itself that is lost during homogenization. In the proposed approach, materials are not homogenized. The FHES does not have a constant (homogenized) material property and can ‘change stiffness’ depending on its deformation behavior. This reduces error due to coarse?graining and allows us to calculate the macroscopic deformation behavior with sufficient accuracy even if a large gradient of strain is generated in the macroscopic field. The novelty of the research is the development of rational heterogeneous finite elements. The paper presents the theory behind the FHES and its practical application to a linear elastic problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite element method in magnetohydrodynamic channel flow problems

The finite element method has been applied to the steady-state fully developed magnetohydrodynamic channel flow of a conducting fluid in the presence of transverse magnetic field. Simple elements have been used to obtain the numerical values of velocity and induced magnetic field. To test the efficiency of the method, three different geometries, viz., rectangle, circle and triangle, are taken as the section of the pipe whose walls are non-conducting. Comparison is made with those cases in which exact solutions are available. Apart from giving good results, the FEM makes it possible to solve the problem for a pipe with arbitrary cross-section which was not possible by the other methods.     

Finite element method for in-plane vibrations of rotating Timoshenko rings and sectors

A simple and logical finite element formulation is presented for the analysis of rings and sectors, with Timoshenko effects. The elemental properties are derived from the governing differential equation of motion using the Galerkin method. A quintic polynomial, satisfying the compatibility of derivatives up to second order, has been used for the ring finite element. The interpolating function is the same as for a thin ring. The efficiency of the formulation has been illustrated by the numerical results presented.     

Finite element method simulation of the molding process for thermal nano-imprint lithography

We made a numerical study on the deformation of a viscoelastic polymethyl methacrylene (PMMA) resist when a rigid SiO2 stamp with a rectangular line pattern is imprinted into the PMMA resist for thermal nano-imprint lithography (NIL). The stress distribution in the polymer resist during the molding process is calculated by a finite element method (FEM). Our simulation results reveal that the asymmetric von Mises stress is distributed over the polymer around the external line, which seems to be due to the squeezing flow under the flat space. The stress seems to be concentrated at the sidewall close to the centerline of the whole structure. Our simulation also reveals that a micro gap is formed between the replicated structure and the outer wall of the mold.