A.F. Saleeb

On the thermodynamic framework of generalized coupled thermoelastic-viscoplastic-damage modeling

A complete potential based framework utilizing internal state variables is put forth for the derivation of reversible and irreversible constitutive equations. In this framework, the existence of the total (integrated) form of either the (Helmholtz) free energy or the (Gibbs) complementary free energy are assumed a priori. Two options for describing the flow and evolutionary equations are described, wherein option one (the fully coupled form) is shown to be over restrictive and the second option (the decoupled form) provides significant flexibility. As a consequence of the decoupled form, a new operator, that is, the compliance operator, is defined, which provides a link between the assumed Gibbs and complementary dissipation potential and ensures a number of desirable numerical features, for example, the symmetry of the resulting consistent tangent stiffness matrix. An important conclusion reached is that although many theories in the literature do not conform to the general potential framework outlined, it is still possible in some cases, by slight modifications of the employed forms, to restore the complete potential structure.

On the hybrid-mixed formulation of C 0 curved beam elements

Abstract Two C 0 curved beam elements based on the hybrid-mixed formulation are studied in the form of membrane-shear locking, mesh convergence, and stress predictions. At the element level, both the displacement and stress fields are approximated separately. The stress parameters are then eliminated from the stationary condition of the Hellinger-Reissner variational principle so that the standard stiffness equations are obtained. The stress functions are chosen from two important considerations: (i) kinematic deformation modes must be avoided, and (ii) the constraint index counting of the element, when applied to limiting cases, must be equal to or greater than one. Based on these considerations, two curved beam elements are derived by including the effect of shear deformation and with linear and quadratic displacement fields. The elements are found to be lock-free for thin-walled beams. Several numerical examples are given to demonstrate the performance of the two curved elements.

A quadrilateral shell element using a mixed formulation

A simple quadrilateral shell element consisting of five nodes, four corner nodes and a central node, is developed for linear elastic analysis of thin as well as moderately thick shells. Based on a modified Hellinger-Reissner principle, finite element equations are derived from the assumed displacement and strain fields. By carefully choosing appropriate strain terms, all kinematic deformation modes are suppressed. Although the present element is similar to a displacement-based degenerated shell, no locking is experienced when it is applied to thin shell problems. Five examples are given to illustrate the analysis capability of the shell element. Numerical results indicate that the element shows fast mesh convergence and gives excellent stress predictions.

A general hereditary multimechanism-based deformation model with application to the viscoelastoplastic response of titanium alloys

Abstract The formulation of a general model for the hereditary behavior of materials, in the viscoelastic and viscoplastic regimes, is presented. In this, we utilize the complete-potential structure as a general framework, together with the notion of strain- and stress- partitioning in terms of separate contributions of several submechanisms (viscoelastic and viscoplastic) to the thermodynamic functions (stored energy and dissipation). Detailed numerical treatments are given for both (i) the implicit integration algorithm for the governing flow and evolutionary rate equations of the model, and (ii) the automated parameter-estimation methodology (using the software code COMPARE) for characterization. For illustration, a specific form of the model presented is characterized for the TIMETAL 21S material using a very comprehensive test matrix, including creep, relaxation, constant strain-rate tension tests, etc. Discussion of these correlations tests, together with comparisons to several other experimental results, are given to assess the performance and predictive capabilities of the present model as well as the effectiveness and practical utility of the algorithms proposed.

Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle

A mixed finite element method is presented for the large strain analysis of rubber-like materials, which are considered to be nearly incompressible. Two types of constitutive relations are included: generalized Rivlin and Ogdens models. The finite element equations are derived on the basis of a perturbed Lagrangian variational principle from which both the displacement and pressure fields are independently approximated by appropriate shape functions. A physically meaningful pressure parameter is introduced in the expression of complementary energy. In the paper, a special effort is made to split the deformation energy into two distinct parts: isochoric and hydrostatic parts. By doing this, a quadratic convergence rate of nonlinear iterative solution is achieved, particularly for problems deformed in the large strain range. The finite element equations are specialized for a two-dimensional 9-node Lagrange element with three-term pressure parameters. Five examples are given to demonstrate the application of the proposed numerical algorithm.

Generalized thin-walled beam models for flexural-torsional analysis

Abstract With non-uniform warping being an important mode of deformation, supplementary to the other six modes of stretching, shearing, twisting, and bending, we utilize a fairly comprehensive one-dimensional beam theory for the development of a simple finite element model for the analysis of arbitrary thin-walled beams under general loadings and boundary conditions. The formulation is valid for both open- and closed-type sections, and this is accomplished by using a kinematical description accounting for both flexural and warping torsional effects. To eliminate the shear/warping locking in this C0-element, a generalized mixed variational principle is utilized, in which both displacement and strain fields are approximated separately. In this, the strain parameters are of the interelement-independent type, and are therefore eliminated on the element level by applying the relevant stationarity conditions of the employed ‘modified’ Hellinger-Reissner functional, thus leading to the standard form of stiffness equations for implementation. A rather extensive set of numerical simulations are given to demonstrate the versatility of the models in practical applications involving usage of such components in their stand-alone forms as well as in plate/shell stiffening.

Deformation and life analysis of composite flywheel disk systems

Abstract In this study an attempt is made to put into perspective the problem of a rotating disk, be it a single disk or a number of concentric disks forming a unit. An analytical model capable of performing an elastic stress analysis for single/multiple, annular/solid, anisotropic/isotropic disk systems, subjected to pressure surface tractions, body forces (in the form of temperature-changes and rotation fields) and interfacial misfits is summarized. Results of an extensive parametric study are presented to clearly define the key design variables and their associated influence. In general the important parameters were identified as misfit, mean radius, thickness, material property and/or load gradation, and speed; all of which must be simultaneously optimized to achieve the ‘best’ and most reliable design. Also, the important issue of defining proper performance/merit indices (based on the specific stored energy), in the presence of multiaxiality and material anisotropy is addressed. These merit indices are then utilized to discuss the difference between flywheels made from PMC and TMC materials with either an annular or solid geometry. Finally two major aspects of failure analysis, that is the static and cyclic limit (burst) speeds are addressed. In the case of static limit loads, a lower (first fracture) bound for disks with constant thickness is presented. The results (interaction diagrams) are displayed graphically in designer friendly format. For the case of fatigue, a representative fatigue/life master curve is illustrated in which the normalized limit speed versus number of applied cycles is given for a cladded TMC disk application.

Finite element solutions of two-dimensional contact problems based on a consistent mixed formulation

Abstract A consistent mixed finite element method for solving two-dimensional contact problems is presented. Derivations of stiffness equations for contact elements are made from a perturbed Lagrangian variational principle. For a contact element, both the displacement and pressure fields are independently assumed. In order to achieve a consisent formulation, thus avoiding any numerical instability, the pressure function is assumed in such a way that all non-contact modes in deformations must be excluded. Stiffness equations for four-noded and six-noded contact elements are given. Four numerical examples are included to demonstrate the methodology.

A Fully Associative, Nonlinear Kinematic, Unified Viscoplastic Model for Titanium-Based Matrices

Specific forms for both the Gibbs and complementary dissipation potentials are chosen such that a complete (that is, fully associative) potential-based multiaxial, unified viscoplastic model is obtained. This model possess one tensorial internal state variable that is associated with dislocation substructure, with an evolutionary law that has nonlinear kinematic hardening, and both thermal- and strain-induced recovery mechanisms. A unique aspect of the present model is the inclusion of nonlinear hardening through the use of a compliance operator, derived from the Gibbs potential, in the evolution law for the back stress. This nonlinear tensorial operator is significant in that it allows both the flow and evolutionary laws to be fully associative (and therefore easily integrated) and greatly influences the multiaxial response under nonproportional loading paths. In addition to this nonlinear compliance operator, a new, consistent, potential-preserving, internal strain unloading criterion has been introduced to prevent abnormalities in the predicted stress-strain curves that are present with nonlinear hardening formulations during unloading and reversed loading of the external variables. Specification of an experimental program for the complete determination of the material functions and parameters for characterizing a metallic matrix, for example, TIMETAL 21S, is given. The experiments utilized are tensile, creep, and step-creep tests. Finally, a comparison of this model and a commonly used Bodner-Partom model is made on the basis of predictive accuracy and numerical efficiency.

A mixed element for laminated plates and shells

Formulation and numerical evaluation of a simple shear-flexible four-noded quadrilateral laminated composite plate/shell finite-element is presented. The element developed is based on a generalized mixed variational principle with independently assumed displacement and laminate internal strain fields. For the latter, a layer-number-independent polynomial interpolation is utilized, together with a judicious selection of the in-plane (spanwise) distributions for the strain components. This was facilitated by the use of a set of bubble functions as additional kinematic degrees of freedom, which was shown to be crucial for eliminating the locking phenomenon. For dynamic applications, a simplified lumped mass matrix is employed. Finally, an extensive number of critical test cases are given to access the elements preformance and to illustrate its effectiveness in static as well as vibration problems for anisotropic laminated plates and shells.