Finite elements on the basis of continuum mechanics

The formulation of finite element models on the basis of different variational principles is reviewed. The degrees-of-freedom of the elements are defined in an abstract way without the help of nodal points. In this manner it is possible to describe elements of arbitrary shape and accuracy. The formulation is confined to linear elasto-statics. For two-dimensional structures two hybrid element models are developed using Legendre polynomials on the element boundaries. Examples of plane stress problems are used to test the generation of convergence by increasing the accuracy of the elements vs. by increasing the number of elements.     

Finite rotations in dynamics of beams and implicit time-stepping schemes

We examine theoretical and computational aspects of three-dimensional finite rotations pertinent to the dynamics of beams. The model problem chosen for consideration is the Reissner beam theory capable of modelling finite strains and finite rotations in geometrically exact manner. Special emphasis is placed on clarifying the geometry aspects, finite rotation updates and the associated linearization procedure pertaining to different choices of rotation parameters. The latter is shown to play an important role in constructing the optimal implementation of a time-stepping scheme. © 1998 John Wiley & Sons, Ltd.     

Toward the description of uniaxial deformation of metals

Effectiveness of using the endochronic theory of plasticity for describing unidimensional deformation processes under cyclic loading is investigated. A method is developed for determining parameters of the equations of state which depend on the amplitude of plastic deformation and length of strain path. A sufficiently good agreement is shown between calculated and experimental data in describing the kinetics of the stress-strain state of specimens of steels 316 and 45 subjected to variable loads.Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev Polytechnic Institute. Translated from Problemy Prochnosti, No. 11, pp. 65–69, November, 1989.     

A fiber-bundle model for the continuum deformation of brittle material

The deformation of brittle material is primarily accompanied by micro-cracking and faulting. However, it has often been found that continuum fluid models, usually based on a non-Newtonian viscosity, are applicable. To explain this rheology, we use a fiber-bundle model, which is a model of damage mechanics. In our analyses, yield stress was introduced. Above this stress, we hypothesize that the fibers begin to fail and a failed fiber is replaced by a new fiber. This replacement is analogous to a micro-crack or an earthquake and its iteration is analogous to stick–slip motion. Below the yield stress, we assume that no fiber failure occurs, and the material behaves elastically. We show that deformation above yield stress under a constant strain rate for a sufficient amount of time can be modeled as an equation similar to that used for non-Newtonian viscous flow. We expand our rheological model to treat viscoelasticity and consider a stress relaxation problem. The solution can be used to understand aftershock temporal decay following an earthquake. Our results provide justification for the use of a non-Newtonian viscous flow to model the continuum deformation of brittle materials.     

Finite element solution for the continuum traffic equilibrium problems

In this paper, we consider a city with a highly compact Central Business District (CBD), and the commuters’ destinations from the CBD are dispersed over the whole city. The street network is approximated as a continuum and commuters’ movements in the city are measured by the flow intensity, and the local travel cost depends on the location and the traffic flow intensity. We extend the continuum user equilibrium problem to deal with the case of variable demand, in which the traffic demand from any destination in the city to the CBD is assumed to be a function of both the destination location and the total travel cost to the CBD. An equivalent mathematical model is formulated and proved to satisfy the user equilibrium conditions, which is then solved by a finite element solution algorithm. Numerical examples are given to illustrate the effectiveness of the proposed method. © 1998 John Wiley & Sons, Ltd.     

Hierarchical description of deformation in block copolymer TPEs

We report on the deformation behavior of commercially relevant lamellar and cylindrical tri-block copolymers poly (styrene-b-ethylene-co-butylene-b-styrene) (SEBS) with two different compositions. The structural changes that occur at various length scales have been studied using a simultaneous small- and wide-angle X-ray scattering (SAXS/WAXS) during uni-axial tensile deformation of the polygrain samples. SAXS provides information about changes that occur upon deforming the glassy cylindrical or lamellar PS domains. WAXS, on the other hand, is sensitive to the crystallographic structure of the rubbery mid-block. Deformation calorimetry has been used to determine the energetics involved. The combined results from the various techniques indicate that the deformation takes place in three stages. First, at small strains, dilation occurs in the rubbery phase. At intermediate strains, the hard lamellar or cylindrical domains undergo micro-buckling, which is associated with a downturn in the stress–strain curve. Finally, we interpret that at higher strains, the bent lamellar/cylindrical domains rotate in the stretching direction resulting in a significant shear on the rubbery mid-block. This in turn leads to strain-induced crystallization in these materials. Although we could not prove it by WAXS, deformation calorimetry (which is more sensitive than the WAXS) was utilized to show its presence.     

Mathematical description of extracontact deformation in the simple process of rolling

To describe the reduction in height in the prefocus zone, the article suggests functions of a special form and the conditions making it possible to find their parameters. That the choice of the function is substantiated is confirmed by comparison with experimental data.Translated from Problemy Prochnosti, No. 11, pp. 45–49, November, 1993.     

An analytic description of the creep deformation of metals in the ultrasonic field

The effect of superimposed ultrasonic vibration on the primary creep of metals is modeled in terms of the synthetic theory of irrecoverable deformation. We consider two sonication modes: (i) the ultrasound acts continuously during the deformation, and (ii) the ultrasound is periodically on and off. Whereas both cases show a significant increase in primary creep, the periodical sonication leads to higher deformation values. To catch the phenomenon of ultrasound-assisted creep, we extend the flow rule equation by a term that accounts for the process occurring on the microlevel of material induced by ultrasound.

     

A theoretical description of large viscoplastic shear deformation in metals

In this work, a new 3-dimensional viscoplastic model based on a previous plasticity theory is presented. The proposed constitutive model anticipates the contribution of the main features of plastic behavior, such as yielding, rate effect, isotropic and kinematic hardening, through a new approximation of the constitutive equation with a viscoplastic term, as well as a new consideration of the functional form of the rate of plastic deformation. A high accuracy simulation of shear experimental data at various rates and temperatures for a variety of materials, as well as the sign inversion of normal stress has been postulated.     

On the description of cyclic creep and rate dependent plastic deformation

Acta Mechanica - A constitutive model is proposed to describe cyclic creep and other rate dependent plastic deformation. The model is general in concept and thus applicable to a wide variety of...     

Application of the concept of finite displacements and rotations for determining nonlinear deformation of sloping shells

An algorithm for computing the coefficients of the first and second energy variations of a nonlinear system is formulated using local approximation of the deformations of shells and canonic representation of the energy. These energy variations are used in the iterative process of solution. Application of the method to calculate for sloping shells and shells with cuts under the action of concentrated and distributed loads is presented.Translated from Problemy Prochnosti, No. 3, pp. 113–116, March, 1990.     

Prediction of creep deformation in ductile two-phase alloys by a continuum mechanics model

The creep deformation of the ductile two-phase alloys was analysed on the basis of the continuum mechanics model which incorporated the projection concept proposed by Evans and Wilshire. The calculated creep curves were compared with the experimental ones in ferrjte-pearlite steels. It was found that the continuum mechanics model was able to predict the whole creep deformation process of the ductile two-phase alloys from the onset of creep loading to the final rupture, if the creep-deformation and creep-rupture data of the individual phases which constituted the two-phase alloys were known. A steady-state creep in the ductile two-phase alloys was predicted by the continuum mechanics model to occur, even if the constituent phases did not have the inherent steady-state creep. This was caused by the internal stresses arising from the creep-strength difference between second-phase and matrix in the two-phase alloys. This steady-state creep was observed in ferrite-pearlite steels during creep at 873 K. The predicted rupture life on the basis of the continuum mechanics model was correlated well with the experimental results in ferrite-pearlite steels, although the former was somewhat shorter than the latter under higher creep stresses. The continuum mechanics model was able to apply to the life prediction and the creep-strength design of the ductile two-phase alloys.     

Finite strain fracture of plates and shells with configurational forces and edge rotations

We propose a simple and efficient algorithm for FEM‐based computational fracture of plates and shells with both brittle and ductile materials on the basis of edge rotation and load control. Rotation axes are the crack front nodes, and each crack front edge in surface discretizations affects the position of only one or two nodes. Modified positions of the entities maximize the modified mesh quality complying with the predicted crack path (which depends on the specific propagation theory in use). Compared with extended FEM or with classical tip remeshing, the proposed solution has algorithmic and generality advantages. The propagation algorithm is simpler than the aforementioned alternatives, and the approach is independent of the underlying element used for discretization. For history‐dependent materials, there are still some transfer of relevant quantities between elements. However, diffusion of results is more limited than with tip or full remeshing. To illustrate the advantages of our approach, three prototype models are used: tip energy dissipation linear elastic fracture mechanics (LEFM), cohesive‐zone approaches, and ductile fracture. Both the Sutton crack path criterion and the path estimated by the Eshelby tensor are employed. Traditional fracture benchmarks, including one with plastic hinges, and newly proposed verification tests are solved. These were found to be very good in terms of crack path and load ∕ deflection accuracy. Copyright © 2013 John Wiley & Sons, Ltd.     

Isogeometric analysis for flexural behavior of composite plates considering large deformation with small rotations

A large deformation theory, so-called Green strains with small rotations, is proposed and employed for flexural analysis of composite plates. Isogeometric analysis cooperated with first-order shear deformation theory is used to derive finite element models. Strain-displacement relations in the sense of von-Kármán theory and the proposed theory are formulated. Shear locking phenomenon is avoided by using reduced integration technique. Newton–Raphson method is employed for nonlinear analysis procedure. Numerical examples, including isotropic and laminated composite plates under different boundary conditions, are investigated. The results have been verified with those available in the literature and show the advantages of the proposed strain theory.     

Finite plastic deformation for metals under torsion

Abstract

Making use of the concept of plastic spin, endochronic theory is applied to describe the torsional deformation in the finite strain range. Deformation of tubular specimens of 70:30 brass, Ni‐200 and Al‐1100 are discussed. Special attention is given to torsion of tubular specimens with constrained and unconstrained ends. In the finite strain range, axial stress or elongation may be observed in addition to the shear stress and strain, depending on the end condition of the tube.     

Finite element analysis of deformation of strain-softening materials

A finite element analysis of strain-softening materials is presented in which the shear band of prescribed thickness is assumed to exist within elements where maximal stress intensity is reached. The incremental stiffness matrix of the element is derived including shear band deformation. Examples presented in the Paper demonstrate that the load-displacement curve and the displacement field are not sensitive to the mesh size used in the solution.