Modeling large strain anisotropic elasto-plasticity with logarithmic strain and stress measures

In this paper we present a model and a fully implicit algorithm for large strain anisotropic elasto-plasticity with mixed hardening in which the elastic anisotropy is taken into account. The formulation is developed using hyperelasticity in terms of logarithmic strains, the multiplicative decomposition of the deformation gradient into an elastic and a plastic part, and the exponential mapping. The novelty in the computational procedure is that it retains the conceptual simplicity of the large strain isotropic elasto-plastic algorithms based on the same ingredients. The plastic correction is performed using a standard small strain procedure in which the stresses are interpreted as generalized Kirchhoff stresses and the strains as logarithmic strains, and the large strain kinematics is reduced to a geometric pre- and post-processor. The procedure is independent of the specified yield function and type of hardening used, and for isotropic elasticity, the algorithm of Eterovi? and Bathe is automatically recovered as a special case. The results of some illustrative finite element solutions are given in order to demonstrate the capabilities of the algorithm.     

Postprocessing of ADINA strains

A procedure has been developed for processing the strains available within ADINA to provide strain output in any desired orientation and to calculate and output strains on the surface of a 3-D solid shell element. To this end, proper additions and modifications have been made to ADINA to implement the procedure directly within the program to allow the processing and output of strains during the normal execution stage. This includes reading in additional input data (of minimum amount), transformation of strains from the global system to specified coordinate directions, the output of transformed strains, and the calculation and output of surface strains based on the interpolation functions available in ADINA. At the present time such capabilities are provided for a limited number of elements on a selective basis, in order to minimize the amount of computer storage required. Some typical structures have been analyzed to compute and output strains both at interior points and on exterior surfaces of elements, for a set of elements specified by the user during the input stage. Results demonstrate the feasibility of the procedure for the transformation and extrapolation of strains.     

A simplified method of solution for the short cycle creep-plasticity problem

This paper describes a method for estimating the long-term effects on structures under cyclic changes of loading. For loads less than a certain critical amplitude (shakedown limit), the stress in the structure will a symptote to a cyclic stationary state consisting of an elastic part in response to the cyclic loading, plus a system of self-equilibrating residuals constant in time. It is shown that corresponding to this cyclic stationary state, the creep energy dissipation per cycle of loading is a maximum. Instead of following the exact time history to reach this state, in this paper it is found by a procedure of successive approximations. It corrects the admissible residual stress distribution at the beginning of a cycle by the creep and plastic strains accumulated over an entire cycle, which are in general not compatible, and requires additional self-equilibrating stresses to give an elastic strain distribution such that the total strain satisfies compatibility. The steady state is reached when no further correction is necessary. Convergence may be accelerated by a suitable choice of initial starting value, and by an artificial choice of the cycle time for the best computational convenience, upon which the steady-state solution can be proved to be independent. The procedure is a powerful device to obtain the cyclic steady-state solution, which will give an upper bound to the creep deformation per cycle and may also be used to find the shakedown limit. The formulation of the procedure in conjunction with the finite element method is given in detail and results of a few examples of the analysis are shown.     

Semi-analytical solution of time-dependent electro-thermo-mechanical creep for radially polarized piezoelectric cylinder

History of strains, stresses, deformations and electric potentials of hollow cylinders made from PZT_5 have been investigated using successive elastic solution method. A differential equation containing creep strains for displacement is obtained. Since creep strains are time, temperature and stress dependent, the closed form solution cannot be found for this constitutive differential equation. Hence, a semi-analytical method is proposed. Electric potentials increase with time (similar to the radial stress histories), since it is induced by the radial stress histories during creep deformation of the cylinder, justifying industrial application of such a material as efficient actuators and sensors.     

Membrane locking and assumed strain shell elements

An explanation of membrane locking behaviour in shell elements and also the use of reduced and selective integration is described. To overcome the conflict between the locking and mechanism problems the author proposed the degenerated shell elements with assumed transverse shear and membrane strains. The location of sampling points for the assumed strain fields is given in the present work. In the formulation of the new elements, assumed transverse shear strains in the natural coordinate system are used to overcome the shear locking problem. Also, assumed membrane strains in the orthogonal curvilinear coordinate system are applied to avoid membrane locking behaviour. Some numerical tests are presented to illustrate the good performance of the assumed strain shell elements.     

Polar decomposition and appropriate strains and stresses for nonlinear structural analyses

Work conjugacy, objectivity, vector expressions, directions and limitations of major pairs of stress and strain measures used in geometrically nonlinear and elastoplastic analyses of structures are investigated. Polar decomposition is examined in detail. For geometrically nonlinear analysis, Jaumann strains and stresses are the most appropriate pair of measures because Jaumann strains are proved to be objective (invariant under rigid-body rotations) corotated engineering strains. It is shown that Jaumann strains can be easily derived by using a new concept of local displacements without using the complex polar decomposition procedure. Moreover, the use of Jaumann stresses and strains results in a direct correlation between energy and Newtonian approaches and makes all structural energy terms interpretable in terms of vectors. For elastoplastic analysis, corotated Cauchy stresses and corotated Eulerian strain rates are shown to be the most appropriate pair of measures. Moreover, if the deformation follows the minimum work path, it is proved that corotated Eulerian strain rates correspond to the simultaneous variation of true stretches at a fixed ratio along the three fixed principal material lines. The derived corotated Cauchy stresses and Eulerian strain rates are useful in analyzing elastoplastic, viscoelastic and viscoplastic materials in deformation processing, such as metal forming.     

Stress,buckling and vibration of hybrid bodies of revolution

The analysis is applicable to bodies of revolution composed of thin shell segments, thick segments and discrete rings. The thin shell segments are discretized by the finite difference energy method and the thick or solid segments are treated as assemblages of 8-node isoparametric quadrilateral finite elements of revolution. Suitable compatibility conditions are formulated through which these dissimilar segments are joined without introduction of large spurious discontinuity stresses. Plasticity and primary or secondary creep are included. Axisymmetric prebuckling displacements may be moderately large. The nonlinear axisymmetric problem is solved in two nested iteration loops at each load level or time step. In the inner loop the simultaneous nonlinear equations corresponding to a given tangent stiffness are solved by the Newton-Raphson method. In the outer loop the plastic and creep strains and tangent stiffness are calculated by a subincremental procedure. The linear response to nonaxisymmetric loading is obtained by superposition of Fourier harmonics. Many examples are given to demonstrate the scope of the computer program, BOSOR6, derived from the analysis and to illustrate certain stress concentration effects in shell-type structures which cannot adequately be treated with use of thin shell theory.     

Finite element elastic-plastic-creep analysis of two-dimensional continuum with temperature dependent material properties

A technique is presented for performing finite element elastic-plastic-creep analysis of two-dimensional continuum composed of material with temperature dependent elastic, plastic, and creep properties. The plastic analysis utilizes the Prandtl-Reuss flow equations assuming isotropic material properties and linear strain-hardening. A power creep flow law formulated by Odquist is used to determine the steady state creep strain rate. The plastic and creep flow laws are employed to derive a ‘softened’ plastic-creep stress-strain matrix. These modified stress-strain relations are then used to formulate the element stiffness matrix in the usual manner. The differences in the elastic, plastic, and creep properties of the material due to the temperature change during the increment result in the formation of pseudo stresses, which in turn lead to load terms that appear on the right hand side of the equilibrium equations. The load terms resulting from these pseudo stresses not only keep the solution on the temperature dependent stress-strain curve of the material, but also correct for the elastic ‘overshoot’ that occurs when an element changes from an elastic to a plastic state. The effect of large displacements is included by the formulation of the geometric stiffness matrix for each element being used in the computer code. With this procedure it becomes economically feasible to perform elastic-plastic-creep stress analysis of two-dimensional continuum subjected to transient thermal and mechanical loadings. Several examples of both elastic-plastic and creep analyses are presented, and the finite element solutions are compared to either other theoretical solutions or experiment.     

A universal integration algorithm for rate-dependent elastoplasticity

An algorithm is developed for integrating rate-dependent constitutive equations of elstoplasticity including isotropic and kinematic hardening, as well as thermal softening and non-coaxiality of the plastic strain rate and the driving stress. The method is unconditionally stable and accurate for large time steps and all possible ranges of rate-dependency. Under a constant loading rate the algorithm gives exact results at arbitrary step sizes for rate-independent materials without hardening, and in proportional loading for rate-independency with hardening, and linear viscosity without hardening. The present method is an extension of a recently proposed integration algorithm for stiff equations to domains of high rate-sensitivity like, for example, in power-law creep. The algorithm employs a plastic predictor-elastic corrector scheme, which, in general, requires less numerical effort in the return mapping process than the assumption of an elastic predictor. Numerical examples underline the efficiency of this integration algorithm in comparison to gradient techniques and an extended radial return method for rate-dependent plasticity.     

Calculation of turbine rotors in secondary creep range

Turbine rotors in power plants are exposed to triaxial stresses by centrifugal forces at high temperatures which induce long time creep effects. In the first step, we implemented an algorithm for centrifugal volume forces in ADINA. In the next step, we tested the numerical behaviour of the modified Newton-algorithm in ADINA within the long time secondary creep range for simplified examples. Isotropie strain hardening was assumed in most cases of creep calculations.

The estimate of creep properties is based on the minimum least square method. The numerical stability in creep calculations is dependent on the magnitude of time step size and on a good fit between creep law properties and the real experimental material data.

In the examples of turbine rotor models with rotational symmetry we were able to estimate the state of stress and strain under creep conditions for life time periods up to 2 × 10⁵ hr.     

A multi-director formulation for nonlinear elastic-viscoelastic layered shells

A C⁰ finite element formulation for nonlinear analysis of multi-layered shells comprised of elastic and viscoelastic layers is presented for applications involving small strains but finite rotations. The elastic and viscoelastic layers may occupy arbitrary layer locations and the formulation is applicable to thick and thin shells. The formulation utilizes a three-dimensional variational approach in which the layered shell is represented as a multi-director field. The incorporated kinematic theory describes, within individual layers, the effects of transverse shear and transverse normal strain to arbitrary orders in the layer thickness coordinate. Stresses are computed through the three-dimensional constitutive equations and the usual “zero normal stress” shell hypothesis is not employed. Sufficiently general constitutive equations for the viscoelastic layers are proposed in objective rate form and a product algorithm, based on an operator split in the complete set of constitutive equations, is used for the temporal integration of the rate equations. The definition of the tangent operator, used in Newton's method for the solution of the nonlinear equations, is derived consistently from the product algorithm. Observations on the use of reduced/selective integration in the presence of high order kinematics are made and a number of numerical examples are presented to illustrate the capability of the formulation.     

Error control in viscoelastic stress analysis

Time step monitoring algorithms are presented for controlling errors which arise during the numerical integration of a viscoelastic stress analysis. With the algorithms presented, a user specifies tolerable stress and creep strain errors and the algorithms monitor integration time steps to keep errors within the limits set. The effects of using numerical integration formulae, of stress error buildup and of creep strain error buildup are considered. The algorithms are tested on a radially thick ring problem.     

One point quadrature shell element with through-thickness stretch

A general purpose one point quadrature shell element accounting for through-thickness deformation is developed. In the shell, a complete 3-D constitutive law is introduced, leading to a 7-parameter theory which explicitly accounts for thickness change and also for a linear variation of thickness stretch. An interpolation scheme for the shell director is developed to avoid thickness locking. The developed shell element covers flexible warping behavior by using a local nodal coordinate system at each node, which is updated with second order accuracy. A physical stabilization scheme for zero energy modes is employed based on the decomposition of the strain field into constant and linear terms with respect to the natural coordinates. The rigid body projection is applied to treat rigid body rotations effectively. Linear and nonlinear patch tests including elasto-plasticity and contact are performed and the results are compared with analytical or previously reported results. The results are also compared with those of 5-parameter shell elements, in order to show that there is no significant deterioration in accuracy, especially for thin shell applications.     

An implicit incrementally objective formulation for the solution of elastoplastic problems at finite strain by the F.E.M.

The mechanical formulation presented in this paper is based on an incremental updated Lagrange procedure using the principle of virtual work at the end of each load increment and an implicit incremental flow rule obtained by an approximate time integration of the objective rate constitutive equations. The approximate time integration is carried out along a particular path in the deformation and rotation space. This path ensures the incremental objectivity and minimizes the equivalent strain over the increment among all the possible paths, consequently avoiding an artificial increase of the plastic equivalent strain during the interpolation. The mechanical formulation presented leads to a fixed set of nonlinear equations, whose unknowns are the nodal displacements of the structure. A numerical algorithm based on a quasi Newton-Raphson method is then proposed to solve this system. The separation of the mechanical formulation from the resolution algorithm ensures the path independence. Numerical tests are carried out for a material obeying an isotropic with work-hardening von Mises criterion and associated flow rule. Single element tests show that this approach gives a very accurate solution even when the strain increment reaches twenty times the elastic strain up to yield. A structural test on a beam measures the influence of the incremental objectivity on the displacements, the equivalent plastic strain and the stresses.     

Transverse shear stiffness of laminated anisotropic shells

The additional constitutive equations required by transverse shear deformation theory of anisotropic heterogeneous shells are derived without the usual assumption of thickness distribution for either transverse shear stresses or strains. The derivation is based on Taylor series expansions about a generic point for stress resultants and couples which identically satisfy plate equilibrium equations. These equations give the in-surface stress resultants and couples in terms of the transverse shear stress resultants at the point and arbitrary constants, which may be interpreted as redundant “forces”. Starting from these expressions, we derive statically correct expressions (in terms of the transverse shear stress resultants and redundants) of the following variables: (1) in-surface stresses, using the stretching-bending constitutive equations and the Kirchhoff distributions of in-surface strains, (2) transverse shear stresses, by integration in the normal direction of the three-dimensional equilibrium equations, and (3) the area density of transverse shear strain energy, by integration in the normal direction of the corresponding volumetric density. Finally, by applying Castigliano's theorem of least work, the shear strain energy is minimized with respect to the redundants, thereby leading to the desired constitutive equations. Corresponding transverse shear stiffnesses are presented for several laminated walls, and reasonable agreement is obtained between transverse shear deformation plate theory using these stiffnesses and exact three-dimensional elasticity solutions for the problem of cylindrical bending of a plate.     

Numerical analysis of anisotropic ductile continuum damage

The paper deals with the numerical analysis of large elastic–plastic deformation behavior of anisotropically damaged ductile solids based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. Estimates of the stress and strain histories are obtained from a straightforward numerical integration algorithm based on operator split methodology which employs an inelastic (damage–plastic) predictor followed by an elastic corrector step. The finite element method is used to approximate the linearized variational problem. Furthermore, identification of material parameters is discussed. Numerical simulation of the elastic–plastic deformation behavior of damaged tension specimens demonstrate the efficiency of the formulation.     

Variation of stress resultants in concrete structures due to time-dependent creep and shrinkage strains of concrete

A general procedure for calculating the variation with time of the internal stress resultants and hence the stresses, in concrete structures is discussed. In particular, a study is made of the changes in the stress resultants due to time-dependent creep and shrinkage strains of concrete.A general procedure of calculating the variation in the stress resultants due to differential creep strains in concrete structures has been proposed by the author[1]. A similar procedure is followed in this paper to study these variations when creep and shrinkage strains take place simultaneously.The method leads to a system of n-linear differential equations of the form: X = AXt + B the solution of which is performed by a computer using Runge-Kutta numerical procedures.A reinforced concrete portal frame exhibiting creep and shrinkage strains is solved by the proposed method and the results are given in tabular and graphical form.     

Size-dependent effects on critical flow velocity of a SWCNT conveying viscous fluid based on nonlocal strain gradient cylindrical shell model

This article investigates vibration and instability analysis of a single-walled carbon nanotube (SWCNT) conveying viscous fluid flow. For this purpose, the first-order shear deformation shell model is developed in the framework of nonlocal strain gradient theory (NSGT) for the first time. The proposed model is a conveying viscous fluid in which the external force of fluid flow is applied by the modified Navier–Stokes relation and considering slip boundary condition and Knudsen number. The NSGT can be reduced to the nonlocal elasticity theory, strain gradient theory or the classical elasticity theory by inserting their specific nonlocal parameters and material length scale parameters into the governing equations. Comparison of above-mentioned theories suggests that the NSGT predicts the greatest critical fluid flow velocity and stability region. The governing equations of motion and corresponding boundary conditions are discretized using the generalized differential quadrature method. Furthermore, the effects of the material length scale, nonlocal parameter, Winkler elastic foundation and Pasternak elastic foundation on vibration behavior and instability of a SWCNT conveying viscous fluid flow with simply supported and clamped–clamped boundary conditions are investigated.