Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an iterative solution technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by iteratively revising the right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified nonlinear stress-strain relations are all met within a given tolerance.     

Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an interative technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by interatively revising and right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified non-linear stress-strain relations are all met within a given tolerance.     

Three dimensional finite element modeling of thermomechanical frictional contact between finite deformation bodies using R-minimum strategy

An algorithm for analyzing the transient thermal coupling with the frictional contact between the multiple elastic–plastic bodies in finite deformation is presented using the R-minimum strategy. An arbitrarily shaped contact element strategy, named as node-to-point contact element strategy, is proposed to handle the thermomechanical frictional contact between finite deformation bodies. Assuming the material properties to be temperature dependent, the constitutive equations for both the thermomechanical frictional contact and the thermal-elastic–plastic materials are deduced respectively and applied in our finite element code. Finally, two examples are presented to show the efficiency and usefulness of this algorithm.     

A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation

A modified finite element formulation to study the post-buckling behaviour of elastic circular plates is presented in this paper. A discussion on the derivation of nonlinear stiffness matrix for post-buckling analysis is included and the present results are compared with continuum solutions.     

An Unstructured Finite Volume Approach for Structural Dynamics in Response to Fluid Motions

A new cell-vortex unstructured finite volume method for structural dynamics is assessed for simulations of structural dynamics in response to fluid motions. A robust implicit dual-time stepping method is employed to obtain time accurate solutions. The resulting system of algebraic equations is matrix-free and allows solid elements to include structure thickness, inertia, and structural stresses for accurate predictions of structural responses and stress distributions. The method is coupled with a fluid dynamics solver for fluid-structure interaction, providing a viable alternative to the finite element method for structural dynamics calculations. A mesh sensitivity test indicates that the finite volume method is at least of second-order accuracy. The method is validated by the problem of vortex-induced vibration of an elastic plate with different initial conditions and material properties. The results are in good agreement with existing numerical data and analytical solutions. The method is then applied to simulate a channel flow with an elastic wall. The effects of wall inertia and structural stresses on the fluid flow are investigated.     

Fracture mechanics J-integral calculations in thermo-elasto-plasticity

A finite element post-processor has been developed to calculate an incremental plasticity-based J-integral for fracture mechanics evaluations. The post-processor accounts for elastic-plastic deformations and thermal strains. The ADINA finite element computer program, with minor modifications by Babcock and Wilcox, was used with the Ramberg-Osgood stress-strain law and provides through its “porthole” files the required results of stresses, strains, displacements, and elastic and plastic strain energies.

The numerical results of the post-processor indicate that the thermal J-integral, which consists of a line integral for the isothermal case and an additional area integral for the thermal effect, can be considered path-independent even in the presence of plastic and thermal strains.     

Elastic stability and buckling loads of multi-span nonuniform beams,shafts and frames on rigid or elastic supports by finite element method using planar uniform line elements

A numerical computer method using planar flexural finite line element for the determination of buckling loads of beams, shafts and frames supported by rigid or elastic bearings is presented. Buckling loads and the corresponding mode vectors are determined by the solution of a linear set of eigenvalue equations of elastic stability. The elastic stability matrix is determined as the product of the bifurcation sidesway flexibility matrix and the second order bifurcation sidesway stiffness matrix which is formed using the element bifurcation sidesway stiffness matrices. The bifurcation sidesway flexibility matrix is determined by partitioning the inverse of the global external stiffness matrix of the system which is formed from the element data using the element stiffness matrices. The method is directly applicable to the determination of the buckling loads of beams and frames partially or fully supported by elastic foundations where the foundation stiffness is approximated by a discrete set of springs. The method of the article provides means to consider complex boundary conditions in buckling problems with ease. Four numerical examples are included to illustrate the industrial applications of the contents of the article.     

Nonlinear finite element analysis of reinforced concrete plates and shells under monotonic loading

Plane stress constitutive models are proposed for the nonlinear finite element analysis of reinforced concrete structures under monotonic loading. An elastic strain hardening plastic stress-strain relationship with a nonassociated flow rule is used to model concrete in the compression dominating region and an elastic brittle fracture behavior is assumed for concrete in the tension dominating area. After cracking takes place, the smeared cracked approach together with the rotating crack concept is employed. The steel is modeled by an idealized bilinear curve identical in tension and compressions. Via a layered approach, these material models are further extended to model the flexural behavior of reinforced concrete plates and shells. These material models have been tested against experimental data and good agreement has been obtained.     

Nonlocal large deformation and localization behavior of metals

The present paper deals with a nonlocal continuum plasticity model which includes the dependence of the yield function on a nonlocal equivalent plastic strain measure. Particular attention is focused on the formulation of a generalized I₁–J₂ yield criterion to describe the effect of hydrostatic stress on the plastic flow properties of metals, and the nonlocal equivalent plastic strain is defined as a weighted average of the corresponding local measure taken over the neighboring material points of the body. The nonlocal yield condition leads to a partial differential equation which is solved using the finite difference method at each iteration of a loading step. Since this requires no additional boundary conditions, the displacement-based finite element procedure is governed by the standard principle of virtual work, and the associated linearized variational equations are obtained in the usual manner from a consistent linearization algorithm. Numerical simulations of the elastic–plastic deformation behavior of ductile metal specimens show the influence of the various model parameters on the deformation and localization prediction. The proposed nonlocal theory preserves well-posedness of the governing equations in the post-localization regime and prevents pathological mesh sensitivity of the numerical results. The internal length scale incorporated in the model determines the size of the localized shear bands.     

A numerical algorithm for endochronic plasticity and comparison with experiment

A numerical algorithm based on the finite element method of analysis of the boundary value problem in a continuum is presented, in the case where the plastic response of the material is given in the context of endochronic plasticity. The relevant constitutive equation is expressed in incremental form and plastic effects are accounted for by the method of an induced pseudo-force in the matrix equations.

The results of the analysis are compared with observed values in the case of a plate with two symmetric notches and loaded longitudinally in its own plane. The agreement between theory and experiment is excellent.     

基于刚体运动与弹性运动的机械系统动力学建模研究

讨论了具有刚体运动与柔性变形的机械系统的动力学建模,将刚体自由度与弹性变形自由度看作广义坐标,利用有限元法对具有刚性运动与弹性变形的机械系统的运动与变形进行了描述,得到了以刚体位移与弹性变形位移表示的单元的广义惯性力;从应力应变入手,得到了表示单元弹性变形与几何非线性变形的结构刚度矩阵与几何非线性刚度矩阵,使用Kane方程推导了弹性连杆机构的单元运动方程,这种建模方法,可以使用在任意结构的机械系统。     

Finite element analysis of laminated shells of revolution

A finite element formulation for the analysis of axisymmetric fibre reinforced laminated shells subjected to axisymmetric load is presented. The formulation includes arbitrary number of bonded layers each of which may have different thicknesses, orientation of elastic axes, and elastic properties. Superparamatric curved elements[17] having four degrees of freedom per node including the normal rotation, are used. Stress-strain relation for an arbitrary layer is obtained from the consideration of three dimensional aspect of the problem. The element stiffness matrix has been obtained by using Gauss quadrature numerical integration, even though the elasticity matrix is different for different layers. The formulation is checked for a cylindrical tube subjected to internal pressure and axial tension, and the results are found to compare very well with the elastic solution [9].     

超声波在各向同性固体中传播的数值模拟

介绍以应用动量守恒和应力-应变关系为各向同性弹性介质进行微变形建立超声波传播方程,并利用一种有限元方法和差分法,分析超声波(P波)在各向问性介质中传播,及在裂缝上的散射结果.并且通过计算机程序模拟二维超声波的传播,散射的数值模拟与几何理论结果是相符合的.因此超声检测仿真软件是可以预测超卢检测过程中的波形.超声探头的建模发展趋势是利用FDM和FEM方法研究超声检测的仿真软件,分析结果将广泛应用到检测过程中.     

Post-buckling strength of thin-walled members

The finite element displacement method considering both geometrical nonlinearity and material non-linearity has been used to investigate the post-buckling behaviour and the ultimate strength of thin-walled nonplanar (three-dimensional) structural members. The two types of nonlinearities are based on Lagrangian coordinates and the flow theory of plasticity, and the formulations are developed using the variational principle and the incremental variational principle. The tangent stiffness matrix which is derived explicitly up to a point prior to volume integration, has been found to be quite efficient. The cases of a hat-section beam under a concentrated load for a web crippling study and a channel section subjected to combined bending and torsion are used to show the capabilities of the computer program. Results indicate that the conventional linear, elastic analysis over-estimates the strength of thin-walled members and may not even be a useful approximation and that the structure may be excessively deformed when approaching the ultimate load. The study also demonstrates the merit of using the finite element method for detailed investigations of particular problems.     

An extension of 3-D procedure to large strain analysis of shells

A numerical stress integration procedure for general 3-D large strain problems in inelasticity, based on the total formulation and the governing parameter method (GPM), is extended to shell analysis. The multiplicative decomposition of the deformation gradient is adopted with the evaluation of the deformation gradient practically in the same way as in a general 3-D material deformation. The calculated trial elastic logarithmic strains are transformed to the local shell Cartesian coordinate system and the stress integration is performed according to the GPM developed for small strain conditions. The consistent tangent matrix is calculated as in case of small strain deformation and then transformed to the global coordinate system.A specific step in the proposed procedure is the updating of the left elastic Green–Lagrangian deformation tensor. Namely, after the stresses are computed, the principal elastic strains and the principal vectors corresponding to the stresses at the end of time step are determined. In this way the shell conditions are taken into account appropriately for the next step.Some details are given for the stress integration in case of thermoplastic and creep material model.Numerical examples include bulging of plate (plastic, thermoplastic, and creep models for metal) and necking of a thin sheet. Comparison of solutions with those available in the literature, and with solutions using other type of finite elements, demonstrates applicability, efficiency and accuracy of the proposed procedure.     

Validated finite element models of the proximal femur using two-dimensional projected geometry and bone density

Two-dimensional finite element models of cadaveric femoral stiffness were developed to study their suitability as surrogates of bone stiffness and strength, using two-dimensional representations of femoral geometry and bone mineral density distributions. If successfully validated, such methods could be clinically applied to estimate patient bone stiffness and strength using simpler and less costly radiographs. Two-dimensional femur images were derived by projection of quantitative computed tomography scans of 22 human cadaveric femurs. The same femurs were fractured in a fall on the hip configuration. Femoral stiffness and fracture load were measured, and high speed video was recorded. Digital image correlation analysis was used to calculate the strain distribution from the high speed video recordings. Two-dimensional projection images were segmented and meshed with second-order triangular elements for finite element analysis. Elastic moduli of the finite elements were calculated based on the projected mineral density values inside the elements. The mapping of projection density values to elastic modulus was obtained using optimal parameter identification in a set of nine of the 22 specimens, and validated on the remaining 13 specimens. Finite element calculated proximal stiffness and strength correlated much better with experimental data than areal bone mineral density alone. In addition, finite element calculated strain distributions compared very well with strains obtained from digital image processing of the high speed video recordings, further validating the two-dimensional projected subject-specific finite element models.     

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.     

Contact between a thermal flying height control slider and a disk asperity

Contact between a thermal flying height control slider and an asperity on a disk is investigated using finite element analysis. The finite element model developed accounts for transient elastic–plastic deformation and heat generation due to frictional heating. Plastic deformation and temperature rise of the read/write element are determined as a function of flying height of the slider, location of the read/write element as well as material properties of typical disk asperities. The model shows good agreement with experimental data. Significant plastic deformation and temperature rise were observed in the shield and alumina regions of the slider. Hard and stiff disk asperities, such as alumina asperities, result in more damage to the slider than soft and compliant nickel-phosphorus ones.     

Finite element stress formulation for dynamic elastic-plastic analysis

The solution to wave propagation problems in solids with elastic-plastic material properties is obtained by using the finite element method directly in terms of the stresses. A variational principle due to Gurtin is modified by including a plastic strain tensor in the constitutive relationship. The resulting finite element equations, which represent the strain-displacement equations written in terms of the stresses, are simultaneous integral equations in time. With a transformation of variables, a set of simultaneous differential equations is obtained of the form$\text{H}\text{s}\text{?}\text{+ Qs+ Ve}{}_{\mathrm{p}}\text{= q(t)}$, where $\text{H}$ is a symmetric positive-semidefinite matrix, and $\text{Q}$ is a symmetric positive-definite matrix. The stresses and the plastic strains are represented by $\text{s}\text{?}$ and $\text{e}{}_{\mathrm{p}}$, respectively.Finite element equations are developed for an axisymmetric ring element with an arbitrary quadrilateral cross section in which the stresses and the plastic strains vary linearly along the sides of the elements. The equations are numerically integrated with respect to time by Newmark's generalized acceleration method.An iterative procedure is presented, which uses the finite element strain-displacement equations and the plasticity relationships, to determine the state of stress at the end of the time step. Several examples are used to demonstrate the solution technique for elastic and elastic-plastic problems.     

Large amplitude free flexural vibrations of thin plates of arbitrary shape

A finite element formulation is developed for analyzing large amplitude free flexural vibrations of elastic plates of arbitrary shape. Stress distributions in the plates, deflection shape and nonlinear frequencies are determined from the analysis. Linearized stiffness equations of motion governing large amplitude oscillations of plates, quasi-linear geometrical stiffness matrix, solution procedures, and convergence characteristics are presented. The linearized geometrical stiffness matrix for an eighteen degrees-of-freedom conforming triangular plate element is evaluated by using a seven-point numerical integration. Nonlinear frequencies for square, rectangular, circular, rhombic, and isosceles triangular plates, with edges simply supported or clamped, are obtained and compared with available approximate continuum solutions. It demonstrates that the present formulation gives results entirely adequate for many engineering purposes.