基于非保守体力的大变形固结有限元法

考虑土体固结过程中体力的非保守特点,完善了T.L.描述的大变形固结平衡方程,并以伴生体力代替非保守体力,提出了相应的大变形固结增量有限元列式. 当忽略土体体积应变的影响时,改进的大变形固结平衡方程及其有限元法均可退化为基于保守体力的传统形式. 采用自主开发的程序对大变形自重固结算例进行分析,结果初步表明:传统方法低估了地基固结度,但高估了地基沉降量和超静孔隙水压力. 基于非保守体力的大变形固结分析在理论上能够更合理地预测地基的固结性状.      

Texture gradient simulations for extrusion and reversible rolling of FCC metals

A texture simulation method is described for some complex plane strain deformation paths during hot shaping of FCC metals. The method employs both finite element calculations and a polycrystal plasticity model based on the Relaxed-Constraints (RC) Taylor hypothesis and a viscoplastic constitutive law. We have considered the {111}<110> slip systems and the {100}, {110}, {112} <110> non-octahedral slip systems. The finite element codes simulate the strain paths of material flow during a shaping process. The local velocity gradients, expressed in the macroscopic reference coordinates, are rewritten in the local flow line coordinates using a kinematic analysis for steady-state flow. Secondly, for the different deformation paths, the RC polycrystal plasticity model is used to numerically simulate the local deformation texture evolutions as a function of depth. Texture simulations are carried out for two deformation processes combining hot compression and shear: extrusion and reversible rolling. For extrusion, the simulated pole figures and ODFs show the typical texture variations through the thickness of an extruded 6082 aluminium alloys, i.e. (β-fibre in the centre and a TD rotated copper component near the surface. It is shown that hot reversible rolling should develop a strong pure shear texture {001}<110> near the surface.     

多尺度材料模拟的自适应有限元方法研究

研究了材料模拟中一类新型耦合多尺度的自适应有限元方法. 采用微观分子动力学耦合宏观有限元的桥尺度方法来模拟材料破坏的前期行为,其中宏观有限元计算推广到了一般非结构三角形网格. 材料破坏形成后,停止微观尺度的计算,它的进一步发展和演化通过一个宏观模型来描述,采用自适应有限元方法来求解这一宏观模型. 其中,后验误差估计的基础是变分多尺度理论,即自适应网格加密是基于粗尺度上残差分布和细尺度上单元Green's函数. 计算中采用了破坏准则来模拟材料的断裂. 数值实验表明了方法的有效性.      

Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation

In this paper we suggest a new phenomenological material model for shape memory alloys. In contrast to many earlier concepts of this kind the present approach includes arbitrarily large deformations. The work is motivated by the requirement, also expressed by regulatory agencies, to carry out finite element simulations of NiTi stents. Depending on the quality of the numerical results it is possible to circumvent, at least partially, expensive experimental investigations. Stent structures are usually designed to significantly reduce their diameter during the insertion into a catheter. Thereby large rotations combined with moderate and large strains occur. In this process an agreement of numerical and experimental results is often hard to achieve. One of the reasons for this discrepancy is the use of unrealistic material models which mostly rely on the assumption of small strains. In the present paper we derive a new constitutive model which is no longer limited in this way. Further its efficient implementation into a finite element formulation is shown. One of the key issues in this regard is to fulfil “inelastic” incompressibility in each time increment. Here we suggest a new kind of exponential map where the exponential function is suitably computed by means of the spectral decomposition. A series expansion is completely avoided. Finite element simulations of stent structures show that the new concept is well appropriate to demanding finite element analyses as they occur in practically relevant problems.     

On the difficulties and remedies in enforcing the div = 0 condition in the finite element analysis of thermal plumes with strongly temperature-dependent viscosity

A finite element (FE) analysis of experimentally observed creeping thermal plumes in a medium whose viscosity is strongly temperature-dependent is performed. Such plumes are considered to play an important role in numerous geological processes and numerical modelling is often the only option to study their physics. Initial simulations by means of the general-purpose Galerkin finite element package NACHOS-II demonstrated serious deficiencies of the method in modelling plumes with large viscosity contrasts, in spite of several options for the solution (mixed or penalty formulation) and the elements (continuous or discontinous pressure). In agreement with observations from FE simulations of isothermal Stokes flow in other studies, we have isolated the violation of the div = 0 or incompressibility constraint as the major culprit in the failure of the FE method. It is demonstrated that the a posteriori computed discrete divergence (DDIV) can be used as a diagnostic tool to evaluate the reliability of the FE solution and to rank the solution and element options provided in the NACHOS code. On the basis of these considerations, the combination of the mixed method with a Q₂-P₁ (discontinuous pressure) element turns out to be the most suitable for the present plume problem but is still unable to sufficiently enforce the div = 0 condition. With a goal to remedy this detrimental behaviour, several FE modifications and new approaches have been taken. These include: (i) use of a new scaling option for the governing equations which has the effect of equilbrating the stiffness matrices and thus improving their condition; (ii) implementation of several iterative solution techniques such as the iterated penalty and the Uzawa algorithm for the augmented Langrangian to better accommodate the dual role of the pressure; (iii) use of a multistep Newton method to better handle the high non-linearity of the coupled flow/transport problem. Although each of these options (or a combination of them) is able to improve on the quality of FE solution, the most startling amelioration has been gained with option (iii). Use of the latter resulted in very satisfactory modelling of the experimentally observed plumes.     

A new discontinuous FE formulation for crack path prediction in brittle solids

In this paper a new finite element (FE) formulation to simulate embedded strong discontinuity for the study of the fracture process in brittle or quasi-brittle solids is presented. A homogeneous discontinuity is considered to be present in a cracked finite element with the possibility to take into account the opening and the sliding phenomena which can occur across the crack faces. In such a context a new simple stress-based implementation of the discontinuous displacement field is proposed by an appropriate stress field correction introduced at the Gauss points level in order to simulate, in a fashion typical of an elastic–plastic classical FE formulation, the mechanical effects of the bridging and friction stresses due to crack faces opening and sliding which can occur during the loading–unloading process structural component or solid being analysed. The proposed formulation does not need to introduce special or modified shape functions to reproduce discontinuous displacement field but simply relaxes the stress field in an appropriate fashion. Both linear elastic and elastic–plastic behaviour of the non-cracked material can be considered. Several 2D problems are presented and solved by the proposed procedure in order to predict load–displacement curves of brittle structures as well as crack patterns that develop during the loading process.The proposed discontinuous new FE formulation gives the advantages to be simple, computationally economic and to keep internal continuity of the numerical FE model; furthermore the developed algorithm can be easily implemented in standard FE programs as a standard plasticity model.     

Non-linear finite element analysis of cone penetration in layered sandy loam soil – Considering precompression stress state

Axisymmetric finite element (FE) method was developed to simulate cone penetration process in layered granular soil. The FE was modeled using ABAQUS/Explicit, a commercially available package. Soil was considered as a non-linear elastic plastic material which was modeled using variable elastic parameters of Young’s Modulus and Poisson’s ratio and Drucker–Prager criterion with yield stress dependent material hardening property. The material hardening parameters of the model were estimated from the USDA-ARS National Soil Dynamics Laboratory – Auburn University (NSDL-AU) soil compaction model. The stress–strain relationship in the NSDLAU compaction model was modified to account for the different soil moisture conditions and the influence of precompression stress states of the soil layers. A surface contact pair (‘slave-master’) algorithm in ABAQUS/Explicit was used to simulate the insertion of a rigid cone (RAX2 ABAQUS element) into deformable and layered soil medium (CAX4R ABAQUS element). The FE formulation was verified using cone penetration data collected on a soil chamber of Norfolk sandy loam soil which was prepared in two compaction treatments that varied in bulk density in the hardpan layer of (1) 1.64 Mg m⁻³ and (2) 1.71 Mg m⁻³. The FE model successfully simulated the trend of cone penetration in layered soils indicating the location of the sub-soil compacted (hardpan) layer and peak cone penetration resistance. Modification of the NSDL-AU model to account for the actual soil moisture content and inclusion of the influence of precompression stress into the strain behavior of the NSDL-AU model improved the performance of FE in predicting the peak cone penetration resistance. Modification of the NSDL-AU model resulted in an improvement of about 42% in the finite element-predicted soil cone penetration forces compared with the FE results that used the NSDL-AU ‘virgin’ model.     

Measurement of deformation gradients in hot rolling of AA3004

In this paper we describe an experimental technique developed to measure the deformation gradients and temperature in a single hot rolling pass of an AA3004 sample that was fitted with an insert. The insert had been previously hand engraved with a 1×1 mm grid pitch, and the analysis of the data digitally captured from the image of the deformed grid enabled the calculation of the components of the deformation gradient tensor. Four steel pins prevented relative motion between the insert and the rest of the sample. No detachment was observed between insert and sample after rolling. The temperature was measured during rolling using two embedded thermocouples, one close to the surface and the other in the centerline. The commercial finite element code ABAQUS was used to create a three-dimensional model of the rolling process. The recorded temperature was compared to the numerical values evaluated after tuning the heat transfer coefficient. The shape of the grid after rolling was checked against the deformed mesh using different fricition coefficients in order to obtain the optimum match. The unusually large length of the insert enabled the rolling process to be stopped halfway so that a picture of the roll-gap area could be obtained. This provided a partially deformed grid that represented the transient state during rolling. The experimentaily determined deformation gradient in this area as well as in the steady-state area agreed well with the finite element oredictions.     

Modeling the effects of material non-linearity using moving window micromechanics

In the analysis of materials with random heterogeneous microstructure the assumption is often made that material behavior can be represented by homogenized or effective properties. While this assumption yields accurate results for the bulk behavior of composite materials, it ignores the effects of the random microstructure. The spatial variations in these microstructures can focus, initiate and propagate localized non-linear behavior, subsequent damage and failure. In previous work a computational method, moving window micromechanics (MW), was used to capture microstructural detail and characterize the variability of the local and global elastic response. Digital images of material microstructure described the microstructure and a local micromechanical analysis was used to generate spatially varying material property fields. The strengths of this approach are that the material property fields can be consistently developed from digital images of real microstructures, they are easy to import into finite element models (FE) using regular grids, and their statistical characterizations can provide the basis for simulations further characterizing stochastic response. In this work, the moving window micromechanics technique was used to generate material property fields characterizing the non-linear behavior of random materials under plastic yielding; specifically yield stress and hardening slope, post yield. The complete set of material property fields were input into FE models of uniaxial loading. Global stress strain curves from the FE–MW model were compared to a more traditional micromechanics model, the generalized method of cells. Local plastic strain and local stress fields were produced which correlate well to the microstructure. The FE–MW method qualitatively captures the inelastic behavior, based on a non-linear flow rule, of the sample continuous fiber composites in transverse uniaxial loading.     

Nonlinear mechanics of a ring structure subjected to multi-pairs of evenly distributed equal radial forces

Combining the elastica theory,finite element(FE) analysis,and a geometrical topological experiment,we stud-ied the mechanical behavior of a ring subjected to multi-pairs of evenly distributed equal radial forces by looking at its seven distinct states.The results showed that the theoretical predictions of the ring deformation and strain energy matched the FE results very well,and that the ring deformations were comparable to the topological experiment.Moreover,no mat-ter whether the ring was compressed or tensioned by N-pairs of forces,the ring always tended to be regular polygons with 2N sides as the force increased,and a proper compressive force deformed the ring into exquisite flower-like patterns. The present study solves a basic mechanical problem of a ring subjected to lateral forces,which can be useful for studying the relevant mechanical behavior of ring structures from the nano-to the macro-scale.     

Coarse-graining atomistic dynamics of brittle fracture by finite element method

In this work we present the finite element (FE) implementation of an atomistic formulation of balance equations and its application to coarse-grained (CG) simulation of dynamic fracture. First, we simulate a notched specimen that contains about 1.8 million atoms by the CG-FE method, and we compare the CG-FE results with that by all-atom molecular dynamics (MD) simulations. We find that CG-FE simulations with about 5% degrees of freedom of the MD simulation can capture the essential dynamic features, not in exact correspondence, but qualitatively and quantitatively similar to that obtained by MD simulations. We then proceed to simulate a series of micron-sized specimens by the CF-FE method. We find that it is the interaction of the forward propagating crack with the stress waves being reflected back by the boundaries of the specimen that triggers the dynamic instability and hence the branching of cracks in micron-sized specimens. The potential application of the method and future work for improvements are discussed.     

高密度封装倒装焊的新简化模型及疲劳寿命分析

高密度封装结构中存在大量焊球,在进行有限元建模时,考虑到焊球数量多且尺寸小的特点,需简化焊球层模型．本文针对高密度封装结构中倒装焊及底填胶,提出了一种新的简化模型,采用非重点部位简化和材料均匀化相结合的方法,将内部焊球层和底填胶简化为均匀层,只保留外圈几层焊球．通过建立代表性体积单元,计算得到均匀化层的材料参数．讨论了外层焊球圈数对焊球层危险点应力的影响,发现保留外圈两层焊球就能得到非常精确的结果．利用新简化模型计算了高密度封装结构的疲劳寿命,所获得的结果与未均匀化模型结果误差在0.3%以内．     

Simulation of forming processes by FEM with a Bingham fluid model

We model the forming process as a fluid flow. A finite element program, FIDAP, which analyses flow problems, was used to calculate velocity and strain rates at points throughout the material during the deformation process. This allows predictions to be made on the shape and quality of the resulting part. The stress-strain relation we used models the plastic flow of metals (Bingham fluids). The FEM approximation of such a fluid is tested by comparing results for a simple analytical example. In forming processes provision must be made for friction between dye and workpiece, and the program was modified accordingly. Two classical ring forming simulations are compared to published results.     

Integration of self-consistent polycrystal plasticity with dislocation density based hardening laws within an implicit finite element framework: Application to low-symmetry metals

We present an implementation of the viscoplastic self-consistent (VPSC) polycrystalline model in an implicit finite element (FE) framework, which accounts for a dislocation-based hardening law for multiple slip and twinning modes at the micro-scale grain level. The model is applied to simulate the macro-scale mechanical response of a highly anisotropic low-symmetry (orthorhombic) crystal structure. In this approach, a finite element integration point represents a polycrystalline material point and the meso-scale mechanical response is obtained by the mean-field VPSC homogenization scheme. We demonstrate the accuracy of the FE-VPSC model by analyzing the mechanical response and microstructure evolution of α-uranium samples under simple compression/tension and four-point bending tests. Predictions of the FE-VPSC simulations compare favorably with experimental measurements of geometrical changes and microstructure evolution. Specifically, the model captures accurately the tension–compression asymmetry of the material associated with twinning, as well as the rigidity of the material response along the hard-to-deform crystallographic orientations.     

A coupled IEM/FEM approach for solving elastic problems with multiple cracks

In this paper, the detailed two-dimensional infinite element method (IEM) formulation with infinite element (IE)–finite element (FE) coupling scheme for investigating mode I stress intensity factor in elastic problems with imbedded geometric singularities (e.g. cracks) is presented. The IE–FE coupling algorithm is also successfully extended to solve multiple crack problems. In this method, the domain of the primary problem is subdivided into two sub-domains modeled separately using the IEM for the multiple crack sub-domain, and the FEM for the uncracked sub-domain. In the IE sub-domain, the similarity partition concept together with the IEM formulation are employed to automatically generate a large number of infinitesimal elements, layer by layer, around the tip of each crack. All degrees of freedom related to the IE sub-domain, except for those associated with the coupling interface, are condensed and transformed to form a finite master IE for each crack with master nodes on sub-domain boundary only. All of the stiffness matrices constructed in the IE sub-domains are assembled into the system stiffness matrix for the FE sub-domain. The resultant FE solution with a symmetrical stiffness matrix, having the singularity effect of imbedded cracks in IEs, is required only for solving multiple crack problems.Using these efficient numerical techniques a very fine mesh pattern can be established around each crack tip without increasing the degree of freedom of the global FEM solution. One is easily allowed to conduct parametric analyses for various crack sizes without changing the FE mesh. Numerical examples are presented to show the performance of the proposed method and compared with the corresponding known results where available.     

Numerical simulation of transpiration cooling through porous material

Transpiration cooling using ceramic matrix composite materials is an innovative concept for cooling rocket thrust chambers. The coolant (air) is driven through the porous material by a pressure difference between the coolant reservoir and the turbulent hot gas flow. The effectiveness of such cooling strategies relies on a proper choice of the involved process parameters such as injection pressure, blowing ratios, and material structure parameters, to name only a few. In view of the limited experimental access to the subtle processes occurring at the interface between hot gas flow and porous medium, reliable and accurate simulations become an increasingly important design tool. In order to facilitate such numerical simulations for a carbon/carbon material mounted in the side wall of a hot gas channel that are able to capture a spatially varying interplay between the hot gas flow and the coolant at the interface, we formulate a model for the porous medium flow of Darcy–Forchheimer type. A finite‐element solver for the corresponding porous medium flow is presented and coupled with a finite‐volume solver for the compressible Reynolds‐averaged Navier–Stokes equations. The two‐dimensional and three‐dimensional results at Mach number Ma = 0.5 and hot gas temperature T_HG=540 K for different blowing ratios are compared with experimental data. Copyright © 2014 John Wiley & Sons, Ltd.     

A spectrally formulated finite element for wave propagation analysis in functionally graded beams

In this paper, spectral finite element method is employed to analyse the wave propagation behavior in a functionally graded (FG) beam subjected to high frequency impulse loading, which can be either thermal or mechanical. A new spectrally formulated element that has three degrees of freedom per node (based upon the first order shear deformation theory) is developed, which has an exact dynamic stiffness matrix, obtained by exactly solving the homogeneous part of the governing equations in the frequency domain. The element takes into account the variation of thermal and mechanical properties along its depth, which can be modeled either by explicit distribution law like the power law and the exponential law or by rule of mixture as used in composite. Ability of the element in capturing the essential wave propagation behavior other than predicting the propagating shear mode (which appears only at high frequency and is present only in higher order beam theories), is demonstrated. Propagation of stress wave and smoothing of depthwise stress distribution with time is presented. Dependence of cut-off frequency and maximum stress gradient on material properties and FG material (FGM) content is studied. The results are compared with the 2D plane stress FE and 1D Beam FE formulation. The versatility of the method is further demonstrated through the response of FG beam due to short duration highly transient temperature loading.     

On the interplay between strain rate and strain rate sensitivity on flow localization in the dynamic expansion of ductile rings

In this work a stability analysis on flow localization in the dynamic expansion of ductile rings is conducted. Within a 1-D theoretical framework, the boundary value problem of a radially expanding thin ring is posed. Based on a previous work, the equations governing the stretching process of the expanding ring are derived and solved using a linear perturbation method. Then, three different perfectly plastic material constitutive behaviours are analysed: the rate independent material, the rate dependent material showing constant logarithmic rate sensitivity and the rate dependent material showing non-constant and non-monotonic logarithmic rate sensitivity. The latter allows to investigate the interaction between inertia and strain rate sensitivity on necking formation. The main feature of this work is rationally demonstrate that under certain loading conditions and material behaviours: (1) decreasing rate sensitivity may not lead to more unstable material, (2) increasing loading rate may not lead to more stable material. This finding reveals that the relation between rate sensitivity and loading rate controls the unstable flow growth. Additionally a finite element model of the ring expansion problem is built in ABAQUS/Explicit. The stability analysis properly reflects the results obtained from the numerical simulations. Both procedures, perturbation analysis and numerical simulations, allow for emphasizing the interplay between rate sensitivity and inertia on strain localization.     

Continuum-Based Shape Sensitivity Analysis for 2D Coupled Atomistic/Continuum Simulations Using Bridging Scale Decomposition

In this paper, we propose the first attempt to perform shape sensitivity analysis for two-dimensional coupled atomistic and continuum problems using bridging scale decomposition. Based on a continuum variational formulation of the bridging scale, the sensitivity expressions are derived in a continuum setting using both direct differentiation method and adjoint variable method. To overcome the issue of discontinuity in shape design due to the discrete nature of the molecular dynamics (MD) simulation, we define design velocity fields in a way that the shape of the MD region does not change. Another major challenge is that the discrete finite element (FE) mass matrix in bridging scale is not continuous with respect to shape design variables. To address this issue, we assume an evenly distributed mass density when evaluating the material derivative of the FE mass matrix. In order to support accuracy verification of sensitivity results using overall finite difference method, we use regular-shaped finite elements and only allow shape change in one direction in our example problems, so that design perturbations can be made to the discrete FE mass matrix. However, the sensitivity formulation is sufficiently general to support irregular-shaped finite elements and arbitrary design velocity fields. The sensitivity analysis results, verified using overall finite difference method, reveal the impact of macroscopic shape design changes on microscopic atomistic responses.     

Elasto-plastic material parameter identification by inverse methods: Calculation of the sensitivity matrix

Inverse methods offer a powerful tool for the identification of elasto-plastic material properties of metals. The basic principle of the inverse method we are studying, is to compare an experimentally measured strain field with a strain field computed by a finite element (FE) model. The material parameters in the FE model are iteratively tuned in such a way that the numerically computed strain field matches the experimentally measured field as closely as possible. One of the building blocks in this identification procedure is the optimization algorithm for the material parameters in the numerical model. The key problem of this optimization algorithm is the determination of a sensitivity matrix, which expresses the sensitivities of the strains with respect to the material parameters. This paper presents an analytical method for the calculation of this sensitivity matrix in the case of a tensile test with non-rotating principal axes of strain.