Modeling fracture in large scale shell structures

We consider a problem of modeling fracture and failure preceded by large scale yielding of ductile shells from the point of view of large-scale structural analysis. We place a special emphasis on the computational efficiency of the constitutive formulation. In this context, we seek the formulation embedded in the shell mechanics framework, which is both theoretically sound and easily implementable into a large-scale explicit dynamic finite element code without precluding vectorization or parallelization. This is achieved through the elasto-plastic damage constitutive model for finite-element analysis of plates and shells. The proposed damage model is purely phenomenological with a scalar damage parameter, which has no physical interpretation, except that it represents on a global scale the micromechanical changes the material undergoes during the process of necking and fracture. The localization leading to softening and fracture is represented by the damage calibration function with exponential damage growth after the onset of necking. The proposed phenomenological damage model uses a general plasticity and shell mechanics frameworks which makes it general and easily implementable into existing finite element codes. The proposed formulation has been implemented into the explicit dynamic finite element software code EPSA (Atkatsh et al., 1980, Atkatsh et al., 1983).     

饱和多孔介质中的混合有限元法和有限应变下应变局部化分析

对基于Biot理论的饱和多孔介质中动力-渗流耦合分析提出了一个耦合场混合元．固相位移、应变和有效应力以及流相压力、压力梯度和Darcy速度在单元内均处理为独立变量分别插值．基于胡海昌-Washizu三变量广义变分原理给出的饱和多孔介质动力-渗流耦合问题控制方程的单元弱形式，导出了单元公式．进一步导出了考虑压力相关非关联塑性的非线性单元公式和发展了相应的一致性算法．对几何非线性分析，采用了共旋公式途径．数值结果例题显示所发展耦合场混合元模拟大应变下由应变软化引起以应变局部化为特征的渐进破坏现象的性能．     

Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation

Under the frame of multibody dynamics, the contact dynamics of elasto-plastic spatial thin beams is numerically studied by using the spatial thin beam elements of absolute nodal coordinate formulation (ANCF). The inter-nal force of the elasto-plastic spatial thin beam element is derived under the assumption that the plastic strain of the beam element depends only on its longitudinal deformation. A new body-fixed local coordinate system is introduced into the spatial thin beam element of ANCF for efficient con-tact detection in the contact dynamics simulation. The linear isotropic hardening constitutive law is used to describe the elasto-plastic deformation of beam material, and the classical return mapping algorithm is adopted to evaluate the plastic strains. A multi-zone contact approach of thin beams previ-ously proposed by the authors is also introduced to detect the multiple contact zones of beams accurately, and the penalty method is used to compute the normal contact force of thin beams in contact. Four numerical examples are given to demonstrate the applicability and effectiveness of the pro-posed elasto-plastic spatial thin beam element of ANCF for flexible multibody system dynamics.     

Associative coupled thermoplasticity at finite strain with temperature-dependent material parameters

The paper deals with associative coupled thermoplasticity at finite strains. J2 plasticity model is based on hyperelastic formulation with multiplicative decomposition of deformation gradient into elastic and plastic parts. As a novel aspect, temperature dependence of all material parameters is introduced. For such model, variational procedure is carried out and discretization by the mixed finite element method is performed. Consistent linearization is carried out and algorithmic elasto-plastic tangent moduli are given. Verification of algorithm is provided by two examples.     

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.     

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.     

Size effects in the conical indentation of an elasto-plastic solid

The size effect in conical indentation of an elasto-plastic solid is predicted via the Fleck and Willis formulation of strain gradient plasticity (Fleck, N.A. and Willis, J.R., 2009, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045–1057). The rate-dependent formulation is implemented numerically and the full-field indentation problem is analyzed via finite element calculations, for both ideally plastic behavior and dissipative hardening. The isotropic strain-gradient theory involves three material length scales, and the relative significance of these length scales upon the degree of size effect is assessed. Indentation maps are generated to summarize the sensitivity of indentation hardness to indent size, indenter geometry and material properties (such as yield strain and strain hardening index). The finite element model is also used to evaluate the pertinence of the Johnson cavity expansion model and of the Nix–Gao model, which have been extensively used to predict size effects in indentation hardness.     

Postbuckling analysis and imperfection sensitivity of general shells by the finite element method

Buckling and imperfection sensitivity are the primary considerations in analysis and design of thin shell structures. The objective here is to develop accurate and efficient capabilities to predict the postbuckling behavior of shells, including imperfection sensitivity. The approach used is based on the Lyapunov–Schmidt–Koiter (LSK) decomposition and asymptotic expansion in conjunction with the finite element method. This LSK formulation for shells is derived and implemented in a finite element code. The method is applied to cylindrical and spherical shells. Cases of linear and nonlinear prebuckling behavior, coincident as well as non-coincident buckling modes, and modal interactions are studied. The results from the asymptotic analysis are compared to exact solutions obtained by numerically tracking the bifurcated equilibrium branches. The accuracy of the LSK asymptotic technique, its range of validity, and its limitations are illustrated.     

A reexamination of plasticity-induced crack closure in fatigue crack propagation

The crack closure concept is often used to consider the R-ratio and overload effects on fatigue crack growth. The presumption is that when the crack is closed, the external load produces negligible fatigue damage in the cracked component. The current investigation provides a reassessment of the frequently used concept with an emphasis on the plasticity-induced crack closure. A center cracked specimen made of 1070 steel was investigated. The specimen was subjected to plane-stress mode I loading. An elastic–plastic stress analysis was conducted for the cracked specimens using the finite element method. By applying the commonly used one-node-per-cycle debonding scheme for the crack closure simulations, it was shown that the predicted crack opening load did not stabilize when the extended crack was less than four times of the plastic zone size. The predicted opening load was strongly influenced by the plasticity model used. When the elastic–perfectly plastic (EPP) stress–strain relationship was used together with the kinematic hardening plasticity theory, the predicted crack opening load was found to be critically dependent on the element size of the finite element mesh model. For R = 0, the predicted crack opening load was greatly reduced when the finite element size became very fine. The kinematic hardening rule with the bilinear (BL) stress–strain relationship predicted crack closure with less dependence on the element size. When a recently developed cyclic plasticity model was used, the element size effect on the predicted crack opening level was insignificant. While crack closure may occur, it was demonstrated that cyclic plasticity persisted in the material near the crack tip. The cyclic plasticity was reduced but not negligible when the crack was closed. The traditional approaches may have overestimated the effect of crack closure in fatigue crack growth predictions.     

各向异性弹塑性中厚度板壳的有限元分析

本文在文献［２，３］的基础上，提出了一个解各向异性弹塑性中厚度板壳问题的有限元方法。考虑材料各向异性的特点，采用了Ｈｉｌｌ推广的Ｈｕｂｅｒ－Ｍｉｓｅｓ屈服准则；借用Ｏｗｅｎ的剪切修正系数，正确计及了叠层复合材料壳体的横向剪切效应；为了避免“自锁”现象，文中采用了９节点的Ｈｅｔｅｒｏｓｉｓ二次壳单元；特别是本文利用插值外推的思想，提出了一个带预测的弧长增量控制法，显著提高了确定变形路径的计算效率。几个数值算例表明本文给出的有限元方法对于各向异性中厚度板壳的弹塑性分析有较好的精度，尤其是对具有复杂变形路径的结构计算，收敛速度提高更快。     

Homogenization of two-phase elasto-plastic composite materials and structures: Study of tangent operators,cyclic plasticity and numerical algorithms

We develop homogenization schemes and numerical algorithms for two-phase elasto-plastic composite materials and structures. A Hill-type incremental formulation enables the simulation of unloading and cyclic loadings. It also allows to handle any rate-independent model for each phase. We study the crucial issue of tangent operators: elasto-plastic (or “continuum”) versus algorithmic (or “consistent”), and anisotropic versus isotropic. We apply two methods of extraction of isotropic tangent moduli. We compare mathematically the stiffnesses of various tangent operators. All rate equations are discretized in time using implicit integration. We implemented two homogenization schemes: Mori–Tanaka and a double inclusion model, and two plasticity models: classical J₂ plasticity and Chaboche’s model with non-linear kinematic and isotropic hardenings. We consider composites with different properties and present several discriminating numerical simulations. In many cases, the results are validated against finite element (FE) or experimental data. We integrated our homogenization code into the FE program ABAQUS using a user material interface UMAT. A two-scale procedure allows to compute realistic structures made of non-linear composite materials within reasonable CPU time and memory usage; examples are shown.     

Material characterization based on dual indenters

Extensive large strain-large deformation finite element analyses were carried out to investigate the response of elasto-plastic materials obeying power law-strain hardening during the loading and unloading process of instrumented indentation with conical indenters of different apex angles. The relationships between the characteristics of the indentation load–displacement curves and the elasto-plastic material properties were computationally established. A reverse analysis algorithm based on load–displacement curves obtained from dual indenters was presented. It was demonstrated that the proposed reverse analysis algorithm can uniquely recover the elasto-plastic material properties from the load–displacement curves of two conical indenters with different apex angles. The numerical results obtained are in good agreement with published values.     

Numerical implementation of composite Koiter shells including membrane/bending coupling coefficientsImplémentation numérique des coques composites de Koiter prenant en considération les coefficients de couplage membrane/flexions

We propose a way for determining the generalized coefficients of rigidity – some of which are membrane/bending coupling coefficients – which appear in the deformation energy of the Koiter model of thin shells. This is concerned with a heterogeneous material in the thickness direction. A new program to compute these coefficients is implemented in the finite element code Modulef, in order to simulate problems of thin multilayered shells with linearly elastic anisotropic layers. We propose an example of an inhibited multilayered thin shell, with hyperbolic middle surface, involving a composite material with unidirectional fibres. To cite this article: H. Ranarivelo, C. R. Mecanique 330 (2002) 273–278.     

VIBRO-ACOUSTIC BEHAVIOR OF SUBMERGED CYLINDRICAL SHELLS: ANALYTICAL FORMULATION AND NUMERICAL MODEL

We propose both an analytical formulation and a numerical model to study the hydroelastic or vibroacoustic behaviour of cylindrical thin shells immersed in an unbounded, inviscid and heavy fluid. The analytical solution allows us to calculate the dynamic response and the pressure radiated in the far field by a baffled cylinder. This formulation uses the truncated modal basis of the dry structure to expand the displacements of the submerged shell. The analytical model is used as a reference in order to validate a numerical model which couples the finite element method (FEM) to the boundary element method (BEM). As opposed to the analytical formulation which is dedicated to baffled circular cylinders only, the numerical model allows us to treat any axisymmetric shell, such as cylindrical and spherical shells, or more complex shells of revolution. The structure is idealized by two-node ring finite elements and the boundary equation is solved using the method of singularities.     

Elasto-plastic analysis of Reissner-Mindlin plates by the use of hybrid stress FEM with layered model

Based on a _mR -type variational formulation featuring a cross-depth layered model in conjunction with a mechanical sub-element for simulating the material constitution, the cross-depth plasticity development of the Reissner-Mindlin plate is investigated by following the loading process. A 4-node quadrilateral hybrid-stressc ⁰-continuous plate-bending element HPT-9 is formulated. Numerical examinations demonstrate its remarkable characteristic behavior in being free from spurious kinematic mode, capable of alleviating locking difficulties as the thin plate limit is approached and providing numerical results with remarkable accuracy and computational efficiency over Spilker's counterpart LH₄. An elasto-plastic analysis of Reissner-Mindlin plates has justified the validity and effectiveness of the present scheme in depicting the cross-depth plasticity development following the loading process.The Project is Supported by National Natural Science Foundation of China.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Stress function method in shakedown analysis of axisymmetric shells

A self-equilibrated stress obtained from the stress functions of thin shells is used for the static shakedown theorem as a residual stress. In combination with the finite element method, a linear programming formulation of the shakedown analysis of axisymmetric shells is derived. The physical meaning of the stress function method is clear and its computing amount is small. Some examples of the plates and shells show that the method is reasonable and efficient.The project was supported by the National Natural Science Foundation of China     

Simulation of fabric drape using a thin plate element with finite rotation

The draping behavior of fabric is simulated by using four node quadrilateral thin plate elements with finite rotation. The finite element formulation is based on the total Lagrangian approach. An exact representation of finite rotation is introduced. The strain energy function accounting for the material symmetry is obtained by the tensor representation theory. To avoid shear locking, the assumed strain technique for transverse shear is adopted. The conjugate gradient method with a proposed line search algorithm is employed to minimize energy and reach the final shape of fabric. The draping behavior of a rectangular piece of fabric over a rectangular table is simulated.     

单元类型和尺寸对裂尖疲劳塑性数值模拟的影响研究

本文采用Jiang-Sehitoglu循环塑性模型和多轴疲劳准则对紧凑拉伸式样裂尖的循环塑性变形、裂纹扩展速率和残余应力进行了有限元数值模拟,着重考察了单元的类型和最小单元尺寸对裂尖循环塑性和裂纹扩展速率的影响.紧凑拉伸试样的材料为1070钢,数值模拟采用了线性单元(四节点)和二次单元(八节点)两种单元,裂尖附近有限元单元的最小尺寸从0.007mm到0.24mm不等.文中将裂纹扩展速率的预测值与实验值进行了比较,通过对裂纹扩展速率的比较,确定在疲劳塑性分析时对单元类型和尺寸进行合理选取.