An augmented multicrack elastoplastic damage model for tensile cracking

In this paper we present the general formulation and numerical aspects of an augmented multicrack elastoplastic damage model aiming to reflect the crack induced anisotropy in concrete like quasi-brittle materials. Consistent evolution laws for the involved internal variables are derived based on the augmented Lagrangian method. The (time) discrete formulation and the corresponding variational structure are investigated, with the Euler–Lagrangian equations defining the closest-point projection approximation of the proposed model. The numerical aspects, such as the stress updating algorithm and the algorithmic consistent tangent moduli, are also discussed in details. It is found that in the developed numerical algorithm the active loading surfaces are determined in such a posterior manner that potential numerical problems due to the iteratively updating procedure in classical algorithms can be avoided. The proposed model is applied to the modeling of tensile cracking in concrete. The behavior of a single crack is characterized by an elliptical cracking surface and a hyperbolic softening function, with the orientations of potential cracks determined by Mohr’s postulate. The model is verified by calculating the single point stress vs. strain relations of concrete under several typical proportional and non-proportional loading cases. Finally, two benchmark tests of concrete structures, i.e. four-point bending beam under cyclic loading (Hordijk, 1992) and double edge notched specimens under mixed tension/shear forces (Nooru-Mohamed, 1992), are numerically simulated. Both predicted load vs. displacement curves and crack patterns agree well with the experimental data.     

金属材料层裂破坏的内聚力模型

本文把内聚力单元嵌入到连续介质有限元单元之间,构建了一个层裂破坏的内聚力模型,以计及层裂破坏过程中能量耗散行为。采用该模型对平板撞击条件下的20号钢层裂实验进行了数值模拟研究,重点讨论了内聚力模型特征参数对计算结果的影响规律。研究结果表明采用指数型损伤演化行为的内聚力模型可以较好地描述弹塑性材料层裂破坏过程中的非线性能量耗散行为。利用一发实测自由面速度波剖面对计算结果进行对比校准,可确定内聚力模型特征参数。该特征参数可同样成功地预示不同撞击速度下的层裂实验,获得的模拟曲线与实验曲线之间符合程度很好,特别是自由面速度“回跳”后波形振荡周期和幅值与实验结果非常接近。这表明了内聚力模型在描述层裂过程中能量耗散行为方面具有较好适用性,并且不难由简单实验标定相关的特征参数。     

Direct Extraction of Cohesive Fracture Properties from Digital Image Correlation: A Hybrid Inverse Technique

The accuracy of an adopted cohesive zone model (CZM) can affect the simulated fracture response significantly. The CZM has been usually obtained using global experimental response, e.g., load versus either crack opening displacement or load-line displacement. Apparently, deduction of a local material property from a global response does not provide full confidence of the adopted model. The difficulties are: (1) fundamentally, stress cannot be measured directly and the cohesive stress distribution is non-uniform; (2) accurate measurement of the full crack profile (crack opening displacement at every point) is experimentally difficult to obtain. An attractive feature of digital image correlation (DIC) is that it allows relatively accurate measurement of the whole displacement field on a flat surface. It has been utilized to measure the mode I traction-separation relation. A hybrid inverse method based on combined use of DIC and finite element method is used in this study to compute the cohesive properties of a ductile adhesive, Devcon Plastic Welder II, and a quasi-brittle plastic, G-10/FR4 Garolite. Fracture tests were conducted on single edge-notched beam specimens (SENB) under four-point bending. A full-field DIC algorithm was employed to compute the smooth and continuous displacement field, which is then used as input to a finite element model for inverse analysis through an optimization procedure. The unknown CZM is constructed using a flexible B-spline without any “a priori” assumption on the shape. The inversely computed CZMs for both materials yield consistent results. Finally, the computed CZMs are verified through fracture simulation, which shows good experimental agreement.     

复合材料粘聚区模型的强度参数预测

提出了一种基于周期性代表性单元（RVE）的细观模型,用于预测粘聚区模型的强度参数,以提高基于粘聚区模型的有限元法模拟复合材料分层的精度。模型从复合材料的细观结构出发,采用最大主应力准则判断粘聚层的初始裂纹的萌生,从而建立了粘聚强度与基体强度的关系;并以宏观正交各向异性为假设条件,确定了RVE的周期性位移边界条件。用该模型预测了AS4/PEEK和T700/QY8911层合板不同铺层间的粘聚强度。采用所预测的粘聚强度值对混合模式弯曲（MMB）和六点弯曲试验进行了仿真,仿真结果与实验值具有较好的一致性。     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets

Magnesium alloy sheets have been extending their field of applications to automotive and electronic industries taking advantage of their excellent light weight property. In addition to well-known lower formability, magnesium alloys have unique mechanical properties which have not been thoroughly studied: high in-plane anisotropy/asymmetry of yield stress and hardening response. The reason of the unusual mechanical behavior of magnesium alloys has been understood by the limited symmetry crystal structure of HCP metals and thus by deformation twinning. In this paper, the phenomenological continuum plasticity models considering the unusual plastic behavior of magnesium alloy sheet were developed for a finite element analysis. A hardening law based on two-surface model was further extended to consider the general stress–strain response of metal sheets including Bauschinger effect, transient behavior and the unusual asymmetry. Three deformation modes observed during the continuous in-plane tension/compression tests were mathematically formulated with simplified relations between the state of deformation and their histories. In terms of the anisotropy and asymmetry of the initial yield stress, the Drucker–Prager’s pressure dependent yield surface was modified to include the anisotropy of magnesium alloy. The numerical formulations and characterization procedures were also presented and finally the correlation of simulation with measurements was performed to validate the proposed theory.     

A multi-scale enriched model for the analysis of masonry panels

A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels.     

Analytical and numerical analysis of frictional damage in quasi brittle materials

Frictional sliding and crack growth are two main dissipation processes in quasi brittle materials. The frictional sliding along closed cracks is the origin of macroscopic plastic deformation while the crack growth induces a material damage. The main difficulty of modeling is to consider the inherent coupling between these two processes. Various models and associated numerical algorithms have been proposed. But there are so far no analytical solutions even for simple loading paths for the validation of such algorithms. In this paper, we first present a micro-mechanical model taking into account the damage-friction coupling for a large class of quasi brittle materials. The model is formulated by combining a linear homogenization procedure with the Mori–Tanaka scheme and the irreversible thermodynamics framework. As an original contribution, a series of analytical solutions of stress–strain relations are developed for various loading paths. Based on the micro-mechanical model, two numerical integration algorithms are exploited. The first one involves a coupled friction/damage correction scheme, which is consistent with the coupling nature of the constitutive model. The second one contains a friction/damage decoupling scheme with two consecutive steps: the friction correction followed by the damage correction. With the analytical solutions as reference results, the two algorithms are assessed through a series of numerical tests. It is found that the decoupling correction scheme is efficient to guarantee a systematic numerical convergence.     

Numerical simulation of quasi-brittle fracture using damaging cohesive surfaces

The cohesive surface methodology is used in a numerical study of fracture of concrete. The traction vs. separation response is governed by an isotropic damage law in which damage evolves according to a prescribed one-dimensional linear or exponential softening law. Cohesive surfaces are immersed in the continuum to allow for a maximum freedom of crack path selection. The single edge notched four point shear beam and the double edge notched tensile bar are used to study: (i) the influence of the tangential cohesive response on the development of the fracture path and (ii) the mesh alignment sensitivity. It is shown that in the present formulation, the tangential cohesive response has a minor influence on both crack path and global characteristics. Mesh alignment does have a significant influence on the outcome of the numerical analysis.     

A nonlinear homogenization procedure for periodic masonry

The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material.     

A DUAL-SURFACE DAMAGE MODEL AND EVALUATION FOR NATURAL SOILS WITHIN THE THERMOMECHANICAL FRAMEWORK

Naturally deposited or residual soils exhibit more complicated behavior than remolded clays. A dual-surface damage model for structured soils is developed based on the thermodynamics framework established in our first paper. The shift stresses and the transformation between the generalized dissipative stress space and actual stress space are established following a systematic procedure. The corresponding constitutive behavior of the proposed model is determined, which reflects the internal structural configuration and damage behavior for geomaterials. Four evolution variables κj^i（i=D, R;j=V, S） and the basic parameters λ, s, v and e0 are introduced to account for the progressive loss of internal structure for natural clays. A series of fully triaxial tests and isotropic compression tests are performed for structured and reconstituted samples of Beijing and Zhengzhou natural clays. The validation of the proposed model is examined by comparing the numerical results with the experimental data.     

Numerical analysis of progressive damage in nonwoven fibrous networks under tension

Understanding a mechanical behaviour of polymer-based nonwoven materials that include large-strain deformation and damage can help to evaluate a response of nonwoven fibrous networks to various loading conditions. Here, a nonwoven felt made by thermal bonding of polypropylene fibres was used as a model system. Its deformation and damage behaviour was analysed by means of experimental assessment of damage evolution based on single-fibre failure and finite-element simulations. Tensile tests of nonwoven fabrics were carried out to characterise their damage behaviour under in-plane mechanical loading. It was found that progressive failure of fibres led to localization of damage initiation and propagation, ultimately resulting in failure of the nonwoven felt. To obtain the criteria that control the onset and propagation of damage in these materials, tensile tests on single fibres, extracted from the felt with bond points attached to their ends, were performed. A finite-element model was developed to study damage initiation and propagation in nonwovens. In the model, structural randomness of a nonwoven fibrous network was implemented by means of direct introduction of fibres according to the orientation distribution function. The evolution of damage in the network was controlled by a single-fibre failure criterion obtained experimentally. The proposed numerical model not only captured the macroscopic response of the felt successfully but also reproduced the underlying mechanisms involved in deformation and damage of nonwovens.     

A thermodynamically-consistent microplane model for shape memory alloys

In microplane theory, it is assumed that a macroscopic stress tensor is projected to the microplane stresses. It is also assumed that 1D constitutive laws are defined for associated stress and strain components on all microplanes passing through a material point. The macroscopic strain tensor is obtained by strain integration on microplanes of all orientations at a point by using a homogenization process. Traditionally, microplane formulation has been based on the Volumetric–Deviatoric–Tangential split and macroscopic strain tensor was derived using the principle of complementary virtual work. It has been shown that this formulation could violate the second law of thermodynamics in some loading conditions. The present paper focuses on modeling of shape memory alloys using microplane formulation in a thermodynamically-consistent framework. To this end, a free energy potential is defined at the microplane level. Integrating this potential over all orientations provides the macroscopic free energy. Based on this free energy, a new formulation based on Volumetric–Deviatoric split is proposed. This formulation in a thermodynamic-consistent framework captures the behavior of shape memory alloys. Using experimental results for various loading conditions, the validity of the model has been verified.     

A tensorial description of the transformation kinetics of the martensitic phase transformation

Based on a local examination of the phase transition front, a macroscopic second order tensor describing the thermodynamic force for the phase transformation is proposed. Consequently, an associated thermodynamic flux is introduced. These tensorial variables are embedded into a material law which describes the behavior of steels during the austenite–martensite phase transformation. The material law is implemented into a finite element formulation. Homogeneous tests in pure tension/compression and torsion are performed to verify the behavior of the material law. Due to the independent modeling of the behavior of the phases, the influence of the yield stress of the austenite on the transformation kinetics can be verified. A classical example is presented to show the ability of the model to calculate large structural problems.     

A direct experimental–numerical method for investigations of a laboratory under-platform damper behavior

Under-platform dampers are commonly adopted in order to mitigate resonant vibration of turbine blades. The need for reliable models for the design of under-platform dampers has led to a considerable amount of technical literature on under-platform damper modeling in the last three decades.Although much effort has been devoted to the under-platform damper modeling in order to avail of a predictive tool for new damper designs, experimental validation of the modeling is still necessary. This is due to the complexity caused by the interaction of the contacts at the two damper-platform interfaces with the additional complication of the variablity of physical contact parameters (in particularly friction) and their nonlinearity. The traditional experimental configuration for evaluating under-platform damper behavior is measuring the blade tip response by incorporating the damper between two adjacent blades (representing a cyclic segment of the bladed disk) under controlled excitation. The effectiveness of the damper is revealed by the difference in blade tip response depending on whether the damper is applied or not. With this approach one cannot investigate the damper behavior directly and no measurements of the contact parameters can be undertake. Consequently, tentative values for the contact parameters are assigned from previous experience and then case-by-case finely tuned until the numerical predictions are consistent with the experimental evidence. In this method the physical determination of the contact parameters is obtained using test rigs designed to produce single contact tests which simulate the local damper-platfom contact geometry. However, the significant limitation of single contact test results is that they do not reveal the dependence of contact parameters on the real damper contact conditions. The method proposed in this paper overcomes this problem.In this new approach a purposely developed test rig allows the in-plane forces transferred through the damper between the two simulated platforms to be measured, while at the same time monitoring in-plane relative displacements of the platforms. The in-plane damper kinematics are reconstructed from the experimental data using the contact constraints and two damper motion measurements, one translational and one rotational. The measurement procedures provide reliable results, which allow very fine details of contact kinematics to be revealed. It is demonstrated that the highly satisfactory performance of the test rig and the related procedures allows fine tuning of the contact parameters (local friction coefficients and contact stiffness), which can be safely fed into a direct time integration numerical model.The numerical model is, in turn, cross-checked against the experimental results, and then used to acquire deeper understanding of the damper behavior (e.g. contact state, slipping and sticking displacement at all contact points), giving an insight into those features which the measurements alone are not capable of producing. The numerical model of the system is based on one key assumption: the contact model does not take into account the microslip effect that exists in the experiments.Although there is room for improvement of both experimental configuration and numerical modeling, which future work will consider, the results obtained with this approach demonstrate that the optimization of dampers can be less a matter of trial and error development and more a matter of knowledge of damper dynamics.     

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).     

Characterization of the Orthotropic Elastic Constants of a Micronic Woven Wire Mesh via Digital Image Correlation

Woven structures are steadily emerging as excellent reinforcing components in dual-phase composite materials subjected to multiaxial loads, thermal shock, and aggressive reactants in the environment. Metallic woven wire mesh materials in particular display good ductility and relatively high specific strength and specific resilience. While use of this class of materials is rapidly expanding, a significant gap in property characterization remains. This research classifies the homogenized, orthotropic material properties of a representative twill dutch woven wire mesh through the use of in-plane uniaxial tensile experiments incorporating a Digital Image Correlation (DIC) strain measurement technique. Values for elastic modulus and Poisson’s ratio are calculated from the experimental data, and shear modulus values are identified by means of constitutive modeling. This approach establishes a reproducible method for characterizing the in-plane elastic response of micronic metallic woven materials via macro-scale uniaxial tensile tests, and shows that a homogenous orthotropic constitutive model may be employed to describe the macro-scale elasticity of this class of materials with reasonable accuracy.     

Strong discontinuity analysis of a class of anisotropic continuum damage constitutive models – Part II: Concrete material application

On the basis of the strong discontinuity analysis, a discrete model expressed in terms of traction vector-displacement jump has been constructed from a continuous model expressed in terms of stress–strain law. In the first part of the paper, this approach has been extended to a class of anisotropic continuum damage constitutive models [1]. In this second part of the paper, the proposed class of discrete anisotropic damage constitutive models is particularized. More precisely, a micromechanical-based anisotropic damage constitutive model is derived. This model accounts in a natural manner for particular crack families orientation. The aims of this paper are (i) to illustrate the capabilities of the proposed discrete enhanced model in reproducing the induced anisotropy appearing in quasi-brittle materials when cracking and (ii) to assess the numerical robustness of the time integration scheme. For this purpose, two numerical examples at the material point level are exposed after a brief presentation of the time integration scheme. The correspondence between the continuous and the discrete model as well as the induced anisotropy features are emphasized.     

Residual Stress Analysis of Orthotropic Materials by the Through-hole Drilling Method

The present study deals with the development and the application of the through-hole drilling method for the residual stress analysis in orthotropic materials. Through a systematic theoretical study of the stress field present on orthotropic plates with a circular hole, the relationships between the relaxed strains measured by a rectangular strain gauge rosette and the Cartesian components of the unknown residual stresses are obtained. The theoretical formulas of each influence coefficient allow the user an easy application of the method to the analysis of uniform-residual stresses on a generic homogeneous orthotropic material. Furthermore, to extend the method to the analysis of the residual stresses on orthotropic laminates, caused by initial in-plane loadings, an alternative formulation is implemented. The accuracy of the proposed method has been assessed through 3D numerical simulations and experimental tests carried out on unidirectional, cross-ply and angle-ply laminates.     

Non-linear response of curved sandwich panels – extended high-order approach

The geometrical non-linear behavior a curved sandwich panel with a stiff or compliant core when subjected to a pressure load using the Extended High-Order Sandwich Panel theory (EHSAPT), is presented. The formulation follows the EHSAPT procedure where the in-plane. i.e circumferential rigidity of the core is considered and the distribution of the displacements through the depth of the core are presumed. These displacement distributions are the closed-form solutions of the 2D governing equations of the curved core without circumferential rigidity that appear in the HSAPT curved sandwich panel model. The mathematical formulation includes the field equations along with the appropriate boundary and continuity conditions that take into account the high-order stress resultants in the core due to the presumed distributions. Finally a numerical study is conducted for a panel loaded by a distributed pressure at the upper face sheet. It reveals that the post-buckling response of a curved sandwich panels is associated with shallow to deep wrinkling deformations of the upper face sheet in the case of a simply-supported panel or a general non-linear pattern without wrinkles in the case of pinned supports with a short span. In both cases a stable post-buckling response is observed similar to that of a plate one.