Micromechanical approach to damage mechanics of composite materials with fabric tensors

The purpose of this study is to apply continuum damage mechanics – introduced through the concept of fabric tensors – to composite materials within the framework of the theory of elasticity. A directional data model of damage mechanics for composite materials will be developed using fabric tensors. The introduction of fabric tensors into the analysis of damage of composite materials will allow for an enhanced and better understood physical meaning of damage. The micromechanical approach will be used here to relate the damage effect through fabric tensors to the behavior of composite materials. In this approach, damage mechanics is introduced separately to the constituents of the composite material through different constituents’ damage effect tensors. The damaged properties of the composite system as a whole can then be obtained by proper homogenization of the damaged properties of the constituents.

The derivation of a generalized formulation of damage evolution will be shown here in a mathematically consistent manner that is based on sound thermodynamic principles. Numerical examples will be presented to show applicability. In addition, damage evolution for the one-dimensional tension case is also illustrated.     

On analysis of the elasto-viscoplastic response of single crystals with anisotropic damage: constitutive modelling and computational aspects

Based on the concept of continuum damage mechanics, an anisotropic damage model for single crystals under the theory of crystal plasticity is presented. Damage and inelastic deformations are incorporated in the proposed model which is developed within the framework of thermodynamics with internal state variables. The dependence of the plastic anisotropy on the damage evolution has been considered. The anisotropic damage is characterized kinematically here through a second-order damage tensor which is physically based. The proposed model can successfully describe the interaction between the evolution of micro-structure of single crystals such as lattice orientation and the hardness development of each slip system and the process of material degradation. The Newton–Raphson iterative scheme is used to integrate the constitutive equations that work directly with the evolution equations for the elastic deformation gradient. The consistent algorithmic tangent stiffness for the present algorithm is formulated. The prescribed algorithm together with the consistent algorithmic tangent stiffness has been implemented into the ABAQUS finite element code by using user subroutine. Using the loading processes with homogeneous deformations and simulation of the classical tensile test of a notched bar illustrate the basic aspects of the model described. Numerical simulations show the validation and performance of the present model and algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

A finite strain model of combined viscoplasticity and rate-independent plasticity without a yield surface

A new internal variable formulation dealing with mechanisms with different characteristic times in solid materials is proposed within a finite deformation framework. The framework relies crucially on the consistent combination of a general viscoplastic theory and a rate-independent theory (generalized plasticity) which does not involve the yield surface concept as a basic ingredient. The formulation is developed initially in a material setting and then is extended to a covariant one by applying some basic elements and results from the tensor analysis on manifolds. The covariant balance of energy is systematically employed for the derivation of the mechanical state equations. It is shown that unlike the case of finite elasticity, for the proposed formulation the covariant balance of energy does not yield the Doyle–Ericksen formula, unless a further assumption is made. As an application, by considering the material (intrinsic) metric as a primary internal variable accounting for both elastic and viscoplastic (dissipative) phenomena within the body, a constitutive model is proposed. The ability of the model in simulating several patterns of the complex response of metals under quasi-static and dynamic loadings is assessed by representative numerical examples.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.     

基于CMSE理论和小生境遗传算法的结构多损伤检测方法研究

本文提供了一种基于交叉模态应变能简称CMSE（cross modal strain energy）和小生境遗传算法的结构多损伤位置和严重程度的有效检测方法。交叉模态应变能方法可以使用完好结构和损伤结构的任意一阶模态信息，并且可以使用极少量（例如只用一阶测量模态）的测量信息。不仅与传统的模态应变能方法相比有了很大的改进，而且对实际检测过程中的信息测量降低了要求。在损伤位置的检测中引入了小生境遗传算法，降低了计算量，提高了检测效率，有利于大型复杂结构的损伤位置检测。复合材料机翼盒段算例结果表明，CMSE方法与小生境遗传算法相结合是检测结构多损伤的有效方法之一。     

Extended Analysis of a Damage Prognosis Approach Based on Interval Arithmetic

Abstract: In the previous work by the authors, an approach to damage prognosis which incorporates the effects of uncertainties has been proposed. This approach is based on the idea of integrating the laws for damage progression within the framework of interval arithmetic, a framework that naturally accommodates uncertainty. The modest purposes of this study are to extend/modify the approach in a small but significant manner and also to consider a more complex problem than the previous benchmark. The developments in the paper are illustrated through two case studies. The first case study revisits the benchmark of the initial paper – an isotropic finite plate under harmonic uniaxial loading, where the damage is assumed to be a central, mode I, through‐crack. The damage propagation model for the cracked plate is the Paris‐Erdogan law. The second example considers the growth of internal delaminations in composite plates subjected to cyclic compression. Under compressive loading, laminated composite plates experience repeated buckling‐unloading of the delaminated layer with a consequent reduction in interlayer resistance. The state of stress near the delamination tip is of mixed mode I and II. A graphite‐epoxy unidirectional specimen has been assumed here and the thin‐film model of Chai and colleagues is used; the latter is a closed‐form solution for the initial post‐buckling and growth behaviour in the ideal case of a surface delamination in an infinitely thick plate. The delamination growth law considered is the one proposed by Kardomateas and co‐workers. As the objective of the study is to examine the effects of uncertainties, the various models are considered within the framework of interval arithmetic; Monte Carlo analysis is also carried out to provide a basis for comparison.     

An anisotropic damage model with dilatation for concrete

An anisotropic damage model for concrete is developed within the general framework of the internal variable theory of thermodynamics. The rate of change of the compliance tensor is described in terms of kinetic relations involving a damage parameter whose increment is governed by the consistency equation associated with a pressure-dependent damage surface in stress space. The use of the compliance tensor implies that damage is reflected through a fourth-order tensor. Dilatation is obtained as a consequence of damage, and permanent deformation due to damage is addressed via a simple evolution equation. The theory is capable of accommodating the anisotropy induced by microcracking and is very suitable for computer implementation.     

Numerical implementation of a multiple-ISV thermodynamically-based work potential theory for modeling progressive damage and failure in fiber-reinforced laminates

A thermodynamically-based work potential theory for modeling progressive damage and failure in fiber-reinforced laminates is presented. The current, multiple-internal state variable (ISV) formulation, referred to as enhanced Schapery theory, utilizes separate ISVs for modeling the effects of damage and failure. Damage is considered to be the effect of any structural changes in a material that manifest as pre-peak non-linearity in the stress versus strain response. Conversely, failure is taken to be the effect of the evolution of any mechanisms that results in post-peak strain softening, resulting in a negative tangent stiffness. It is assumed that matrix microdamage is the dominant damage mechanism in continuous fiber-reinforced polymer matrix laminates, and its evolution is controlled with a single ISV. Three additional ISVs are introduced to account for failure due to mode I transverse cracking, mode II transverse cracking, and mode I axial failure. Typically, failure evolution (i.e., post-peak strain softening characterized through a negative tangent stiffness) results in pathologically mesh dependent solutions within a finite element (FE) framework. Therefore, consistent characteristic lengths are introduced into the formulation to govern the evolution of the three failure ISVs. Using the stationarity of the total work potential with respect to each ISV, a set of thermodynamically consistent evolution equations for the ISVs are derived. The theory is implemented in association with the commercial FE software, Abaqus. Objectivity of total energy dissipated during the failure process, with regards to refinements in the FE mesh, is demonstrated. The model is also verified against experimental results from two laminated, T800/3900-2 panels containing a central notch and different fiber-orientation stacking sequences. Global load versus displacement, global load versus local strain gage data, and macroscopic failure paths obtained from the models are compared against the experimental results.     

Finite deformation plasticity and viscoplasticity laws exhibiting nonlinear hardening rules

 This paper deals with plasticity and viscoplasticity laws exhibiting nonlinear kinematic hardening as well as nonlinear isotropic hardening rules. In Tsakmakis (1996a, b) a constitutive theory has been formulated within the framework of finite deformations, which is based on the concept of so-called dual variables and associated time derivatives. Within two families of dual variables, two different formulations have been proposed for kinematic hardening, referred to as Models 1 and 2. In particular, rigid plastic deformations without isotropic hardening have been considered. In the present paper, the constitutive theory of Tsakmakis (1996a, b) is appropriately extended to take into account isotropic hardening as well as elastic deformations. Care is taken that the evolution equations governing the hardening response fulfill the intrinsic dissipation inequality in every admissible process. For the case of small elastic strains combined with a simplification concerning kinematic hardening, to be explained in the paper, an efficient, implicit time-integration algorithm is presented. The algorithm is developed with a view to implementation in the ABAQUS Finite Element code. Also, explicit formulas for the consistent tangent modulus are derived. Received 22 September 1999     

Evaluating the strengths of thick aluminum-to-aluminum joints with different adhesive lengths and thicknesses

Composite joints are often the weakest elements in composite structures. In this paper, we propose a modified version of the damage zone theory based on the yield strain ratio. We use this framework to predict failure loads for various adhesive joints. Thick aluminum-to-aluminum joint specimens with eight different adhesive lengths and four adhesive thicknesses were manufactured and tested. The strengths of different adhesive lengths could be predicted to within 15.4% using the damage zone ratio method. In addition, the strengths of joints that feature different adhesive thicknesses were predicted to within 16.3% using the damage zone ratio method.     

Parameter estimation of the generalized extreme value distribution for structural health monitoring

Structural health monitoring (SHM) can be defined as a statistical pattern recognition problem which necessitates establishing a decision boundary for damage identification. In general, data points associated with damage manifest themselves near the tail of a baseline data distribution, which is obtained from a healthy state of a structure. Because damage diagnosis is concerned with outliers potentially associated with damage, improper modeling of the tail distribution may impair the performance of SHM by misclassifying a condition state of the structure. This paper attempts to address the issue of establishing a decision boundary based on extreme value statistics (EVS) so that the extreme values associated with the tail distribution can be properly modeled. The generalized extreme value distribution (GEV) is adopted to model the extreme values. A theoretical framework and a parameter estimation technique are developed to automatically estimate model parameters of the GEV. The validity of the proposed method is demonstrated through numerically simulated data, previously published real sample data sets, and experimental data obtained from the damage detection study in a composite plate.     

A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete

An existing viscoelastic constitutive model which accounts for the effects of rate-dependent damage growth is described and applied successfully to characterize the uniaxial stress, constant strain rate behavior of asphalt concrete. The special case of an elastic continuum damage model with multiaxial loading, which is based upon thermodynamics of irreversible processes with internal state variables, is first reviewed and then it is shown how this model has been extended to a corresponding viscoelastic damage model through the use of an elastic-viscoelastic correspondence principle. The general mathematical model is next specialized to uniaxial loading. A rate-type evolution law, similar in form to a crack growth law for a viscoelastic medium, is adopted for describing the damage growth within the body. Results from laboratory tests of uniaxial specimens under axial tension at different strain rates are then shown to be consistent with the theory. The discussion of data analysis describes the specific procedure used here to obtain the material parameters in the constitutive model for uniaxial loading and how the method may be generalized for multiaxial loading.     

Viscoplastic microstructural processes in solids under impulsive thermal loading

The stress–strain state of a disk caused by a thermal pulse is investigated within the framework of a dynamic coupled thermomechanics problem accounting for microstructural phase transformations caused by the heating and subsequent cooling of the material. The solution of the axisymmetric problem is obtained numerically from a thermodynamically consistent theory for the inelastic behavior of the material which takes into account the thermal dependencies of the material properties using the finite element method. The influence of the microstructural transformations on the dynamic and quasistatic response of the material as well as the residual stress–strain state within the irradiated zone are also studied in detail.     

Identification of damage mechanisms in concrete under high level creep by the acoustic emission technique

Many researchers have proposed hypotheses concerning the physical mechanisms that govern creep and among them the development of microcracks is well recognized. For high load levels, microcracking may initiates at the moment of load application and begins to grow to form a time-dependent crack path. An experimental investigation is proposed here in order to provide interesting insight into the coupling between creep and damage with specimens loaded in flexure. The acoustic emission (AE) technique is used as a tool to provide information on the pertinence of the physical hypothesis that microcracks appear during creep. An original test is performed to accelerate the creep phenomenon by submitting concrete beams to desiccation after a basic creep period. The results show a good proportionality between the creep deformation and the AE activity and thus the efficiency of acoustic measurements for the estimation of the state of damage. In addition, an unsupervised pattern recognition analysis is used as a tool for the classification of the monitored AE signatures. The cluster analysis shows two clusters during basic creep and three clusters during desiccation creep indicating different damage mechanisms.     

A damage-mechanics-based approach for modelling decohesion in adhesively bonded assemblies

A cohesive interface model formulated within the framework of damage mechanics is presented for the simulation of decohesion in adhesively bonded assemblies. Characteristic features of the model are: the introduction of a single energy-based damage variable for describing the damage state of the interface; use of a decohesion propagation condition relying upon the linear elastic fracture mechanics (LEFM) energy balance; a treatment for the mixed-mode situation based on the definition of an equivalent energy release rate whose expression is consistently derived from the formulation. The comparisons between numerical and experimental responses obtained for typical test problems illustrate the capabilities of the proposed approach.     

Relative kinematics exploited in Kane's approach to describe multibody systems in relative motion

A general constitutive framework is presented capable of representing different irreversible deformation modes, like plasticity, elastic damage, complex evolution of the hardening properties and the induced coupling effects. The formulation can be framed in the generalized standard material models with internal variables and multiple dissipative activation functions. The formulation is thermodynamically consistent and the state laws, the structure of the dissipation and of the activation functions are all derived complying with the principles of thermodynamics. The generalized flow rules are derived under the hypothesis of generalized associativity. The main aspect of the proposed model is the definition of an internal damage variable which is appended as a factor of the internal energy and is then able to describe a progressive degradation of an isotropic hardening modulus. In particular, the one-dimensional stress-strain case, for a ductile material is examined in some detail. The complex evolution of the strength which follow the first post yielding phase is modelled as an irreversible degradation of an hardening component. This case is examined neglecting the elastic damage effects, since for metals the degradation of the elastic properties is produced at large strains and can be considered a subsequent state. The discrete step problem is presented and a resolution strategy based on the Euler-backward difference scheme is proposed. The numerical results for an assigned, monotonically increasing, total strain show that the proposed model possesses the feature to represent complex hardening evolutions including the hardening saturation conditions and some unstable branches which characterize softening transition phases.     

Coupled damage tensors and weakest link theory for the description of crack induced anisotropy in concrete

Crack induced anisotropy in concrete is modeled within the general framework of damage mechanics. The damage state, formulated by using the effective stresses, is described by two second order tensors for direct and for induced cracks, respectively. These variables are deduced from different causes of deterioration by the generalized weakest link theory. The major contributions of this paper are the development of a failure criterion in multiaxial compression loading, and the definition of a projection method for assessing the effects of an initial oriented damage in any loading direction. The model is applied to the numerical analysis of concrete structures and the results are compared to available experimental data.     

Deflection dynamics of rock beam caused by ultrasound

A new method for monitoring the time dependent dynamics of materials is proposed and implemented. By completely separating the conditioning (here ultrasound), the probing (here gravity), and the material state indicator (here deflection), the details of this dynamic process becomes apparent. The method allows both continuous monitoring of the material state without cross-interaction by the measuring process on the results, as well as complete freedom of conditioning and probing. It was successfully tested for sensitivity and repeatability when applied on a horizontally suspended beam of gabbro rock, which was observed to sag when subjected to ultrasound. These introductory tests have given new insights. The beam rises back, against the force of gravity, after the ultrasound is turned off. The deflection motions are fast both at the onset and at the termination of ultrasound, with the subsequent continuations being much slower. This new method is able to provide the higher accuracy needed for the advancement of the theoretical framework for material property time dependent dynamics.