Cohesive fracture modeling of crack growth in thick-section composites

This paper presents a combined method for modeling the mode-I and II crack growth behavior in thick-section fiber reinforced polymeric composites having a nonlinear material response. The experimental part of this study includes crack growth tests of a thick composite material system manufactured using the pultrusion process. It consists of alternating layers of E-glass unidirectional roving and continuous filament mats in a polymeric matrix. Integrated micromechanical and cohesive finite element (FE) models are used to simulate the crack growth response in eccentrically loaded single-edge-notch, (tension), ESE(T) and notched butterfly specimens. Micromechanical constitutive models for the mat and the roving layers are used to generate the effective nonlinear material behavior from the in situ fiber and matrix responses. The validity of the numerical modeling approach before the onset of crack growth is investigated using an infrared thermal method. Cohesive FE models are calibrated and used to simulate the complete crack growth behavior for different crack configurations. The proposed integrated framework of multi-scale material models with cohesive fracture models is shown to be an effective method for predicting the structural and material responses including failure load and crack growth in thick-section fiber reinforced polymeric composites.     

Progressive damage and nonlinear analysis of pultruded composite structures

This study combines a simple damage modeling approach with micromechanical models for the progressive damage analysis of pultruded composite materials and structures. Two micromodels are used to generate the nonlinear effective response of a pultruded composite system made up from two alternating layers reinforced with roving and continuous filaments mat (CFM). The layers have E-glass fiber and vinylester matrix constituents. The proposed constitutive and damage framework is integrated within a finite element (FE) code for a general nonlinear analysis of pultruded composite structures using layered shell or plate elements. The micromechanical models are implemented at the through-thickness Gaussian integration points of the pultruded cross-section. A layer-wise damage analysis approach is proposed. The Tsai–Wu failure criterion is calibrated separately for the CFM and roving layers using ultimate stress values from off-axis pultruded coupons under uniaxial loading. Once a failure is detected in one of the layers, the micromodel of that layer is no longer used. Instead, an elastic degrading material model is activated for the failed layer to simulate the post-ultimate response. Damage variables for in-plane modes of failure are considered in the effective anisotropic strain energy density of the layer. The degraded secant stiffness is used in the FE analysis. Examples of progressive damage analysis are carried out for notched plates under compression and tension, and a single-bolted connection under tension. Good agreement is shown when comparing the experimental results and the FE models that incorporate the combined micromechanical and damage models.     

Nonlinear constitutive models for FRP composites using artificial neural networks

This paper presents a new approach to generate nonlinear and multi-axial constitutive models for fiber reinforced polymeric (FRP) composites using artificial neural networks (ANNs). The new nonlinear ANN constitutive models are complete and have been integrated with displacement-based FE software for the nonlinear analysis of composite structures. The proposed ANN constitutive models are trained with experimental data obtained from off-axis tension/compression and pure shear (Arcan) tests. The proposed ANN constitutive model is generated for plane–stress states with assumed functional response in some parts of the multi-axial stress space with no experimental data. The ability of the trained ANN models to predict material response is examined directly and through FE analysis of a notched composite plate. The experimental part of this study involved coupon testing of thick-section pultruded FRP E-glass/polyester material. Nonlinear response was pronounced including in the fiber direction due to the relatively low overall fiber volume fraction (FVF). Notched composite plates were also tested to verify the FE, with ANN material models, to predict general non-homogeneous responses at the structural level.     

Efficient techniques for predicting viscoelastic behavior of sublaminates

Even for linear elastic behavior, stress analysis of thick laminated composites can be very computation intensive if every lamina is modeled discretely. In such cases, modeling of individual lamina is impractical and the homogenization method for sublaminates becomes essential. In the current work, 3D homogenization formulas for an elastic sublaminate, which were derived by the authors in previous work, were utilized to determine the 3D effective properties for a viscoelastic sublaminate. The properties were determined by three methods that exploited the 3D elastic homogenization formulas: (i) quasi-elastic method, (ii) correspondence principle, and (iii) direct time integration of the incremental viscoelastic equations. The finite element method with discrete modeling of the plies was used to obtain reference solutions. The effective viscoelastic properties obtained using the three methods based on the elastic homogenization formulas were in very good agreement with the reference solution. Among these methods, the quasi-elastic method was found to be both accurate and the simplest method in determining the effective properties. The methods were also used to predict the stress response of a sublaminate to different strain histories. The direct time integration method using the 3D elastic homogenization formulas performs accurately and efficiently for this problem.     

Finite element modeling of impact, damage evolution and penetration of thick-section composites

Impact, damage evolution and penetration of thick-section composites are investigated using explicit finite element (FE) analysis. A full 3D FE model of impact on thick-section composites is developed. The analysis includes initiation and progressive damage of the composite during impact and penetration over a wide range of impact velocities, i.e., from 50 m/s to 1000 m/s. Low velocity impact damage is modeled using a set of computational parameters determined through parametric simulation of quasi-static punch shear experiments. At intermediate and high impact velocities, complete penetration of the composite plate is predicted with higher residual velocities than experiments. This observation revealed that the penetration-erosion phenomenology is a function of post-damage material softening parameters, strain rate dependent parameters and erosion strain parameters. With the correct choice of these parameters, the finite element model accurately correlates with ballistic impact experiments. The validated FE model is then used to generate the time history of projectile velocity, displacement and penetration resistance force. Based on the experimental and computational results, the impact and penetration process is divided into two phases, i.e., short time Phase I - shock compression, and long time Phase II - penetration. Detailed damage and penetration mechanisms during these phases are presented.     

Nonlinearly viscoelastic analysis of asphalt mixes subjected to shear loading

This study presents the characterization of the nonlinearly viscoelastic behavior of hot mix asphalt (HMA) at different temperatures and strain levels using Schapery’s model. A recursive-iterative numerical algorithm is generated for the nonlinearly viscoelastic response and implemented in a displacement-based finite element (FE) code. Then, this model is employed to describe experimental frequency sweep measurements of two asphalt mixes with fine and coarse gradations under several combined temperatures and shear strain levels. The frequency sweep measurements are converted to creep responses in the time domain using a phenomenological model (Prony series). The master curve is created for each strain level using the time temperature superposition principle (TTSP) with a reference temperature of 40°C. The linear time-dependent parameters of the Prony series are first determined by fitting a master curve created at the lowest strain level, which in this case is 0.01%. The measurements at strain levels higher than 0.01% are analyzed and used to determine the nonlinear parameters. These parameters are shown to increase with increasing strain levels, while the time–temperature shift function is found to be independent of strain levels. The FE model with the calibrated time-dependent and nonlinear material parameters is used to simulate the creep experimental tests, and reasonable predictions are shown.     

A numerical integration approach for fractional-order viscoelastic analysis of hereditary-aging structures

In this paper, a novel numerical integration scheme is proposed for fractional-order viscoelastic analysis of hereditary-aging structures. More precisely, the idea of aging is first introduced through a new phenomenological viscoelastic model characterized by variable-order fractional operators. Then, the presented fractional-order viscoelastic model is included in a variational formulation, conceived for any viscous kernel and discretized in time by employing a discontinuous Galerkin method. The accuracy of the resulting finite element (FE) scheme is analyzed through a model problem, whose exact solution is known; and the most significant variables affecting the solution quality, such as the number of Gaussian quadrature points and time subintervals, are then investigated in terms of error and computational cost. Moreover, the proposed FE integration scheme is applied to study the short- and long-term behavior of concrete structures, which, due to the severe aging exhibited during their service life, represents one of the most challenging time-dependent behavior to be investigated. Eventually, also the Euler implicit method, commonly used in commercial software, is compared.     

Multiscale design of a rectangular sandwich plate with viscoelastic core and supported at extents by viscoelastic materials

This work presents a multiscale model of viscoelastic constrained layer damping treatments for vibrating plates/beams. The approach integrates a finite element (FE) model of macroscale vibrations and a micromechanical model to include effects of microscale structure and properties. The FE model captures the shear deformation of the viscoelastic core, rotary inertial effects of all layers, and viscoelastic boundaries of the plate. Comparison with analytical and FE results validates the proposed FE model. A self-consistent (SC) model makes the micro to macro scale transition to approximate the effective behavior a heterogeneous core. Modal damping resulting from the presence of voids and negative stiffness regions in the core material is modeled. Results show that negative stiffness regions in the viscoelastic core material, even at low volume fractions, yield superior macroscopic damping behavior. The coupled SC and FE models provide a powerful multiscale predictive design tool for sandwich beams and plates.     

Responses of viscoelastic polymer composites with temperature and time dependent constituents

This study formulates a concurrent micromechanical model for predicting effective responses of fiber reinforced polymer (FRP) composites, whose constituents exhibit thermo-viscoelastic behaviors. The studied FRP composite consists of orthotropic unidirectional fiber and isotropic matrix. The viscoelastic material properties for the fiber and matrix constituents are allowed to change with the temperature field. The composite microstructures are idealized with periodically distributed square fibers in a matrix medium. A unit-cell model, consisting of four fiber and matrix subcells, is generated to obtain effective nonlinear thermo-viscoelastic responses of the composites. A time-integration algorithm is formulated to link two different thermo-viscoelastic constitutive material models at the lowest level (homogeneous fiber and matrix constituents) to the effective material responses at the macro level, and to transfer external mechanical and thermal stimuli to the constituents. This forms a concurrent micromechanical model, which is needed as the material properties of the constituents depend on the temperature field. Consistent tangent stiffness matrices are formulated at the fiber and matrix constituents and also at the effective composite level to improve prediction accuracy. The thermo-viscoelastic responses obtained from the concurrent micromodel are verified with available experimental data. Detailed finite element (FE) models of the FRP microstructures are also generated using 3D continuum elements for several fiber volume fractions. Thermo-viscoelastic responses of the concurrent micromodel are also compared to the ones of the detailed FRP microstructures.     

Coupled heat conduction and deformation in a viscoelastic composite cylinder

We study the coupled problem of deformation due to mechanical and thermal loading of a composite cylinder made up of two layers of linear isotropic viscoelastic materials. The effect of a time-varying temperature field due to unsteady heat conduction on the short term and long term material response is examined in terms of the stress, displacement, and strain fields. The material properties of the two layers of the composite cylinder at any given location and time are assumed to depend on the temperature at that location at that given instant of time. Sequentially coupled analyses of heat conduction and deformation of the viscoelastic composite cylinder are carried out. Analytical solutions for the stress, strain and displacement fields of the viscoelastic composite cylinder are obtained from the corresponding solution of the linear elasticity problem by applying the Correspondence Principle. We examine the discontinuity in the hoop stress and the radial strain at the interface of the two layers caused by mismatches in material properties, during transient heat conduction. We find that the discontinuities change over time as the mismatch in the moduli of the two layers changes due to the material properties which are time-dependent. We also investigate the effect of the thermal field on the time-dependent field variables in the composite body.     

Affordable processing of thick section and integral multi-functional composites

The use of multi-functional integral armor is of current interest in armored vehicles and military carriers. In the present study, thick-section laminated composites and multi-layered integrated composites have been processed/manufactured with the aim of providing multi-functionality including easy reparability, quick deployment, enhanced ballistic damage and fire protection, as well as lightweight advantages. The design of an integral armor utilizes a combination of thick-section structural composite, ceramic tiles, resilient rubber, fire retardant laminate liner and a composite durability cover. Processing techniques such as automated fiber placement and/or autoclave molding are traditionally used to process dissimilar multi-layered structure, but prove to be expensive.This work focuses on emerging cost-effective liquid molding processes such as vacuum assisted resin transfer/infusion molding (VARTM) for the production of thick-section and integral armor parts (up to 50 mm thick). While thick-section composites have applications in a variety of structures including armored vehicles, marine bodies, civil infrastructure, etc. in the present work they refer to the structural laminate within the integral armor. The processing steps of thick-section composite panels and integral armor have been presented. The integrity of the interfaces has been evaluated through scanning electron microscopy (SEM). Representative results on static and dynamic response (high strain rate, HSR and ballistic impact) of the VARTM processed thick-section composite panels are presented. Wherever applicable, comparisons are made to conventional closed-mold resin transfer molding (CMRTM). Process sensing by way of flow and cure monitoring of the resin in the fiber perform has been conducted using embedded direct current (DC)-based sensors in the thick-section preform and integral armor interfaces. The feasibility of cost-effective VARTM for producing thick-section composites and integral armor has been demonstrated.     

Active constrained layer damping of geometrically nonlinear vibration of rotating composite beams using 1-3 piezoelectric composite

In this paper, an analysis for active constrained layer damping (ACLD) of rotating composite beams undergoing geometrically non linear vibrations has been carried out. Commercially available vertically/obliquely reinforced 1-3 piezoelectric composite (PZC) material has been used as the material of the constraining layer of the ACLD treatment. A finite element (FE) model has been derived to carry out the analysis. The substrate beam is considered thin and hence, first order shear deformation theory (FSDT) and von-Karman type nonlinear strain–displacement relations are used to derive the coupled electromechanical nonlinear FE model. The rotary effect has been suitably modelled by incorporating extensional strain energy due to centrifugal force. The Golla–Hughes–McTavish method has been employed to model the constrained viscoelastic layer of the ACLD treatment in the time domain. The numerical responses revealed that the ACLD treatment with 1-3 PZC constraining layer efficiently performs the task of active damping of geometrically nonlinear vibrations of the rotating composite beams. The effects of the fibre orientation angles of the angle-ply substrate beams and the 1-3 PZC constraining layer on the ACLD of the geometrically nonlinear vibrations have been investigated. Also, the effect of the thickness variations of the 1-3 PZC layer and the viscoelastic constrained layer on the damping characteristics of the overall rotating composite beams has been studied.     

Coupled heat conduction and thermal stress analyses in particulate composites

This study introduces two micromechanical modeling approaches to analyze spatial variations of temperatures, stresses and displacements in particulate composites during transient heat conduction. In the first approach, a simple micromechanical model based on a first order homogenization scheme is adopted to obtain effective mechanical and thermal properties, i.e., coefficient of linear thermal expansion, thermal conductivity, and elastic constants, of a particulate composite. These effective properties are evaluated at each material (integration) point in three dimensional (3D) finite element (FE) models that represent homogenized composite media. The second approach treats a heterogeneous composite explicitly. Heterogeneous composites that consist of solid spherical particles randomly distributed in homogeneous matrix are generated using 3D continuum elements in an FE framework. For each volume fraction (VF) of particles, the FE models of heterogeneous composites with different particle sizes and arrangements are generated such that these models represent realistic volume elements “cut out” from a particulate composite. An extended definition of a RVE for heterogeneous composite is introduced, i.e., the number of heterogeneities in a fixed volume that yield the same expected effective response for the quantity of interest when subjected to similar loading and boundary conditions. Thermal and mechanical properties of both particle and matrix constituents are temperature dependent. The effects of particle distributions and sizes on the variations of temperature, stress and displacement fields are examined. The predictions of field variables from the homogenized micromechanical model are compared with those of the heterogeneous composites. Both displacement and temperature fields are found to be in good agreement. The micromechanical model that provides homogenized responses gives average values of the field variables. Thus, it cannot capture the discontinuities of the thermal stresses at the particle-matrix interface regions and local variations of the field variables within particle and matrix regions.     

A two-scale homogenization framework for nonlinear effective thermal conductivity of laminated composites

In this study, we formulate the effective temperature-dependent thermal conductivity of laminated composites. The studied laminated composites consist of laminas (plies) made of unidirectional fiber-reinforced matrix with various fiber orientations. The effective thermal conductivity is obtained through a two-scale homogenization scheme. A simplified micromechanical model of a unidirectional fiber-reinforced lamina is formulated at the lower scale. Thermal conductivities of fiber and matrix constituents are allowed to change with temperature. The upper scale uses a sublaminate model to homogenize temperature-dependent thermal conductivities of only a representative lamina stacking sequence in laminated composites. The effective thermal conductivity of each lamina, in the sublaminate model, is obtained using the simplified micromechanical model. The thermal conductivities from the micromechanical and sublaminate models represent average nonlinear properties of fictitiously homogeneous composite media. Interface conditions between fiber and matrix constituents and within laminas are assumed to be perfect. Experimental data available in the literature are used to verify the proposed multi-scale framework. We then analyze transient heat conduction in the homogenized composites. Temperature profiles, during transient heat conduction, in the homogenized composites are compared to the ones in heterogeneous composites. The heterogeneous composites, having different fiber arrangements and sizes, are modeled using finite element (FE) method.     

A combined experimental and numerical approach to study ballistic impact response of S2-glass fiber/toughened epoxy composite beams

A combined experimental and 3D dynamic nonlinear finite element (FE) approach was adopted to study damage in composite beams subject to ballistic impact using a high-speed gas gun. The time-histories of dynamic strains induced during impact were recorded using strain gages mounted on the front of the composite beam specimen. During ballistic impact tests, the impact velocity was also measured. The commercially available 3D dynamic nonlinear FE code, LS-DYNA, modified with a proposed user-defined nonlinear-orthotropic damage model, was then used to simulate the experimental results. In addition, LS-DYNA with the Chang–Chang linear-orthotropic damage model was also used for comparison. Good agreement between experimental and FE results was found from the comparisons of dynamic strain and damage patterns. Once the proposed nonlinear-orthotropic damage model was verified by experimental results, further FE simulations were conducted to predict the ballistic limit velocity (V₅₀) using either the number of damaged layer approach or a numerically established relation between the projectile impact velocity versus residual velocity or energy similar to the classical Lambert–Jonas equation for metals.     

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

This study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.     

A micromechanical approach for the prediction of the time-dependent failure of high temperature polymer matrix composites

In the present study, a novel micromechanical approach is introduced to study the time-dependent failure of unidirectional polymer matrix composites. The main advantage of the present micromechanical model lies in its ability to give closed-form solutions for the effective nonlinear response of unidirectional composites and to predict the material response to any combination of shear and normal loading. The creep failure criterion is expressed in terms of the creep failure functions of the viscoelastic matrix material. The micromechanical model is also used to calculate these creep failure functions from the knowledge of the creep behavior of the composite material in only transverse and shear loadings, thus eliminating the need for any further experimentation. The composite material used in this study is T300/934, which is suitable for service at high temperatures in aerospace applications. The use of micromechanics can give a more accurate insight into the failure mechanisms of the composite materials in particular at high temperatures where the general behavior of the polymer matrix composite is governed by matrix viscoelasticity and the time-dependent failure of the matrix is a localized phenomenon. The obtained creep failure stresses are found to be in reasonable agreement with the experimental data.     

Deformation in viscoelastic sandwich composites subject to moisture diffusion

This study analyzes the effect of moisture diffusion on the deformation of viscoelastic sandwich composites, which are composed of orthotropic fiber-reinforced laminated skins and viscoelastic polymeric foam core. It is assumed that the elastic and time-dependent (transient) moduli at any particular location in the foam core depend on the moisture concentration at that location. Sequentially coupled analyses of moisture diffusion and deformation are performed to predict overall performance of the studied viscoelastic sandwich systems. Time and moisture dependent constitutive model is used for the polymer foam core, while skins are assumed linear elastic. The overall time-dependent responses of the sandwich composites subject to moisture diffusion are analyzed using finite element (FE) method. Experimental data available in the literature and analytical solutions are used to support convergence studies in the FE analyses. Contributions of moisture dependent elastic and the time-dependent moduli to the overall stress, strain and displacement field are studied. FE analyses of the delamination between skins and core in sandwich composite under combined moisture diffusion and mechanical loading are also performed.     

A Simple Locking-Alleviated 4-Node Mixed-Collocation Finite Element with Over-Integration,for Homogeneous or Functionally-Graded or Thick-Section Laminated Composite Beams

In this study, a simple 4-node locking-alleviated mixed finite element (denoted as CEQ4) is developed, for the modeling of homogeneous or functionally graded or laminated thick-section composite beam structures, without using higher-order (in the thickness direction) or layer-wise zig-zag theories of composite laminates which are widely popularized in current literature. Following the work of [Dong and Atluri (2011)], the present element independently assumes a 5-parameter linearly-varying Cartesian strain field. The independently assumed Cartesian strains are related to the Cartesian strains derived from mesh-based Cartesian displacement interpolations, by exactly enforcing 5 pre-defined constraints at 5 pre-selected collocation points. The constraints are rationally defined to capture the basic kinematics of the 4-node element, and to accurately model each deformation mode of tension, bending, and shear. A 2 by 2 Gauss quadrature is used when each element is used to model a piece of a homogeneous material or structure, but over-integration (using a higher-order Gauss Quadrature, a layer-wise Gauss Quadrature, or a simple Trapezoidal Rule in the thickness direction) is necessary if functionally-graded materials or thick-section laminated composite structures are considered. Through several numerical examples, it is clearly shown that the present CEQ4 is much more accurate than the well-known Pian-Sumihara (1984) element as well as the primal four-node element, for the modeling of homogeneous beams. For functionally-graded materials, the presently-developed element can accurately capture the stress distribution even when very few elements are used; but the Pian-Sumihara element fails, because the assumption of linearly-varying stressfield is generally invalid unless a very fine mesh is used in the thickness direction. For thick-section laminated composite beams, reasonably accurate solutions (for axial as well as transverse stresses) are obtained even when only one CEQ4 element is used in the thickness direction. Without using higher-order theories or layer-wise zig-zag assumptions for displacement or stress fields in the thickness direction, for thick-section laminates, the present method can accurately compute the jumps in axial stresses at the interfaces of layers. Extension of the present CEQ4 concept to C0 elements of higher-order, for plates and shells as well as for multi-physics will be pursued in future studies.