Impact experiments of thick fabric carbon/epoxy laminate specimens, with small thickness ratio, are conducted at distinct energy levels and thicknesses to characterise the damage process. These specimens and loading conditions are representative of a new generation of critical structural components in aviation, such as wing spars, landing gear beams and fittings, that are increasingly being made entirely from composites. The tests address the need to better understand the damage process for specimens with a small thickness ratio since existing experimental impact data for large thickness ratio (thin laminates) may not be directly applicable. Two energy levels, two different fabric layups and two impact methods (drop-weight and gas-cannon) were used. Data from high-speed cameras were processed in a novel way, providing the force during impact. C-scans and micrographs were used to characterise damage. The results show that specimens with a thickness ratio of 5 (20 mm thick) experience more bending compared to specimens with a ratio 2.5 (40 mm thick). For gas-cannon impacts, this results in a higher delaminated area. The drop-weight impacts show almost no differences in damage size for the thickness range analysed. The influence of layup on the global impact response is negligible, but locally it can result in significant variations in dent depth. The dent depth scales linearly with the impact energy and the delaminated area linearly with the impact velocity. There is no clear correlation between the compression-after-impact failure mechanisms and the residual strength. Impact damage, at the current energy levels, showed a minimal reduction of residual strength.

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Computationally-efficient surrogate models based on a Polynomial Chaos Expansion (PCE) are developed to quantify the uncertainties in the fracture behavior and lifetime of a self-healing thermal barrier coating system (SH-TBC) and a benchmark conventional TBC system. The surrogate models are built using deterministic information from micromechanical finite element simulations of thermal cycling of the systems, which are conducted until failure by spallation. Fracture and healing events are simulated using a cohesive-zone based crack healing model. The thermally-grown oxide layer (TGO) interface amplitude and its growth rate, the diameter and volume fraction of healing particles, and the mean distance of particles from the interface are used as training variables. Statistical characteristics and sensitivity indices are obtained from the trained models. It is found that the interface amplitude is the most significant contributor to the variance in the TBC lifetime, with other parameters displaying a relatively minor influence. Healing particles extend the expected value of TBC lifetime, however they also increase the uncertainty of thermal fatigue life. The analysis of self-healing TBCs exemplifies how PCE-based surrogate models can serve as a powerful tool for deriving design insights in complex material systems.

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The influence of the cohesive zone length on the crack driving force is quantified and analyzed in a representative system of particles dispersed in a matrix of a composite material. For heterogeneous material systems, e.g. particulate composites, it is known that as a crack approaches the particles, the crack driving force may increase (shielding) or decrease (anti-shielding) depending on the relative stiffness of the particles. These results have been established in numerous studies using the classical linear elastic fracture mechanics approach (LEFM). The cohesive zone method (CZM) introduces a length scale parameter, referred to as the cohesive zone (or fracture process zone) length scale, into the formulation of fracture mechanics. It is generally established that fracture mechanics predictions using the CZM are similar to those obtained using LEFM in the limit case where the process zone is very small relative to a suitable characteristic dimension of the problem. However, the influence of the length scale parameter has not been clearly demonstrated for crack propagation in a heterogeneous material system, especially when the cohesive zone length is not negligible. By considering a simple crack-particle-matrix system, it is shown that, in addition to the elastic properties, the process zone length scale parameter exhibits a critical influence on the crack driving force. For this study, the concept of configurational forces is utilized and the eXtended Finite Element Method (XFEM) is employed as a tool to simulate crack propagation. Through numerical simulations, it is shown that (i) the magnitude of the driving force vector directly depends on the length scale parameter and (ii) the direction of the driving force is largely influenced by the presence of a cohesive zone. This, in turn, alters the crack trajectory in the particulate system if the criterion for the direction of crack propagation depends on the orientation of the driving force vector. Towards this end, two different criteria for direction of crack propagation, namely maximum principal stress and maximum energy dissipation, are compared in the presence of a cohesive zone and the results are reported. The study reveals the crucial influence of the inherent length scale associated with the cohesive zone method when applied to crack propagation in particulate composite systems and elucidates important differences when comparing predictions from distinct theories of fracture mechanics.

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The thermal cyclic behavior of self-healing thermal barrier coatings (SH-TBC) is analyzed numerically to develop a lifetime prediction model. Representative microstructures are studied adopting a unit cell based multiscale modeling approach along with a simplified evolution model for the thermally-grown oxide layer (TGO) to study the evolution of damage and healing in a self-healing TBC system. The fracture and healing process is modeled using the cohesive zone-based healing model along with a crack tracking algorithm. The microstructural model includes splat boundaries and a wavy interface between the Top Coat and the Bond Coat, typical of Air Plasma Sprayed TBCs. A particle-based self-healing mechanism is accounted for with a random distribution of healing particles subjected to a numerically accelerated thermal cyclic loading condition. Lifetime extension of the self healing TBCs is quantified by conducting thermal cyclic analyses on conventional TBCs (benchmark system without self-healing particles). Parametric analyses on healing parameters such as crack filling ratio and strength recovery of the healed crack are also conducted. The results are presented in terms of the evolution of the crack pattern and the number of cycles to failure. For self-healing TBCs with a suitable healing reaction (i.e., cracks being partially filled and a minimal local strength after healing), an improvement in TBC lifetime is observed. In contrast, if the healing mechanism is not activated, the presence of the healing particles is actually detrimental to the lifetime of the TBC. Correspondingly, in addition to superior crack filling ratio and healed strength, significant improvement in lifetime is achieved for self healing TBCs with a higher probability of crack-healing particle interaction. This highlights the importance of a robust activation mechanism and a set of key material requirements in order to achieve successful self-healing of the TBC system.

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Two distinct length scale transition methodologies are developed to establish effective traction-separation relations for fracture in composite materials within a hierarchical multiscale framework. The two methodologies, one kinetics-based and the other kinematics-based, specify effective fracture properties that satisfy a surface-based Hill-Mandel consistency condition. Correspondingly, the total amount of energy dissipated is the same whether a crack is described in detail with micro quantities or in terms of an effective macroscopic crack. Though both methods guarantee consistency in terms of energy rates across length scales, they provide in general distinct effective traction-separation relations. Several representative samples of fiber reinforced composites are analyzed numerically, including the formation and propagation of cracks at mid-ply locations as well as (idealized) ply interfaces. Through post-processing of the microscale results, it is shown that the kinematics-based averaging method provides a macroscopic traction that is prone to rapid fluctuations while the kinetics-based averaging method shows a more smooth response but with openings that can deviate from the surface average of the microscale openings. The two methods are also compared with a previously-proposed scale transition methodology, which is a hybrid method that only satisfies the Hill-Mandel condition approximately. The suitability of the three methods is discussed in light of the results obtained from the simulations.

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In this paper, design strategies are developed to explore better approaches of enforcing local layer-wise curvature constraints in the optimization of variable stiffness laminates in order to ensure the manufacturability of optimized designs based on the limitations of automated fiber placement. The methods developed here aim to improve an existing approach of imposing the curvature constraint directly on the fiber angles (i.e., direct control method) and are suitable for a design framework that uses lamination parameters as primary design variables. One approach developed here, termed the indirect control method, enforces the curvature constraint indirectly with better computational efficiency through the spatial gradient of the lamination parameters. It is shown that the curvature constraint on the actual fiber angles can also be satisfied with a sufficiently stringent upper bound albeit it produces overly conservative designs. Alternatively, an enhanced approach, termed the hybrid control method, is developed by combining the direct method and a relaxed version of the indirect control method. The case studies of minimum compliance design indicate that it provides the best manufacturable design among the three methods in the context of variable stiffness laminates using lamination parameters.

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Purpose: When simulating fluid-structure interaction (FSI), it is often essential that the no-slip condition is accurately enforced at the wetted boundary of the structure. This paper aims to evaluate the relative strengths and limitations of the penalty and Lagrange multiplier methods, within the context of modelling FSI, through a comparative analysis. Design/methodology/approach: In the immersed boundary method, the no-slip condition is typically imposed by augmenting the governing equations of the fluid with an artificial body force. The relative accuracy and computational time of the penalty and Lagrange multiplier formulations of this body force are evaluated by using each to solve three test problems, namely, flow through a channel, the harmonic motion of a cylinder through a stationary fluid and the vortex-induced vibration (VIV) of a cylinder. Findings: The Lagrange multiplier formulation provided an accurate solution, especially when enforcing the no-slip condition, and was robust as it did not require “tuning” of problem specific parameters. However, these benefits came at a higher computational cost relative to the penalty formulation. The penalty formulation achieved similar levels of accuracy to the Lagrange multiplier formulation, but only if the appropriate penalty factor was selected, which was difficult to determine a priori. Originality/value: Both the Lagrange multiplier and penalty formulations of the immersed boundary method are prominent in the literature. A systematic quantitative comparison of these two methods is presented within the same computational environment. A novel application of the Lagrange multiplier method to the modelling of VIV is also provided.

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A multiscale fracture model is developed to study the influence of defects appearing at a microscale in a fiber-reinforced composite laminate. The model establishes a link between the geometrical characteristics of sub-ply imperfections that may be created during manufacturing and the overall fracture strength and fracture energy of the composite. In particular, a recently-developed multiscale theory is expanded to account for microvoids inside the matrix and gaps between closely-spaced fibers that prevent filling. These defects are explicitly incorporated in finite element simulations to study their influence on the onset and propagation of cracks at the sub-ply level. To connect these microcracks to the effective fracture behavior at a ply-level, a computational homogenization technique is applied to extract the energetically-equivalent macroscopic fracture properties. Through a parametric analysis of configurations, the influence of the void content (porosity), void type and void shape on the effective fracture strength and the effective fracture energy of a composite are quantified. Results show that the porosity is the main parameter influencing fracture properties while the shape of the defects and their type (matrix or interfiber) only play a secondary role. Furthermore, the influence of voids on the fracture properties appears to be strongly dependent on the loading conditions. In particular, for the range of porosity analyzed (up to 8%), the influence of voids in mode I on the transverse fracture strength is not significant but the transverse fracture energy decreases approximately linearly down to about 50% of its original value. In contrast, in mode II, the transverse fracture strength is significantly affected with increasing porosity. Furthermore, the transverse fracture energy depends nonlinearly on the porosity and the reduction is relatively more pronounced than for mode I.@en

The effect of splat interfaces on the fracture behavior of air plasma-sprayed thermal barrier coatings (APS-TBC) is analyzed using finite element modeling involving cohesive elements. A multiscale approach is adopted in which the explicitly resolved top coat microstructural features are embedded in a larger domain. Within the computational cell, splat interfaces are modeled as being located on a sinusoidal interface in combination with a random distribution of pores. Parametric studies are conducted for different splat interface waviness, spacing, pore volume fraction and fracture properties of the splat interface. The results are quantified in terms of crack nucleation temperature and total microcrack length. It is found that the amount of cracking in TBCs actually decreases with increased porosity up to a critical volume fraction. In contrast, the presence of splats is always detrimental to the TBC performance. This detrimental effect is reduced for the splat interfaces with high waviness and spacing compared to those with low waviness and spacing. The crack initiation temperature was found to be linearly dependent on the normal fracture properties of the splat interface. Insights derived from the numerical results aid in engineering the microstructure of practical TBC systems for improved resistance against thermal fracture.

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The influence of microstructural pore defects on fracture behaviour of Thermal Barrier Coatings (TBC) is analysed using finite element analysis involving cohesive elements. A concurrent multiscale approach is utilised whereby the microstructural features of the TBC are explicitly resolved within a unit cell embedded in a larger domain. Within the unit cell, a random distribution of pores is modelled along with three different layers in a TBC system, namely, the Top Coat (TC), the Bond Coat (BC) and the Thermally Grown Oxide (TGO). The TC/TGO and the TGO/BC interfaces are assumed to be sinusoidal of specified amplitude and frequency extracted from experimental observations reported in the literature. To simulate fracture in the TBC, cohesive elements are inserted throughout the inter-element boundaries in order to enable arbitrary crack initiation and propagation. A bilinear traction-separation relation with specified fracture properties for each layer is used to model the constitutive behaviour of the cohesive elements. Parametric studies are conducted for various pore geometrical features, porosity, fracture properties of Top Coat layer and Thermally Grown Oxide layer thicknesses. The results are quantified in terms of crack initiation and evolution. It is found that the presence of pores has a beneficial effect on the fracture behavior up to a certain value of porosity after which the pores become detrimental to the overall performance. Insights derived from the numerical results can help in understanding the failure behavior of practical TBC systems and further aid in engineering the TBC microstructure for a desired fracture behavior.

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A computational fracture analysis is conducted on a self-healing particulate composite employing a finite element model of an actual microstructure. The key objective is to quantify the effects of the actual morphology and the fracture properties of the healing particles on the overall mechanical behaviour of the (MoSi2) particle-dispersed Yttria Stabilised Zirconia (YSZ) composite. To simulate fracture, a cohesive zone approach is utilised whereby cohesive elements are embedded throughout the finite element mesh allowing for arbitrary crack initiation and propagation in the microstructure. The fracture behaviour in terms of the composite strength and the percentage of fractured particles is reported as a function of the mismatch in fracture properties between the healing particles and the matrix as well as a function of particle/matrix interface strength and fracture energy. The study can be used as a guiding tool for designing an extrinsic self-healing material and understanding the effect of the healing particles on the overall mechanical properties of the material.@en

Despite the increased use of thick fabric Carbon Fibre Reinforced Polymer (CFRP) materials in highly loaded aerospace structures (e.g., 20-50mm thick CFRP structures), a comprehensive characterisation of damage due to impact events on these structures remains an elusive and challenging task. To address this issue, three methods with varying degrees of computational complexity are developed to simulate and predict a representative impact problem on a thick composite plate. These simulation methods range from a low-fidelity 1D semi-analytical impact response model to high-fidelity 2D and 3D numerical impact damage models. The accuracy, in terms of impact response and damage prediction, is assessed by comparison with experimental results. It was found that the higher fidelity does not directly translate to a higher accuracy due to challenging modelling strategies for the 2D and 3D numerical impact damage models. The 1D semi-analytical impact response model was found to be the most accurate in predicting the force and displacement histories of both large-mass and small-mass impact events. However, this model is not capable of predicting the extent of damage. Comparison of the resulting impact damage from the 2D and 3D numerical impact damage models with experiments showed that improvements are required to capture the correct damage mechanisms. These damage mechanisms are complex and increasing the model complexity requires an extensive evaluation of the modelling strategies and numerical variables. Improvements are suggested that should increase the accuracy of these models. Dynamics are important for thick laminates as there is almost no plate bending and most of the impact energy is absorbed in local damage and deformations. Thinner composites generally experience more bending and can therefore be evaluated quasi-statically.

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A multiscale framework for the analysis of fracture is developed in order to determine the effective (homogenized) strength and fracture energy of a composite material based on the constituent's material properties and microstructural arrangement. The method is able to deal with general (mixed-mode) applied strains without a priori knowledge of the orientation of the cracks. Cracks occurring in a microscopic volume element are modeled as sharp interfaces governed by microscale traction-separation relations, including interfaces between material phases to account for possible microscale debonding. Periodic boundary conditions are used in the microscopic volume element, including periodicity that allows cracks to transverse the boundaries of the volume element at arbitrary orientations. A kinematical analysis is presented for the proper interpretation of a periodic microscopic crack as an equivalent macroscopic periodic crack in a single effective orientation. It is shown that the equivalent crack is unaffected by the presence of parallel periodic replicas, hence providing the required information of a single localized macroscopic crack. A strain decomposition in the microscopic volume element is used to separate the contributions from the crack and the surrounding bulk material. Similarly, the (global) Hill–Mandel condition for the volume element is separated into a bulk-averaged condition and a crack-averaged condition. Further, it is shown that, though the global Hill–Mandel condition can be satisfied a priori using periodic boundary conditions, the crack-based condition can be used to actually determine the effective traction of an equivalent macroscopic crack.

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The performance of a self-healing Thermal Barrier Coating (TBC) containing dispersed healing particles depends crucially on the mismatch in thermomechanical properties between the healing particles and the TBC matrix. The present work systematically investigates this phenomenon based on numerical simulations using cohesive element-based finite element analysis. The effect of the mismatch in Coefficient of Thermal Expansion (CTE) and fracture strength between the healing particles and the matrix on the fracture characteristics is quantified in detail. Unit cell-based analyses are conducted on a representative self-healing TBC system under a thermal loading step typically experienced by TBC systems in jet turbines. Two different simulation setups are considered within the TBC unit cell namely (i) a single pair of healing particles and (ii) a randomly distributed array of healing particles. The results of the simulations are reported in terms of the fracture pattern, crack initiation temperature and crack length for various CTE mismatch ratios. Correlations are established between the results obtained from the two simulation setups essentially revealing the effect of spatial distribution and proximity of healing particles on the fracture pattern. The results obtained from the analyses can be utilised to achieve a robust design of a self-healing TBC system.@en

We develop the general form of the variational multiscale method in a discontinuous Galerkin framework. Our method is based on the decomposition of the true solution into discontinuous coarse-scale and discontinuous fine-scale parts. The obtained coarse-scale weak formulation includes two types of fine-scale contributions. The first type corresponds to a fine-scale volumetric term, which we formulate in terms of a residual-based model that also takes into account fine-scale effects at element interfaces. The second type consists of independent fine-scale terms at element interfaces, which we formulate in terms of a new fine-scale "interface model." We demonstrate for the one-dimensional Poisson problem that existing discontinuous Galerkin formulations, such as the interior penalty method, can be rederived by choosing particular fine-scale interface models. The multiscale formulation thus opens the door for a new perspective on discontinuous Galerkin methods and their numerical properties. This is demonstrated for the one-dimensional advection-diffusion problem, where we show that upwind numerical fluxes can be interpreted as an ad hoc remedy for missing volumetric fine-scale terms.@en

A wedge loaded testing methodology to determine the fracture energy and strength of (semi-) brittle (metallo-)ceramics is presented. The methodology combines a tailored specimen geometry and a comprehensive finite element analysis based on cohesive zone modelling. The use of a simulation-based approach to extract both fracture strength and energy from experimental data avoids the inherent inaccuracies found in closed-form expressions that rely on a priori assumptions about the deformation field. Results from wedge splitting tests on Ti3SiC2 and Ti2AlC (MAX phase) materials are used to illustrate the procedure. The simulation-based approach is further validated by comparing the fracture strength and fracture energies predicted by the proposed method and those indicated by a conventional four-point bending fracture toughness test (ASTM standard). The new protocol offers the possibility to measure not only the fracture properties of brittle material in its pristine state but also in the healed state.@en

We initiate the study of the discontinuous Galerkin residual-based variational multiscale (DG-RVMS) method for incorporating subgrid-scale behavior into the finite element solution of hyperbolic problems. We use the one-dimensional viscous Burgers equation as a model problem, as its energy dissipation mechanism is analogous to that of turbulent flows. We first develop the DG-RVMS formulation for a general class of nonlinear hyperbolic problems with a diffusion term, based on the decomposition of the true solution into discontinuous coarse-scale and fine-scale components. In contrast to existing continuous variational multiscale methods, the DG-RVMS formulation leads to additional fine-scale element interface terms. For the Burgers equation, we devise suitable models for all fine-scale terms that do not use ad hoc devices such as eddy viscosities but instead directly follow from the nature of the fine-scale solution. In comparison to single-scale discontinuous Galerkin methods, the resulting DG-RVMS formulation significantly reduces the energy error of the Burgers solution, demonstrating its ability to incorporate subgrid-scale behavior in the discrete coarse-scale system.@en

A cohesive zone-based constitutive model, originally developed to model fracture, is extended to include a healing variable to simulate crack healing processes and thus recovery of mechanical properties. The proposed cohesive relation is a composite-type material model that accounts for the properties of both the original and the healing material, which are typically different. The constitutive model is designed to capture multiple healing events, which is relevant for self-healing materials that are capable of generating repeated healing. The model can be implemented in a finite element framework through the use of cohesive elements or the extended finite element method (XFEM). The resulting numerical framework is capable of modeling both extrinsic and intrinsic self-healing materials. Salient features of the model are demonstrated through various homogeneous deformations and healing processes followed by applications of the model to a self-healing material system based on embedded healing particles under non-homogeneous deformations. It is shown that the model is suitable for analyzing and optimizing existing self-healing materials or for designing new self-healing materials with improved lifetime characteristics based on multiple healing events.@en

This work is concerned with the development of a framework to solve shape optimization problems for transient heat conduction problems within the context of isogeometric analysis (IGA). A general objective functional is used to accommodate both shape optimization and passive control problems under transient conditions. An adjoint sensitivity analysis, which accounts for possible discontinuities in the objective functional, is performed analytically and subsequently discretized within the context of IGA. The gradient of the objective functional is used in a descent algorithm to solve optimization problems. Numerical examples are presented to validate and demonstrate the capacity to manage thermal fields under transient conditions.

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In isogeometric shape optimization, the use of the search direction directly predicted from the discrete shape gradient makes the optimization history strongly dependent on the discretization. This discretization-dependency can affect the convergence and may lead the optimization process into a sub-optimal solution. The source of this discretization-dependency is traced to the lack of consistency with the local steepest descent search direction in the continuous formulation. In the present contribution, this inconsistency is analyzed using the shape variation equations and subsequently illustrated with a volume minimization problem. It is found that the inconsistency originates from the NURBS discretization which induces a discrete quadratic norm to represent the continuous Euclidean norm. To fix this inconsistency, three normalization approaches are proposed to obtain a discretization-independent normalized descent search direction. The discretization-independence of the proposed approaches is verified with a benchmark problem. The superiority of the proposed search direction and its suitability for numerical implementation is illustrated with examples of shape optimization for mechanical and thermal problems. Although the present work focuses on a NURBS-based discretization usually used in conjunction with isogeometric analysis, the proposed methodology may also be applied to alleviate the “mesh-dependency” in (traditional) Finite Element-based shape optimization.

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