To date, the failure criteria, applied predominantly to braided composites, are still developed from classical damage theories of laminated composites. Garnich and Akula reviewed some of the most commonly applied criteria for UD fibre-reinforced polymers and classified them into either mode-dependent or mode-independent criteria.

Nowadays, two categories of failure criteria are actively pursued. Mode-independent failure criteria use mathematical expressions to depict a damage surface as a function of strength of materials. All the polynomial and tensorial criteria belong to such a category. Tsai-Wu criteria are the most well-known and general one for composites, belonging to a type of Tensor Polynomial Criterion Tsai and Wu For practical proposes, the polynomial criterion is expressed in tensor notation as Hart-Smith, The parameters F i and F ij are related to the composite strength in the principal directions.

Then, the explicit form of the general expression is:. In recent studies, the Tsai-Wu tensor polynomial failure criterion was used by McLendon and Whitcomb and Wang et al. Jiang et al. Wan et al. Besides the Tsai-Hill criterion , several other similar quadratic criteria have been proposed by Hoffman and Chamis These criteria can be considered as generalized Tsai-Wu type criteria.

Traditional ply-based failure criteria, such as Tsai-Wu and Tsai-Hill, consider a yarn-matrix system as a whole and, therefore, they are not suitable to predict whether the failure occurs inside a yarn, a matrix, or at their interfaces Tolosana et al. When characterising failure of composites, researches focus on their homogeneity rather than anisotropic nature.

This is inappropriate since internal unique structures of composites influence their properties and failure character Paris and Jackson Moreover, the polynomial criteria may be not suitable in design, particularly for bi-axial tensile loading Edge Considering a non-homogeneous character of braided composites, mode-dependent criteria were proposed. Mode-dependent criteria are generally established in terms of mathematical expressions based on material strengths.

They consider different failure modes of the constituents. Because of this advantage, these criteria are adequate for PFA. Two of the simplest examples are the maximum-stress and maximum-strain criteria. The former criterion predicts that composites fail when the stress exceeds the maximum tolerance value. Three different conditions of failure are considered for a maximum stress in a longitudinal direction, a transversal direction and for shear stresses:.

In Eqs. Indices 1, 2 and 3 are used to describe X, Y and Z directions, respectively. Hence, S 12 , S 13 and S 23 signify in-plane and two out-of-plane shear strengths, respectively. Similarly, the maximum-strain criterion means that when the strain exceeds the given allowable value, the constitutive materials fail. These maximum criteria can be used for homogeneous textile composite models Sevkat et al. As simple methods to analyses composites failure, the major limitation of the maximum-stress and maximum-strain criteria is that they ignore the interaction between stresses and strains in the composites.

Therefore, they were mostly applied to specific constitutive material elements, such as failure of fibre Mao et al.

- Inspired by Your Shopping History;
- Transparency: How Leaders Create a Culture of Candor (J-B Warren Bennis Series)!
- Modelling of Damage Evolution in Braided Composites: Recent Developments.

Hashin proposed different failure modes associated with the fibre tow and the matrix, considering, in both modes, differences in tension and compression Hashin , as shown in Fig. Illustration of failure modes described in Hashin-type failure criteria Doitrand and Fagiano Li et al. When the braided composites were regarded as orthotropic materials, failure modes in the thickness direction, as shown below, should be considered Kang et al. In some studies, criteria for thickness-direction failure modes were presented in other forms Fang et al. Thus, for the matrix mode, Hashin proposed a quadratic criterion because, on the one hand, a linear criterion underestimated strength of the material and, on the other hand, a polynomial of higher degree would be too complicated to manage Hashin ; Paris and Jackson Although Hashin himself limited the scope of his proposal to UD composites, the criteria were widely applied to braided composites in recent years Binienda ; Zhang et al.

So far, the mode-dependent failure criteria were proved to be more suitable for analysis of failure initiation in braided composites. It should be noted that the mode-dependent failure criteria can be also presented in a strain-based form, e. Micro-mechanics of failure is a theory that links constitutive materials individual fibre, matrix and their interface and a macroscopic stress response of composites Ha et al. It is believed that failure of fibrous composites can be assessed with micro-scale analysis. No difference between tension and compression failure modes at constituent levels is considered, and failure of fibre-matrix interface is incorporated:.

A fibre is a transversely isotropic material, and two possible failure criteria are needed for its failure. The first is a simple maximum-stress criterion; the other is the Tsai-Wu criterion. It was argued that the adoption of quadratic failure criteria, such as the Tsai-Wu, required the values of transverse tensile and compressive as well as shear strengths, which were difficult to obtain in experiments. So, a simplification of the quadratic criteria to the maximum-stress criteria was preferred Ha et al.

The epoxy matrix is regarded as isotropic, and has a higher strength value under uniaxial compression than under tension. For the matrix, a Christensen Criterion was applied, which is a modified version of the von Mises failure criterion Christensen Finally, the fibre-matrix interface can be considered to follow a traction-separation failure criterion Ha et al. MMF was reported to be able to predict successfully both the initial and final failures for all the 12 specified test cases Kaddour and Hinton MMF is different from conventional methods primarily in two ways.

On the other hand, the conventional macro-level methods generally require one or more interaction parameters in order to capture the interaction of stress components in the matrix and fibres, while MMF uses a micromechanical model to account for the stress interaction, so that the interaction parameter is not needed Xu et al.

A modified MMF scheme was proposed to improve predictions of shear strength by adding shear component in the criteria Ha et al. To model progressive failure of braided textile composites, numerous studies combined two damage-evolution theories for inter- and intra-laminar damages, respectively. The first theory was a cohesive zone model CZM , widely used to capture inter-laminar delamination Xie et al.

Application of CZM requires a-priori knowledge of an intended crack path and a use of cohesive elements. Another theory to evaluate intra-laminar failure was continuum damage mechanics CDM Camanho et al. In CDM, damage is described by introducing internal state variables D ij into an algorithm of continuum mechanics to represent micro defects in a damage process in the material. Stiffness values of composites degraded with the growing damage variables DVs D ij homogeneously when a material met its failure criteria.

Although many methods were developed, it is still an open question how to define DVs considering complicated failure modes of braided composites. In this section, some stiffness-degradation approaches most broadly used in recent investigations are discussed. An instantaneous stiffness-degradation method was initially developed by Blackketter et al. In this empirical stiffness-reduction scheme, DVs are usually constants. When stresses at an integration point of a finite element satisfy the damage-initiation criterion, damage at this integration point happens and stiffness is reduced to a specific value according to the relevant failure mode.

The scheme was widely used for damage prediction in composites without any convergence difficulties. It also showed good capability to simulate the mechanical performance of non-crimp fabric NCF composite structural parts associated with different failure modes of yarns subjected to tension loading Tserpes and Labeas, A degradation factor associated with particular failure mode was used to multiply the corresponding modulus as soon as that particular mode of failure was detected.

The computed damage initiation and accumulation had consistent results with experiments. However, the results indicated that the final failure load depended on the mesh size and time increment.

- The Learning Alliance: Systems Thinking in Human Resource Development?
- Modern Surface Technology?
- Differential Geometrical Methods in Theoretical Physics;
- Our Bestsellers.
- 100 Careers in the Music Business.

Recently, this degradation scheme was further developed for textile composites. Mao et al. Microscale damage modes of the composite material were determined by using the maximum-stress criterion for fibres and the MMF for a matrix. The mechanical response was assumed to be linear elastic until initiation of failure. The instantaneous stiffness-degradation method was applied, as shown in Fig. For an element that reached the failure criterion, the elastic constants were reduced as follows:. The softening factor n is computed to control the softening process. In addition, d ij are the degradation factors, prescribed in terms of reduction of effective stiffness along different directions.

Failure analysis was carried out to provide the influence of each damage mechanism on overall laminates stiffness, and, thus, the values of d ij were determined by means of meso-mechanical failure analysis with six cases of boundary conditions. Additionally, quantitative analysis based on virtual tests was conducted to determine the degradation factors for every damage mechanism to control the stress reduction in the damage-evolution law. The reported degradation factors are listed in Table 1.

Longitudinal stress-strain curve a and stiffness-reduction of woven-composite unit cell b in instantaneous stiffness-degradation method Mao et al. Warren et al. The model combined the well-established Hashin failure criteria and the Matzenmillere-Lublinere-Taylor damage model to capture bearing damage; failure indicators were used to calculate DVs. In this model, DVs affected directly elastic properties of the material by means of modification of diagonal components of a compliance matrix at each material point.

For brittle materials, four DVs and all functions of the specific failure indicators, as seen in Table 2 , were used to modify this material compliance matrix:. Its off-diagonal terms remained unaffected by DVs in order to satisfy physical and thermodynamic conditions. A similar method was applied by Wang et al. Nobeen et al. And such DVs Eqs.

They were stored as Solution Dependent Variables and can be monitored throughout the progression of the analysis. Predicted stress-strain curve of bi-axial braided composites and damage contours Wang et al. Although reasonable numerical results were obtained using the instantaneous stiffness-degradation method in many works, magnitudes of the stiffness-reduction factors were somewhat arbitrarily chosen by researchers based on the types of failure criteria and different failure modes. Therefore, the advanced failure criteria and damage factors for braided composites need to be investigated further, and more efforts are unquestionably needed in the future.

Evolution of DVs in the continuum stiffness-degradation method is based on a thermodynamic framework or an energy-dissipation theory. In the early stage, CDM was built to study damage development for single-ply or laminate composites because the damage mechanisms of UD composites were relatively easy to quantify.

Nowadays, various evolution laws based on the continuum stiffness-degradation method are also suitable for braided composites. In these studies, the evolution of DVs could be presented in a linear or an exponential form. When constituents of a material fail in an element, it dissipates energy equal to its elastic energy. Thus, degradation of the stiffness tensor is characterized by a damage matrix, C D , defined by internal DVs d I associated to different failure modes I Fang et al.

When the Hashin failure criteria are applied, the DV of each failure mode is expressed by the following equivalent displacement:. Figure 6 shows the linear evolution of DVs. Linear damage-evolution law in bilinear equivalent stress-displacement relationship Li, X Ii eq and X If eq could be calculated with the following equations Kang et al. The equivalent displacements of damage initiation listed in Table 3 have the similar forms with Eq. Therefore, the damage-evolution equation is associated with the characteristic element length, local strain and fracture energy of the braided-composite constituents.

The damaged stiffness matrix C D can be expressed in a matrix form by using the components of undamaged stiffness matrix and the principal values of the damage tensor D I according to the Murakami-Ohno damage model Murakami, In the studies conducted by Li et al. Local failure mechanisms, such as fibre splitting and matrix cracking, were taken into consideration. In a FE model, the braiding architecture of the fibre tows was modelled at meso-scale.

Fang and Liang investigated the compressive properties of 3D braided composites using this damage theory. In these studies, only planar stress components were considered, otherwise, a fully three-dimensional damaged stiffness matrix was developed as. Zhang et al. The model also showed the capability of predicting a free-edge effect besides the primary damage evolution in tri-axial braided composites, as shown in Fig.

Free-edge warping behaviour: a out-of-plane displacement; b scheme of warping locations and directions Zhang et al. Zhou et al. In this model, a similar bilinear damage evolution approach Zako et al. The results showed that tensile strength decreased for off-axis angles see Fig. Such a scheme was also applied by Zhang et al. Strength of woven composites for different off-axis angles Zhou et al. The linear damage-evolution method was also adopted by Matveev et al. The multi-scale approach demonstrated that a wide distribution of fibre strength led to a narrow distribution of composite strength accompanied a shift to lower mean values.

Another multi-scale approach for PFA of braided composites at coupon-level was elaborated and validated by Xu et al. Starting from elastic constants of constituents i. In their study, the damage evolution was determined by the equivalent strain, a scalar measure of the strain components. Using the equivalent strain and equivalent stress, a multi-linear stress-strain damage model was proposed for the matrix in fibre tows, as illustrated in Fig. Multi-linear stress-strain damage model Xu et al. As shown in Fig. After this, the material exhibited hardening behaviour followed by softening, depending on the damage state.

In order to predict the strength, mesoscale FE models of representative unit cells of bi- and tri-axial braided composites were developed Xu et al. Another example was presented by Zhao et al. They proposed simplified methods for progressive-failure prediction in braided composites mainly subjected to uniaxial in-plane loading and bending. The global stress-strain behaviour and principle damage modes in bending were obtained without losing much accuracy.

Apparently, most schemes were based on linear damage models. A viscous model is usually applied to mitigate the convergence difficulties associated with strain-softening constitutive models. Therefore, the non-linear damage evolution law is usually presented as an exponential expression in the following general form:.

### About This Item

In recent investigations, the non-linear damage evolution approach was applied to capture progressive damage evolution in braided composites in static and quasi-static loading regimes. Lu et al. In their numerical model, a 3D Linde failure criterion was adopted to describe a progressive-damage process in yarns, which was a strain-based continuum damage formulation with different failure criteria for fibre and matrix failure modes, respectively. A gradual degradation of the material properties was assumed, which was controlled by the individual fracture energies of fibre and matrix.

DVs for fibre d f and matrix d m modes were defined as follows:. After failure initiation, the damaged stiffness matrix could be defined as. Therefore, the DVs d i could be regularized as:. Computing with this damage-evolution approach, the tensile modulus and strength of 3D braided composites were obtained with a good agreement to the experimental data. The calculated results showed that the effect of interfacial elastic modulus on the tensile modulus of 3D braided composites was prominent, while the strength of braided composites was not very sensitive to that parameter, as shown in Fig.

Effect of interfacial elastic modulus on stress-strain curves of 3D braided composites Lu et al. Zhong et al. Except for the longitudinal tension failure modes, the exponential damage evolution laws were still adopted for the fibre yarn in the model. The Puck failure criteria Puck et al. The damage process was irreversible, and the mechanical properties degraded as soon as any criterion was activated.

The DVs of the fibre yarn could be expressed as. These parameters were defined by other material-property inputs. The DVs d 4 and d 6 represent the effect of damage on shear stiffness due to fibre and inter-fibre fracture, and d 5 due to inter-fibre fracture. Accordingly, S d is the damaged compliance matrix of fibre yarns, which could be written as:. The failure modes of the matrix are different from those of the fibre yarn. However, the exponential evolution law could also be adopted based on mechanical response of the matrix material. Using this method, as seen in Fig.

The obtained results showed that mechanical properties of 3D woven composites were influenced by their thickness. The slope of stress-strain curve and the failure strength increased with the thickness growing. Predicted stress-strain curves of woven composites and experimental results Zhong et al. Although complexities of modelling mechanical behaviour of braided composites are present in many aspects, the improvement of modelling techniques is generally driven by two purposes. One is to account for all the physical phenomena observed in experiments in FE modelling; the other is to resolve numerical limitations of the FE method and balance its efficiency and accuracy.

Recently, nonlinear problems of mechanics of textile composites were considered in many studies. In experimental observations, two reasons of nonlinearities can be identified: a geometrical nonlinearity caused by a fabric structure and a material nonlinearity caused by micro-cracks evolving in the material inducing a loss of stiffness, indicating that the nonlinearity is related to progressive failure of the material.

As mentioned above, the CDM approach generally uses a damage parameter characterizing the damage evolution responsible for the loss of stiffness due to micro-cracks. Besides, the nonlinearity can be considered as macroscopic behaviour of the material independently from damage evolution. The nonlinearity was represented by the Ramberg-Osgood formulation. This formulation was developed by Bogetti et al.

The nonlinear behaviour was linked to a unidimensional plasticity definition to account for permanent deformation without coupling between different material directions. Failure was predicted using the maximum-stress or Tsai-Wu failure criterion. A smeared formulation involving DVs was used to avoid undesirable localization phenomena Pinho et al. In this study, the model represented nonlinear material behaviour very well; a good agreement between experimental and numerical results was obtained under tensile and compressive loadings.

However, the permanent deformation was still too high and stiffness degradation for the unloading process should be developed. With development of the multi-scale modelling approach, another key problem for improving the accuracy of FE simulations is a link between micro- and meso-scale models. In a multi-scale approach, micro- and meso-scale models are usually carried out subsequently. In some studies Xu et al. The correlation can be expressed by the following formula:.

The SAFs for the meso stress M i and for the temperature increment A i were determined by means of direct finite-element analysis of a micro-scale unit cell with proper boundary conditions. The MCT provided a way to link the results provided by the meso-scale and macro-scale models with a good computational efficiency. In this approach, constituent properties were used to obtain composite properties employing a micromechanical model. Then, an analytical relationship between composite stresses and strains and those in constituents was obtained based on an assumption that a continuum field existed for each constituent.

This study also demonstrated capability of predicting initial matrix failure in textile composites with n constituent based on extension of the MCT. Moreover, both micro- and meso-scale models can be developed with the GMC method. A comparison of meso- and micro-scale approaches to modelling progressive damage in plain-weave-reinforced polymer-matrix composites was carried out by Bednarcyk et al.

As illustrated in Fig. The micro-scale approach utilized the GMC semi-analytical micromechanical theory to represent a nonlinear response of the tows. The same damage model was used to model the matrix material within the tows. FE predictions were made for the shear and tensile responses of the plain-weave composite. Very similar results were obtained using the two approaches, for both effective behaviour of the woven composites and local damage fields predicted.

For example, in the tensile simulations, both approaches predicted similar additional local damage coupling in the tow cross-over regions due to the multi-axiality of the stress fields. However, the FE results were considerably mesh-dependent because of softening present in the damage model. This mesh-dependence could be reduced with a crack-band approach, which regularized the energy dissipated in the damage model.

Schematic of micro-scale a and meso-scale b approaches to model plain-weave composite using GMC micromechanics model with repeating unit cell Bednarcyk et al. Recently, more advanced studies of mechanical properties of 3D braided composites were considered to include some microscopic effects, such as defects. Dong and Huo developed a two-scale FE model for fibre tows and 3D braided composites to predict their elastic properties and micro stresses.

Two basic types of defects - voids in a resin matrix and dry patches in fibre tows -were taken into account. The obtained results showed that the volume fraction of voids in fibre tows had a more notable effect on the elastic constants than those of the matrix. However, the effect of internal defects on the strength was not included. In this study, a digital element approach was used to model the textile structure with a near micro-scale resolution for the purpose of reinforcement analysis. The textile unit-cell generation method enabled a process-dependent modelling and, therefore, textile-specific simulations of composites with excellent agreement with real manufactured samples.

Convergence analyses were carried out, and the textile model was directly used to build up a virtual composite model suitable for virtual mechanical tests, as illustrated in Fig. A comparison of simulation results and test data demonstrated their very good agreement. The introduced model had a capability to be a part of a complete virtual process chain for development of high-performance composites. To summarise, FE-based methods showed their capabilities and advantages in predicting the mechanical response of woven and braided composites.

Numerous failure criteria and damage-evolution laws can be effectively applied in models of braided composite. Based on the above discussions, the main purpose of recent studies was to improve simulation accuracy, which depends on two important considerations. The first one is that various failure modes observed in experiments should be accounted for in the FE simulations.

The other is that advanced analytical approaches should be carried out to connect scales in multi-scale models. However, taking these considerations inevitably reduces the computational efficiency. Although some modelling attempts showed good accuracy when compared with experimental results, usually they were only demonstrated for a single structure or applicable only to some specific cases. More studies are needed to analyse the effect of braiding parameters, boundary conditions and complicated loading conditions in the future. During manufacturing, service life-time, maintenance etc.

In such regimes, small weak point in a composite part can lead to catastrophic consequences. Therefore, a response of braided composites to such conditions should be clearly understood. An overview of general impact behaviour of fibre-reinforced polymeric composites FRPC is given in some review papers Davies and Olsson ; Crookston et al. In most studies, the composites were impacted at different velocities, and the resulting damage was evaluated. The effects of various parameters such as impact energy and impactor shape on impact responses of composites were also studied.

However, little attention was paid to braided composites. Impact damage in structural textile composites was introduced into consideration recently Binienda, ; Ma and Gao, , and most of these efforts were based on experimental studies, rather than numerical simulations. This section introduces some recent numerical attempts to predict damage in braided composites under impact loadings, with impact energy from relatively low levels to high ones.

In a low-velocity impact LVI , contact duration between an impactor and a target is long enough for entire structure to respond and, hence, absorb more elastic energy. Low-velocity impacts with sufficient energy can cause various types of barely visible impact damage BVID , such as matrix failure, delamination, fibre breakage, fibre-matrix debonding and fibre pull-out. The CDM approach was investigated extensively in recent years and its application to impact-damage modelling proved to be very effective for UD laminates.

Compared with laminates, few authors focused on a LVI response of woven and braided composites. LVIs are commonly encountered in personal sports protection and some other structural fortifications. The maximum-stress criterion with the instantaneous stiffness-degradation method was successfully used in modelling LVIs in braided composites. Sun et al. In this FE-based approach, impact damage was determined by maximum-stress criteria in several failure modes.

As the damage area increases, the modulus of the fibre tows degrades as. CDA is the critical damage area.

## Lloyd V. Smith | WSU School of Mechanical and Materials Engineering | Washington State University

The instantaneous stiffness degradation scheme is shown in Table 4. The user-defined subroutine VUMAT is usually used to define the mechanical constitutive relationship of the 3D braided composite under drop-weight impact. The damage morphologies of 3D braided composite specimens showed that the main failure mode was resin cracks, debonding and fibre-tow breakage. The corresponding load-displacement histories and failure modes were obtained from numerical simulation and verified by experimental results. It was observed that as the impact energy increased, both the maximum impact load and deflection also increased.

Sevkat et al. Evolution of dynamic force, strain and absorbed energy as well as post-impact damage patterns obtained from experiments and FE simulations were in a good agreement. Colombo and Vergani characterized a textile carbon-fibre-reinforced composite in undamaged and damaged conditions, with numerical and analytical micromechanical approaches, in order to provide a method for assessing its residual stiffness after impact.

An extent of degradation of the damaged composite was estimated by Blackketter-type reduction factors applied to elastic properties of unit cells. With this simple general approach, it was possible to predict a range of stiffness reduction for braided composites. However, in these studies, neither progressive damage nor plastic effect was accounted for in the FE models. Besides, values of the maximum impact displacement and interface delamination were not well captured.

## Dynamic response and damage of composite shell under impact

Usually, such approaches do not aim to run explicit simulations of the impact. Continuous stiffness degradation associated with the Hashin failure criteria was also applied to dynamic problems. Since this approach may lead to excessive element distortions and other related numerical difficulties, element deletion was utilised in simulation. Gideon et al. Schwab et al. In these studies, damage and failure behaviours of the textile composites was modelled using an orthotropic energy-based CDM approach, with DVs depending on an equivalent stress-displacement relationship, as presented in Section 2.

The simulations showed that a stress distribution during the impact event depended mostly on the reinforcing phase of the laminates. The behaviour of composites under low-velocity impact was defined mainly by the initial impact energy, rather than impact velocity or impactor mass. The proposed modelling strategy provided the ability to predict the overall energy absorption of a composite subjected to a transverse impact as well as energy contributions of individual mechanisms. Furthermore, shell elements were applied in these models to increase computational efficiency and stability.

Hence, these approaches were suitable to simulate complete perforation of the composite, as shown in Fig. However, damage and failure within shell elements representing individual plies resulted from in-plane stress and strain components only. Therefore, damage due to transverse shear and out-of-plane tension was not accounted for. A comparative study was carried out by Wang et al. In this study, a multi-scale computational approach was developed and the 3D Hashin damage criteria were applied to capture main damage modes of a braided textile composite.

The results show that both surface- and element-based CZM can be applied as interface between composite layers. When shell elements were used as composite plies, the absorbed energy was underestimated. The progressive failure model with 3D stress elements provided more precise results for the delamination areas and energy dissipation capacity, at a higher computational cost. In most of the modelling studies, the applied strategies provided a good representation at a homogeneous level, but could not represent the local damage modes related to the weaving and braiding patterns.

Pascal et al. In this model, ply bundles were represented by 1D rod elements connected by nodes to 4 edges of a quadrilateral shell element. Properties of the rods were adapted to represent local strain concentration. For delamination modelling, each ply was connected with a shell-to-shell interface element with a cohesive law. The bundles were temporally and locally relaxed until, carrying all the loads, they failed in tension. Damage evolution was given by. Besides, a third independent DV d 12 was implemented to model the final in-plane shear rupture. The bundle rupture in tension was assumed to be brittle.

Therefore, the classic maximum-tensile strain criterion was used for rupture of the rods. Comparisons with experimental data from drop-weight and gas-gun tests showed good accuracy of the force history and damage size and shape. With the numerical simulations, the influence of microstructure parameters on the impact behaviours could be revealed and the microstructure could be optimized.

Load-displacement curves of drop-weight impact test and bottom-face damage patterns experimental and numerical results at specific points Pascal et al.

## Department of Mechanical Engineering

A drawback of the continuum-element discretisation within these models was rather large computing times, even for models at a coupon level. Therefore, although the advanced strategy was able to represent local strain and stress fields related to the interlacing braided and woven architectures, it could hardly be used for large-scale structures. Different from a drop-weight scenario, the kinetic energy of a projectile under ballistic impact is observed to dissipate in the form of distinctive mechanisms. The predominant energy absorption mechanisms of laminates under high-velocity impact are related to kinetic energy of moving a cone formed on a distal side of the target, frictional losses during penetration and energy absorption related to failure modes, such as shear plugging, tensile fibre failure of primary yarns, fibre debonding, fibre pull-out, matrix cracking intra-laminar and inter-laminar delamination.

For braided composites, although advanced numerical models were employed to predict their mechanical properties and failure modes, some of them are not suitable to study their ballistic behaviour because of high strain rates and high pressure conditions in the impact area in high velocity-impacts.

Qiao et al. In their review, constitutive models of strain rate-dependent polymeric composites and their implementation in micromechanics models were briefly introduced. They also developed a nonlinear finite-element code e. At an early stage, the main purpose of modelling the ballistic response of composites was not their damage-evolution mechanisms. Instead, research was mostly aimed at macro ballistic parameters including residual velocities and maximum levels of dynamic displacement Gower et al.

Specifically, in terms of textile composites, the focus was on different composite features like a fabric type and its multi-layer structure , projectile geometry, impact velocity and effect of friction of fabric yarns on the response of composites Zeng et al. Therefore, analysis of failure mechanisms still remains challenging in terms of accuracy of results and efficiency of the methodology. The most popular approach to modelling the ballistic impact is a macro-homogeneous method, meaning that every composite layer is modelled as a homogeneous orthotropic material without making a distinction between the yarns and the matrix, but considering the whole as a single part.

A good agreement between experimental and FE results was found from comparisons of dynamic strains and damage patterns.

- Marine Natural Products - Diversity and Biosynthesis?
- Nonlinear Finite Element Analysis and Adina. Proceedings of the 4th ADINA Conference.
- A Companion to African American History;
- Agreement Systems (Linguistik Aktuell Linguistics Today, Volume 92).
- Ethical Questions and International NGOs: An exchange between Philosophers and NGOs.
- Better Off Dead: A Casey Jones Mystery (Casey Jones Mysteries, Book 5).

Gu and Ding refined the so called fibre inclination model to analyse ballistic penetration properties of 3D braided composite. The inclined UD lamina was established, containing a braided yarn with the same diameter and fibre volume fraction as in the 3D braided composite at the actual microstructure. The obtained results indicated that the refined quasi-microstructure model could approximately simulate the real ballistic-impact damage of 3Dâ€”braided composites. Cui et al. In both approaches, the instantaneous stiffness-degradation scheme was used while the strain-rate effect was underestimated.

Applying the continuum damage mechanics, Goldberg and co-workers developed a sub-cell model for tri-axial braided composite Littell et al. In this method, a unit cell of the braided composite was modelled as a series of shell elements, with each element modelled as a laminated composite. By defining integration points of these shell elements, a sequence and angles of layer could be assigned easily, and ballistic simulations could be performed with high efficiency with LS-DYNA. Although all the research works mentioned above employed shell elements to model braided composites, those modelling schemes could not reproduce properly the deflection and global deformation of a composite plate due to a transverse impact, especially for thick plate sections.

The interaction between layers and delamination failure modes could not be simulated as well. In simulations of impact tests on flat panels, although the calculated penetration velocity correlated reasonably well with experimentally obtained values, the predicted damage patterns did not reflect experimental observations.

This difference was possibly attributed to the use of quasi-static mechanical properties of the modelled composites. Since anisotropic stiffening is probably attributable to strain-rate effect, further investigation should be carried out. Liu et al. This material model was based on the Hashin failure criteria with five failure modes: tensile and compressive fibre failure, fibre crushing, through-thickness matrix failure and delamination.

This constitutive description could simulate progressive damage in composite laminates by controlling strain softening after failure in high-velocity impacts. The CDM formulation took into consideration post-failure mechanisms in a composite plate characterized by an exponential reduction in material stiffness, as mentioned in Section 2. The effect of the strain rate on ply strength was then modelled with strain-rate-dependent functions expressed as. The evolution of a penetration resistance force, energy absorption and damage with time during the impact process was predicted. Deformation of the composite plate and tension of carbon fibres were effective energy-absorption modes.

The authors also addressed differences between the impact processes and damage mechanisms related to blade-like and cylindrical projectiles, as shown in Fig. Delamination evolution mechanism under ballistic impact with cylindrical a and blade-like b projectiles. In the composite plate, the blue colour represents intact material and the red colour-delamination damage Lulu et al. The above models were generally created using a macro-homogeneous approach in order to obtain reliable results with a reduced computational cost and effort.

However, detailed damage patterns observed in experiments were missing.

For instance, cracks and damage propagate along fibre-yarn directions when braided composites are subject to impact. Therefore, it is better to include the braid architecture directly into a FE model. A difference between reproducing yarns around the impact area with a square shape and primary yarns in their entire length was evaluated. In addition, these works focused on guaranteeing the continuity of physical properties between different areas.

Hence, the continuity between a meso-mechanical region and an orthotropic shell zone was analysed from stress-wave propagation. The yarns were reproduced individually, using shell elements instead of 3D elements. Recently, some studies Pan et al. The meso-structure model was based on real architecture of the 3D braided composite. Ductile and shear criteria with an isotropic plastic-damage model considering the strain-rate effect were selected in the FE models to simulate the failure and damage processes at the meso-scale levels. A damage-evolution law was based on energy dissipated during the damage process.

The variable D was evaluated up to a limit of 0. The meso-scale model effectively predicted mechanical properties and failure morphologies of the 3D braided composites with detailed information on distributions of deformation and stress in braiding yarns. Further, the FE results revealed that the braiding structure had a significant influence on thermomechanical failure.

A multi-scale approach and the CDA theory were also employed in the study of high-strain-rate compression Wan et al. Both marco- and meso-scale models were built and compared by Bresciani et al. In the macro-homogeneous model, the equivalent mechanical properties were employed while in the meso-scale model, fabrics were simulated with their specific architectures and individual mechanical properties. These two numerical approaches provided a deep insight into a stress state of a target during ballistic impact showing different behaviours of the layers.

The results clearly demonstrated that the first layers during impact were subjected to high shear stresses and that the yarns in the rear layers underwent severe in-plane tension stresses. Therefore, a multilayer composite plate with different materials and structures across its thickness can be a potential solution to improve its resistance. In general, the meso-scale model provided more accurate results, as shown in Fig. Comparison of experimental tests with damage morphology obtained with meso-heterogeneous model Bresciani et al. Besides drop-weight and ballistic impacts, extensive modelling studies were carried out to deal with impact problems in other situations, such as Izod-type impact Ullah and Silberschmidt , tube crush McGregor et al.

In most of these impact scenarios, a strain-rate dependency and a homogeneous method were considered due to high impact energy so that geometrical features and damage modes of textile composites were ignored. Therefore, such investigations applying relatively arbitrary damage-evaluation schemes are beyond the scope of this review.

Generally, the failure criteria and damage-evolution models used in these cases were similar to those in studies of static loading. According to this progress, the overall response of braided composites under impact was better captured with FE method than before, including such features as BVID, impact force, duration time, maximum displacement and residual properties of targets. However, improvements are still needed to overcome various limitations. For instance, the accuracy of predictions is based on material parameters, which are obtained mostly from complicated experimental studies or from the literatures.

Furthermore, these schemes are still very expensive in terms of computational time, since explicit analyses are necessary to provide detailed information about impacted regions. Finally, it is not possible to generalize results from these studies, since each research is based on impact tests on specific type of textile composites. Fatigue deformation, damage and failure are important subjects to investigate long-time service of braided composite materials.

For decades, FE methods have been used extensively in fatigue prediction for composite materials. Reviews of an early fatigue-damage modelling work were published by Degrieck and Van Paepegem and Post et al. Most of these methodologies were established on the basis of experimental studies of laminates with specific lay-up sequences under specific testing conditions, resulting in distinctive fatigue behaviours and properties. Therefore, some of those methods were difficult to extend to textile composites because of their different structures and failure modes.

For braided and woven composites, comprehensive literature reviews Xu, ; Sevenois and Van Paepegem, ; Wang et al. The entire fatigue process can be divided into different stages, and each stage involves distinct damage modes. It was also known that different 3D structures induced different fatigue behaviours. The response of both periodic unit cells and random stochastic volume elements SVEs is analysed; the fibres are assumed to behave as linear elastic isotropic solids while the matrix is taken as a linear viscoelastic solid.

Monte Carlo analyses are conducted to determine the probability distributions of all viscoelastic properties. Simulations are conducted on SVEs of increasing size in order to determine the suitable size of a representative volume element RVE. The predictions of the FE simulations are compared to those of existing theories and it is found that the Mori-Tanaka and Lielens models are the most effective in predicting the anisotropic viscoelastic response of the RVE.

Titanium foams of relative density in the range 0. The response is ductile in compression but brittle, and weaker, in shear and tension. Virtual foam microstructures are generated by an algorithm based on Voronoi tessellation of three-dimensional space, capable of reproducing the measured size distribution of the pores in the foam. Finite Element FE simulations are conducted to explore the mechanical response of the material, by analysing the elasto-plastic response of a statistical volume element SVE.

The simulations correctly predict the ductile compressive response and its dependence on relative density.

As the mechanical behavior and performance of composites varies under different dynamic loading regimes and velocities, the book is divided into sections that examine the different loading regimes and velocities. A periodic Representative Volume Element RVE is constructed based on measurable statistical descriptors of the microstructure, and its response is simulated by concurrent electrical and mechanical FE analyses; the scatter of the predictions and their sensitivity to RVE size are explored.