Constitutive modelling for composite forming 25 crossings of the yarns in the fabric.The energy contribution in the particles consists of thread repelling,thread stretching,thread bending,thread trellising and gravity. The total energy in the cloth is simply the sum of the energy of all particles.The modelling strategy for particle based solutions is generally time dependent.In the first time step,the model accounts for the gravity and the collision between the cloth and the drape surface.In the next step a stochastic energy-minimising technique is used to find the local energy minima for the cloth.Finally, permutations are introduced to produce a more asymmetric final configuration. Similarly to the energy based functions,force based functions were also developed for the interactions of the particles (Colombo et al.,2001).This representation method is applied in commercially available software,since it is computationally more attractive than the energy based particle interaction functions. Cordier and Magnenat-Thalmann (2002)simulated the cloth behaviour on dressed virtual humans in real time.They proposed a hybrid drape algorithm combining the advantages of physically (particle)based and geometric deformations,avoiding the computationally expensive collision calculations as much as possible.The cloth is segmented into three sections in their simulations. Cloths that remain at constant distance to the drape body are modelled in the first section.Typically,these are stretch cloths.In the second layer,the loose cloth follows predefined discs,representing the limbs.Finally,floating cloth such as skirts is represented in the third section.A force based particle method is used for modelling the floating cloth,incorporating the collision with the underlying body and the cloth itself.Real-time modelling of cloth behaviour is feasible with this approach using middle range computers (up to 1GHz PCs). The method requires the mechanical properties of the cloth and the product shape as input.Typically,the method is used for modelling the shape of hanging cloth on objects or humans in the fashion industry.The emphasis is therefore not 3the deformation and stresses within the fabric but the resulting shape of the cloth as a whole.Possibly,this is why no implementation in the technical industry has been found for this method in the literature. Truss based schemes Fabrics are woven using a periodic arrangement of fibre bundles.These periodic arrangements are called Representative Volume Elements(RVE),or unit cells. The fibres in these unit cells can be represented using trusses.The fibre interaction,such as shear-locking of the fabric,is modelled with diagonal stringers.Kato et al.(1999)proposed a unit cell representation based on such a fabric lattice model in 1999. In 2003,Tanov and Brueggert modelled the inflation of a car side airbag in an FE(Finite Element)simulation,using a loosely woven fabric model.The yarns in the fabric were represented by pinned-joined bars with two locking springs on
crossings of the yarns in the fabric. The energy contribution in the particles consists of thread repelling, thread stretching, thread bending, thread trellising and gravity. The total energy in the cloth is simply the sum of the energy of all particles. The modelling strategy for particle based solutions is generally time dependent. In the first time step, the model accounts for the gravity and the collision between the cloth and the drape surface. In the next step a stochastic energy-minimising technique is used to find the local energy minima for the cloth. Finally, permutations are introduced to produce a more asymmetric final configuration. Similarly to the energy based functions, force based functions were also developed for the interactions of the particles (Colombo et al., 2001). This representation method is applied in commercially available software, since it is computationally more attractive than the energy based particle interaction functions. Cordier and Magnenat-Thalmann (2002) simulated the cloth behaviour on dressed virtual humans in real time. They proposed a hybrid drape algorithm combining the advantages of physically (particle) based and geometric deformations, avoiding the computationally expensive collision calculations as much as possible. The cloth is segmented into three sections in their simulations. Cloths that remain at constant distance to the drape body are modelled in the first section. Typically, these are stretch cloths. In the second layer, the loose cloth follows predefined discs, representing the limbs. Finally, floating cloth such as skirts is represented in the third section. A force based particle method is used for modelling the floating cloth, incorporating the collision with the underlying body and the cloth itself. Real-time modelling of cloth behaviour is feasible with this approach using middle range computers (up to 1 GHz PCs). The method requires the mechanical properties of the cloth and the product shape as input. Typically, the method is used for modelling the shape of hanging cloth on objects or humans in the fashion industry. The emphasis is therefore not the deformation and stresses within the fabric but the resulting shape of the cloth as a whole. Possibly, this is why no implementation in the technical industry has been found for this method in the literature. Truss based schemes Fabrics are woven using a periodic arrangement of fibre bundles. These periodic arrangements are called Representative Volume Elements (RVE), or unit cells. The fibres in these unit cells can be represented using trusses. The fibre interaction, such as shear-locking of the fabric, is modelled with diagonal stringers. Kato et al. (1999) proposed a unit cell representation based on such a fabric lattice model in 1999. In 2003, Tanov and Brueggert modelled the inflation of a car side airbag in an FE (Finite Element) simulation, using a loosely woven fabric model. The yarns in the fabric were represented by pinned-joined bars with two locking springs on Constitutive modelling for composite forming 25 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 6:57:44 PM IP Address: 158.132.122.4
26 Composites forming technologies Side truss (a) Diagonal truss (b) 2.7 Schematic representation of the unit cell by truss elements:(a)one diagonal spring element,(b)two diagonal spring elements. the diagonal of their unit cell.A schematic representation of this unit cell is depicted in Fig.2.1(b). Sharma and Sutcliffe (2003)also represented the unit cell in a network of pinned-joined trusses.The edge trusses represent the fibres in the fabric.To introduce shear stiffness in the unit cell,one diagonal spring is introduced,as peay poo presented in Fig.2.1(a).An FE analysis was performed with this pinned-joined net of trusses to predict the draping process. The advantages of these models are the simple mechanics and ease of use. Little input is required and the simulation time is relatively short compared to continuum based approaches.The resin behaviour is not incorporated in these models.Forming multi-layered composites can be simulated just as with the mapping based schemes since these models are also based on the representation of a single layer of fabric.Through-thickness shear interaction between the fabric layers during forming is not accounted for. 0 2.2.2 Continuum models Several constitutive models have been proposed for fabric drape modelling.A distinction can be made between elastic material models,viscous material models and multi-component models.Most of these constitutive models are formulated in a plate or shell theory and implemented in Finite Element formulations. Elastic models Finite Element drape simulations by means of elastic models have been applied since the mid 1990s.One of the earliest models was presented by Chen and Govindaraj in 1995.They developed an elastic orthotropic continuum based model to represent the fabric drape behaviour.The material model was based on a flexible shell theory.A non-linear FE formulation was used to predict the forming of a fabric onto a table
the diagonal of their unit cell. A schematic representation of this unit cell is depicted in Fig. 2.1(b). Sharma and Sutcliffe (2003) also represented the unit cell in a network of pinned-joined trusses. The edge trusses represent the fibres in the fabric. To introduce shear stiffness in the unit cell, one diagonal spring is introduced, as presented in Fig. 2.1(a). An FE analysis was performed with this pinned-joined net of trusses to predict the draping process. The advantages of these models are the simple mechanics and ease of use. Little input is required and the simulation time is relatively short compared to continuum based approaches. The resin behaviour is not incorporated in these models. Forming multi-layered composites can be simulated just as with the mapping based schemes since these models are also based on the representation of a single layer of fabric. Through-thickness shear interaction between the fabric layers during forming is not accounted for. 2.2.2 Continuum models Several constitutive models have been proposed for fabric drape modelling. A distinction can be made between elastic material models, viscous material models and multi-component models. Most of these constitutive models are formulated in a plate or shell theory and implemented in Finite Element formulations. Elastic models Finite Element drape simulations by means of elastic models have been applied since the mid 1990s. One of the earliest models was presented by Chen and Govindaraj in 1995. They developed an elastic orthotropic continuum based model to represent the fabric drape behaviour. The material model was based on a flexible shell theory. A non-linear FE formulation was used to predict the forming of a fabric onto a table. 2.1 Schematic representation of the unit cell by truss elements: (a) one diagonal spring element, (b) two diagonal spring elements. 26 Composites forming technologies Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 6:57:44 PM IP Address: 158.132.122.4
Constitutive modelling for composite forming 27 At the same time,Kang and Yu (1995)developed a similar shell based drape model.They used a convective coordinate system in a total Lagrangian formulation with an orthotropic continuum based elastic material model.The nonlinear incremental formulation of the total Lagrangian scheme was solved using Newton's method.Again,the draping of a cloth onto a table was simulated. A few years later Boisse et al.(1997)modelled the bi-axial fabric behaviour in forming processes.The undulation of the yarns in the fabric was accounted for in the bi-axial weave model,assuming fibres with stiffness in the fibre direction only.A refined version of this material model was presented by Boisse etal.(2001). Ivanov and Tabiei (2002)developed an elastic material model based on the RVE of a plain weave fabric.A homogenisation technique accounts for the weave's microstructure,where the yarns are assumed to be transversely iso- tropic.The shear properties of the fabric are neglected up to the locking angle. On further shear the yarn shear properties are used for the fabric response. Elastic material laws are fairly simple.The implementation of these models in FE packages is therefore reasonably simple as well,compared to more advanced material models.Generally,the matrix material behaviour is viscoelastic.The properties of the matrix cannot be taken into account accurately with a purely elastic material law.Only single layers of fabric were draped using these models in the literature. 、3月Viscous moder5 Spencer (2000)modelled the behaviour of impregnated woven fabrics as a viscous fluid.The fibres were assumed inextensible in his model,effectively restraining the deformation of the fluid in the fibre directions.The fluid was also gassumed incompressible.In the plane stress situation,the model simplifies to a single parameter model and is able to simulate the draping behaviour of the fabric.Similarly,Spencer(2001)proposed a viscoplasticity model for draping fabric reinforced composites. The elastic behaviour of the fabric itself is not incorporated in these material models.Therefore,processing-induced fibre stresses are unaccounted for in these models.The models account for the drape behaviour of one layer of fabric only. Multi-component models Multi-component models are a combination of several material models.The fibres are often represented as elastic materials in these models.Some models incorporate the fabric shear behaviour,others account for the resin behaviour using viscous material laws
At the same time, Kang and Yu (1995) developed a similar shell based drape model. They used a convective coordinate system in a total Lagrangian formulation with an orthotropic continuum based elastic material model. The nonlinear incremental formulation of the total Lagrangian scheme was solved using Newton's method. Again, the draping of a cloth onto a table was simulated. A few years later Boisse et al. (1997) modelled the bi-axial fabric behaviour in forming processes. The undulation of the yarns in the fabric was accounted for in the bi-axial weave model, assuming fibres with stiffness in the fibre direction only. A refined version of this material model was presented by Boisse et al. (2001). Ivanov and Tabiei (2002) developed an elastic material model based on the RVE of a plain weave fabric. A homogenisation technique accounts for the weave's microstructure, where the yarns are assumed to be transversely isotropic. The shear properties of the fabric are neglected up to the locking angle. On further shear the yarn shear properties are used for the fabric response. Elastic material laws are fairly simple. The implementation of these models in FE packages is therefore reasonably simple as well, compared to more advanced material models. Generally, the matrix material behaviour is viscoelastic. The properties of the matrix cannot be taken into account accurately with a purely elastic material law. Only single layers of fabric were draped using these models in the literature. Viscous models Spencer (2000) modelled the behaviour of impregnated woven fabrics as a viscous fluid. The fibres were assumed inextensible in his model, effectively restraining the deformation of the fluid in the fibre directions. The fluid was also assumed incompressible. In the plane stress situation, the model simplifies to a single parameter model and is able to simulate the draping behaviour of the fabric. Similarly, Spencer (2001) proposed a viscoplasticity model for draping fabric reinforced composites. The elastic behaviour of the fabric itself is not incorporated in these material models. Therefore, processing-induced fibre stresses are unaccounted for in these models. The models account for the drape behaviour of one layer of fabric only. Multi-component models Multi-component models are a combination of several material models. The fibres are often represented as elastic materials in these models. Some models incorporate the fabric shear behaviour, others account for the resin behaviour using viscous material laws. Constitutive modelling for composite forming 27 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 6:57:44 PM IP Address: 158.132.122.4
28 Composites forming technologies Sidhu et al.(2001)proposed a bi-component FE analysis to model dry fabric forming.The yarns were represented as trusses,using a linear elastic material law.A layer of shell elements accounted for the friction between the yarns and for the locking of the fabric.The material behaviour in the shells was non- linearly elastic and orthotropic.The nodes of the truss and the shell elements were connected in the FE formulation.Friction between the fabric and the tooling was not incorporated in the simulation of the stamping of a spherical shape. Cherouat and Billoet (2001)presented a model for draping thermoplastic composite materials.The yarns in the fabric were represented using truss elements.The material law representing the yarns was non-linear elastic and based on three deformation mechanics:straightening,relative rotation and tensile stretching.The resin was modelled as an isotropic viscoelastic medium in a layer of shell elements.Again the nodes of the two meshes are connected in the FE simulations.Coulomb friction between the tooling and the fabric was assumed in the forming simulation of a hemispherical product. McEntee and O Bradaigh (1998)modelled the drape behaviour of a multi- layered thermoplastic composite on tools with a single curvature.Two- dimensional elements were stacked through the thickness of the sheet,each ply represented by a row of elements.The constitutive relation in each ply was based on the 'ideal fibre reinforced fluid'model by Rogers (1989).A layer of 。s9 contact elements was placed between the ply elements.Experiments demonstrated the presence of a resin rich layer between the individual plies during forming,justifying a viscous contact behaviour. De Luca et al.(1998)modelled the drape behaviour of composite laminates in a dynamic explicit FE scheme.Each single fabric layer in the laminate was represented by a layer of elements.Per layer,a bi-phase material model was applied,decoupling the behaviour of the elastic fibres and the viscous matrix. The shell element layers were stacked in the thickness direction of the sheet. Between the shell elements,a 'specialised viscous-friction law'was applied. The model predicts a clear interaction between the laminate lay-up and the drapeability.Experimental results confirm the importance of this interlaminar shear effect.The method provides good results but becomes quite slow by expanding the problem computationally. Recapitulating,draping can be modelled using the combination of continuum based material models and the FE method.A drape simulation is non-linear due to the large deformations of the fabric during draping and the evolving contact conditions. The required input consists of the tool and laminate geometry definition,the material property data and the appropriate boundary conditions.The results of the simulation combine the information on the material deformation with the loading required for shaping.The interlaminar shear effects during forming can play a significant role in the drapeability of multi-layered composite com-
Sidhu et al. (2001) proposed a bi-component FE analysis to model dry fabric forming. The yarns were represented as trusses, using a linear elastic material law. A layer of shell elements accounted for the friction between the yarns and for the locking of the fabric. The material behaviour in the shells was nonlinearly elastic and orthotropic. The nodes of the truss and the shell elements were connected in the FE formulation. Friction between the fabric and the tooling was not incorporated in the simulation of the stamping of a spherical shape. Cherouat and BilloeÈt (2001) presented a model for draping thermoplastic composite materials. The yarns in the fabric were represented using truss elements. The material law representing the yarns was non-linear elastic and based on three deformation mechanics: straightening, relative rotation and tensile stretching. The resin was modelled as an isotropic viscoelastic medium in a layer of shell elements. Again the nodes of the two meshes are connected in the FE simulations. Coulomb friction between the tooling and the fabric was assumed in the forming simulation of a hemispherical product. McEntee and OÂ BraÂdaigh (1998) modelled the drape behaviour of a multilayered thermoplastic composite on tools with a single curvature. Twodimensional elements were stacked through the thickness of the sheet, each ply represented by a row of elements. The constitutive relation in each ply was based on the `ideal fibre reinforced fluid' model by Rogers (1989). A layer of contact elements was placed between the ply elements. Experiments demonstrated the presence of a resin rich layer between the individual plies during forming, justifying a viscous contact behaviour. De Luca et al. (1998) modelled the drape behaviour of composite laminates in a dynamic explicit FE scheme. Each single fabric layer in the laminate was represented by a layer of elements. Per layer, a bi-phase material model was applied, decoupling the behaviour of the elastic fibres and the viscous matrix. The shell element layers were stacked in the thickness direction of the sheet. Between the shell elements, a `specialised viscous-friction law' was applied. The model predicts a clear interaction between the laminate lay-up and the drapeability. Experimental results confirm the importance of this interlaminar shear effect. The method provides good results but becomes quite slow by expanding the problem computationally. Recapitulating, draping can be modelled using the combination of continuum based material models and the FE method. A drape simulation is non-linear due to the large deformations of the fabric during draping and the evolving contact conditions. The required input consists of the tool and laminate geometry definition, the material property data and the appropriate boundary conditions. The results of the simulation combine the information on the material deformation with the loading required for shaping. The interlaminar shear effects during forming can play a significant role in the drapeability of multi-layered composite com- 28 Composites forming technologies Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 6:57:44 PM IP Address: 158.132.122.4
Constitutive modelling for composite forming 29 ponents.The drape behaviour of multi-layered composites can be modelled by stacking multiple element layers through the thickness of the sheet and connecting them by friction laws. 2.2.3 Multi-layered models Current drape predictions are based on single fabric layer models or an assembly of single fabric layer models.In some production processes the fabric layers are formed sequentially onto the mould.Interlaminar shear between the individual fabric layers is small in such a production process.Modelling each single fabric layer sequentially suffices for such a process. However,the interaction between the layers in the sheet is important when draping multi-layered composites.FE simulations with multiple elements through-the-thickness of the sheet are used to account for this interaction between the layers.A friction law accounts for the interlaminar shear between the individual fabric layers.The drawback of using multiple sheet elements on top of each other is the increase of the complexity of the FE model.The total number of degrees of freedom(DOFs)grows linearly with the number of layers in the model and so does the number of contact conditions to be evaluated.A non-linear system of equations has to be solved in the FE representation.The computation times to solve such a non-linear system of equations will easily increase at least quadratically with the increasing number of DOFs.As a result the computation 155 time with increasing layers in the drape simulation behaves correspondingly. A computationally more efficient method is preferred to predict the drape behaviour of uti-ayered fabric comosites.This cabe achieved in a muti layer drape material model as presented by Lamers in 2004.This drape model incorporates the inter-ply and intra-ply shear behaviour of multi-layered fabric reinforced composites.The use of multiple elements through-the-thickness of theinterlaminar shear within the multi-layer model.The same type of FE the laminate (and the corresponding contact logic)is avoided by accounting for element can be used for the single layer and the multi-layer material model.The number of DOFs in an FE simulation with this multi-layer material model will therefore be equal to the number DOFs of the single layer model.Hence,the computation time for solving the non-linear system of equations will be comparable. 2.3 Continuum based laminate modelling Continuum mechanics defines the kinematics,stresses,strains and the con- servation laws for arbitrary continuous media.Forming simulations require a constitutive relation in addition to these conservation laws. Here,constitutive relations are presented for single layer and multi-layered composites,based on linear elastic fibres and a Newtonian viscous resin.An
ponents. The drape behaviour of multi-layered composites can be modelled by stacking multiple element layers through the thickness of the sheet and connecting them by friction laws. 2.2.3 Multi-layered models Current drape predictions are based on single fabric layer models or an assembly of single fabric layer models. In some production processes the fabric layers are formed sequentially onto the mould. Interlaminar shear between the individual fabric layers is small in such a production process. Modelling each single fabric layer sequentially suffices for such a process. However, the interaction between the layers in the sheet is important when draping multi-layered composites. FE simulations with multiple elements through-the-thickness of the sheet are used to account for this interaction between the layers. A friction law accounts for the interlaminar shear between the individual fabric layers. The drawback of using multiple sheet elements on top of each other is the increase of the complexity of the FE model. The total number of degrees of freedom (DOFs) grows linearly with the number of layers in the model and so does the number of contact conditions to be evaluated. A non-linear system of equations has to be solved in the FE representation. The computation times to solve such a non-linear system of equations will easily increase at least quadratically with the increasing number of DOFs. As a result the computation time with increasing layers in the drape simulation behaves correspondingly. A computationally more efficient method is preferred to predict the drape behaviour of multi-layered fabric composites. This can be achieved in a multilayer drape material model as presented by Lamers in 2004. This drape model incorporates the inter-ply and intra-ply shear behaviour of multi-layered fabric reinforced composites. The use of multiple elements through-the-thickness of the laminate (and the corresponding contact logic) is avoided by accounting for the interlaminar shear within the multi-layer model. The same type of FE element can be used for the single layer and the multi-layer material model. The number of DOFs in an FE simulation with this multi-layer material model will therefore be equal to the number DOFs of the single layer model. Hence, the computation time for solving the non-linear system of equations will be comparable. 2.3 Continuum based laminate modelling Continuum mechanics defines the kinematics, stresses, strains and the conservation laws for arbitrary continuous media. Forming simulations require a constitutive relation in addition to these conservation laws. Here, constitutive relations are presented for single layer and multi-layered composites, based on linear elastic fibres and a Newtonian viscous resin. An Constitutive modelling for composite forming 29 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 6:57:44 PM IP Address: 158.132.122.4