Modeling and simulation of material distribution in the sequential co-injection molding process

Qingsheng Liu,Youqiong Liu,Chuntao Jiang

School of Mathematics and Statistics,Xinyang Normal University,Xinyang 464000,China

Keywords:Co-injection molding Finite volume method Simulation Level set Material distribution

ABSTRACT In co-injection molding,the properties and distribution of polymers will affect the application of products.The focus of this work is to investigate the effect of molding parameters on the skin/core material distribution based on three-dimensional (3-D) flow and heat transfer model for the sequential coinjection molding process,and the flow behaviors and material distributions of skin and core melts inside a slightly complex cavity(dog-bone shaped cavity)are predicted numerically.The governing equations of fluids in mold are solved by finite volume method and Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm on collocated meshes,and the domain extension technique is employed in numerical method for this cavity to assure that the numerical algorithm is implemented successfully.The level set transport equation which is used to trace the free surfaces in co-injection molding is discretized and solved by the 5th-order Weighted Essentially Non-Oscillatory (WENO) scheme in space and 3rd-order Total Variation Diminishing Runger-Kutta (TVD-R-K) scheme in time respectively.Numerical simulations are conducted under various volume fraction of core melt,skin and core melt temperatures,skin and core melt flow rates.The predicted results of material distribution in length,width and thickness directions are in close agreement with the experimental results,which indicate that volume fraction of core melt,core melt temperature and core melt flow rate are principal factors that have a significant influence on material distribution.Numerical results demonstrate the effectiveness of the 3-D model and the corresponding numerical methods in this work,which can be used to predict the melt flow behaviors and material distribution in the process of sequential co-injection molding.

1.Introduction

Co-injection molding which is also known as sandwich injection molding has been widely used for about forty years.In coinjection molding process,two kinds of polymers which may have different properties are injected into mold cavity in a sequential or simultaneous mode.The advantages of different materials can be combined flexibly in part design by this molding technology,and materials with specific functions can be used to improve physical properties of the parts[1,2].The recycled polymers can be utilized as core layer material and virgin polymer for skin layer material in order to produce less expensive parts while the product quality remains unchanged,which makes the co-injection molding technology particularly attractive [3].

In co-injection molding process,the dynamic interactions of polymer melts are very complex under the condition of different material properties and processing parameters,which makes it difficult to design and control the molding process accurately.The quality and performance of product not only depend on the design of products,but also the choices of skin and core materials;to a large extent,it rely on the processing parameters in the process of molding.The materials distribution and interface shape in product are the major factors that affect the quality of the product.Since the molding process is influenced by many factors,such as material characteristics,processing parameters and geometric dimensioning,it is difficult to realize the required interface shape precisely.Generally speaking,it is important to get a homogeneous skin and core materials distribution,which relies on the complex flow behavior of polymer melts during the filling process.The core melt does not penetrate sufficiently deep into the skin melt will resulting in a part with poor skin thickness uniformity.On the contrary,if the core melt tends to completely penetrate through the skin melt,breakthrough phenomenon occurs,which will caused a defective part.A successful design of co-injection molding process involves a long trial and error route,and accurate simulation of melt flow behaviors can help engineers to reach optimal designs and obtain the suitable processing parameters.Therefore to reduce the product cost and shorten development cycles.

Mold filling investigations of co-injection molding by experiment have been reported in the literatures [4–15].Most of these researchers focused on the influence of processing parameters and material properties on the material distribution,mechanical properties,internal structure,breakthrough phenomenon,residual stresses and birefringence for different mold geometry.So far,attempts were made in all kinds of ways for modeling coinjection molding[1,2,13–19].Schlatteret al.[13]proposed a multifluid model for sequential co-injection molding based on the lubrication approximations,which the displacement of the interface between the two fluids are characterized by a transport equation;the thermomechanical equations are solved by a modified finite volumes scheme while the transport equation is solved by discontinuous Galerkin formulation.Chenet al.[14] proposed a simulation method that based on the control-volume/finite element method,the dual-filling-parameter technique were adopted in each layer of thickness direction in order to trace the flow front of skin and core material in sequential co-injection molding.Kim and Isayev [15] took into account the elasticity of polymers and investigated a 2-D non-isothermal transient flow in sequential co-injection molding based on a nonlinear viscoelastic constitutive equation for the first time,the interface distribution,the flowinduced and thermally induced residual stresses were predicted using a hybrid finite-element/finite-difference/control-volume method.It is the first numerical investigation of thermally and flow-induced birefringence of sequential co-injection molding.Few studies were focused on the simultaneous co-injection molding[16–18].Liet al.[16,17]formulated a 2-D simulation of the process of simultaneous co-injection molding,and investigated the effects of rheological properties and processing conditions on the material distribution,penetration behavior and breakthrough phenomena.The hybrid finite-element/finite-difference/control-volu me method and volume of fluid (VOF) method are employed to solve the governing equations and transport equations respectively.All of the abovementioned authors use the Hele-Shaw approximation to model and predict the melt flow behaviors in the filling process of co-injection molding.However,Hele-Shaw approximation is obtained based on creeping flows confined between thin walls,this 2.5-D approach is not suitable for simulating the flows in mold that presenting thick sections,expansions,contractions,sudden changes in direction,etc.[19].In addition,the 2.5-D approach can’t predict the material distribution and certain important physical quantities in thickness direction.A threedimensional (3-D) finite element analysis algorithm for the filling process of sequential co-injection molding in a center-gated rectangular plate and a C-shaped plate was proposed by Ilincaet al.[1,19].The evolution of air/skin melt and skin/core melt interfaces are captured by a pseudo-concentration method and two separate transport equations are solved by a streamline upwind Petrov–Galerkin method,the continuity and momentum equations are solved by Galerkin least-squares method,the energy equation is solved by Galerkin least-squares/Galerkin gradient least-squares method.Liuet al.[2,18] developed a 3-D flow model and the relative finite volume method for the filling stage of sequential and simultaneous co-injection molding in rectangular plate and center-gated disk plate,which the governing equations for different fluids in the whole computational domain are written in a unified form,then the numerical algorithm can be implemented more conveniently.The two kinds of free surfaces are governed by one level set transport equation and captured by level set method.The commercially available simulation software Moldflow,CMold and Moldex3D are also used by some authors to investigate the flow behaviors and material distribution in co-injection molding under different processing conditions in recent years [12,20–22].

The objective of this article is to explore the effect of molding parameters on the skin/core material distribution of sequential co-injection molding based on the 3-D flow and heat transfer model proposed by us,apply 3D model and finite volume method on a slightly complex cavity (dog-bone shaped cavity) furtherly,since simple cuboid cavities were used in our previous work.The model and the relative numerical methods include finite volume method,level set method and domain extension method are validated by comparing the numerical results with experimental results and Moldflow results by Patcharaphun and Mennig [12]under different processing conditions.The values of core melt volume fraction,skin and core melt temperatures,skin and core melt flow rates are varied in order to obtain detail information of material distribution in length,width and thickness directions.It can deepen understanding of the flow behaviors in the filling process of co-injection molding.All fluids in the mold are governed by mass,momentum and energy conservation equations,and these equations are coupled with level set transport equation.The governing equations for fluids are discretized by finite volume method on collocated meshes and solved by Semi-Implicit Method for Pressure Linked Equations(SIMPLE)algorithm.Domain extension technique is used to cope with irregular computational domain so that the algorithm can be executed conveniently on collocated meshes.The level set and reinitialization equations which are used to trace the air/skin melt and skin/core melt interfaces in co-injection molding are discretized and solved by the 5th-order Weighted Essentially Non-Oscillatory (WENO) scheme and in space and 3rd-order Total Variation Diminishing Runger-Kutta (TVD-R-K)scheme in time respectively.

2.Mathematical Model

2.1.Mass,momentum and energy conservation equations

Two kinds of polymeric melts in mold cavity are assumed to be incompressible fluid under nonisothermal condition.The effect of surface tension can be neglected,and the elastic effect of polymer melts is not to be taken into account.The gas can freely leave the mold cavity while the polymer melts advance[23].A 3-D flow and heat transfer model is presented here to depict the flow and heat transfer behaviors of polymer melts in co-injection molding,in which the material parameters in different fluid domain are written in a unified form.Then,the mass,momentum and energy conservation equations in the whole computational domain are given by:

where u,t,p,ρφ,ηφ,Cφ,κφ,d denote velocity,time,pressure,density,viscosity,specific heat,thermal conductivity and rate of deformation tensor respectively.

The schematic diagram of free surfaces in standard injection molding and co-injection molding are shown in Fig.1(a) and (b)respectively.The evolution of free surfaces in sequential coinjection molding process in this paper are traced by level set method,and level set function φ=0 in the interfaces.For standard injection molding,there are two kinds of fluids (air and polymer melt) and one free surface in the filling process of molding.The density,viscosity,specific heat and thermal conductivity are constant in each fluid,and the material parameters of fluids in the mold are rewritten as a unified formviathe smooth Heaviside functionHεφ（）,which are given by

where the subscripts m and g denote melt and air respectively.

For co-injection molding,there are three kinds of fluids (air,skin and core polymer melts) and two kinds of free surfaces in the filling process of molding (air/skin melt and skin/core melt interfaces).Based on the idea of two phase flow used in the standard injection molding,the material parameters of air,skin polymer and core polymer can be written as a unified form in the whole computational domain by introducing an indicator function,which are given as follows:

Fig.1.Schematic diagram of free surfaces (a) standard injection molding;(b) coinjection molding.

where the subscripts s,cand g denote skin material melt,core material melt and gas respectively.The indicator functionslDand the smooth HeavisidefunctionHε（φ ）are given by

where parameter ε denotes interface thickness.The momentum and energy equations for three-fluid flow can be written as a unified form in the whole computational domain similarly,which are shown in Eqs.(2) and (3).Then,the numerical method and algorithm can be implemented conveniently.

Numerical results in this paper are shown using the dimensionless variables,and the relative equations are non-dimensionalized via:

whereRe,Pe,Brare the Reynolds,Peclet and Brinkman numbers respectively.(Re=ρsUL/ηs,Pe=ρsCs/κs,Br=ηsU2/κsT0).

2.2.Level set equation

The level set method which was devised by Osher and Sethian[24] was used to capture air/skin melt and skin/core melt interfaces in filling process of sequential co-injection molding in this article.The air/skin melt interface and skin/core melt interface are caputured by only one level set equation simultaneously,which is more convenient for computation.However,two separate transport equations are needed to govern and capture the two different kinds of moving interface using a pseudoconcentration method.The zero level set of a smooth function φ（x,y,z,t）denotes the interface between two different fluids,φ（x,y,z,t）is positive inside the domain of one fluid and negative inside the domain of another fluid.The level set equation is given by Eq.(12),and the reinitialization equation [25] which usually keeps φ（x,y,z,t）as the algebraic distance to the interfaces is given by Eq.(13).

wheretris artificial time and sign（φ0）has the following form

2.3.Cross-WLF viscosity model

The polymer viscosity is usually described by Cross-WLF model in injection molding simulation at present [1–3,26–28].The equation of Cross model is given by

where η0,,nand τ*are zero shear viscosity,shear rate,model constant and the critical stress level between Newtonian and power law behavior respectively.If η0in Eq.(15) is described by the WLF zero shear viscosity model,then,we obtain Cross-WLF viscosity model.The WLF model has the following form:

whereD1,D2,D3,A1andA2are material parameters.

Fig.2.Sketch map of the collocated meshes and control volume(the uppercase and lowercase letters denote node and interface respectively).

3.Numerical Method

3.1.Discretization of finite volume method

The numerical solution of flow and heat transfer model (mass,momentum and energy equations) in this paper are obtained by finite volume method on collocated meshes together with SIMPLE algorithm.Fig.2 shows the sketch map of the collocated meshes and control volume in which all physical quantities are stored on the same nodes.The mass,momentum and energy equations can be rewritten in the general conservative form as follows:

where θ1,θ2,and χ are constants,ξ andSξare determined by different equations under consideration.These parameters are shown in Table 1.The pressure–velocity decoupling problem is usually solved by interpolation technique.More details about the interpolation technique and numerical method.

Table 1 Definition of the constants and functions in Eq.(17)

The discretization of continuity equation can be obtained by integrating over a control volume,which has the following form

Similarly,the discretization of momentum equation is given by

whereSξis the source term,andaP,aE,aW,aN,aS,aT,aBare coefficients which are close related to convection and diffusion terms.Here,the convection terms are discretized by the central difference scheme.

The discretization of energy equation is given by

whereSTdenotes source term,and the relative coefficients are given by

The level set and reinitialization equations are discretized by the 5th-order WENO scheme in space,which have the following forms:

The level set and reinitialization equations are discretized by 3rd-order TVD-R-K scheme [29] in time,which is given by

where φ（0）=φn,φ（3）=φn+1;L（φ ）=uφx+vφy+wφzandLφ（）=sign（φ0）（1 -|?φ|）for the level set and reinitialization equations respectively.

3.2.Domain extension method

Domain extension method is usually used to cope with irregular computational domain in the field of fluid flow and heat transfer[30–34].The basic idea of this method is that irregular computational domain needs to be extended to a minimum rectangle or cuboid containing the irregular computational domain just right for two-dimension and three-dimension respectively.The dynamic viscosityandthermalconductivityoffluidinextendedregionshould be set to a very large number(1025–1030)and a very small number(10-30–10-25) respectively,then the numerical method and algorithm can be conducted conveniently on the collocated meshes.

4.Results and Discussion

Numerical simulations of material distribution in sequential coinjection molding were implemented for a dog-bone shaped cavity which is shown in Fig.3 under different processing parameters.All numerical results in this paper were obtained and shown in the dimensionless variables.The typical length isL=0?02 m,while the velocity and temperature scales are,U=1 m ?s-1andT0=1 K respectively.The polymer used for both skin and core layer is polystyrene,and the co-injection molding of the dogbone shaped parts was carried out by Patcharaphun and Mennig[12] on an machine (Model:320S 500–350).The temperature of mold wall was set at 40°C.The material properties and constants of Cross-WLF model for polystyrene are given in Table 2.The SIMPLE algorithm for the model in this paper is implemented in FORTRAN code.

Table 2 Material properties and Cross-WLF model constants of PS

4.1.Filling process of co-injection molding in a dog-bone shaped cavity

Fig.3.Schematic diagram of dog-bone shaped geometry.

The dog-bone shaped cavity can be extended to a minimum cuboid based on the domain extend method with size of 170 mm×24 mm×4 mm,and the non-dimensional computation domain is 8?5×1?2×0?2. The grid size Δx=Δy=Δz=0?0333 was used for computation.In this section,the volume fraction of core material is 60%,the temperature of both skin and core melts are 230°C,and the flow rates of skin and core melts are 18?5 cm3?s-1and 27 cm3?s-1respectively.

At the skin melt filling stage,a certain amount of skin melt is injected into the mold cavity.The initial air/skin melt interface was set to be a semiellipsoid for convenience,and the evolution of air/skin melt interface is presented in Fig.4.When the volume fraction of skin melt injected into the mold is attained 40%,the core melt penetration stage starts and core melt is injected into the mold,in this stage,the injection of skin melt is stopped.Fig.5 presents the evolution of air/skin melt and skin/core melt interfaces at the core melt penetration stage,in which the red area denotes skin layer and the blue area denotes core layer that encapsulated by skin melt.Here,a top view is shown for the sake of observation.From Fig.5,we can see that core melt penetrates rapidly into the skin melt as the continuous injection of core melt,the skin/core melt interface expands and runs after the air/skin melt interface.As the skin melt is pushed forward and touches the ends of mold wall by core melt,the cavity is filled and filling process is finished.

Fig.4.Evolution of air/skin melt interface at the skin melt filling stage.

Fig.5.Evolution of air/skin melt and skin/core melt interfaces at the core melt penetration stage.

Fig.6.Material distribution in the length and width directions for different core volume fraction:(a) experimental results [12];(b) present simulation results.

4.2.Effect of melt volume fraction

Fig.7.Material distribution at the cross-sections for different core volume fraction.

Fig.8.The obtained core material thickness fraction at different core volume fraction:(a)experimental results[12];(b)simulation results[12];(c)present simulation results.

Fig.9.Material distribution in the length and width directions for different skin temperature:(a) experimental results [12];(b) present simulation results.

Numerical simulations were implemented under different processing parameters.The core melt volume fraction varies from 55%to 65%,while the initial temperature of skin and core melt injected into the mold cavity are 230°C,and the injection flow rates for skin and core melt are 18?5 cm3?s-1and 27 cm3?s-1respectively.Fig.6 illustrates skin/core material distribution at the last moment of the filling process for different core volume fraction between experimental results obtained by Patcharaphun and Mennig [12]and present simulation results.As can be observed in Fig.6,the core melt penetrates more deep in the skin melt as more core melt is injected into the mold cavity,particularly in the flow direction.When the core melt volume fraction varies from 55% to 60%,the core material melt is encapsulated in skin melt,i.e.the co-injection molded parts produced without breakthrough.When the core melt volume fraction attains 65%,the skin/core melt interface overtakes the air/skin melt interface,and the breakthrough phenomenon occurs,i.e.the defective parts will be generated.Fig.7 shows material distribution at the cross-sectionsx=2,7,12,15 and 16 mm respectively for different core volume fraction.Fig.8 shows influence of core melt volume fraction on core material thickness fraction which is defined as the ratio of core material thickness δ to cavity thicknessb.Fig.8(a) and (b) are experimental results by optical microscopy and simulation results by Moldflow respectively [12],and Fig.8(c) is the predicted results based the present model and numerical method,From Figs.7 and 8,we can see that the core melt penetrates deeper inside the skin melt in all directions as the core melt volume fraction varies from 55% to 65%.The core material thickness fraction predicted in this paper approach to experimental results at the front and middle parts of the cavity.Small difference existed between present results and experimental results at the far end of the mold cavity,the probable cause for this is the mesh size used in finite volume method,which is not taken to be fine enough in view of computational cost.But as a whole,the model and numerical method can be used to predict the material distribution in dog-bone shaped cavity of coinjection molding.

Fig.10.The obtained core material thickness fraction at different skin temperature:(a)experimental results[12];(b)simulation results[12];(c)present simulation results.

Fig.11.Material distribution in the length and width directions for different core temperature:(a) experimental results [12];(b) present simulation results.

4.3.Effect of melt temperatures

The effect of skin and core melt temperatures on the material distribution of core melt in length,width and thickness directions are shown in Figs.9–12.The core melt volume fraction was set at 60%,and the injection flow rates for skin and core melt are 18?5cm3?s-1and 27cm3?s-1respectively.Fig.9 shows comparison of material distribution in length and width directions at the last moment of the filling process between experiment and simulation when skin melt temperature varies from 210 to 260°C and core melt temperature maintains in 230°C.Fig.10 shows material distribution(in the form of core material thickness fraction)in thickness direction corresponding to the processing parameters of Fig.9.It can be seen from Figs.9 and 10 that lower skin melt temperature lead to a thinner core melt close to the sprue region and therefore core penetration is longer in length direction.Accordingly,higher skin melt temperature determines a thicker core melt close to the sprue region and shorter core penetration length.Fig.11 shows comparison of material distribution in length and width directions at the last moment of the filling process between experiment and simulation when core melt temperature varies from 210 to 260°C and skin melt temperature maintains in 230°C.Fig.12 shows core material thickness fraction corresponding to the processing parameters of Fig.11.Figs.11 and 12 illustrate higher core melt temperature determines a thicker core melt and longer penetration length.Nevertheless,the penetration width at the far end of the mold cavity is getting shorter and shorter as increasing of core melt temperature.The influence of core melt temperature on material distribution is more conspicuous than that of skin melt temperature.As is well known,a higher temperature determines a lower viscosity of melt.Hence,the viscosity ratio of skin to core melt decreases with increasing skin temperature while the core temperature remain unchanged;and the viscosity ratio of skin to core melt increases with increasing core temperature while the skin temperature remain unchanged.It can be seen from Figs.9–12 that the present numerical results of material distribution in dog-bone shaped cavity under different injection temperatures are in good agreement with experimental results,only the penetration width of core melt is a little shorter than the experimental results at the far end of mold cavity.Fig.13 presents the predicted results of material distribution at the cross-sectionsx=2,7,12,15 and 16 mm respectively for different skin and core temperatures.

Fig.12.The obtained core material thickness fraction at different core temperature:(a)experimental results[12];(b)simulation results[12];(c)present simulation results.

Figs.14 and 15 show the predicted temperature distributions ofx–ymidplane at the last moment of the filling process under different processing conditions corresponding to Figs.9 and 11 respectively,from which we can see that the melt temperature becomes higher and higher from the mold walls to the centerline in thickness direction.The maximum temperature is becomes lower and lower from the sprue to the end of mold cavity in length direction.The maximum melt temperature in cavity is mainly determined by core melt temperature.In Fig.14,only skin melt temperature varies from 210 to 260°C and core melt temperature keeps in 230°C in processing conditions,the maximum melt temperature in cavity maintains in 230°C all the time,although skin melt temperature exceed the core temperature in some cases.The reason is that when the skin melt at higher temperature was injected into mold and touches the mold walls at lower temperature,it decrease rapidly due to heat conduction.Hence,in Fig.15,when core melt temperature varies from 210 to 260°C and skin melt temperature keeps in 230°C in processing conditions,the maximum melt temperature in cavity is consistent with temperature of core melt injected into the mold initially.

4.4.Effect of melt flow rate

Fig.13.Material distribution at the cross-sections for different skin and core temperatures.

Fig.14.The predicted temperature distribution of x–y midplane in Kelvin scale at the last moment of filling stage for different skin melt temperature;(a)skin/core=210/230°C;(b) skin/core=230/230°C;(c) skin/core=260/230°C.

Fig.15.The predicted temperature distribution of x–y midplane in Kelvin scale at the last moment of filling stage for different core melt temperature;(a)skin/core=230/210°C;(b) skin/core=230/230°C;(c) skin/core=230/260°C.

Fig.16.Material distribution in the length and width directions for different skin flow rate:(a) experimental results [12];(b) present simulation results.

Fig.17.The obtained core material thickness fraction at different skin flow rate:(a) experimental results [12];(b) simulation results [12];(c) present simulation results.

Fig.18.Material distribution in the length and width directions for different core flow rate:(a) experimental results [12];(b) present simulation results.

Fig.19.The obtained core material thickness fraction at different core flow rate:(a) experimental results [12];(b) simulation results [12];(c) present simulation results.

The influence of skin and core melt flow rates on the material distribution of core melt in length,width and thickness directions are shown in Figs.16–19.The core melt volume fraction was set at 60%,and the initial temperature of skin and core melt injected into the mold cavity are 230°C.Fig.16 shows comparison of material distribution in length and width directions at the last moment of the filling process between experiment and simulation when skin melt flow rate varies from 9?25 to 37 cm3?s-1and core melt flow rate maintains in 27 cm3?s-1.Fig.17 shows material distribution(in the form of core material thickness fraction)in thickness direction corresponding to the processing parameters of Fig.16.Figs.16 and 17 show that lower skin melt flow rate leads to a slightly longer core penetration in length direction,and numerical results indicate that there is not much difference of core material distribution in thickness and width directions under different skin melt flow rates.Fig.18 demonstrates comparison of material distribution in length and width directions at the last moment of the filling process between experiment and simulation when core melt flow rate varies from 13?5 to 54 cm3?s-1and skin melt flow rate maintains in 18?5 cm3?s-1,and core material thickness fraction is shown in Fig.19.As can be seen clearly from Figs.18 and 19,the change of core melt flow rate has a great influence on material distribution.Higher core melt flow rate lead to a thicker core melt near the sprue region and shorter core penetration length.When the core melt has a higher flow rate,it can pushes more skin melt not only in the length direction,but also in the thickness and width directions,which cause a good deal of skin melt increased at the far end of the mold cavity and hence the core penetration length becomes shorter.The present numerical results are accordance with the experimental results by Patcharaphun and Mennig [12].Fig.20 presents the predicted results of material distribution at the cross-sectionsx=2,7,12,15 and 16 mm respectively for different skin and core flow rate.All the obtained results show that core melt flow rate has a great influence on the material distribution.In fact,the flow rate variation will affect the cooling time and the shear heating of melt,and cause the variations of temperature and viscosity finally.With increasing of the core melt flow rate,the shear rate of melt near the mold wall is becoming higher,and the shear thinning behavior becomes more pronounced.So,the variations of temperature and flow rate will influence the viscosity ratio of the skin/core melt,and it is the most important factor to influence the final result.

Fig.20.Material distribution at the cross-sections for different skin and core flow rate.

5.Conclusions

A 3-D flow and heat transfer model is presented in this paper so as to investigate the flow behaviors of polymer melts and material distributions in co-injection molding,in which the governing equations for different fluids in the whole computational domain can be written in a uniform form.Numerical method and algorithm for the model are expounded in details,which the aforementioned equations are discretized by finite volume method on collocated meshes and level set equations are discretized by the 5th-order WENO scheme in space and 3rd-order TVD-R-K scheme in time respectively.Numerical simulations of co-injection molding are carried out for a dog-bone shaped cavity under different processing conditions,so as to investigate the effect of melt volume fraction,temperature and flow rate on material distribution.Numerical results show that the volume fraction,temperature and flow rate of core melt are principal factors that have a significant effect on material distribution.The skin melt temperature and skin melt flow rate have little effects than that of core melt temperature and core melt flow rate.The core melt penetrates deeper inside the skin melt in all directions as the core melt volume fraction increase.Higher core melt temperature determines a thicker core melt and longer penetration length.Nevertheless,the penetration width at the far end of the mold cavity is getting shorter and shorter as increasing of core melt temperature.Higher core melt flow rate lead to a thicker core melt near the sprue region and shorter core penetration length.Whether change core melt temperature or flow rate will cause the variation of skin melt viscosity on account of heat transfer and shear effects,and influence material distribution ultimately.The present simulation results are in close agreement with the published experimental results,which demonstrate the effectiveness of the flow and heat transfer model and the corresponding numerical methods in this paper.The model can be used to predict the melt flow behaviors and material distribution during the co-injection molding process.

Based on the theoretical assumptions and numerical findings made in the present investigation,some interesting research topic to investigate in the future could be recommended:(1) The material used for both skin and core layer is polystyrene in present study,and different materials can be used for skin and core layers in the future;(2) the elastic effect of polymer melts is not to be taken into account in present study,and the viscoelastic constitutive equation can be used in 3D model;(3) residual and birefringence distribution in both skin and core layer could be discussed in the future;(4) the present study focus on the filling stage of sequential co-injection molding,and the packing stage could be take into account in the future.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by Science and Technology Research Key Project of the Education Department of Henan Province(20A430023,20B130002,20A110031),Natural Science Foundation of Henan Province (202300410340),National Natural Science Foundation of China (11901504) and Nanhu Scholars Program for Young Scholars of Xinyang Normal University.