Warpage prediction of the injection-molded strip-like plastic parts☆

Chaofang Wang,Ming Huang*,Changyu Shen,Zhenfeng Zhao

National Engineering Research Center for Advanced Polymer Processing Technology,Zhengzhou University,Zhengzhou 450002,China

1.Introduction

Injection molding is a widely used plastic processing method and plays an important role in plastic industry due to its high production rate,economical efficiency and ability to produce complex articles with high precision[1-3].In recent years,traditional method of the mold design and injection molding practices based on past experience is no longer adequate[4].Numerical simulation techniques have now become helpful tools for mold designers and process engineers in injection molding[5-9].During the last decade,various authors studied the numerical simulation of injection molding,and some related CAE software have also been developed,such as Mold flow,Moldex,HSCAE and Z-mold,which helped engineers to design part and mold,select parameters of injection molding,and so on[10-12].

CAE software is becoming indispensable tool for the engineers to design the mold for injection-molded parts.However,it does not work well for some parts with special shape.The strip-like part is a common one of them.The decorative/reinforced strips in cars and the boundary frames of solar panels are some examples.Unfortunately,general commercial CAE tools cannot work well for strip-like parts due to their insufficient accuracy.For most parts with shell-like geometry,the well known 2.5 dimensional(2.5D)model,namely Hele-Shaw model,can provide very good solution[13].Based on the 2.5D model and shell theory,Zhou et al.[14,15]predicted the warpage of the shield plate of airconditioners.Ozcelik et al.[16]and Zhao et al.[17]used Mold flow software to simulate the warpage of the cell phone cover and the LCD TV front shell.The representative thickness for these parts is 2 to 3 mm and the ratio of part surface size to thickness is above 10.Although the strip-like part is not thick,its cross section has only low ratio of height/width to thickness,so the advantage of Hele-Shaw model cannot be taken.

In addition to the Hele-Shaw model,three-dimensional(3D)model[18-22]is also often used to simulate plastic injection molding process.Yan etal.[23]investigated the 3D computational model and successfully predicted the warpage of the blower cover.Hakimian and Sulong[24]simulated the warpage of injection-molded micro gears by using Autodesk Mold flow Insight.However,we still have to face some difficulty if 3D model is employed to deal with the strip-like part.In the Hele-Shaw model,the thickness is usually divided into 20 layers.This means that the elemental size in 3D model should be about 0.1-0.2 mm if the accuracy same as the Hele-Shaw model is expected.Because the strip-like part is so long,around 108to 109elements,or even more,are needed,which is beyond the capacity of general PCs.Of course,coarse mesh and some simplification in computing that the current commercial software might use allow the simulation executed,but the error is unavoidable,and not acceptable to the strip-like parts.

A new solution to predict the warpage of the strip-like injection molded parts was suggested,which was composed of two steps.Firstly,the part was macroscopically considered as a beam with smoothly changed cross section and this beam then was divided into a few one-dimensional(1D)elements.Next,at each 1D elemental node,the thermal and strain analysis of the cross section of the part was made using 2D finite element method(FEM);after these calculations,the curvatures at all beam nodes were obtained,and the part warpage prediction became very simple.In this way,the time-consuming 3D finite element analysis(FEA)was turned into several 2D FEA and a 1D FEA,so that the warpage of the strip-like injection-molded part was reasonably estimated.The numerical examples showed that the predicted result was in good agreement with practical cases.

2.Basic Idea

A typical example of the strip-like plastic part is shown in Fig.1.No matter what kind of cross section it might be,its size is much smaller in comparison with its length.Therefore,it is generally a little-curved beam macroscopically.To predict the final deformation of the part,the beam theory is employed to calculate the moment and curvature firstly.The moment here refers to the one that the part is subjected to when it is still in the mold.When the part is ejected,it is liberated from the mold,the moment is released,and resulted in partial deformation.It is obvious that the crucial work is the calculation of the in-mold moment or curvature.So long as it is done,the left job is straightforward.

Fig.1.Chafing strip of car.

It is well known that the in-mold stress has two sources,i.e.,thermal and flow-induced.In fact,although there is shear stress during the filling stage,it disappears when the mold is filled up.The so-called flow induced stress more refers to the one caused by the flow-induced material property change.As it is common knowledge that the flow-induced stress is much smaller than the thermal one[25],the former will be neglected in our discussion.There are three determinant factors in the stress development.The first one is packing induced.When the melt cools down to a certain degree,the packing pressure will be solidified in the part.The second factor is temperature variation.When the temperature becomes lower,the thermal stress appears.When the stress is larger than packing-induced stress(the reverse case not considered here),the part tends to shrink.However,the mold prohibits this shrinkage,so that the in-mold stress appears which will produce the strain to balance the thermal strain.The last factor is stress relaxation,or equivalently,strain creep.Because of non-uniform temperature distribution,different areas in the part have different time-points of solidification.On the other hand,except for final residual stress,all stresses located at different areas will be released at same time in demolding,which causes the different stress relaxation periods.Generally speaking,the area with longer relaxation period tends to have less stress,vice versa.It is the fundamental reason of the part warpage.

The modeling will be done on the basis of the above discussion.To realize our purpose,the following assumptions are specified:

·The part must be long enough so beam theory can be used.

·The geometry of the cross section does not change too much along the axial(x)direction.In the analysis,the part will be cut into several elements.This means that the curvature in the element can be interpolated.

·The heat does not transfer along the x-direction,but only on the cross section of the part.

·Nowhere there is part movement in the mold,i.e.,the strain being zero everywhere before the ejection.

·Only the packing pressure and thermal stress are considered,and flow-induced stress is neglected.

·The material is isotropic,i.e.,the crystal or fiber orientation is not taken into consideration.

·The twist of the part is not considered.

On the basis of the above assumptions,the procedure to simulate the deformation of the injection-molded strip-like part is as follows.(1)The part is considered as a beam and will be cut into a few elements connected by nodes along the x-direction.The element length is about 5 times the part width,depending on the section shape.If the section varies less,the element is chosen longer.(2)At each node position,the cross section perpendicular to x-axis is taken and the finite element mesh is drawn on it,as shown in Fig.2.Different sections have their own mesh.(3)The thermal analysis is made on all sections.(4)According to the cooling history of each section,the in-mold elastic strain is calculated on the nodal bases,which is further used to get curvature of the beam node.(5)Finally,curvature distribution in each beam element is interpolated and the deflection can be integrated based on the beam theory.

Fig.2.Sample of finite element mesh of a cross section.

3.Temperature Field

According to the assumption,the heat conduction along x-direction is neglected.So the heat transfers only on the y-z plane.Please note that the 1D analysis as in the Hele-Shaw model is not acceptable because the ratio of dimension of the section to the part thickness is not large enough.The end time of the filling stage is taken as starting time of the temperature analysis for all sections.The standard FEM is used in this study.

In our study,there exists no internal heat source and material property is isotropic,thus the 2D heat conduction equation can be simply[26,27]

where ρ is density,c specific heat,k heat conductivity,hinheat transfer coefficient between melt and mold,houtheat transfer coefficient between part and air,Tmeltinitial melt temperature,Tmoldmold temperature and Tairroom temperature.

With the standard FEM formulation[28],the system of equations that gives the discrete solution of Eqs.(1)-(3)can easily be derived:

in which K is stiffness matrix,M the mass matrix and q heat loading coming from the third boundary condition.The detail of derivation can be found in any reference about heat transfer FEM and will not be repeated here.

Using finite difference,Eq.(4)can be rewritten as

In this work,K and M do not change.To speed up the calculation,a constant time step is taken so that the left side matrix just needs to be split once.Although the more accurate computing model might be used in future work,we assume that the part is filled instantly,and mold temperature is constant at this moment.In this assumption,the part starts to cool down synchronously in every cross section.The calculation stops when temperatures in all elements drop to the room temperature.

4.In-mold Strain and Stress

The development of the in-mold strain and stress has a fewstages.In the first stage,the temperature is higher than solidification temperature,and stress and strain are zero.At the melt-solid transform,the packing pressure,if there is,will be solidified in the part to form the initial stress and strain.This is the beginning of the second stage.At the point of packing pressure release,the in-mold stress and elastic strain has a sudden jump,and the third stage begins which ends when the part is ejected.In the second and third stages,the mold imposes constraint on the part so total strain keeps unchanged.The elastic stress and strain develop along with part-cooling to balance the thermal strain.

The difficulty of the in-mold stress/strain calculation is that the material mechanical data keep changing with temperature,which makes the development of the stress/strain in mold very complicated[29,30].To overcome the problem,a special computing model is taken,as shown in Fig.3,which was presented in our previous research[31].For every node in the section mesh,the tensile strain of the part in x direction consists of three ingredients,in which εerepresents the elastic strain that determines final part deformation,εpbased on a dashpot model is the creep or unrecoverable plastic strain,and εhis the thermal strain.The strain starts to grow until the melt temperature drops to transition one.At this point,because of packing pressure,if there is,there is an initial strain in the part:

where E is Young's modulus and ν is Poisson ratio.Before the ejection,all strains should meet the condition:

Fig.3.Computing model for in-mold strain.

Using the same time step as the above temperature calculation,the equations of strain calculation can be written as

where the superscript represents time step,andare zero,α is coefficient of thermal expansion and μ is the dashpot coefficient.When packing completes,the pressure is released and εewill have a sudden jump.In this model,material data E(T)and α(T)can be determined in lab test or approximately with PVT data,and μ(T)can be measured in the method of ref.[32].

5.Out-mold Strain and Stress

When the part is ejected from the mold,all constraints on the part are released immediately,so strain-based condition of Eq.(7)is no longer applicable.Because there is no external force on the part,the resultant force must be zero.The calculation goes in two steps[33].Firstly,only shrinkage is considered.On the basis of the above calculation,the in-mold elastic strains just before the ejection are known for any node on any section.When the part is ejected,the shrinkage will happen to release those strains.Assuming that all the nodes on a section keep coplanar before and after demolding,the average strain ε on the section should meet the requirement:

in which subscript i is related to elements and j to elemental nodes,A is elemental area and superscript 0 for strain refers to one before the ejection.From Eq.(11),can be expressed as

After the ejection and without part-bending,the elastic strain at node k becomes

In addition to the shrinkage,the part also has warpage after the ejection to release the moment.Assuming there is curvature κyaround y axis on the x-z plane at the position of a certain section,it should meet the requirements:

From now on,the part will continue cooling down until room temperature,along with continuous shrinkage and warpage without any constraint.In this stage,the stress in the part becomes so small that the effect of the stress relaxation is negligible.Different from the in mold stress calculation,the elastic strain cannot accumulate.When any little thermally induced strain appears,it must conform to Eq.(11).Similar to the derivation of Eqs.(11)-(17),the elastic strain at node k and curvature can be written as

When the part temperature gets to room temperature everywhere,the above calculation is completed and the curvatures at all section locations are obtained which then are used for the next warpage prediction.If constant elasticity is used,the above computations do not need to be done step by step,but just one time calculation.In this case,superscript n+1 in Eqs.(18)-(24)represents final results,n represents the result at ejection time.Tn+1in Eq.(18)takes air temperature,but Tntakes lower value between the temperature at the ejection time and transition temperature.

6.Deformation

As mentioned above,the strip-like part can be treated as a beam in macro-level.This beam has been cut into a few elements and the curvatures κzand κyat all beam nodes are known by the calculation of Eqs.(6)-(24).Different from general structural analysis,the case here is simpler.According to the theory of structural mechanics,the curvature-deflection equation of the beam can be simply expressed as

where v and w are the deflections in y and z directions respectively.Not losing the generality,only the deflection w is discussed here.To a certain element,assuming that the local coordinates of its ends is 0 and L respectively,and the curvatures are κy0and κyL,the inner curvature in the element can be expressed as

For our problem,there is no certain constraint.To get the solution,a reference point has to be given where the known rotation and deflection are specified.To simplify the calculation,the node close to the middle of the beam is taken as the reference point with zero deflection and rotation.Starting from this node and its adjacent elements,the solutions at its neighbor nodes are obtained and then more nodes are handled.In this way,the deflections on all nodes can be calculated.

7.Case Studies

All the above formulations were coded by the authors using Visual C++.To test the correctness,two practical cases,provided by Shanghai GuoKaiMold Co.,Ltd.,were introduced.The material ABS430 is used for the examples given here.The data to be used in the calculation are as follow:density 1054 kg·m-3;specific heat 1652 J·kg-1·°C-1;heat conductivity 0.18 W·m-1·°C-1;Poisson ratio 0.38;transition temperature 87°C;heat transfer coefficient between melt and mold 25,000 W·m-2·°C-1;heat transfer coefficient between part and air 35 W·m-2·°C-1.Coefficient of thermal expansion and Young's modulus are provided by the following PVT equations:

in which b1sis 9.73×10-4m3·kg-1,b2s2.27×10-7m3·kg-1·°C-1,b3s2.11×108Pa,b4s4.10×10-4°C-1,b590°C,B 5.9×109Pa·s,C 2.5°C and T060°C.The processing controlling parameters are:melt temperature 220°C,mold temperature 60°C,air temperature 25°C,packing pressure is 40 MPa,packing time is 9 s and cooling time is 18 s.In the current code,the constant elastic data is used.

7.1.Chafing strip of car(1)

The chafing strip is shown in Fig.1.Its length is 910 mm.At the middle,its width is 39 mm and thickness 3 mm.The strip section is uniformly reduced from one end to the other.At the end of x=-454 mm,width is 32 mm and at the end of x=454 mm width is 43 mm.Fig.4 shows the sections at x=±310 mm.In this case,a total of 7 sections are taken out to do the analysis which are located at x=0 mm,±155 mm,±310 mm,±454 mm.To ensure the accuracy of calculation,the mesh size is determined according to the section thickness,which is not more than 1/6 of the thickness.

Fig.4.Strip sections in different locations.

The part is produced by an injection machine,and the mold is shown in Fig.5.The real warpage of this product is measured by the gauge in Fig.6.The predicted and real deflection in the z direction is given in Table 1.The deflection in the y direction is too small and meaningless for analysis.

Fig.5.Injection mold.

7.2.Chafing strip of car(2)

Fig.6.Warpage measuring.

Table 1 Warpage prediction(Case 1)

Fig.7.Strip section of Case 2.

Its length is 700 mm and its section geometry is shown in Fig.7.The section changes very little from one end to the other,and it is hard to be found to the naked eye.The difference can be detected by comparing geometrical moment of inertia.At x=-300 mm,the moment is 2.765×10-10m4and at x=300 mm,it becomes 2.523×10-10m4.The part width is about 29 mm and thickness 3 mm.In this example,a total of 5 sections are taken out to do the analysis which are located at x=0 mm,±200 mm,±349 mm.The same measuring method as in Case 1 is used.Again,only the deflection in the z direction is considered and the result is given in Table 2.It can be found that the error in the current case is larger than the first one.The reason is that the constant mold temperature is used in both cases.In fact,the mold temperature is nonuniform.If a part has some ribs,like Case 2,the mold temperature in that area of ribs tends to become higher.If more accurate mold temperature is used in the future work,the error will hopefully be reduced.

Table 2 Warpage prediction(Case 2)

8.Conclusions

A special algorithm to predict the deformation of the injection molded plastic strip-like part was suggested.As mentioned above,the Hele-Shaw model is not suitable to predict the warpage of this kind of parts due to its limitation.On the other hand,the general 3D model is too time consuming and not acceptable for users.In fact,there is no commercial CAE software available to handle this kind of parts up to date.The present work solved the problem to a certain degree.The 3D FEA was replaced by several 2D FEA and one 1DFEA,which enormously reduced the computing time.This two-step FEM was proved to be very simple and efficient,and the predicted warpage was also reasonable.Although most strip-like parts may not be straight and the mold temperature might not be uniform either,the present job provides a new thought to solve the problem.More complicated cases could be handled as well.For any non-straight strip-like part,the section curvature could be obtained as this work demonstrates,and then the part deformation would be easily calculated by FEM used in the structural analysis.This algorithm is believed to be a promising tool for the mold design of strip-like plastic parts.

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