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Modeling warpage in reinforced polymer disks.


The control of warpage in plastic parts is an important consideration in material selection. When a thermal or mechanical load is applied, a part can warp to minimize internal energy. Cooling a semicrystalline polymer part below its crystallization end point is an important thermal load that can cause warpage.

The impact of thermal loads on part warpage varies. The simplest case is the uniform cooling or heating of an isotropic, homogenous part. The distorted shape will be similar to the original, enlarged or reduced in size depending on the direction of the temperature change. Another familiar case is a bimetalic strip consisting of a thin strip of two isotropic homogenous materials with different coefficients of thermal expansion cemented together. Cooling or heating the strip will result in its curling, an effect frequently employed to calibrate thermometers. Similar warpage can occur in a thin polymer part if internal stress development is influenced by a varying cooling rate through its thickness. Another case of interest is the change in curvature of a thin part caused by a difference between the thermal expansion in its plane and through its thickness. This anisotropy can cause shape distortion in thin injection molded box-shaped containers due to induced angle changes at the edges joining faces. Residual stress development and relaxation resulting from these thermal loads and other post-filling packing stresses for injection molded parts have been discussed extensively in the literature (1-8) where polymer models are also described and their predictions compared with experimental data. These references discuss the complex residual stress development during the cooling of an injection molded part and explain why although progress is being made, the prediction of part warpage and shrinkage still remains a challenging and interesting one.

For fiber reinforced injection molded thin plastic parts, often the most dominant cause of part distortion during cooling is in-plane thermal contraction anisotropy; it is focus of this paper. The anisotropy is induced by a preferential alignment of the reinforcing material relative to the flow pattern during mold filling and the lower thermal expansion coefficient of the reinforcement relative to the surrounding polymer. Other effects inducing warpage such as temperature non-uniformities during mold filling and cooling, packing pressure, (degree of polymer crystallization for semicrystalline polymers), also play a role, but are not considered here in the formula derived to predict disk warpage. Without adequate data to quantify some of these more complex polymer phenomena, estimating part warpage from estimates of material anisotropy can be a useful design guide. For example to estimate part warpage, local part anisotropy is sometimes calculated by relating it to orientation of the reinforcing material. Models have been presented in the literature, (9-12), that give estimates of fiber orientation during injection and compression molding, and these models have been employed to deduce part anisotropy and thus relate part warpage to part design (13-17). This approach has been a useful guide in die design even though it neglects a number of stress relaxation effects during cooling. The role of polymer stress relaxation after "solidification" is also neglected in this paper, not to imply these effects can be ignored or that they cannot contribute to part warpage, but to focus instead on part warpage induced from in-plane material anisotropy induced by fiber orientation. A fiat circular disk geometry is chosen to analyze the in-plane expansion anisotropy because of its simplicity and use in appraising the tendency of different materials to induce warpage; and its value in providing focus on the warpage mechanism of interest.

Two disk idealizations are created to analyze the role of in-plane expansion anisotropy and its impact on warpage in injection molded disks. The first idealization is a metal reinforced polymer disk wheel that is examined analytically to develop general warpage formulae relating disk temperature to warpage initiation, and its subsequent increase as the thermal load on the disk is increased. This disk is elastically very anisotropic. The second disk idealization is analyzed numerically and considers the disk's elasticity isotropic, with the usual Young's modulus and Poisson's ratio characterizing its response; the impact of thermal expansion anisotropy induced by fiber orientation within the disk is quantified by taking the thermal expansion in the radial direction different to that in the tangential direction.

The stress pattern within these disks prior to buckling is discussed along with the stress distribution's impact on crack propagation. For simplicity, the analysis assumes a uniform disk temperature during cooling and the impact of polymer yield is not included. The conditions that create saddle warpage [ILLUSTRATION FOR FIGURE 1 OMITTED], often observed in edge-gated fiber reinforced injection molded polymer disks, are analyzed with the two disk idealizations. Also the development of disk cup warpage, observed in center-gated fiber reinforced injection molded disks, is analyzed. The build up of compressive polymer stress in the tangential or hoop direction is shown to give rise to saddle warpage, while the build-up of extensional polymer stress in the same direction is shown to be the dominant cause for a disk warping into a cup shape.

The commercial structural analysis software, ABAQUS (18), provides the basis for the numerical analysis. The analogy developed between the model disks and thin circular edge-gated and center-gated injection molded disks makes the resultant formulae useful for designers estimating property data from warpage measurements. This warpage analogy is explored for an edge-gated disk injection molded with 30% by weight glass filled polyester, where experimental warpage measurements are compared with those calculated from the derived formulae.


The two model disk designs created to mathematically model warpage are now described. The first, a circular polymer disk wheel is useful for analysis of the warpage mechanism and is referred to as the disk wheel. The second model disk, referred to as the continuous disk, develops stress patterns more similar to fiber reinforced injection molded disks.

The disk wheel model [ILLUSTRATION FOR FIGURE 2 OMITTED], designed to minimize the impact of radial stress, is shaped like a circular disk wheel with metal spokes that are contoured into flat strips, laid in the plane of the disk and emanating from a very small rigid circular metal disk placed at the wheel center. The metal spokes are spread evenly around the wheel but cover a small surface area; they are fixed in their angular position due to their central attachment and have high resistance to in-plane extension, compression or shear. The metal spokes are visualized as running through thicker strips of concentric polymer rings that lie in the disk plane and cover most of the wheel surface, but are sufficiently separated to ensure they do not touch during wheel deformation, thus preventing any significant radial stress development within the polymer. The polymer strip thickness, h, is thin compared to its radial width, which is small compared to the disk radius, R. The metal spoke area covered by the polymer is sufficiently thin to ensure no impact on the polymer out-of-plane bending modulus, while the very small spoke area not covered by polymer is sufficiently thick to ensure a uniform out-of-plane bending modulus over the disk surface is a reasonable approximation. The metal radial reinforcement anchors the polymer in place allowing no in-plane radial or shear movement. Out-of-plane bending and in-plane polymer hoop stress development are the two mechanisms the disk has to store strain-energy, and these energies will be analyzed and their impact on disk warpage quantified.

The continuous disk model consists of a circular elastic disk of thickness, h. radius, R. and whose elastic properties are isotropic and described by a Young's modulus, E, and Poisson's ratio, [Sigma]. The radial direction at any point on the disk is along the line connecting the point to the center of the disk and is directed outward from the center. The tangential direction [ILLUSTRATION FOR FIGURE 3 OMITTED] is perpendicular to the radial direction and in the counter clockwise direction looking down on the disk surface. The material thermal expansion coefficient along radial direction at any point, [[Alpha].sub.r], has a different value from the thermal expansion coefficient, [[Alpha].sub.[Theta]], in the tangential direction perpendicular to the radial direction. Thermal expansion anisotropy and its impact on warpage is the focus of this model.


While filling a thin edge-gated injection molded disk cavity with fiber reinforced polymer, a predominant fiber orientation pattern often develops in the flow direction [ILLUSTRATION FOR FIGURE 4 OMITTED]; principally the disk tangential direction (9-12). Thus, the disk's thermal expansion properties can be compared to the continuous disk model with the reinforcement in the tangential direction. The further development of this analogy to fiber reinforced injection molded disks requires considerable simplification of the cooling process during injection molding: an "instant freeze" model is employed where the impact of temperature differences through the part thickness and over its surface are neglected, and cooling is assumed to take place fast enough to dwarf the impact of stress relaxation below a critical temperature; for the edge-gated glass filled polyester disk selected for model comparison, the critical freeze temperature will be called the "crystallization end temperature." In this way, the focus is on the impact of material anisotropy; a more refined model for polymer cooling is circumvented and the consideration of packing pressure after mold filling is avoided. There is an extensive literature evaluating the impact of cooling and packing during injection molding, (2-4, 6, 7). Warpage after disk ejection from the mold is estimated with the formulae derived from the continuous disk model where the coefficient of expansion in the flow direction during mold filling is set equal to the model tangential thermal expansion coefficient, and the thermal expansion perpendicular to the flow direction set equal to the model radial thermal expansion. The center-gated disk develops a divergent radial flow pattern, tending to sometimes align fibers perpendicular to the flow direction; then disk warpage will have a saddle shape. In other instances, the fiber motion during filling is governed mostly by wall shear producing an orientation primarily in the radial direction and a cup warpage shape. The continuous disk model, with a radial thermal expansion coefficient set equal to the thermal expansion in the flow direction, and the tangential thermal expansion coefficient set to the thermal expansion perpendicular to the flow direction is a model analog for these center-gated disks.

Thus warpage formulae derived for the continuous model disk can be applied to injection molded disks with modification of the formulae constants. When augmented with experimental warpage data, the warpage formulae can be employed to infer "effective" thermal expansion properties for fiber reinforced injection molded parts. Such a formulae application to a fiber reinforced edge-gated injection molded disk is discussed later. Also, although not pursued in this paper, the formulae help further clarify statistical correlations reported in (14-16). where the fiber orientation and warpage of a center-gated injection molded disk are analyzed in the manner described in the Introduction and Overview section, and conditions leading to cup or saddle warpage are related to fiber orientation and accompanying disk anisotropy for the polymer disk.


The Disk Wheel

When the disk wheel model is heated or cooled, the annular polymer strips covering its surface attempt to expand or contract. The polymer strips are separated from their neighbors by a small distance allowing them to relax their radial stress to essentially zero. However, they are anchored to the radial metal spokes, designed to constrain in-plane movement. The polymer strips width-to-length ratio that gives them beam-like properties, thus,

[T.sub.[Theta][Theta]] = -E([[Alpha].sub.0] - [[Alpha].sub.r])[Delta]T (1)

relates the polymer hoop stress, [T.sub.[Theta][Theta]], to the temperature change, [Delta]T (measured from the thermally unstressed disk temperature), polymer and metal reinforcement expansion coefficients, [[Alpha].sub.0] and [[Alpha].sub.r] respectively, and polymer Young's modulus, E. The impact of the metal thermal expansion could be ignored sInce it is much lower than the polymer's thermal expansion.

The Continuous Disk.

Pre-warpage stress buildup in the continuous disk model varies over its surface and requires the equilibrium equations from classical elasticity and plate theory to describe its development; (19, 20) and the Appendix. It can then be shown that the disk radial and hoop stress, [T.sub.rr], [T.sub.[Theta][Theta]], change with radial position, r, on the disk surface as follows:

[T.sub.rr] = E [Delta]T ([[Alpha].sub.r] - [[Alpha].sub.[Theta]])/2 ln(r/R) (2)

[T.sub.[Theta][Theta]] = E [Delta]T ([[Alpha].sub.r] - [[Alpha].sub.[Theta]])/2 (ln(r/R) + 1) (3)

with the natural logarithm function denoted by ln. The disk radius is R; and the temperature change, [Delta]T, is measured from the thermally unstressed disk temperature. The thermal expansion coefficients In the radial and tangential directions are [[Alpha].sub.r] and [[Alpha].sub.[Theta]], respectively.

For simplicity, these stress formulae assume thermal expansion coefficients that are Independent of temperature and a polymer that does not yield. In practice for polymer parts, an integral over the temperature range of interest would be required to account for thermal expansion and a correction needed if material yielding occurs. If the polymer part has changing orientation through its thickness (21), a thermal expansion averaged through the part thickness is required and the derived expressions then yield average stresses through the part cross section. The formulae yield infinite stress at the disk center and thus should not be applied there. The stress singularity can be interpreted physically by considering the same disk but with a small circular cutout of radius b, at its center. The circular cutout gives rise to zero radial stress at radial positions r = b, and it can be shown analytically by considering the disk equilibrium equations that its impact to the remaining disk stress is confined close to positions at r = b. As the inner radius decreases to zero, the stress in the disk approaches the solution given above. Thus the above solution is a useful mathematical form to analyze the stress pattern in an axially symmetric disk prior to buckling. For example, if the coefficient of thermal expansion in the radial direction is greater than in the tangential direction, the stress at the disk center could cause cracking during cooling. This type of cracking is frequently seen in the center of tree branches after cutting, where the contraction mechanism is governed by moisture loss. Also when freeze-off occurs at the gate of an injection mold after filling the mold cavity with a fiber reinforced polymer, the formulae of Eqs 2 and 3, although derived with the isotropic elasticity assumption, can provide some understanding of the stress buildup around the gate.

Crack Susceptibility in the Continuous Disk

Consider a polymer disk with fiber reinforcement aligned mostly in the tangential direction. The reinforcement lowers the disk expansion along the fiber length (22), yielding a positive disk anisotropy, [[Alpha].sub.r] - [[Alpha].sub.[Theta]]. Thus for disk cooling, [Delta]T in the previous formulae is negative and

[[Theta].sub.[Theta][Theta]] [less than] 0 at r = R (5)

or the hoop stress at the outer radius of the disk is compressive, discouraging the growth of crack in this area.

Moving towards the center of the disk, the radial stress grows positive, indicating the polymer is being stretched in the radial direction, and thus tending to promote crack propagation in the tangential direction if the material was defective. The hoop stress changes sign at radial position, r = R/e, where e = 2.718, indicating an increased tendency for radial crack propagation towards the disk center.

For a polymer disk with the fibers predominantly orientated in the radial direction, the stress pattern developed has the opposite sign to that described above. Cooling inhibits crack growth towards the disk center. However, a crack initiated at the outer boundary of the disk at r = R, could tend to propagate into a radial position, r = R/e; and beyond due to the resultant outer disk stress relaxation.


A thin elastic disk will warp if its strain-energy in the warped state is less than if it were constrained to stay flat. When a disk with thermal expansion anisotropy changes temperature, significant strain-energy can develop within it because of in-plane compression or extension on the disk mid-plane. Plate theory will be employed to analyze the impact of this material compression or extension and will show when the strain-energy increases beyond a certain value, disk warpage results. Linearly elastic disk behavior is assumed while recognizing that effects such as stress relaxation after polymer solidification would require adjustments in this purely elastic approach.

Classical plate theory of thin structures shows how the disk strain-energy can be uncoupled into two components: in-plane extension-compression energy, calculated by noting the relative movement of points in the central plane of the disk (an imaginary plane that halves the disk into two thinner disks); and out-of-plane bending energy that causes curvature changes on the disk surface, but no relative movement of neighboring points in the disk central plane; (19, 20) and the Appendix. This strain-energy decomposition is a key to understanding the warpage mechanism; it will be applied to the disk wheel model where the rate of increase in bending energy with warpage remains constant, but the rate of reduction of in-plane extension-compression energy with possible warpage increases with "temperature load." Thus a critical temperature is reached where if the disk warps, lower strain-energy results. The relation of this critical temperature to disk geometry is determined for the disk wheel and continuous disk models.

After warpage commences, its increase with temperature load is governed by the disk expansion anisotropy, [[Alpha].sub.r] - [[Alpha].sub.[Theta]], but also by how warpage reduces in-plane strain over the disk surface; a more uniform release promotes faster warpage. This will be demonstrated analytically for the disk wheel model where less tangential restraint on polymer strip movement is shown to substantially increase saddle warpage growth.


Saddle Warpage

The saddle shape the disk deforms to as its temperature is increased will be approximated by the equation:

Z(R, [Theta]) = A [r.sup.2] sin(2[Theta]) (6)

with the out-of-plane movement, z, varying with the cylindrical position coordinate on the disk surface, (r, [Theta]) [ILLUSTRATION FOR FIGURE 1 OMITTED]. The unknown constant, A, determines the magnitude of the disk warpage and is employed in this section to derive saddle warpage formulae. In practice, although the warped saddle disk shape does not always take this symmetric form, an examination of disks warped into a saddle shape shows they can be adjusted into this shape with relatively small external forces, indicating this expression is a reasonable one to estimate warpage employing the strain-energy minimization argument to be described below. [The term given in Eq 6 can also be considered the first of a series of terms in a Galerkin type expansion, (23), approximating the shape of the disk. Note, constraining forces on the disk during the temperature rise can induce higher order warpage modes in exceptionally thin parts, complicating the analysis and thus not considered below.] The maximum out of plane movement of the disk, W, is given by

W = A [R.sup.2], (7)

where the disk radius is given as R.

Plate theory formulae relate disk bending energy to disk curvatures; (19) and the Appendix. They show for the saddle shape bending mode described, the bending energy per unit area of disk surface, [U.sub.B], does not vary over the surface and is given by:

[U.sub.B] = 4[D.sub.B] (1 - [Sigma]) [A.sup.2] (8)

with the disk bending modulus related to the disk Young's modulus, E, Poisson's ratio, [Sigma], and disk thickness, h, as follows:

[D.sub.B] = E [h.sup.3]/(12(1 - [[Sigma].sup.2])) (9)

Thus the bending energy depends on the cube of the disk thickness.

Disk warpage will reduce the thermally induced in-plane strain-energy as follows. The hoop strain, [e.sub.[Theta][Theta]], accompanying the disk bending is derived from the geometry of the warped disk and is:

[e.sub.[Theta][Theta]] = B([Theta]) [(Ar).sup.2] (10)


B([Theta]) = 2(1 - 4/3 [sin.sup.2](2[Theta])) (11)

The hoop strain has mostly a positive value indicating it will on average relax the thermally induced compressive hoop strain except in areas with angular positions near [Theta] = [+ or -] 45[degrees] and [+ or -] 135 [degrees]. Relaxing the constraining action of metal spokes on the annular polymer strips covering its surface will be explored by allowing them to slide in the tangential direction, thus equalizing the stress over their length. Then,

B([Theta]) = 2/3 (12)

In either case, the hoop stress, [T.sub.[Theta][Theta]], for the warped disk is:

[T.sub.[Theta][Theta]] = E(([[Alpha].sub.r] - [[Alpha].sub.[Theta]])[Delta]T + B([Theta])[(Ar).sup.2]) (13)

where the thermal expansion anisotropy, [[Alpha].sub.r] - [[Alpha].sub.[Theta]], is negative and thus heating induces polymer compression. The temperature change measured from the thermally unstressed disk temperature is given as [Delta]T.

The resultant disk mid-plane strain-energy per unit area of disk surface, [U.sub.ext], varies linearly with disk thickness, h, and is:

[Mathematical Expression Omitted] (14)

The total disk strain-energy, U(total), is the sum of the in-plane strain-energy and bending energy,

[Mathematical Expression Omitted] (15)

the integral of the strain-energy per unit area over the disk surface. The warpage parameter, A, determining warpage magnitude, depends on the disk temperature, T. Thus,

A = A(T) (16)

and the change in disk strain-energy, U(total), with temperature, T, is:

[Mathematical Expression Omitted] (17)

[Mathematical Expression Omitted] (18)

where the first integral is the rate of increase in strain-energy from the thermal expansion of the disk, and

[Mathematical Expression Omitted] (19)

is the rate of change with warpage of disk in-plane strain-energy plus bending energy which are directly related to the warpage motion.

The fundamental law governing the motion of all elastic structures is that deformation occurs so as to minimize the strain-energy stored. Thus, as the disk temperature is increased, it will deform to minimize the increase in strain-energy, and thus the term [I.sub.1] d [A.sup.2]/dT Eq 18. Thus for a stable disk response to temperature increase, either:

d[A.sup.2]/dT = 0 and [I.sub.1] [greater than] 0 (20)


A [greater than] 0 and [I.sub.1] = 0 (21)

Thus for warpage to commence, the energy integral [I.sub.1], must decrease to zero as temperature is increased by a critical temperature, [Delta][T.sub.min], given by the equation:

[I.sub.1] = 0. (22)


[Mathematical Expression Omitted] (23)

= [(h/R).sup.2] (1/([[Alpha].sub.0] - [[Alpha].sub.r])) (1/(1 + [Sigma])) (24)

and after warpage starts,

d[I.sub.1]/dT = 0 (25)


[Mathematical Expression Omitted] (26)

= 3/4 ([[Alpha].sub.[Theta]] - [[Alpha].sub.r]) (27)

provides an expression of the rate of increase of warpage with temperature for the more rigid disk wheel model.

For the more relaxed wheel disk discussed earlier in the section: the annular polymer strips covering the wheel can slide to provide uniform strain over their length; the metal spokes still prevent any in-plane radial movement. Then, although the minimum temperature to initiate warpage is the same as the unmodified disk wheel model, subsequent warpage growth is greater for the more relaxed wheel with

[R.sup.2] d[A.sup.2]/dT = 9/4 ([[Alpha].sub.[Theta]] - [[Alpha].sub.r]) (28)

To summarize, the saddle warpage equation for the disk wheel relates the square of the out-of-plane disk movement, [W.sup.2], with disk temperature change, [Delta]T, measured from the thermally unstressed disk temperature, as follows:

[W.sup.2] = C ([[Alpha].sub.[Theta]] - [[Alpha].sub.r]) ([Delta]T - [Delta][T.sub.min]) [R.sup.2] (29)

Thus, warpage is proportional to the square root of the thermal expansion anisotropy, ([[Alpha].sub.[Theta]] - [[Alpha].sub.r]) ([Delta]T-[Delta][T.sub.min]). The disk constant, C, governing the growth of warpage with temperature, is equal to 3/4 for the disk wheel with the concentric annular polymer strips rigidly bonded to the radial metal spokes; and equal to 9/4 for the more relaxed disk wheel model.

The minimum temperature rise, measured from the thermally unstressed disk temperature, required to warp the disk is:

[Delta][T.sub.min] = [(h/R).sup.2] (1/([[Alpha].sub.[Theta]] - [[Alpha].sub.r])) 1/(1 + [Sigma]) (30)

It is proportional to the square of the ratio of the disk thickness to diameter, and inversely proportional to the disk thermal expansion anisotropy, [[Alpha].sub.[Theta]] - [[Alpha].sub.r]. The inclusion of the Poisson's ratio, [Sigma], is an outcome of the assumed classical dependence of the disk bending on Poisson's ratio. Note, to simplify the derivation of warpage formulae while focusing on its key features, the disk's polymer stiffness (Young's modulus), is assumed independent of temperature, However, a similar argument leads to the same (saddle and cup) warpage formulae without this stiffness assumption; and, similar warpage formulae also apply if a layered structure, symmetric about a central plane, with each layer having the same general form as the disk wheel model, is considered.

Cup Warpage

If cooled, the disk wheel model will warp into a cup shape [ILLUSTRATION FOR FIGURE 1 OMITTED], due to positive hoop stress development in the polymer strips. A cup shape of the following form will be assumed:

z(r) = A[r.sup.2] (31)

with out-of-plane movement, z, varying only with the radial distance, r, from the disk center. The resultant maximum out-of-plane movement, W, is:

W = A[R.sup.2] (32)

where the disk radius is given as R. The bending energy per unit disk surface area, caused by this warpage is:

[U.sub.B] = (E/(3(1 - [Sigma]))) [h.sup.3][A.sup.2] (33)

and is a constant over the disk surface. The hoop stress on the disk mid-plane accompanying the warpage is:

[T.sub.[Theta][Theta]] = E(([[Alpha].sub.r] - [[Alpha].sub.[Theta]])[Delta]T - (2/3)([Ar.sup.2])) (34)

Thus the disk geometric function, B([Theta]), describing in-plane deformation and employed to analyze the saddle warpage, is for the cup shape:

B([Theta]) = -2/3 (35)

Employing the same disk strain-energy argument as was applied for the saddle warpage, it follows that cooling is required for cup warpage:

[Delta][T.sub.min] = -[(h/R).sup.2](1/([[Alpha].sub.[Theta]] - [[Alpha].sub.r])) 1/(1 - [Sigma]) (36)

is negative and is the minimum cooling required, if measured from the thermally unstressed disk temperature. Subsequent warpage development can be described by the equation:

[W.sup.2] = -(9/4)([[Alpha].sub.[Theta]] - [[Alpha].sub.r])([Delta]T - [Delta][T.sub.min])[R.sup.2] (37)

where [Delta]T ([less than] 0) is the temperature change measured from the thermally unstressed disk wheel temperature.


Saddle Warpage

The mechanism causing warpage in the continuous disk model is very similar to that discussed previously for the disk wheel model. The commercial structural analysis software, ABAQUS, was employed to do the finite element numerical analysis, FEA. In contrast to the analytical approach applied to the disk wheel, FEA can be applied to an arbitrarily shaped part. It is implemented for the continuous disk by subdividing its surface into quadrilateral "shell" elements. The deformation within each quadrilateral is described in terms of the displacement and surface curvature at its vertices, mid-face and center nodes; and the element strain-energy then follows from plate theory; (18-20) and the Appendix. The total disk strain-energy is calculated by adding individual quadrilateral contributions over the disk surface. The principle of virtual work (20) is employed to compute the disk deformation for the temperature change of interest; it is the deformation that yields the minimum disk strain-energy. ABAQUS assembles and solves the resultant large set of nonlinear equations for the disk warped shape. In contrast, the previous analytical approach when applied to the disk wheel model provided the two uncoupled linear Equations 22 and 25, equivalent to this large set of nonlinear equations.

The result of a series of numerical calculations at different disk temperature, property and geometry values can be summarized in a similar form to the disk wheel warpage result as follows. The disk warpage, W, is related to the disk temperature change, [Delta]T, by the formulae:

[W.sup.2] = 1.36([[Alpha].sub.[Theta]] - [[alpha].sub.r]) ([Delta]T - [Delta][T.sub.min]) [R.sup.2] (38)


[Delta][T.sub.min] = [(h/R).sup.2] (1.93-0.81 [Sigma])(1/([[Alpha].sub.[Theta]] - [[Alpha].sub.r])) (39)

is the minimum temperature change required to initiate saddle warpage; all temperatures changes are measured relative to the thermally unstressed disk temperature. The constants in Equations 38 and 39 fit the numerical results with a high degree of correlation, except for the constant 1.36, which although it showed no apparent dependence on Poisson's ratio, did vary by about 5% over the zero to 0.5 Poisson's ratio range selected. If the reinforcement is in the tangential direction, [[Alpha].sub.[Theta]] - [[Alpha].sub.r] is negative and then cooling induces the saddle shape with negative [Delta]T and [Delta][T.sub.min]. If the reinforcement is in the radial direction, [[Alpha].sub.[Theta]] - [[Alpha].sub.r] is positive and heating induces saddle warpage. Note it requires a larger temperature load to initiate warpage for the continuous model than for the disk wheel model. This is to be expected because the earlier pre-warpage stress analysis shows how the continuous model develops half the hoop stress at its outer disk periphery. Subsequent warpage growth with temperature change is faster here than for the disk model with the polymer strips fully attached to the metal radial spokes, but slower here if the polymer strips in the disk wheel model are allowed to slide in the tangential direction as discussed in previous sections. Thus continuous model's warpage growth is about half way between the two conditions examined for the disk wheel model warpage.

Cup Warpage

The formulae for cup warpage have a similar form. In the same manner, results from the ABAQUS structural analysis software are summarized as follows. The maximum out-of-plane warpage, W, is given by the formulae:

[W.sup.2] = 1.64([[Alpha].sub.r] - [[Alpha].sub.[Theta]]) ([Delta]T-[Delta][T.sub.min]) [R.sup.2] (40)


[Delta][T.sub.min] = [(h/R).sup.2] (1.64 + 0.4 [Sigma] + 3.4[[Sigma].sup.2])(1/([[Alpha].sub.r] - [[Alpha].sub.[Theta]])) (41)

Like the saddle warpage formulae, the fitted correlation between these formulae and the numerically calculated warpage is very high. If the reinforcement is in the radial direction, [[Alpha].sub.r] - [[Alpha].sub.[Theta]] is negative and cooling induces the cup shape with [Delta]T and [Delta][T.sub.min] negative. The reverse is true if [[Alpha].sub.r] - [[Alpha].sub.[Theta]] is positive with a temperature increase needed for cup warpage. Similar to the saddle warpage case. cup warpage for the continuous disk model requires a greater temperature load to initiate it than the disk wheel model. In addition, the cup warpage for the continuous model grows slower with increasing temperature load than for the disk wheel model.


The continuous disk model assumes the reinforced polymer has elastic isotropy and examines the impact of thermal expansion anisotropy, However, fiber reinforced injection molded parts are stiffer in the direction the reinforcing fibers predominantly align, and the impact of this mechanical anisotropy will be briefly addressed since the derived warpage formulae have not considered its impact on warpage.

It is first noted that the two model cases considered, the disk wheel model and the continuous disk model, are mechanically quite different. The disk wheel with its metal spokes is designed to have infinite extensional stiffness in the radial direction. and absorbs all its elastic energy through isotropic bending and polymer compression or extension in the hoop direction. On the other hand, the continuous disk model with isotropic elastic properties, can develop internal elastic energy from unrelaxed strains in either the radial or hoop direction on the disk. Despite this difference in mechanical behavior, similar warpage formulae apply.

Fiber reinforced injection molded disks fall somewhere between the latter two mechanical extremes. A numerical exploration of the impact of elastic anisotropy is thus relevant; a complete examination is outside the scope of this paper but the claim that its impact on warpage for fiber orientation injection molded parts is secondary to thermal expansion, needs more substantiation. Thus, an assessment of the accuracy of assuming isotropic elasticity when predicting warpage of a reinforced polymer disk is now described. The approach adopted estimates elastic properties of a composite disk; these properties and the disk thermal expansion anisotropy are employed to numerically calculate warpage; the result is then compared to the warpage calculated with isotropic elasticity and similar disk thermal expansion anisotropy.

The properties of a 30% by weight fiber reinforced polyester polymer injected into a plaque, as shown in Fig. 5, are taken to guide this exploration of mechanical anisotropy. Three point bending tests on these plaques indicate the stiffness ([approximately]8840 MPa), in the predominant direction the fibers align, (flow direction). is about twice that in the direction perpendicular to the fibers ([approximately] 4420 MPa). A fundamental mathematical model developed at the Center for Composite Materials at the University of Delaware is employed to estimate composite elastic properties from those of the underlying polymer resin and fiber reinforcement (27). The composite selected is the same glass reinforced polyester blend from the molded plaque, Fig. 5, and measured elastic properties of the plaque composite discussed above guide the selection of fiber dimensions and orientation taken in the model to yield stiffness relatively close to the measured values reported above. A slightly more pronounced elastic anisotropy is modeled since the goal is to determine its impact on warpage. Thus, the polymer and reinforcement properties yield composite elastic properties reported in the Table 1. These properties are employed to numerically predict saddle warp age of an orthotropic elastic continuous disk model and the result is compared with warpage calculated for the isotropic elastic continuous disk model. The thermal expansion anisotropy for both disks is assumed equal with thermal expansion in the radial direction (= 50.9 x [10.sup.-6]/[degrees]C), twice that in the disk hoop direction (= 26.5 x [10.sup.-6]/[degrees]C), In addition, both disks have identical geometric dimensions with diameter 101.6 mm, and 1.59 mm thickness.
Table 1. Composite Elastic Properties.

Property (MPa) Polyester Glass Composite

Radial (1) Young's modulus 2689 72,387 4,674
Hoop (2) Young's modulus 2689 72,387 10,410
Perpendicular Young's modulus 2689 72,387 4,240
2, 3 Shear modulus 1034 30,196 1,613
1, 3 Shear modulus 1034 30,196 1,510
1, 2 Shear modulus 1034 30,196 2,778
1, 2 Poisson's ratio 0.3 0.2 0.447
1, 3 Poisson's ratio 0.3 0.2 0.324
2, 3 Poisson's ratio 0.3 0.2 0.214

The result comparing the magnitude of saddle warpage for the two disks is displayed graphically in Fig. 6. The temperature selected for the development of internal stress is 205 [degrees] C; the minimum cooling load required to induce warpage (67.5 [degrees] C versus 75 [degrees] C) is about 10% higher for the anisotropic elastic disk. With further disk cooling, elastic anisotropy has little impact on the rate of increase of disk warpage with temperature. Thus, in this case neglecting elastic anisotropy contributes to a 3% higher warpage prediction at 25 [degrees] C, a not very significant impact.

The properties in the above table are now employed to investigate how disk cup warpage is impacted by them during disk cooling. However, the properties in the disk radial and hoop directions are interchanged to induce cup warpage. The result displaying cup warpage under these conditions is shown in Fig. 7 along with the corresponding warpage calculated with disk elastic isotropy. The minimum cooling load required to induce warpage (77 [degrees] C versus 67 [degrees] C) is about 14% lower for the anisotropic elastic disk (different from the saddle result where elastic anisotropy retarded warpage development). However, similar to the saddle warpage result, elastic anisotropy has little impact on the rate of increase of disk warpage with further disk cooling. Neglecting elastic anisotropy contributes to a 9% lower warpage prediction at 25 [degrees] C (3.2 mm versus 3.5 mm), a change greater than the saddle result, but still secondary to the impact of thermal expansion anisotropy. In this sense, for the purposes of evaluating the impact of in-plane anisotropy on warpage during cooling, elastic anisotropy although not always negligible, plays a secondary role when compared with thermal expansion anisotropy. Also, it is noted that changing the isotropic polymer elastic properties employed to calculate isotropic warpage by decreasing Poisson's ratio from 0.3 to 0.1 has almost an identical impact on saddle and cup disk warpage as the pronounced elastic anisotropy considered in this section. The emphasis in this section has been on warpage in reinforced injection molded disks and thus the impact of elastic anisotropy on warpage during heating has not been considered but it is noted the warpage initiation temperature rise, [Delta][T.sub.min], differs from that presented here for disk cooling.


The derived warpage formulae for the disk wheel and continuous disk models will now be applied to analyze warpage of a 101.6 mm diameter, 1.59 mm thick, 30% by weight glass reinforced edge-gated injection molded polyester disk. The flow pattern in the mold immediately before it is filled is shown schematically in Fig. 4. After ejection from the mold, the disk warps into a saddle shape similar to that shown in Fig. 1. When the molded polymer disk is slowly heated in an oven from room temperature to 132 [degrees] C, warpage decreases as shown in Fig. 8. Moreover, this measured warpage decrease provides an opportunity to calculate formulae constants in the previously derived warpage formulae.

Samples from a 76.2 mm x 127 mm, 1.59 mm thick plaque [ILLUSTRATION FOR FIGURE 5 OMITTED], molded from the same fiber reinforced polyester and filled at about the same rate as the disk mold (mold fill time 0.4 s), provide (with an Orton Dilatometer, 23 [degrees] C to 130 [degrees] C) the following experimental estimates of material thermal expansion parallel, [[Alpha].sub.[parallel]], and perpendicular, [[Alpha].sub.[perpendicular], to the flow direction,

[[Alpha].sub.[parallel] = 26.5 X [10.sup.-6]/[degrees] C

[[Alpha].sub.[perpendicular]] = 50.9 X [10.sup.-6]/[degrees] C

Thus, the experimental data from Fig. 8 yields warpage, W:

[W.sup.2] = 1.24 ([[Alpha].sub.[parallel]] - [[Alpha].sub.[parallel]]) [absolute value of [Delta]T] [R.sup.2] (42)

where [Delta]T is the difference between the disk temperature and the critical temperature, [T.sub.crit], the disk first displays a warpage increase during cooldown. The graph from Fig. 8 provides the estimate, [T.sub.crit] = 136 [degrees] C. An estimate of the crystallization end temperature for this semi-crystalline composite in the injection mold is taken at 205 [degrees] C, (1, 24-26). Thus, an estimate of the minimum temperature change below the crystallization end temperature required to initiate a warpage increase, [Delta][T.sub.min], is 69 [degrees] C, yielding the expression:

[Delta][T.sub.min] = -1.73 [(h/R).sup.2] (1/([[Alpha].sub.[perpendicular]] - [[Alpha].sub.[parallel]])) (43)

The experimentally derived formulae constants in Eqs 42 and 43 are quite close to the earlier theoretically derived constants for the continuous disk model. Care should be exercised in employing the formulae, particularly to predict warpage less than the disk thickness where the impact other residual stresses (see Introduction) can dominate. For example, the above formulae would imply zero warpage at room temperature for a disk molded from the same material but with thickness, 3.2 mm, twice the value of the disk considered above. However, a small warpage of about 0.4 mm is observed in such a disk, a result that would be considered unacceptable for certain applications.


In-plane thermal expansion anisotropy is a frequent cause for warpage in reinforced polymer parts. This mechanism has been analyzed for axially symmetric circular disks. A minimum disk temperature change is required to initiate warpage and formulae are developed to relate this critical temperature change to disk geometry and expansion anisotropy. All temperature changes are measured relative a thermally unstressed disk temperature. The general form for subsequent warpage growth is derived for two model axially symmetric disks. The analogy developed to center-gated and edge-gated fiber reinforced injection molded disks recommends similar warpage formulae of the following form apply when these disks are ejected from the mold:

[W.sup.2] = F([[Alpha].sub.[perpendicular]] - [[Alpha].sub.[parallel]]) [absolute value of [Delta]T - [Delta][T.sub.min]] [R.sup.2] (44)


[Delta][T.sub.min] = -G[(h/R).sup.2] (1/([[Alpha].sub.[perpendicular]] - [[Alpha].sub.[parallel]])) (45)

relating out-of-plane disk warpage, W, with the amount of disk cooling, [Delta]T, below the temperature the polymer can develop thermal stresses. Both the temperature drop, [Delta]T, and the minimum temperature drop to initiate warpage, [Delta][T.sub.min], are negative. The disk constants, F and G, have values of order unity and can be determined experimentally. Their value will depend on whether the disk is edge-gated or center-gated, and whether it warps to a saddle or cup (as determined by the value of [[Alpha].sub.[perpendicular]] - [[Alpha].sub.[parallel]]); within certain classes of reinforced polymers, it is expected the constants will have a small dependence on material composition. As a guide, substantial elastic anisotropy in axially symmetric disks is shown to change the constant, G: a 10% increase for saddle warpage and a 15% decrease for cup warpage; little change appeared in the constant, F, a useful result for estimating [[Alpha].sub.[perpendicular]] - [[Alpha].sub.[parallel]] from warpage measurements. The disk expansion anisotropy, [[Alpha].sub.[perpendicular]] - [[Alpha].sub.[parallel]] or difference between the disk thermal expansion coefficient perpendicular to the flow direction and along the flow direction, is the important disk material property. Care should be exercised in employing the formulae if the warpage is less than the disk thickness because the impact of other warpage mechanisms not included in their derivation may then play a more dominant role. An example of a fiber glass reinforced edge-gated injection molded polyester disk is discussed to demonstrate the applicability of the formulae.



The equilibrium equations (10), at any point on the axially symmetric continuous disk model, relate stresses in the radial and tangential directions, [T.sub.rr] and [T.sub.[Theta][Theta]], in the following way:

d[T.sub.rr]/dr + [T.sub.rr] - [T.sub.[Theta][Theta]]/r = 0 (46)

In addition, the radial and tangential strains, [e.sub.rr] and [e.sub.[Theta][Theta]], are related to the stresses, and temperature change, [Delta]T, with the linear elastic equations:

[e.sub.[Theta][Theta]] = ([T.sub.[Theta][Theta]] - [Sigma] [T.sub.rr])/E + [[Alpha].sub.[Theta][Delta]T

[e.sub.rr] = ([T.sub.rr] - [Sigma] [T.sub.[Theta][Theta]])/E + [[Alpha].sub.r][Delta]T (47)

The solution to these equations with zero radial stress at the disk outer boundary, at radius, R, then yields Eqs 2 and 3 given earlier, describing pre-warpage stress in the continuous disk model.

Consider a small surface patch on a thin plate. Set up a rectangular x-y-z coordinate system on it, with the z-axis perpendicular to the surface. The patch center-plane divides its thickness in half in the z-direction and the plate is assumed symmetric about this plane. Let (u,v,w) be the subsequent average displacement through the plate thickness of points in the x-y-z directions. Let [e.sub.xx] and [e.sub.yy] be the extension per unit length from the unstressed state within the patch in the x and y-directions respectively. Similarly, let [e.sub.xy] denote the shear from its unstressed state. Then the strains [e.sub.xx], [e.sub.yy] and [e.sub.xy] are related to patch displacement and the temperature change, through the relations:

[e.sub.xx] = [Delta]u/[Delta]x + 1/2 [([Delta]w/[Delta]x).sup.2] - [[Alpha].sub.x][Delta]T

[e.sub.yy] = [Delta]v/[Delta]y + 1/2 [([Delta]w/[Delta]y).sup.2] - [[Alpha].sub.y][Delta]T

[e.sub.xy] = 1/2([Delta]v/[Delta]x + [Delta]u/[Delta]y + [Delta]w/[Delta]x [Delta]w/[Delta]y) - [[Alpha].sub.xy][Delta]T (48)

where [[Alpha].sub.x], [[Alpha].sub.y] and [[Alpha].sub.xy] denote the thermal expansions and thermal shear coefficients averaged through the plate patch thickness. In addition, small changes in the patch surface curvature [[Kappa].sub.xx], [[Kappa].sub.yy] and twist, [[Kappa].sub.xy], are related to its displacement through the expressions:

[[Kappa].sub.xx] = [[Delta].sup.2]w/[Delta][x.sup.2], [[Kappa].sub.yy] = [[Delta].sup.2]w/[Delta][y.sup.2], [[Kappa].sub.xy] = [[Delta].sup.2]w/[Delta]x[Delta]y (49)

Strain-energy within a plate patch develops as it is deformed or heated. For a thin shell symmetric about its mid-plane and undergoing small deformations, this energy per unit surface area, U, uncouples into the sum of two parts: extensional in-plane strain-energy and flexural strain-energy (19). In an isotropic elastic plate patch with thermal expansion anisotropy, the energy decomposition takes the following form:

U = [D.sub.E]/2 ([([e.sub.xx] + [e.sub.yy]).sup.2] - 2(1-[Sigma])([e.sub.xx][e.sub.yy] - [[e.sup.2].sub.xy]))

+ [D.sub.B]/2([([[Kappa].sub.xx] + [[Kappa].sub.yy]).sup.2] - 2(1 - [Sigma]) ([[Kappa].sub.xx] [[Kappa].sub.yy] - [[[Kappa].sup.2].sub.xy])) (50)

with the plate bending modulus, [D.sub.B], related to its thickness, h, and elastic properties, as given in Eq 9; and the plate extensional modulus, [D.sub.E], (= 12 [D.sub.B]/[h.sup.2]). For injection molded parts, the strain-energy decomposition still holds but its form is more complicated to reflect the reinforcement alignment change with distance from the part central plane: the calculation of bending modulus, [D.sub.B], and its anisotropy should reflect this with customary weighting the elastic properties with distance from the central plane; while the calculation of extensional modulus, [D.sub.E], with its anisotropy, remains a thickness average result.


The author would like to express his appreciation for several fruitful collaborations he had in the 1980s that led him to develop fiber orientation with companion polymer property models applicable to complex shaped injection molded parts at DuPont. These collaborations were with A. E. Hirsch and P. W. Gilmour at DuPont; K. K. Wang and many others at Cornell University Injection Molding Program; C. L. Tucker at the University of Illinois; R. L. McCullough and J. W. Gillespie at the Center for Composite Materials at the University of Delaware. The models make possible warpage calculations for complex shaped injection molded parts; later successfully demonstrated by H. Kikuchi at DuPont Japan.


h - Disk thickness.

R - Disk radius.

r, [Theta] - Polar coordinates of a point on the disk with origin at the disk center.

z(r, [Theta]) - Out-of-plane warpage movement of disk surface.

A - Parameter determining the warpage magnitude.

W - Maximum out-of-plane movement caused by the warpage of a circular disk = A [R.sup.2].

[e.sub.rr], [e.sub.[Theta][Theta]] - Disk mid-plane strain in the radial and tangential direction.

B([Theta]) - Function related to mid-plane strain: [e.sub.[Theta][Theta]] = [A.sup.2][r.sup.2]B([Theta]).

[integral] dS - Surface integral over the disk's circular face.

[T.sub.[Theta][Theta]] - Disk mid-plane tangential or hoop stress.

[T.sub.rr] - Disk mid-plane radial stress.

E - Disk Young's modulus.

F - Warpage constant for injection molded disks.

G - Warpage constant for injection molded disks.

[Sigma] - Disk Poisson's ratio.

[[Alpha].sub.r], - [[Alpha].sub.[Theta]] - Disk thermal expansion coefficients in the polar (r, [Theta]) directions.

[[Alpha].sub.[parallel]], [[Alpha].sub.[perpendicular]] - Disk thermal expansion coefficients along and perpendicular to the flow direction during mold filling.

T - Disk temperature.

[Delta]T - Disk temperature change measured from the thermally unstressed disk temperature.

[Delta][T.sub.min] - Minimum temperature change required to initiate warpage; measured from the thermally unstressed disk temperature.

[U.sub.ext] - Strain-energy per unit part surface area due to mid-plane deformation.

[U.sub.B] - Strain-energy per unit part surface area due to bending.

U(total) - Total disk strain-energy.

[D.sub.B] - Disk bending modulus.

[I.sub.1] - Disk energy integral.


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Author:Fahy, E.J.
Publication:Polymer Engineering and Science
Date:Jul 1, 1998
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