Impact Resistance

ContentsEXECUTIVE SUMMARYINTRODUCTION AND BASIC CONCEPTSSUMMARY OF DESIGN RULES: IMPACTMODELING AND IMPACT MECHANICSFAILURE DEFINITIONIMPACT TESTSDESIGN INFORMATIONREFERENCES

EXECUTIVE SUMMARY

Designing for impact resistance is a common requirement placed on parts to be made from thermoplasticmaterials. The specifications can take a variety of familiar forms:

A thermoplastic bumper must survive an impact load delivered by a 1,134-kg (2,500-lb) pendulum mass at a

velocity of 8 km/hr (5 mi/hr).A computer housing and its internal electronics must survive an impact caused by dropping the component

from a height of 91 cm (3 ft).A thermoplastic container must withstand the impact of a 0.91 kg (2 lb) ball dropped from a height of 1.83 m (6

ft).

In fact, there are many practical applications where thermoplastics are chosen because they are inherently moreimpact resistant than alternative materials. For example, because of its large yield strain, a sheet of impactresistant thermoplastic will absorb substantially more energy before denting than a metal sheet of equalunsupported area and stiffness. However, despite their inherent advantages in many situations, thermoplastic partscan certainly incur damage or break during impact events. Proper materials selection and design can help minimizethese experiences.

The purpose of this engineering manual is to provide information that will be useful for properly designing athermoplastic part to withstand impact. One of the most basic tenets of designing for impact events is that generallythe mass of at least some of the objects involved in the impact is essential to defining the response of thethermoplastic part. Although there are situations where the mass of the plastic must be accounted for to define theresponse of the part, such as a bullet impacting a thermoplastic window, in a great many instances the mass of theplastic may be considered negligible compared to that of stiffer and heavier parts made of other materials.

The first section of this manual presents a summary of the essential design rules and rules-of-thumb. More detailon these rules can be found in the body of the guide. The second section of this text discusses several basicconcepts of dynamics that are important for an analyst trying to translate impact requirements -- defined in terms ofmasses, drop heights, and velocities -- into terms that can then be compared to material property limits. At the endof this section, more advanced analytical discussion is included for those analysts who may be applying detailedfinite-element analyses (FEAs) to thermoplastic components. Different approaches to this type of analysis arediscussed.

Following the discussion of dynamic analysis is a section dealing with modes of failure that may be encountered inthermoplastic part design and the material properties that are necessary to judge whether there is a risk of thatfailure in the part under consideration. Brittle as well as ductile failures are considered. Notch-sensitive materials,which may exhibit either type of failure depending on the temperature loading, rate, and geometry of the part, aregiven special attention. Next, the most common impact tests performed to characterize the impact resistance of amaterial are discussed. Finally, several approaches to local geometric detail are outlined that may be useful indesign and trouble-shooting for parts made of notch-sensitive thermoplastic material.

INTRODUCTION AND BASIC CONCEPTS

In general, an impact problem involves at least 2 objects, each with its own geometry, stiffness, and mass, travelingon a collision course with velocities V1 and V2, as shown in Figure 1. When the 2 objects collide, each will besubjected to forces that are rapidly applied over a short period of time and then removed in an equally rapidfashion. As a result, rapid loading rate and short duration of load are often used to distinguish an impact load froma statically applied load, as illustrated in Figure 2. Furthermore, except for the case of load ordisplacement-controlled laboratory tests, the time history of the impact force is not usually known. Instead, it is 1 of

the variables of the problem and will be dependent on the mass, stiffness, and velocity of both bodies involved inthe collision. References [1] and [2] are 2 general texts dealing with impact problems.

Figure 1General impact event

Figure 2Faster loading rates in impact events can lead to mass effects in structural response and ratedependency of failure

Impact resistance can be considered to be the relative resistance of a component to failure due to stresses appliedat high rates. It should be emphasized that failure can take many different forms depending on the componentunder consideration. Excessive elastic deformation, permanent deformation, tearing, and fracture are all possiblefailure criteria that must be assessed and applied appropriately in the design process.

Rapid deformation rates can have two consequences. First, if the rate of loading is fast enough, then the massproperties of the loaded body become significant. For slower loading rates, mass effects can be ignored asinsignificant. Second, since the mechanical properties of plastics are often rate sensitive, it becomes important toquantify how the rate affects properties critical to the engineering performance of the materials.

Consideration of mass effects due to rapid loading rates is related primarily to the mechanics models used topredict response and will be dealt with first. The most important rate sensitivity observed in plastics is associated

with the criterion used to predict failure.

SUMMARY OF DESIGN RULES: IMPACT

1. Impact problems involve at least 2 objects, each with its own geometry, stiffness, and mass, traveling ona collision course with measurable velocities.

Rapid loading rate and short duration of load distinguish an impact load from a statically applied load.

Impact resistance can be considered to be the relative resistance of a component to failure due to stresses

applied at high rates.Elastic deformation, permanent deformation, tearing, and fracture are all possible failure criteria in an impact

event.If the rate of loading is fast enough, then the mass properties of the loaded body become significant.

It is important to quantify how rate affects properties critical to the engineering performance of thermoplastic

materials, because the mechanical properties of plastics are often rate-sensitive.2. When loading times are short in comparison to a structure's natural period of vibration, the mass of thestructure is important in establishing its maximum response. The problem should be treated in a dynamic,not static fashion.

A loading may be judged as fast or slow by comparing the loading time to the natural period of oscillation of

the dynamic system.For load times considerably larger than the system's natural period, the mass of the system may be ignored

and displacements calculated using static equations. However, for shorter loading times, the inertial effects must beconsidered because the system's dynamic response may be substantially different than a static approximationwould indicate.

As the load durations approach the system's natural period, the maximum dynamic displacement becomes

significantly larger than the equivalent static displacement.As the load time duration is further decreased to values well under the first natural period, the maximum

dynamic response becomes less than the equivalent static response.For a conservative analysis, assuming zero damping is a reasonable approximation for predicting the initial

peak response of the system to an impact load.3. The mass of thermoplastic components can often be ignored in impact problems. Quasi-static analysescan be performed if the relationship between the initial impactor energy and the energy absorbed by theplastic component can be defined. When simple energy balances are not obvious, implicit or explicitdynamic analysis of the impact system may be required.

Although applied load may be well defined in magnitude and duration for some dynamic laboratory tests

performed under load or displacement control, it is not true for most practical impact events. In most cases theimpact event is defined in terms of velocities, masses, or kinetic energies. The load is thus unknown and must becalculated as part of the solution process.

Both the magnitude and the time history of the impact force are not only dependent upon the kinetic energy of

the impactor (Body 2), but also the stiffness of Body 2 as well as the mass and stiffness of the target, Body 1. In themost general of situations, a transient dynamic analysis of the impact event would be necessary. The mass and thestiffness properties of both the impactor and the target bodies would have to be modeled. Furthermore, specialconsideration must be given to modeling the contact conditions between the 2 bodies.

The problem is significantly simplified if the mass and stiffness of the impactor are both significantly larger than

the respective target properties. In many realistic engineering applications, the mass of a thermoplastic componentis negligible in comparison to that of accompanying metallic subcomponents or impactors. Furthermore, mostplastic bodies are quite flexible structurally. As a result, it is often possible to assume that a metal member of acomponent or impactor is rigid in comparison to a plastic component.

If the above conditions are true, it can now be assumed that when the impactor is brought to rest, all of the

original kinetic energy in the projectile is transformed to strain energy in the plastic component. This leads to adescription of the impact force applied to the target mass as slow or quasi-static in nature. As a result, a static load

could be applied to the target structure to approximate the response of the target. Even if significant material orgeometric nonlinearity occurs in the target, the maximum deformation of the target can be identified as the pointwhere the integrated work exerted by the force on the target is equal to the original impact energy of the projectile.4. Finite-element analysis (FEA) is a commonly applied tool for analyzing impact events. There are 3different approaches to consider for the solution of general impact problems using FEA, as follows:A. Quasi-static analysisB. Implicit dynamic analysisC. Explicit dynamic analysis

A fourth approach applied to dynamic problems in general but less efficient for thermoplastic components, is modalsuperposition. This approach requires that the problem be linear, often a severe limitation for analyses involvingthermoplastic componenets. In these cases, strains and rotations frequently lead to geometric nonlinearity, andfrequently the material will undergo yielding, all of which are incompatible with linear assumptions of modalanalysis.In a quasi-static analysis, the mass of the component being analyzed is treated as negligible in theproblem. This is a reasonable assumption in at least 2 situations:

The load history is well defined and its duration is much longer than the lowest natural period of vibration for

the structure. Laboratory impact events where the cross-head rate is controlled are prime examples of this class ofproblems.

The plastic component is part of an impact event in which its mass is negligible in comparison to other masses

in the problem.

In this case it is possible to apply simple energy and momentum balances to define the amount of work that is doneon the plastic part during the impact. In such a case, a static load can be applied to the plastic component whileignoring the mass of the plastic. The impact event is considered to have terminated when the appropriate work hasbeen done on the component. Examples of this class of problems include dropping weights onto plastic parts andpendulum and barrier tests of automotive bumpers.

If it is not straightforward to establish the work done on the plastic component during impact, then it becomesimpossible to define the point when quasi-static analysis should be terminated. When the energy balance in notstraightforward to define, a dynamic analysis of the event will be required.

For linear problems, implicit dynamic analyses are generally recommended.

However, most thermoplastic impact events possess some combination of geometric, material, and contact

nonlinearities. The resulting implicit dynamic analysis will be time consuming. For large problems, measured interms of degrees of freedom, explicit dynamic analyses are more efficient and are recommended.

The presence of instabilities in a structure during loading such as buckling is another reason for considering adynamic analysis, even when dynamic effects may be insignificant. These severe nonlinearities may prevent aquasi-static solution as a result of the unstable nature of equilibrium during portions of the loading. Impact eventsinvolving buckling and post-buckling behavior are often more easily analyzed using an explicit dynamics approach.Thermoplastic automotive bumpers are an example.

5. In most impact studies, failure mode is an important concern. There are 2 failuremodes: ductile and brittle.A. Ductile failure is characterized as a slow and noncatastrophic, yielding or tearing process requiring additionalenergy to further spread the damage zone.

Ductile failures can be classified as either a yielding failure or a tearing failure. Yielding occurs when the materialexperiences permanent deformation, as in necking. Tearing occurs after the material has yielded and sufficientplastic (permanent) strain has accumulated to cause the material to pull or tear apart.

Yielding is defined using a simple uniaxial tensile test. In plastics the yield stress is usually taken to be the initialpeak in a uniaxial stress/strain curve. Multiaxial stress fields may be assessed by comparing an effective multiaxialstress equation (usually the von Mises stress equation) with the uniaxial yield stress of the material. Yielding, i.e.permanent plastic deformation, has initiated when the von Mises effective stress in a part equals or exceeds theuniaxial yield stress of the material.

If some permanent deformation is acceptable, strain to failure criteria may be used as the ductile-failure criteria todetermine if tearing is expected.

Many thermoplastics display pressure-dependent yielding behavior. Tensile hydrostatic stresses tend to decrease

the yield stress, while compressive hydrostatic stresses tend to increase the yield stress. However, because mostthermoplastic parts are thin walled, large hydrostatic fields cannot develop on a macro level. Therefore, ignoringpressure effects will not significantly affect gross part-performance predictions.

If a material's yield stress is significantly pressure-dependent, as in some of the heavily rubber modified materials,and if a part sees large regions of compressive stresses then using a pressure dependent yielding model or aseparate tensile and compressive yield stress in the finite element analysis would yield better part performancepredictions.Yield stresses of polymers depend on rate and temperature in general, higher strain rates and lowertemperatures lead to higher yield stresses.

Operating temperatures are usually known, but strain rates must be calculated.

The strain rate can be approximated by dividing the maximum strain found in the part at a given displacement

by the time it took the part to reach that displacement. For simple geometries and loading, a hand calculation maybe performed to estimate strain rate; however, for more complex geometries and loadings, or for more accurateresults, an elastic finite element analysis would be preferred.

Although strain rates will vary from location to location in a structure, it is usually sufficient in a structural analysis touse a yield stress corresponding to the highest strain rate seen in the structure.

In certain advanced finite element analyses, strain rates are calculated internally and a rate-dependent yieldingmodel can be defined. This eliminates the need for performing an initial elastic FEA to estimate strain rate andallows the yield stress to vary throughout the part based on local strain rates.

Strain-to-failure values determined by correlating mechanical test results and finite element analyses are used asductile tearing criterion. To determine if tearing will occur, equivalent plastic strain predictions from finite elementanalyses are compared to rate-dependent, strain-to-failure values.

B. Brittle failure is characterized by a sudden and complete catastrophic failure which,once initiated, requires no further energy to propagate.

Brittle failure criteria for many polymers have not yet been firmly established. However, maximum principal stress isa logical brittle failure criterion which has been established for certain polymers.

Crazing: Crazing has been established and validated as a brittle failure criterion for polycarbonate andpolyetherimide. Brittle failure mechanisms for opaque materials are not as easy to establish; however, a maximumprincipal stress failure criterion has been established and used successfully for these materials as well.

Ductile-to-brittle transitions: A material that fails ductilely in one test or application may fail brittlely in another,depending on geometry and loading conditions, loading rates, and temperature. Applications experiencing lowertemperatures, higher rates and more severe stress concentrators are more likely to fail brittlely. All three factorsaffect failure behavior.

The combination of stresses at a location within a part defines the stress state at that location. Stress states areclassified into 3 general categories: uniaxial, biaxial, and triaxial. Beam designs, as in grillwork, often experienceuniaxial stresses; flat plate designs, as in fender panels, typically experience biaxial stresses; while areas nearstress concentrators, such as corners of boxes, intersections of ribs or bosses with their base, and so on usuallyare in a triaxial state of stress. Brittle failure of a material in the presence of such local geometric detail is oftenreferred to as notch sensitivity.

For notch-sensitive materials, understanding the stress state within a part will determine whether failure will be

ductile or brittle. In thermoplastic materials, as the stress state moves from a uniaxial state of stress to a tensile,triaxial state of stress, the likelihood of brittle failure increases.

Local geometries that develop triaxial stress states typically yield locally, initially. However, principal stress

levels sufficient to cause brittle failure may be reached before macroscopic plastic flow can develop, with thepossibility of sudden brittle failure.

A geometry that fails ductilely at a given loading rate and temperature may fail brittlely at a different loading rateand temperature. When assessing whether or not a part will fail brittlely, three factors, stress state, strain rate andtemperature must be considered collectively.6. There are 2 general categories of impact tests: disk and notched beam.

Disk tests consist of the Gardner Drop and the Dynatup, with the most often quoted measured quantity being

the energy required for failure. Note that this quantity does not represent any fundamental material property andcannot be used as part of design analysis to predict the performance of a component with a general geometry.These tests are best used to compare materials on a relative basis in a biaxial stress state.

Notched-beam tests consist of the Izod and Charpy tests. Note that the units of measure in these tests are not

those of stress, nor is "impact strength" as defined by these tests a material property. Therefore reported valuescan not be used to design a component. These tests are best used to compare materials on a relative basis in atriaxial stress state.7. Design information

Fracture Maps can be plotted from ductility ratios (defined as the ratio of actual failure load in a notched-beam

geometry to its theoretical maximum, ductile-load carrying capability for that geometry). Fracture maps showregions of ductile versus brittle behavior and can be used to choose a material that will behave ductilely in a givenapplication. Or, if the material is already selected, to determine if brittle failure is a concern.

Geometric severity ratios: geometries producing tensile, triaxial stress states that have a large maximum

principal stress component are more susceptible to brittle failure. One way of characterizing the susceptibility of ageometry to brittle failure is to calculate the geometry's ratio of maximum principal stress to yield stress usingdetailed finite-element analyses. These geometric severity ratios can then be compared to a material's criticalmaximum principal stress to yield stress ratio at the appropriate rate and temperature to determine if brittle failure isa concern. Geometric Severity Ratios have been generated as a function of local geometric parameters for somecommon generic geometries; including notched beams, cantilever flanges, and ribbed plates.

Rib design: Ribs are often added to thermoplastic parts to increase part stiffness, but they may reduce part

strength by promoting brittle failure. Ribs should be positioned such that fillet radii between the ribs and the plateare placed in compression rather than in tension. If that is not practical, keep stresses near the fillet radii belowthose required to cause brittle failure. This can be accomplished by adjusting fillet radii, plate and rib thickness,and/or spacing between ribs. These geometric parameters should be adjusted so that the Geometric Severity Ratiois less than the material's critical maximum principal stress to yield stress ratio, referred to as the stress ratio, at theappropriate strain rate and temperature.

MODELING AND IMPACT MECHANICS

Mass Effects of Rapid Loading RateWhen loading times are short in comparison to a structure's natural period of vibration, the mass of the structure isquantitatively important in establishing its maximum response.

As a beginning to understanding mass effects due to high loading rates, consider the response of a simple,idealized, discrete system, such as the one shown in Figure 3. The dynamic response of such a system, isdiscussed in any elementary text on dynamics, such as Reference [6]. The simple system shown in Figure 3 iscomposed of 3 elements: a mass, m, a spring with stiffness, k, and a dashpot with a coefficient of viscous damping,c. The dynamic equation of equilibrium for such a system is:

(1a)

(1b)

(1c)

The solution to Equation (1) is usually written in terms of 2 parameters w and z, where w is the system's natural

frequency defined as:

(2)

and z is the viscous damping factor defined as:

(3)

Figure 3Idealized 11° of freedom mechanical system

For the time being, the forcing function F(t) will be assumed to be known. Although this is not true in general impactevents, it is often an appropriate assumption for load- controlled laboratory tests.

The forcing function F(t) in Eq. (1a) can be considered to be completely general in nature. However, in the case ofan impact load, the time during which the load is applied is generally short in duration. In order to judge whether theduration of a load has been "short," there must be some standard of time with which the loading time can becompared. One logical measure of time to use for this comparison is the natural period of oscillation for theunforced response of the undamped dynamic system.

If there is no load applied to the system in Figure 3, but the mass is moved away from its equilibrium position by adistance x0 and released, then the response of the mass is a decaying sinusoidal function of time that is shown inFigure 4 for a damping factor z of 0.1. If there is no damping in the system, the dynamic response does not decaywith time, as is also shown in Figure 4. For any small damping factor (z << 1.0), the natural period of the dampedsinusoidal response function is only negligibly different from the undamped response, as shown in the Figure. Thisnatural period can be shown to be:

(4)

Figure 4Periodic response of damped and undamped mechanical systems

The regular oscillation of a free dynamic system shown in Figure 4 and characterized by the natural period definedin Equation (4) establishes a natural "clock" for the system with the time period T as the basic unit of time. Onemethod of judging whether a load application is fast or slow is to compare the loading time to the natural period ofthe dynamic system.

Using the simple system shown in Figure 3, we can more quantitatively define the meaning of a "short" load time aswell as its consequences. Assume that the load F(t), applied in Figure 3, is defined as:

(5)

This load is shown graphically in Figure 5.

Figure 5Idealized loading history.

First, consider the response of this system for a variety of impact load duration times, TL, in order to understandquantitatively what defines a "short" load time. For this comparison, we will assume that the damping coefficient, z,is 0.05, the natural frequency, w, is 2p, and the maximum applied load, F, is (2p)2. Using the definition given inEquation (4), the natural period for this system is:

(6)

For such a system and maximum applied force, the system displacement, x, is illustrated graphically in Figure 6 forloading times, TL, of 5.0, 2.0, 1.0, and 0.5. The ordinate of Figure 6 is the dynamic displacement valuenondimensionalized by the displacement under a static load of equal magnitude, F. If the load duration is 5 thenatural period of the system, the response shown in Figure 6 is nearly static in nature. That is to say, thedisplacement at any time is quite accurately approximated by the force defined in Figure 5 divided by the systemstiffness, k. There is some visible oscillation with a frequency equal to the system natural frequency during theapplication of load and some small free vibration after the dynamic load is removed, but both are negligible. As theload duration, TL, is decreased while the maximum force is held constant, there are noticeable differences in thedynamic response. First, there is noticeable free vibration of the system after the loading is terminated. Second, asthe load duration is shortened and approaches the natural period of the system, the displacement during the load isnoticeably different from the equivalent static displacement. As the load durations approach the system naturalperiod, the maximum dynamic displacement becomes significantly larger than the equivalent static displacement.As the time duration is further decreased to values well under the first natural period, the maximum dynamicresponse becomes less than the equivalent static response. Considering this comparison in Figure 6, mass effectsof a structure are seen to be important if the time over which the load is applied is of the same order or smaller thanthe period of the dynamic system. For load times considerably larger than the system natural period, the mass ofthe system may be ignored and displacements calculated using static equations. However, for shorter loadingtimes, the inertial effects must be considered, because the system dynamic response may be substantially differentthan a static approximation.

Figure 6Response of a 1° of freedom mechanical system to an idealized loading history of varyingduration.

Damping EffectsStructural damping can usually be ignored in impact problems.

In the previous discussion examining the effect of the load application time for an impact event, the nondimensionaldamping coefficient, z, was assumed to be 0.05. Although the stiffness, k, and mass, m, of dynamic systems areusually well defined, the damping coefficient is usually more difficult to quantify. Figure 7 illustrates the effect of thedamping coefficient on the dynamic response of the system. As in Figure 6, the displacement, x, isnondimensionalized by the static displacement under a force F, and the time is nondimensionalized by the naturalperiod of response. For this comparison, the load time, TL, is assumed to be equal to the natural period of thesystem. The system response to this load is illustrated in Figure 7 for 5 values of the damping factor, z ranging from0 to 0.5. For most structural systems, z will have a value between 0.00 and 0.05. In this range, the maximumdisplacement of the system varies by only about 5%. The largest response is associated with a damping coefficientof 0.0. As a result, from the standpoint of conservative analysis, an assumption that the damping is 0 will be areasonable first approximation for predicting the initial peak response of the system to an impact load.

Load Definition in Impact EventsThe mass of thermoplastic components can often be ignored in impact problems. In these situations, quasi-staticanalyses can be performed if the relationship between the initial impactor energy and the energy absorbed by theplastic component can be defined. When simple energy balances are not obvious, implicit or explicit dynamicanalysis of the impact system may be required.

In the previous discussions of dynamic response, the applied load was well defined in terms of both magnitude andduration. Although this may be true for some dynamic laboratory tests performed under load or displacementcontrol, it is not true for most practical impact events. In most cases the impact event is defined in terms ofvelocities, masses, or kinetic energies. The load is unknown and must be calculated as part of the solution process.

As an illustration of this concept, consider a more realistic 2 body impact problem defined schematically in Figure 8.There are 2 bodies associated with this impact, each described by a mass and a spring. Body 1 is attached to the

ground with its spring and is initially at rest. It will be considered to be the target. Body 2, which will be consideredthe impactor, is moving with a velocity V toward Body 1. It will be assumed that the response of Body 1 is ofprimary interest. If the impact force imposed on Body 1 is defined, then its response can be calculated in the samefashion as considered previously for the system in Figure 3. However, in this case, the applied force is not explicitlydefined. Its definition requires knowledge of the response of both bodies. During the time the 2 bodies are incontact, the equations of equilibrium and the initial conditions are described as:

(7a)M1

+ (K1

+ K2)x

1- K

2x

2= 0

(7b)M2

K2x

2- K

2x

1

0

(7c)x1(0) = x

2(0) = 0

(7d)

1(0) = 0

(7e)

2(0) = V

Figure 8An idealized 2° of freedom impact problem.

The impact force exerted on Body 1 by the impactor can be expressed as

(8)

There are several important concepts highlighted by this simple problem. First, both the magnitude and the timehistory of the impact force are not only dependent upon the kinetic energy of the impactor (Body 2), but also thestiffness of Body 2, as well as the mass and stiffness of Body 1. For example, if the mass and stiffness of the 2bodies are respectively equal and have values of 1 unit of mass and 1 unit of stiffness, then the impact force historywill have the form shown in Figure 9 for these parameter values. However, if the stiffness of Body 2 is reduced to1/2 unit, then the impact force time history will have a significantly different form, also shown in Figure 9. Similarly,if either the mass or the stiffness of Body 1 is changed, the impact force is also altered. In the most general ofsituations, a transient dynamic analysis of the impact event would be necessary. The mass and the stiffnessproperties of both the impactor and the target bodies would have to be modeled. Furthermore, specialconsideration would also have to be given to modeling the contact conditions between the 2 bodies.

Figure 9Impact force history for an idealized impact problem with 2 different stiffness ratios.

The problem is significantly simplified if the mass and stiffness of the impactor are both significantly larger than therespective target properties. This approximation is particularly important for plastic structures since they aregenerally light and flexible in comparison to other structures. In many realistic engineering applications, the mass ofa thermoplastic component is negligible in comparison to that of accompanying metallic subcomponents orimpactors. In addition, the modulus of elasticity for common thermoplastics varies from values as low as 28 MPa (4ksi) for some elastomers to values as high as 10.3 GPa (1,500 ksi) for glass-filled plastics. Most unfilledengineering thermoplastics have moduli in the 1.4 - 3.5 GPa (200 - 500 ksi) range.

All of these values are very low relative to other engineering materials. Steel and aluminum, for instance, havemoduli of 207 GPa (30 ' 106 psi) and 69 GPa (10 ' 106 psi) respectively almost an order of magnitude stiffer thanmost thermoplastics. Furthermore, most plastic structures are thin in nature, and, as a result quite flexible from astructural standpoint. Therefore, it is often possible to assume that a metal member of a component or impactor isrigid in comparison to the plastic component. Examples where these assumptions are appropriate include steelballs or pendulum masses impacting such plastic components as computer housings or automotive bumpers.Another similar example would be drop testing of electronic components containing significant, internal metallicmasses with plastic housings.

If the mass of the plastic target can be neglected, then Equation (7a) becomes:

(9)

Substituting Equation (9) into (7b) leads to a single differential equation:

(10)

If

is small in comparison to 1.0 (i.e., the impactor is much stiffer than the target), then Equation (10) simplifies to:

(11)

and the initial conditions of the problem are:

(12a)x2(0) = 0

(12b)

2(0) = V

This simplified governing differential equation is not only easier to solve, but, in addition, energy-balance methodscan be applied to calculate the plastic structure's response. It can now be assumed that when the impactor isbrought to rest, all of the original kinetic energy in the projectile is transformed to strain energy in the plasticcomponent. If this approximation is applied to the 2 body problem just considered, then the energy balance can bewritten as:

(13)

The maximum displacement of the target can then be expressed as

(14)

For the case where

and

, it can be shown by solution of Equation (11) that the time duration over which the impact force is applied isapproximately

(15)

Since this loading time is much longer than the natural period of the target

, the arguments previously made in this section lead to a description of the impact force applied to the target massas slow or quasi-static in nature. As a result, a static load could be applied to the target structure to approximatethe response of the target. Even if significant material or geometric nonlinearity occurs in the target, the maximumdeformation of the target can be identified as the point where the integrated work exerted by the force on the targetis equal to the original impact energy of the projectile.There are 3 different analytical approaches to consider for impact problems.1. Quasi-static analysis2. Implicit dynamic analysis while assuming the mass of the thermoplastic component to be negligible incomparison to other masses in the problem.3. Explicit dynamic analysis of a model accounting for all of the mass in the problem.

A fourth approach, which is also applied to dynamic problems in general, but is less efficient for thermoplasticcomponents, is modal superposition. This approach requires that the problem be linear. This is often a severelimitation for thermoplastic components because the strains and rotations often lead to geometric nonlinearity andthe material may often undergo yielding all of which are incompatible with the linear assumptions of modal analysis.In a quasi-static analysis, the mass of the component being analyzed is treated as negligible. This is areasonable assumption in at least two situations:1. The load history is well defined and its duration is much longer than the lowest natural period of vibration for thestructure. Laboratory impact events where the crosshead rate is controlled are prime examples of this class ofproblems.2. The plastic component is part of an impact event in which its mass is negligible in comparison to other masses inthe problem. Furthermore, it is possible to apply simple energy and momentum balances to define the amount ofwork that is done on the plastic part during the impact. In such a case, a static load is applied to the plasticcomponent while ignoring the mass of the plastic. The impact event is considered to have terminated when theappropriate work has been done on the component. Examples of this class of problems include dropping weightsonto plastic parts and pendulum and barrier tests of automotive bumpers.

The advantage of this analysis approach is that significant nonlinearities can be accounted for while using arelatively small number of solution steps. The size of these solution steps is only dependent upon the significanceof the nonlinearities in the problem. Excessively small time steps associated with the dynamic aspects of theproblem are avoided. In many cases, this can significantly reduce required CPU time.

Implicit Dynamic Analysis with Thermoplastic Mass NeglectedThere are some impact problems where it is not straightforward to establish the work that is done on the plasticcomponent during the impact event. Without that definition, it is not possible to define when the quasi-staticanalysis should be terminated. One example of such a problem might be a drop test where the impact force vectoris not through the center of mass of the component. In such a case, the original kinetic energy of linear motion istranslated into both stored strain energy in the plastic component and as well as kinetic energy associated withrotational motion about the center of mass. When the energy balance is not straightforward to define, a dynamicanalysis of the event will be required. It still may be possible to model the thermoplastic parts as massless whileaccounting for the significant mass in the problem. The advantage in this idealization is that now the incrementaltime steps required to solve the problem will be much larger than if the mass of the thermoplastic part were alsoincluded.

Explicit Dynamic Analysis Including Thermoplastic MassIf the mass of the plastic component is a significant factor in the impact problem, the resulting fully dynamicanalysis will be more time consuming to carry out. Implicit dynamic solution techniques algorithms such as thosediscussed in the last sub-section will still work, however, as the problem size in terms of degrees-of-freedomincreases, it is more likely that explicit equation solving techniques will be more efficient.

In addition to situations where the mass effects of the thermoplastic part is significant to the dynamic response dueto impact, there may be a second reason for considering a fully dynamic problem formulation where the mass ofthe thermoplastic part is included. There are situations where even though the mass of the thermoplastic isquantitatively insignificant to the solution, a dynamic analysis has advantages over quasi-static solutions whichignore thermoplastic mass. Some impact events can lead to significant nonlinearities in the response of thethermoplastic component. At times these non-linearities can lead to buckling and unstable post-buckling behavior.

One example of such a situation is thermoplastic bumpers which are formed into thin walled, built-up beams. Thesebeams can be susceptible to instabilities that make nonlinear static analyses very difficult and time consuming tocarry out. A dynamic analysis, however, which includes the mass of the unstable structure may be numericallymore well behaved, although it will require a greater number of solution steps. As pointed out previously, for largeranalyses, an explicit dynamic analysis often offers advantages in overall solution time in copmarison with implicitmethods and becomes the analysis of choice.

When there is a potential for structural instability (i.e., buckling and collapse) in the thermoplastic component duringthe impact response, it may be advantageous to use explicit dynamics equation solvers. This may insure that asolution is obtained in some problems where severe numerical difficulty due to structural instability prevents asolution under quasi-static assumptions.

FAILURE DEFINITION

When assessing impact performance, failure is an important concern. Most impacted parts are required to absorb acertain impact energy without failing. Automotive bumper impacts and drops of electronic enclosures are goodexamples. Added to this concern is the possibility of two different failure modes: ductile and brittle. In a ductilefailure, the part fails in a slow, noncatastrophic manner in which additional energy is required to further spread thedamage zone. In contrast, a brittle failure is characterized by a sudden and complete failure that, once initiated,requires no further energy to propagate. The failure criteria for the two failure modes differ. Generally an effectivestress such as the von Mises stress is used as the ductile failure criterion indicating when plastic (permanent)deformation has occured. If some permanent deformation is acceptable, then a strain-to-failure criterion may beused as the ductile failure criterion indicating when tearing is expected to occur. Brittle failure criteria for manypolymers have not yet been firmly established and can vary from one material to the next.

Ductile FailureAs previously mentioned, ductile failures are gradual, noncatastrophic failure events. Once initiated, the damagezone associated with a ductile failure will not propagate unless additional energy is supplied through some form ofexternal loading. Ductile failures can be classified as either a yield failure or a tearing failure. Yielding occurs whenthe material experiences permanent deformation, i.e., the material will not return to its original shape uponunloading. Tearing occurs after the material has yielded and sufficient plastic (permanent) strain has accumulatedto cause the material to pull or tear apart. The definition of ductile failure (either yielding or tearing) is subjectiveand depends on the application. In some applications permanent deformation (yielding) may be considered failure,while in other applications some permanent deformation may be allowed. In these later situations tearing woulddefine the ductile failure event.

YieldingYielding is typically defined using a simple tensile test. In plastics the yield stress is usually taken to be the initialpeak in a uniaxial stress-strain curve (Figure 10). Even though this yield stress limit is associated with a uniaxialstress field, effective stress-expressions are available defining yielding for multiaxial stress fields as well. Yieldpredictions can then be made by comparing an effective, multiaxial stress, usually the von Mises stress, Equation(16), with the uniaxial yield stress of the material. Yield occurs when the effective, Mises stress equals the uniaxial,tensile yield stress. Although the von mises yield criterion originated for metals, it has been used successfully topredict load-deflection behavior in thermoplastic parts experiencing yielding.

Figure 10Uniaxial Stress-Stretch curve for Polycarbonate

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where are the principal stresses.

Yielding - Pressure EffectsAs a consequence of using the von Mises criterion, yielding is assumed to be independent of hydrostatic stress orpressure, Equation (17); however, many thermoplastics do display pressure dependent yielding behavior. Tensilehydrostatic stresses tend to decrease the yield stress, while compressive hydrostatic stresses tend to increase theyield stress.

(17)

In most cases pressure effects on yielding are not significant from an engineering viewpoint and can be ignored.Since most thermoplastic parts are thin walled, large hydrostatic stress fields can not develop, except possibly nearsome local stress concentrations. Therefore, ignoring pressure effects will not significantly affect gross, partperformance, predictions. A comparison of load-deflection behavior for a barrier impact of an automotive bumperusing a von Mises yield criterion and using a pressure dependent yield criterion is shown in Figure 11. Littledifference is observed.

Figure 11A comparison of analytical load-displacement responses for a barrier impact of an automotivebumper using the von Mises yield criterion and a pressure dependent yield criterion

For most structural analyses, the dependence of yield stress on hydrostatic pressure can be ignored as negligiblewithout serious error in accuracy.

Yielding - Rate and Temperature EffectsThe yield stress of a polymer depends upon the rate and temperature at which it is tested. In general, higher ratesand lower temperatures lead to higher yield stresses. Examples of temperature and rate effects on the yield stressof LEXAN® polycarbonate are shown in Figure 12.

Figure 12Yield stress of LEXAN® polycarbonate as a function of strain rate and temperature

As this figure shows, the yield stress increases linearly for each order-of-magnitude increase in strain rate.Mathematically, this relationship between yield stress and strain rate can be expressed as shown in Equation (18),where

is the unknown yield stress at a desired strain rate of

. The A term is a temperature dependent material parameter representative of the yield stress increase per decadeincrease in strain rate. Values of A and B are shown in Table I below for a variety of polymers. Plots of yield stressversus strain rate are provided in Figures 13 and 14 at room temperature and -30° C, respectively.

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Table 1Yielding Parameters for Various Plastics

Figure 13Yield stress vs. strain rate for various plastics at 23° C

Figure 14Yield stress vs. strain rate for various plastics at -30° C.

When choosing the yield stress to use in a hand calculation or numerical analysis it is important to choose the yieldstress corresponding to the temperature and strain rate experienced by the part. The temperature experienced bythe part is usually known. However, the strain rate experienced by the part must be calculated. The strain rate canbe approximated by dividing the maximum strain found in the part at a given displacement by the time it took thepart to reach that displacement. If the part geometry and loading is simple enough, the strain may be calculatedusing closed-form solutions. For more complex geometries and loadings or for more accurate results, an elasticfinite-element analysis may be performed to calculate the strains in the part for a given deflection.

Of course the strain rate in the component will vary from one location to the next. Since yielding will occur first inthe most highly strained region, it is recommended that the strain rate be calculated for the region of highest strain.Since the effect of rate on yield stress is only significant for orders-of- magnitude variations in rate, approximatingthe yield stress using the maximum strain rate in the part is usually sufficient. As an example of the accuracy ofsuch an approach, consider the box-like structure and test setup shown in Figure 15. This structure was loaded atvarious cross-head rates and load-displacement data were collected. Next, elastic-plastic finite-element analyses ofthe component were carried out using a single yield stress corresponding to the maximum strain rate in thestructure. A comparison of actual and predicted peak loads is shown in Figure 16 as a function of cross-head rate.The agreement between the analytical and experimental results is very good.

Figure 15Box-like structure subjected to impact load

Figure 16Comparison of predicted and measured maximum loads of the box-like structure as a function ofstrain rate

Although strain rate will vary from location to location in a structure, using a single yield stress corresponding to themaximum strain rate and ambient temperature of the component is usually sufficient for structural analysis.

Table 2 offers some basic information with respect to characteristic strain rates associated with different tests andcomponent geometries.

Impact Event Strain Rate(%/s)

5 mph automotivebumper impact

10-100

15 mph automotiveinstrument panel(dashboard) impact

100-1000

Floor drop ofportable electronics(4 ft.)

10000-30000

Ball Drop on device(UL Test)

5000-15000

Car side-panelimpact

100-500

TearingDepending on the application, any tear in a material may be considered a failure or some tearing may be allowed,provided the structure can absorb the required energy or carry the specified load. In both cases, a strain-to-failurevalue is generally used as the tearing criterion. To determine if tearing will occur, strain predictions are simplycompared to strain-to-failure values. To determine if the part can absorb the required energy or carry the desiredload once tearing has occurred, FEAs can be performed using strain-to-failure damage models found in some finiteelement codes. Unfortunately, strain-to-failure data are not readily available. Since tearing failures are localized, nosimple test procedures exist to determine the true local strain at failure. Methods are being explored to determinestrain-to-failure by correlating finite element strain predictions and ductile failure test data.

A tearing failure criterion has been established in the form of an equivalent plastic strain-to-failure for a number ofmaterials. This has been done by correlating finite element analyses to experimental failure test results. Tests andfinite element analyses were performed on disk, holed disk and notched beam geometries. These analysesdetermined plastic failure strains using a yielding model that included strain rate effects, pressure effects and strainhardening behavior. Equivalent plastic strain-to-failure values are provided as a function of strain rate at roomtemperature and at -30 °C for a number of materials in Figures 17 and 18. As can be seen in these figures higherrates and lower temperatures tend to decrease the strain-to-failure of a material.

Figure 17Equivalent Plastic Strain-to-Failure for various plastics at 21° C

Figure 18Equivalent Plastic Strain-to Failure for various plastics at -30° C

Brittle FailureIn contrast to ductile failures, brittle failures are fast, complete, catastrophic failure events. Once a brittle failure hasinitiated, it can propagate without additional loading. Brittle failures typically result in parts that are broken intoeither a few or many separate pieces. Although brittle failures are more catastrophic than ductile failures,brittle-failure mechanisms and failure criterion are not as well understood nor as readily applied. One brittle failuremechanism, crazing, has been studied extensively and crazing criterion have been established and validated forpolycarbonate (LEXAN® 141) and polyetherimide (ULTEM® 1000) resins. Other brittle failure mechanisms andfailure critera are currently being explored for other materials.

Crazing (Lexan 141 and Ultem® 1000 Resin)In certain glassy amorphous polymers, brittle failures are preceded by the formation of one or more crazes underlarge triaxial stress fields. Crazes are similar to internal voids within a material and once initiated act as an internalcrack or defect. With continued loading these voids grow and often lead to brittle failure. Polycarbonate andpolyetherimide are two such polymers which are susceptible to craze formation and growth and subsequent brittlefailure. Since crazing precedes brittle failure in glassy amorphous polymers, a craze initiation criterion can be usedas a conservative brittle failure criterion.

Crazing criteria have been investigated and validated for LEXAN® polycarbonate and ULTEM® polyetherimide[4-7]. Craze initiation criteria have been established for these two materials in the form of a rate dependent, critical,maximum principal stress criterion. In other words, craze initiation occurs when the maximum principal stress in thepart reaches a critical, rate-dependent, value, given in Equation (18). If maximum principal stress levels within thepart are kept below those required for craze initiation, then brittle failure is not a concern.

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The critical values of maximum principal stress required for craze initiation as a function of rate are shown forpolycarbonate and polyetherimide in Figures 19 and 20 at room temperature and -30° C, respectively. In thesefigures the critical maximum principal stress as a function of rate has been normalized by the rate-dependent yieldstress of the material. This ratio is referred to as the stress ratio. In this plot the relative susceptibility of the two

materials to brittle failure can be compared. The material having the larger ratio of critical maximum principal stressto yield stress is less likely to fail brittlely. As shown in these figures lower values of maximum principal stress arerequired for brittle failure at higher strain rates.

Figure 19Ratio of critical maximum principal stress for brittle failure to yield stress as a function of strainrate at 23° C

Figure 20Ratio of critical maximum principal stress for brittle failure to yield stress as a function of strainrate at -30° C

The stress ratio parameter (s1-c/sy) can be used to rank materials with respect to their susceptibility to brittlefailure. Materials with lower values of this ratio are more susceptible to brittle fracture. Values of this parameter fora material can be established using a notched beam tested under displacement control conditions and standarduniaxial tensile yield data.

As seen in Figure 19 and 20, critical maximum principal stress to yield stress ratios are typically above 1.0 for toughamorphous polymers, indicating that a complex, severe, triaxial stress state is required to initiate crazing andsubsequent brittle failure; therefore, a simple hand calculation cannot be performed to estimate this ratio.Unfortunately, typical finite element structural analyses are not sufficient either. Since most plastic parts are thinwalled, the majority of FEAs employ thin shell elements, which assume zero stress through the shell thickness(plane stress). These elements cannot accurately determine the 3-D stress field near a stress concentrator, suchas a corner. Even if FEAs were performed using 3-D brick elements, the mesh refinement required to accuratelydetermine the 3-D stress fields near each stress concentrator accurately, would make the task unmanageable interms of modeling and CPU time. For example, consider the 3-D mesh of a notched beam shown in Figure 21.Sixteen elements spread over a 2-mm (0.080 in.) depth from the notch surface were required to accuratelydetermine the local, 3-D, stress field. Since maximum principal stress levels cannot be accurately calculated forcomplex part geometries, design aids characterizing common, generic features of thermoplastic parts must be usedto design against brittle failure.

Figure 21Typical 3-D finite element mesh used to determine peak principal stress levels in a notched beamgeometry

The stress ratio parameter (

1-c/

y) is a rate and temperature dependent parameter. A first approximation for determining if brittle failure is a

possibility is to compare the value of (1-c/y) for the material of interest at the temperature and strain rate to which itwill be exposed with values of (1/y)max for the appropriate design detail geometry in Figure 32.

Ductile to Brittle TransitionsA material which fails ductilely in one test or application may fail brittlely in another depending on the part geometryand loading conditions, the rate at which the part is tested, and the temperature. Applications experiencing lowertemperatures, higher rates, or having more severe stress concentrators are more likely to fail in a brittle fashion. Nosingle factor dictates failure behavior. All three factors geometry, rate, and temperature affect failure behavior.

Stress State EffectsA part's geometry and loading determine the stresses it experiences, which in turn determine its failure behavior.The combination of stresses at a location within a part defines the stress state at that location. For simplicity, stressstate will be discussed in terms of principal stresses.

Stress states can be classified into three general categories: uniaxial (

# 0,

2=

3= 0), biaxial (

1# 0,

2# 0,

3= 0), and triaxial (

1# 0,

2# 0,

3# 0). A part can experience any one of these stress states or combinations of all three. A tensile specimen sees a

uniaxial state of stress with the sole principal stress acting in the direction of the load (Figure 22a). Grill work orother beam like sections of a part often are in a uniaxial state of stress. A Dynatup specimen is in a biaxial state ofstress, consisting of circumferential and radial stress components (Figure 22b). Thin-walled plate-like structuressuch as the side of a monitor housing or the panel of a fender would often be in a biaxial state of stress uponloading. The area directly beneath the notch in a notched Izod test sample is in a triaxial state of stress (Figure22c). Triaxial stress states are found primarily near stress concentrators: corners of boxes, intersections of ribs orbosses with their base, intersections of ribs with other ribs, etc. Brittle failure of a material in the presence of suchlocal geometric detail is often referred to as notch sensitivity.

Figure 22aStress state in a tensile test

Figure 22bStress state in a Dynatup test

Figure 22cStress state in a notched Izod test

There are many materials that exhibit ductile failure modes when subjected to uniaxial and biaxial stress states, butcan fail in a brittle fashion because of local geometric details that create triaxial stress states. These materials areoften referred to as notch- sensitive materials.

Understanding the stress state within a part is important because it will determine whether failure is ductile orbrittle. The two competing failure modes, ductile and brittle, have different failure criteria. In thermoplastic materials,as the stress state moves from a uniaxial state of stress to a tensile, triaxial state of stress the likelihood of brittlefailure increases. Under uniaxial stress states, as stress levels increase, stress combinations necessary for ductilefailure are achieved prior to achieving those necessary for brittle failure; therefore, the part fails ductilely. Undertensile, triaxial stress states as stress levels increase, stresses necessary for brittle failure are reached prior toreaching those necessary for ductile failure and hence the part fails in a brittle manner (Figure 23).

Figure 23Competing failure modes

As an example of a stress state affecting failure behavior, consider tests performed on two identical diskgeometries of ULTEM® 1000 resin, one with a rib and one without a rib (Figure 24). The flat disk failed ductilely.

The disk with the rib, however, failed in a brittle manner at a load almost an order of magnitude lower than the flatdisk. The addition of the rib produced a tensile, triaxial stress state necessary for brittle failure in ULTEM® 1000.Ribs will increase the stiffness of a part but may substantially reduce a part's strength.

Figure 24Effect of geometry on failure behavior

Ductile: Failure Load = 15000 N Brittle: Failure Load = 1800 N

Strain Rate and Temperature EffectsA geometry that fails ductilely at a given loading rate and temperature may fail brittlely at a different loading rateand temperature. In a given geometry, the likelihood of brittle failure increases as the loading rate increases or thetemperature decreases.

As an example of the effects of rate on failure behavior, consider a notched LEXAN® 141 resin beam loaded in3-point-bending, as shown in Figure 25. For this geometry, failure loads per unit thickness are plotted versusnotch-root strain rate for two beam thicknesses: 3.2 mm (0.125 in.) and 6.4 mm (0.25 in.) (Figure 26). At strainrates below 1.0 1/s, the 6.4 mm (0.250 in.) thick beams exhibited increased load-carrying capability with increasingstrain rate. These specimens failed ductilely. However, beyond a 1.0 1/s strain rate the specimens failed brittlelyand load-carrying capability continued to drop with increasing loading rate. At room temperature this specificnotched beam geometry underwent a ductile-to-brittle transition at a strain rate of approximately 1.0 1/s. However,it is important to remember that the strain rate at which the transition occurred is also geometry and temperaturedependent. If 3.2-mm (0.125 in.) thick specimens are tested over the same range of rates, no transition is observed(Figure 26). For these thinner specimens the transitional rate is orders of magnitude higher because theyexperience less plane-strain constraint than the thicker specimens and, therefore, develop less severe triaxialstress states. If the same, ductile, 3.2 mm (0.125 in.) thick specimens are tested at -34°C (-29°F), then a ductile tobrittle transition is observed at a strain rate between 1.0 and 10.0 1/s. When assessing whether or not a part will failbrittlely, all 3 factors stress state, strain rate, and temperature, must be considered collectively

Figure 25Notched beam, 3-point-bend configuration

Figure 26Failure load per unit thickness for notched beams in 3-point-bending

IMPACT TESTS

Disc Tests: Gardner Drop and DynatupThe puncture test, illustrated in Figure 27 for example, subjects a plastic plate to severe deformations and strains.This test is one of the most regularly used measures of impact resistance for plastics. A punctured plate, such asthe one shown in Figure 28 and made of a ductile polymer, typically experiences true strains ranging from40-300%, depending on the material.

Figure 27Schematic of a puncture test geometry

Figure 28Punctured polycarbonate plate

ASTM Procedure D3029 [Reference 4] describes the geometry and test conditions recommended when using thisspecimen. The measured quantity most often quoted with respect to this test is energy required for failure. It shouldbe emphasized that this quantity does not represent any fundamental material property. It cannot be used as partof a design analysis to predict the performance of a component with general geometry. It is only useful forcomparing the performance of different polymers in this specific geometry. It is left to the qualitative judgment of theengineer to determine whether puncture resistance is a primary concern in the design of the component.

Notched Beam: Izod and CharpyBecause of the important effect of notch sensitivity on brittle failure at higher rates of loading, the Izod and Charpytests are often used to characterize the impact resistance of plastics. The Izod and Charpy tests, shownschematically in Figure 29, are very similar in that they both use notched beam specimens subjected to bendingmoments. The notch serves to both create a stress concentration as well as produce a constrained triaxial state oftension at a small distance below the bottom of the notch. Both of these effects tend to make the test severe fromthe standpoint of early transition to brittle behavior as a function of both rate and temperature. As a result, behaviorin these tests is often quite different from disc puncture tests that are also used to assess impact resistance.

Figure 29Charpy and Izod test configurations

The Charpy geometry is a simply supported beam with a centrally applied load on the reverse side of the beamfrom the notch. The Izod geometry, on the other hand, is a cantilever beam with the notch located at the root of thebeam. In both cases the load is applied dynamically by a free-falling pendulum of known initial potential energy.The important dimensions of interest for these tests include the notch angle, the notch depth, the notch-tip radius,the thickness of the beam, and the width of the beam. All of these quantities, as well as more detailed information

specifying loading geometry and conditions, are specified in ASTM standards [Reference 5]. As the pendulum fallsin both of these tests, its original potential energy is converted into kinetic energy. Some of this energy is in turnused to break the specimen, which is encountered at the low point of the pendulum arc. If the specimen is broken,the pendulum continues its swing and comes to a halt at a height less than its starting location. The energyexpended to break the specimen can then be calculated as the difference between the initial and final potentialenergies of the pendulum. The value generally reported from these tests is the energy required to break the bardivided by the net cross-sectional area of the notch, typically 3.17 mm (0.125 in). The units of this measure are J/m(ft-lb/in) and the parameter is referred to as impact strength.

It should be emphasized that the units of this measure are not those of stress, nor is "impact strength," as definedby these tests, a material property. In other words, the measurement cannot be used directly to design acomponent. In fact, the values of impact strength are significantly affected by the parameters defining the specimengeometry, like the notch-tip radius and beam width. Even the identification of a transition temperature can besignificantly affected by geometry such as the width of the beam. Wider beams tend to provide more plane-strainconstraint, and transition temperatures often appear more distinct and at higher temperature values than resultsfrom thinner beams. As a result, when comparing materials via the value of "impact strength" it is imperative thatthe test geometries be identical. In addition, the transition temperature identified by plotting the "impact strength" asa function of temperature may or may not be appropriate for the component geometry under consideration.

DESIGN INFORMATION

Fracture MapsBecause both ductile and brittle failure modes exist for most unfilled thermoplastics, both types of failure must beconsidered when designing a part. However, since many thermoplastics behave ductilely over a broad range ofstress states, strain rates, and temperatures, only the ductile failure mode may need to be considered in someapplications. An initial, conservative method to determine whether or not brittle failure is a concern is to look at amaterial's performance under a very severe state of stress at an appropriate strain rate and temperature. If thematerial behaves ductilely under this "worst-case" condition, then brittle failure of the component is unlikely tooccur. A method of describing a material's ductility under a severe stress state is to calculate its "Ductility Ratio." Amaterial's ductility ratio, (Eq. 20), is defined as the ratio of its actual failure load in a notched-beam geometry to itstheoretical maximum ductile, load-carrying capability for that geometry. A ductility ratio of 1.0 corresponds to aductile failure. Ductility ratios < 1.0 correspond to varying levels of brittle failure.

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Ductility Ratios can be plotted as a function of strain rate at different temperatures to create "fracture maps" thatmap out regions of ductile versus brittle behavior. Fracture maps are shown in Figures 30 and 31 for LEXAN® 141and ULTEM® 1000 resins, respectively.Fracture maps can be used in 2 ways:

To choose a material that will behave ductilely in a given application;

To determine if brittle failure is a concern, in cases where a material has already been selected.

Figure 30Fracture map for(LEXAN ® 141) polycarbonate at room temperature

Figure 31Fracture map for (ULTEM 1000) polyetherimide at room temperature

Geometric SeverityAs we have seen, geometries producing tensile, triaxial stress states and having a large maximum principal stresscomponent are more susceptible to brittle failure. One way of characterizing the susceptibility of a geometry tobrittle failure is to numerically calculate the geometry's maximum principal stress to yield stress ratio. Geometrieshaving larger ratios of maximum principal stress to yield stress are more likely to fail in a brittlely. Figure 32provides a list of these ratios for common, generic geometries. Using Figure 32 geometric severity ratios can bechosen which represent features of the part being designed. These geometric severity ratios can then be comparedto a material's critical maximum principal stress to yield stress ratio at the appropriate rate and temperature(Figures 19 and 20) to determine if brittle failure is a concern.

Geometry

Figure 32Geometric severity ratios for common geometric features

There are numerous design details that give rise to stress states promoting brittle failure because of large local (

1/

y)max

value, defined as the geometric severity ratio. Figure 32 illustrates some of these geometries and the valuesof (

1/

y)max

associated with each. For a given material, geometries with higher values of (

1/

y)max

are more susceptible to brittle failure.

Rib Design

Ribs are often added to thermoplastic parts to increase part stiffness. Still, they may reduce part strength and,therefore deserve special consideration. When designing with ribs, many options are available to reduce thelikelihood of brittle failure. The simplest option is to position the ribs so that the fillet radii between the ribs and theplate are in compression rather than in tension (Figure 33). This will create compressive, triaxial stress states nearthe fillet radii rather than tensile triaxial stress states. If this first option is not practical, then stresses near the filletradii should be kept below those required to cause brittle failure. This can be accomplished by adjusting localrib-plate geometry parameters, (e.g. fillet radii and plate thickness) or by adjusting rib spacing. Geometricparameter studies have been performed to quantifying local stress fields near the fillet radius in a ribbed plategeometry (Figures 34-36). In Figures 34-36, ratios of maximum principal stress to yield stress are plotted as afunction of fillet radius and plate thickness for rib width to plate thickness ratios of 0.5, 0.75 and 1.0, respectively.These figures can be used in conjunction with Figures 5 and 6 to assess if brittle failure is a possibility for a givenmaterial and local rib geometry.

Avoid if Possible

Figure 33Better

Figure 34Maximum principal stress to yield stress ratios as a function of fillet radius and plate thicknessfor a rib thickness to plate thickness ratio of 0.5

Figure 35Maximum principal stress-to-yield stress ratios as a function of fillet radius and plate thicknessfor a rib thickness to plate thickness ratio of 0.75

Figure 36Maximum principal stress to yield stress ratios as a function of fillet radius and plate thicknessfor a rib thickness to plate thickness ratio of 1.0.

In some situations, altering design details can have an effect on improving the resistance of a geometry to brittlefailure. Figures 34-36 illustrate the effect of parameters associated with a simple rib geometry on the geometricseverity ratio, (

1/

y)max

for that geometry.

REFERENCES

1. W. Goldsmith, Impact, Richard Clay, Bungay, Suffolk, 1960.

2. J. A. Zukas, T. Nicholas, H. F. Swift, L. B. Greszazuk, and D. R. Curran, Impact Dynamics, Wiley, NY, 1982.

3. P. I. Vincent, Polymer, Vol. 1, p. 427, 1960.

4. "Standard Test Methods for Impact Resistance of a Rigid Plastic Sheeting or Part by Means of a Tup (FallingWeight)," D3029-90, Annual Book of ASTM Standards, Vol. 08.02, American Society for Testing and Materials,Philadelphia, PA, pp. 524-535, 1992.

5. "Impact Resistance of Plastics and Electrical Insulating Materials," D256-90B, Annual Book of ASTM Standards,Vol. 08.01, American Society for Testing and Materials, Philadelphia, PA, pp. 58-74, 1992.

6. N. O. Myklestad, Fundamentals of Vibration Analysis, McGraw-Hill, NY, p.83, 1956.

7. R. P. Nimmer and J. T. Woods, "An Investigation of Brittle Failure in Ductile, Notch-Sensitive Thermoplastics,"Polymer Engineering and Science, Vol. 32, No. 16, pp. 1126-1137, 1992.

8. J. T. Woods and H. G. deLorenzi, "An Assessment of Crazing Criteria for Polyetherimide in 3-DimensionalStress Space," Polymer Engineering and Science, Vol. 33, No. 21, pp. 1431-1437, 1993.

9. H. G. deLorenzi and J. T. Woods, "Strain Rate and Temperature Effects on the Failure of NotchedPolycarbonate and Polyetherimide Beams," Proceedings of the 1993 Society of Plastics Engineers (SPE) AnnualTechnical Meeting, SPE, Brookfield, CT, pp. 1411-1417, 1993.

10.J. T. Woods, R. P. Nimmer, and K. F. Ryan, "The Development and Validation of Rate Dependent Brittle FailureCriterion for Polycarbonate and Polyetherimide," accepted for publication, Proceedings of the 1994 Society ofPlastics Engineers (SPE, Annual Technical Meeting, SPE, Brookfield, CT, 1995.

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