60 Plastics Engineered Product Design overall flow chart that goes from the product concept to product release. Use is made of the optimization theory and its application to problems arising in engineering that follows by determining the material and hbricating process to be used. The theory is a body of mathematical results and numerical methods for finding and identifjmg the best candidate from a collection of alternatives without having to specifjr and evaluate all possible alternatives. Thc process of optimization lies at the root of engineering, since the classical hnction of the engineer is to design new, better, more efficient, and less expensive products, as well as to devise plans and procedures for the improved operation of existing products. To optimize this approach the boundaries of the engineering system are necessary in order to apply the mathematical results and numerical techniques of the optimization theory to engineering problems. For purposes of analysis they serve to isolate the system from its surroundings, because all interactions between the system and its surroundings are assumed to be fixed/frozen at selected, representative levels. However, since interactions and complications always exist, the act of defining the system boundaries is required in the process of approximating the real system. It also requires defining the quantitative criterion on the basis of which candidates will be ranked to determine the best approach. Included will be the selection system variables that will be used to characterize or identifjr candidates, and to define a model that will express the manner in which the variables are related. Use is made of the optimization methods to determine the best con- dition without actually testing all possible conditions, comes through the use of a modest level of mathematics and at the cost of performing repetitive numerical calculations using clearly defined logical procedures or algorithms implemented on computers. This composite activity constitutes the process of formulating the engineering optimization problem. Good problem formulation is the key to the success of an Optimization study and is to a large degree an art. This knowledge is gained through practice and the study of successfd applications. It is based on the knowledge and experience of the strengths, weaknesses, and peculiarities of the techniques provided by optimization theory. Unfortunately at times this approach may result in that the initial choice of performance boundary/requirements is too restrictive. In order to analyze a given engineering system fully it may be necessary to expand the performance boundaries to include other sub-performance systems that strongly affect the operation of the model under study. As an example, a manufacturer finishes products that are mounted on an assembly line and 2 - Design Optimization 61 I_____ decorates. In an initial study of the secondary decorating operation one may consider it separate from the rest of the asscmbly linc. However, one may find that the optimal batch size and method of attachment sequence are strongly influenced by the operation of the plastic fabrication department that produces the fabricated products (as an example problems of frozen stresses, contaminated surface, and other detriments in the product could interfere with applying the decoration). Required is selecting an approach to determine a criterion on the basis of which the performance requirements or design of the system can be evaluated resulting in the most appropriate design or set of operating conditions being identified. In many engineering applications this criterion concerns economics. In turn one has to define economics such as total capital cost, annual cost, annual net profit, return on investment, cost to benefit ratio, or net present worth. There are criterions that involve some technology factors such as plastic material to be used, fabricating process to be used, minimum production time, number of products, maximum production rate, minimum energy utilization, minimum weight, and safety. Problem/Solution Concept In the art of the design concept there is the generation of solutions to meet the product requirements. It represents the sum of all of the subsystems and of the component parts that go to make up the whole system. During this phase, one is concerned with ideas and the generation of solutions. In practice, even with the simplest product design, one will probably have ideas as to how you might ultimately approach the problem(s). Record thcsc ideas as they occur; however avoid the temptation to start engineering and developing the ideas further. This tendency is as common with designers as it is with other professionals in their respective areas of interest. So, record the ideas but resist the temptation to proceed. Target as many ideas as you can possibly generate where single solutions are usually a disaster. While it is recognized that you may have limited experience and knowledge, both of technological and non- technological things, you must work within limits since design is not an excuse for trying to do impossible things outside the limits. Notwithstanding these facts, you need to use what you know and what you can discover. You will need to engineer your concepts to a level where each is complete and recognizable, and technically in balance within the limits and is feasible in meeting product requirements. 62 Plastics Engineered Product Design Design Approach The acquisition of analytical techniques and practical skills in the engineering sciences is important to the design system. Through a study of engineering of any label based on mathematics and physics applied through elemental studies, one acquires an all-round engineering competence. This enables, for example, one to calculate fatigue life, creep behavior, inertia forces, torsion and shaft stresses, vibration characteristics, etc. The list of calculations is limitless if one considers aLl the engineering disciplines and is therefore generally acceptable as the basis for any engineering review. However, the application of such skills and knowledge to engineering elements is partial design. To include the highly optimized, best material and/or shape in any design when it is not essential to the design may involve engineering analysis of the highest order that is expensive and usually not required. Limitations, shortcomings, or deficiencies have to be recognized otherwise potentially misdirected engineering analysis give rise to a poor design. What has been helpfd in many design teams is to include non-engineers or non-technologists (Fig. 2.1). However, this needs a disciplined, structured approach, so that everyone has a common view of total design and therefore subscribes to a common objective with a minimum of misconceptions. Participants should be able to see how their differing partial design contributions fit into the whole project. Model Less Costly When possible the ideal approach is to design products that rely on the formulation and analysis of mathematical models of static and/or dynamic physical systems. This is of interest because a model is more accessible to study than the physical system the model represents. Models typically are less costly and less time-consuming to construct and test. Changes in the structure of a model are easier to implement, and changes in the behavior of a model are easier to isolate and understand in a computer system (Chapter 5). A model often provides an insight when the corresponding physical system cannot, because experimentation with the actual system could be too dangerous, costly, or too demanding. A model can be used to answer questions about a product that has not yet been finalized or realized. Potential problems can provide an immediate solution. A mathematical model is a description of a system in terms of the available equations that are available fkom the engineering books. The 2 - Design Optimization 63 desired model used will depend upon: (1) the nature of the system the product represents, (2) the objectivcs of the designer in developing the model, and (3) the tools available for developing and analyzing the model. Because the physical systems of primary interest are static and/or dynamic in nature, the mathematical models used to represent these systems most ofien include difference or differential equations. Such equations, based on physical laws and observations, are statements of the fundamental relationships among the important variables that describe the system. Difference and differential equation models are expressions of the way in which the current values assumed by the variables combine to determine the hture values of these variables. As reviewed later it is important to relate static and/or dynamic loads on plastic products to operating temperatures. Model Type A variety of models are available that can meet the requirements for any given product. The choice of a particular model always represents a compromise between the accuracy in details of the model, the effort required in model formulation and analysis, and usually the time frame that has to be met in fabricating the product. This compromise is reflected in the nature and extent of simplifylng assumptions used to develop the model. Generally the more faithful or complete the model is as a description of the physical system modeled, the more difficult it is to obtain useful general solutions. Recognize that the best engineering model is not necessarily the most accurate or precise. It is, instead, the simplest model that yields the information needed to support a decision and meet performance requirements for the product. This approach of simplicity also involves the product’s shape to the fabricating method used. Most designed products do not complicate fabricating them, however there are those that can complicate the fabrication resulting in extra cost not initially included and the possibility of defective parts. Recognize that simpler models frequently can be justified, particularly during the initial stages of a product study. In particular, systems that can be described by linear difference or differential equations permit the use of powerful analysis and design techniques. These include the transform methods of classical theory and the state-variable methods of modem theory. Target is to have more than one model in the evaluation. Simple models that can be solved analytically are used to gain insight into the behavior of the system and to suggest candidate designs. These designs 64 Plastics Engineered Product Design are then verified and refined in more complex models, using computer simulation. If physical components are developed during the course of a study, it is often practical to incorporate these components directly into the simulation, replacing the corresponding model components. Computer Sofmare Mathematical models are particularly usell because of the large body of mathematical and computational theory that exists for the study and solution of equations. Based on this theory, a wide range of techniques has been developed. In recent years, computer programs have been written that implement virtually all of these techniques. Computer software packages are now widely available for both simulation and computational assistance in the analysis and design of control systems (Chapter 5). Design Analysis Approach Plastics have some design approaches that differ significantly from those of the familiar metals. As an example, the wide choice available in plastics makes it necessary to select not only between TPs, TSs, reinforced plastics (RPs), and elastomers, but also between individual materials within each family of plastic types (Chapter 1). This selection requires having data suitable for making comparisons which, apart from the availability of data, depends on defining and recognizing the relevant plastics behavior characteristics. There can be, for instance, isotropic (homogeneous) plastics and plastics that can have different directional properties that run fi-om the isotropic to anisotropic. As an example, certain engineering plastics and RPs that are injection molded can be used advantageously to provide extra stiffness and strength in predesigned directions. It can generally be claimed that fiber based RPs offer good potential for achieving high structural efficiency coupled with a weight saving in products, fuel efficiency in manufacturing, and cost effectiveness during service life. Conversely, special problems can arise from the use of RPs, due to the extreme anisotropy of some of them, the fact that the strength of certain constituent fibers is intrinsically variable, and because the test methods for measuring RPs’ performance need special consideration if they are to provide meaningfbl values. Some of the advantages, in terms of high strength-to-weight ratios and high stifhess-to-weight ratios, can be seen in Figs. 2.2 and 2.3, which show that some RPs can outperform steel and aluminum in their ordinary forms. If bonding to the matrix is good, then fibers augment mechanical strength by accepting strain transferred fi-om the matrix, 2 - Design Optimization 65 ~ ~i~ur~ 2.2 Tensile stress-strain curves for different materials Stms Epow 3% gC= Epov c Past to future tensile properties of RPs, steel, and aluminium 0 4 - RaaOpagilasbargm(psi)tOdaPny(Ibslcuh.)XId 3 2 which otherwise would break. This occurs until catastrophic debonding OCCUTS. Particularly effective here are combinations of fibers with plastic matrices, which often complement one another’s properties, yielding products with acceptable toughness, reduced thermal expansion, low ductility, and a high modulus. ~ Viscoe I ast ici ty xx-xx -~~ i-~ Viscoelasticity is a very important behavior to understand for the designer. It is the relationship of stress with elastic strain in a plastic. The response to stress of all plastic structures is viscoelastic, meaning that it takes time for the strain to accommodate the applied stress field. Viscoelasticity can be viewed as a mechanical behavior in which the 66 Plastics Engineered Product Design relationships between stress and strain are time dependent that may be extremely short or long, as opposed to the classical elastic behavior in which deformation and recovery both occur instantaneously on application and removal of stress, respectively. The time constants for this response will vary with the specific characteristics of a type plastic and processing technique. In the rigid section of a plastic the response time is usually on the order of microseconds to milliseconds. With resilient, rubber sections of the structure the response time can be long such as &om tenths of a second to seconds. This difference in response time is the cause of failure under rapid loading for certain plastics. By stressing a viscoelastic plastic material there are three deformation behaviors to be observed. They are an initial elastic response, followed by a time-dependent delayed elasticity that may also be llly recoverable, and the last observation is a viscous, non-recoverable, flow component. Most plastic containing systems (solid plastics, melts, gels, dilute, and concentrated solutions) exhibit viscoelastic behavior due to the long-chain nature of the constituent basic polymer molecules (Chapter 1). This viscoelastic behavior influences different properties such as brittleness. To understand why the possibility for brittle failure does exist for certain plastics when the response under high-speed stressing is transferred fiom resilient regions of a plastic, an analysis of the response of the two types of components in the structure is necessary. The elastomeric regions, which stay soft and rubbery at room temperature, will have a very low elastomeric modulus and a very large extension to failure. The rigid, virtually crosslinked regions, which harden together into a crystalline region on cooling, will be brittle and have very high moduli and very low extension to failure, usually fiom 1 to 10%. If the stress rate is a small fraction of the normal response time for the rubbery regions, they will not be able to strain quickly enough to accommodate the applied stress. As a consequence for the brittle type plastics, virtually crosslinked regions take a large amount of the stress, and since they have limited elongation, they fail. The apparent effect is that of a high stretch, rubbery material undergoing brittle failure at an elongation that is a small fraction of the possible values. A fluid, which although exhibits predominantly viscous flow behavior, also exhibits some elastic recovery of the deformation on release of the stress. To emphasize that viscous effects predominate, the term elastico- viscous is sometimes preferred; the term viscoelastic is reserved for solids showing both elastic and viscous behavior. Most plastic systems, both melts and solutions, are viscoelastic due to the molecules 2 - Design Optimization 67 / ~ -_%,”. becoming oriented due to the shear action of the fluid, but regaining their equilibrium randomly coiled configuration on release of the stress. Elastic effects are developed during processing such as in die swell, melt fracture, and fiozen-in orientation. Polymer Structure The viscoelastic deviations from ideal elasticity or purely viscous flow depend on both the experimental conditions (particularly temperature with its five temperature regions and magnitudes and rates of application of stress or strain). They also depend on the basic polymer structure particularly molecular weight (MW), molecular weight distribution (MWD), crystallinity, crosslinking, and branching (Chapter 1). High MW glassy polymer [an amorphous polymer well below its glass transition temperature (T,) value (Chapter l)] with its very few chain motions are possible so the material tends to behave elastically, with a very low value for the creep compliance of about Pa-’. When well above the T, value (for an elastomer polymer) the creep compliance is about The intermediate temperature region that corresponds to the region of the T, value, is referred to as the viscoelastic region, the leathery region, or the transition zone. Well above the T, value is the region of rubbery flow followed by the region of viscous flow. In this last region flow occurs owing to the possibility of slippage of whole polymer molecular chains occurring by means of coordinated segmental jumps. These five temperature regions give rise to the five regions of viscoelastic behavior. Light crosslinking of a polymer will have little effect on the glassy and transition zones, but will considerably modify the flow regions. Pa-’, since considerable segmental rotation can occur. Viscoelasticity Behavior There is linear and nonlinear viscoelasticity. The simplest type of viscoelastic behavior is linear viscoelasticity. This type of rheology behavior occurs when the deformation is sufficiently mild that the molecules of a plastic are disturbed &om their equilibrium configuration and entanglement state to a negligible extent. Since the deformations that occur during plastic processing are neither very small nor very slow, any theory of linear viscoelasticity to date is of very little use in processing modeling. Its principal utility is as a method for character- izing the molecules in their equilibrium state. An example is in the comparison of different plastics during quality control. 68 Plastics Engineered Product Design In the case of oscillatory shear experiments, for example, the strain amplitude must usually be low. For large and more rapid deformations, the linear theory has not been validated. The response to an imposed deformation depends on (1) the size of the deformation, (2) the rate of deformation, and (3) the kinematics of the deformation. Nonlinear viscoelasticity is the behavior in which the relationship of stress, strain, and time are not linear so that the ratios of stress to strain are dependent on the value of stress. The Boltzmann superposition principle does not hold (Appendix B). Such behavior is very common in plastic systems, non-linearity being found especially at high strains or in crystalline plastics. Relaxation/Creep Analysis Theories have been developed regarding linear viscoelasticity as it applies to static stress relaxation. This theory is not valid in nonlinear regions. It is applicable when plastic is stressed below some limiting stress (about half the short-time yield stress for unreinforced plastics); small strains are at any time almost linearly proportional to the imposed stresses. When the assumption is made that a timewise linear relationship exists between stress and strain, using models it can be shown that the stress at any time t in a plastic held at a constant strain (relaxation test), is given by: o = 0,e-vr where: 0 = stress at any time t 0, = initial stress e = natural logarithmic base number I' = relaxation time The total deformation experienced during creep loading (with the sample under constant stress 0) is given by: E = (o/E,) + (o/8 (1 - e-") + bth) where: E = total deformation, E, = initial modulus of the sample E = modulus after time t, 17 = viscosity of the plastic Excluding the permanent set or deformation and considering only the creep involved, equation (2-2) becomes: (2-3) The term y in Eqs. (2-1) and (2-2) has a different significance than that in equation (2-3). In the first equations it is based on static relaxation and the other on creep. A major accomplishment of this viscoelastic theory is the correlation of these quantities analytically so that creep E = (o/8 + (018 (I - e-") deformation can be predicted from relaxation data and relaxation data from creep deformation data as shown in the following equation: (ao/o) relaxation = (&/&,,I creep (2-4) Creep strains can be calculated using equation (2-4) in the form of: E = Eo (solo) = (oo/o) (oo/€o) (oo/o) (2-5) where ( l/Eo) ((J,/(J) may be thought of as a time-modified modulus, i.e., equal to 1/E, from which the modulus at any time t, is: E = €0 (o,/o> (2-6) that is the value to replace E in the conventional elastic solutions to mechanical problems. Where Poisson's ratio, y, appears in the elastic solution, it is replaced in the viscoelastic solution by: y= (3B-€) 6B (2-7) where B is the bulk modulus, a value that remains almost constant throughout deformations. Stress relaxation and creep behavior for plastics are closely related to each other so that one can be predicted from knowledge of the other. Therefore, such deformations in plastics can be predicted by the use of standard engineering elastic stress analysis formulas where the elastic constants E and y can be replaced by their viscoelastic equivalents given in equations (2-2) and (2-3). If data are not available on the effects of time, temperature, and strain rate on modulus, creep tests can be performed at various stress levels as a hnction of temperature over a prescribed period of time. As an example, for rocket and missile stress relaxations data obtained over a time period of 4 to 5 sec to an hour provide the essential information. For structural applications, such as pipelines, data over a period of years are required. Data from relatively short-term tests can be extrapolated by means of theory to long-term problems. However this approach can have its inherent limitations. Another method used involves the use of the rate theory based on the Arrhenius equation. In the Arrhenius equation the ordinate is the log of the material life. The abscissa is the reciprocal of the absolute temperature. The linear curves obtained with the Arrhenius plot over- come the deficiency of most of the standard tests, which provide only one point and indicate no direction in which to extrapolate. Moreover, any change in any aspect of the material or the environment could alter the slopes of their curves. Therein lies the value of this method. [...]... 6/10 Ferrite 83 2,500 5,000 0.4 63 0.507 0.568 0. 638 0. 732 0.952 540 4 93 784 6 83 440 525 Polypropylene Carbon powder 2,500 5,000 1.100 1.140 1.970 6. 230 6.920 8.660 114 40 87 36 63 29 Nylon 6/6 Glassfiber 15, carbon powder 2,500 5,000 2.160 2.400 2.510 116 104 100 Nylon 6 Glass beads 30 1,250 5,000 0.140 0 .32 0 0 .36 8 0.290 0.650 0.750 8 93 391 862 38 5 34 0 33 3 Plustic Strain (%) hours 7 0 100 1,OOO (I@ psi)... 90 Plastics Engineered Product Design Table 2.2 Examples o f reinforced thermoplastics flexural creep data Apparent modulus Stress,psi Nylon 6/6 Glass fiber 15, mineral 25 2,500 5,000 0.555 0.6 23 0.709 0.8 23 0.967 1.140 450 401 607 517 35 3 439 Polyester (PBTJ Glass fiber 15, mineral 25 2,500 5,000 0.452 0.470 0.482 0.6 93 0.742 0.819 5 53 532 721 674 519 610 Nylon 6/10 Ferrite 83 2,500 5,000 0.4 63 0.507... 80 Plastics Engineered Product Design With certain plastics, particularly high performance RPs, there can be two or three moduli Their stress-strain curve starts with a straight line that results in its highest E, followed by another straight line with a lower S, and so forth To be conservative providing a high safety factor the lowest E is used in a design, however the highest E is used in certain designs... Viscosity In addition to its behavior in viscoelastic behavior in plastic products, viscosity of plastics during processing provides another important relationship to product performances (Chapter 1).Different terms are used to identie viscosity characteristics that include methods to detcrmine 72 Plastics Engineered Product Design viscosity such as absolute viscosity, inherent viscosity, relative... of the material that has provided knowledge about the effect of molecular structure on the properties of plastics, particularly in the case of amorphous plastics in a 74 Plastics Engineered Product Design rubbery state as well as extending knowledge concerning the complex behavior of crystalline plastics Studies illustrate how experimental data can be applied to a practical example of the long-time... (1 + V ) (2-12) This calculation that is true for most metals, is generally applicable to 86 Plastics Engineered Product Design Figure 2.9 Theoretical approaches to shear stress behavior SHEARING LOAD plastics However, this calculation does not apply with the inherently nonlinear, anisotropic nature of most plastics, particularly the fiberreinforced and liquid crystal ones Torsion Stress-Strain Shear... with plastics Residual Stress It is the stress existing in a body at rest, in equilibrium, at uniform temperature, and not subjected to external forces Often caused by the stresses remaining in a plastic part as a result of thermal and/or mechanical treatment in fabricating parts Usually they are not a problem in the finished product However, with excess stresses, the 88 Plastics Engineered Product Design. ..70 Plastics Engineered Product Design This method requires extensive test data but considerably more latitude is obtained and more materials obey the rate theory The method can also be used to predict stress-rupture of plastics as well as the creep characteristics of a material The assumption is made that the physical... higher or lower than compressive strengths Since most plastics exhibit some yielding or nonlinearity in their tensile S-S curve, there is a shift from triangular stress distribution toward rectangular distribution when the product is subject to bending This behavior with plastics is similar to that when designing in steel and also for ultimate design strength in concrete Shifts in the neutral axis... sides, etc.) are used to 84 Plastics Engineered Product Design - I I _ ~ - - ~ - *m_l _u _ "i",f~r? 2.8 Stress-strain tensile and compressive response tends to be similar COMPRESSNE TENSILE eliminate the flowering action that could provide inaccurate readings that in turn influence results by usually making them stronger The majority of tests to evaluate the characteristics of plastics are performed in . 60 Plastics Engineered Product Design overall flow chart that goes from the product concept to product release. Use is made of the optimization. technically in balance within the limits and is feasible in meeting product requirements. 62 Plastics Engineered Product Design Design Approach The acquisition of analytical techniques and practical. insight into the behavior of the system and to suggest candidate designs. These designs 64 Plastics Engineered Product Design are then verified and refined in more complex models, using