Polymer Engineering Science and Viscoelasticity

Development of synthetic polymers and growth of the polymer industry during the last 100 years has been staggering. The commercial success of polymer-based products has generated a demand such that the total production of plastics (by volume) has exceeded the combined production of all metals for more than 20 years. It has been suggested that Polymer Science evolved from the following five separate technologies: (1) Plastics, (2) Rubbers or Elastomers, (3) Fibers, (4) Surface Finishes, and (5) Protective Coatings, each of which evolved separately to become major industries (Rosen 1993). As a result much of the early development of polymers or plastics was focused on these commercial products and other non-structural uses. The need to develop synthetic rubber due to the interruption of trade routes during WW II served as a catalyst to large scale federal funding for polymer research. This increased effort resulted in better understanding of the nature of polymers as well as improved analytical and experimental approaches to their behavior. In more recent years, however, polymers have become an engineering structural material of choice due to low cost, ease of processing, weight savings, corrosion resistance and other major advantages. In fact modern polymeric adhesives and polymer matrix composites (PMC) or fiber-reinforced plastics (FRP) are today being used in many severe structural environments of the aerospace, automotive and other industries.

The engineering design of structures using polymers requires a thorough knowledge of the basic principles of stress and strain analysis and measurement. Readers of this book should have a fundamental knowledge of stress and strain from a course in elementary solid mechanics and from an introductory course in materials. Therefore, we do not rigorously derive from first principles all the necessary concepts. However, in this chapter we provide a review of the fundamentals and lay out consistent notation used in the remainder of the text. It should be emphasized that the interpretations of stress and strain distributions in polymers and the properties derived from the standpoint of the traditional analysis given in this chapter are approximate and not applicable to viscoelastic polymers under all circumstances. By comparing the procedures discussed in later chapters with those of this chapter, it is therefore possible to contrast and evaluate the differences.

Many materials found in nature are polymers. In fact, the basic molecular structure of all plant and animal life is similar to that of a synthetic polymer. Natural polymers include such materials as silk, shellac, bitumen, rubber, and cellulose. However, the majority of polymers or plastics used for engineering design are synthetic and often they are specifically formulated or “designed” by chemists or chemical engineers to serve a specific purpose. Other engineers (mechanical, civil, electrical, etc.) typically design engineering components from the available materials or, sometimes, work directly with chemists or chemical engineers to synthesize a polymer with particular characteristics. Some of the useful properties of various engineering polymers are high strength or modulus to weight ratios (light weight but comparatively stiff and strong), toughness, resilience, resistance to corrosion, lack of conductivity (heat and electrical), color, transparency, processing, and low cost. Many of the useful properties of polymers are in fact unique to polymers and are due to their long chain molecular structure. These issues will be discussed at length in the next chapter. In this chapter, focus will be on general characteristics, applications and an introduction to the mechanical behavior including elementary concepts of their inherent time dependent or viscoelastic nature.

The discussion in previous chapters has provided a glimpse of the relationship between the molecular structure of polymers and their mechanical behavior. In this chapter the intent is to provide more detailed information about the molecular structure of polymers and the relation of such structure to mechanical performance. Typically materials courses taken by engineering students prior to 1980 contained little, if any, information on the structure of polymers that might be useful in the engineering design of polymer based structures. While now most elementary books on materials do include a chapter or two on polymers they are often omitted on the class syllabus due to the pressures of schedules and/or time constraints. As a result, engineering students often do not obtain a knowledge base that allows the safe design of polymeric structures. All too frequently, the engineering design of structural polymers is based upon principles that are best used for metals. The purpose of the present chapter is to provide a framework for understanding the structure of polymers and hence the structure–property relationships that give rise to their unique mechanical behavior with time, temperature and other environmental parameters as discussed in subsequent chapters. Due to the prevalence of polymers in industrial uses, a general understanding of the concepts outlined in this chapter are essential for an engineer to be able to make informed design decisions on polymeric components and, importantly, to be able to discuss on common ground with synthesis people the type of polymer needed to be produced for a given application.

A review of the basic definitions of stress and strain was given in Chap. 2. It was noted that a linear elastic solid in uniaxial tension or pure shear is governed by Hooke’s law given by, where σ (or τ) is the applied stress, ε (or γ) is the resulting strain, and E (or G) is the elastic modulus and is applicable for many materials under certain circumstances of environment for small stresses and small strains.

As discussed previously, the relation between stress and strain for linear viscoelastic materials involves time and higher derivatives of both stress and strain. While the differential equation method can be quite general, a hereditary integral method has proved to be appealing in many situations. This hereditary integral equation approach is attributed to Boltzmann and was only one of his many accomplishments. In the late nineteenth century, when the method was first introduced, considerable controversy arose over the procedure. Now, it is the method of choice for the mathematical expression of viscoelastic constitutive (stress-strain) equations. For an excellent discussion of these efforts of Boltzmann, see Markovitz (1977).

One of the most important functions of engineering design is to be able to predict the performance of a structure over its design lifetime. Necessarily the mechanical behavior of materials used in a structure must also be known over the intended life of the structure. For engineering design based upon linear elasticity, it is assumed that no intrinsic change in mechanical properties occurs over time. However, the molecular structure of polymers gives rise to mechanical properties that do change over time.

The study of polymer engineering science and viscoelasticity is not complete unless attention is given to the stress (or strain) analysis of important structural problems. These include sets of problems related to viscoelastic materials (e.g., polymers) analogous to those in the first course in solid mechanics (often called strength of materials), courses on structural mechanics (including energy methods, Castigliano’s theorems, etc.), the theory of linear elasticity (stress functions, three dimensional problems, etc.), the theory of linear elastic plates and shells, elastic stability and others. While it is not possible to cover all these topics, it is possible to cover selected problems in several areas to demonstrate common methods of approach such that individuals can continue to explore problems unique to their own area of interest. Hopefully, the brief introduction given here can assist one in solving structural analysis problems for viscoelastic materials provided the necessary background to solve a similar structural analysis problem for an elastic material has been mastered.

The various approaches to the solution of viscoelastic boundary value problems discussed in the last chapter for bars and beams carry over to the solution of problems in two and three dimensions. In particular, if the solution to a similar problem for an elastic material already exists, the correspondence principle may be invoked and with the use of Laplace or Fourier transforms a solution can be found. Such solutions can be used with confidence but one must be cognizant of the general equations of elasticity and the methods of solutions for elasticity problems in two and three dimensions as well as any assumptions that might often be applied. To provide all of the necessary information and background for multidimensional elasticity theory is beyond the scope of this text but the procedures needed will be outlined in the following sections.

Because Young’s modulus of most polymers is relatively low compared to other structural materials such as metals, concrete, ceramics, etc., strains and deformations may be relatively large. A casual glance at the stress-strain response of polycarbonate given in Fig. 3.​7 indicates that the strain at yield is about 5 % and at failure is more than 60 %. Further, examination of the creep response of polycarbonate (Brinson 1973) as discussed in Chap. 11 indicates nonlinear behavior for strains larger than about 3 % and the material begins to neck or yield (Luder’s bands form) for strains larger than about 5 %. Obviously, polycarbonate as well as other polymers with similar behavior cannot be considered to be linear for such circumstances. For these reasons, it is appropriate to have basic understanding of nonlinear processes in order to be able to design structures made of polymeric materials. The intent here is to give basic definitions that will assist in identifying nonlinear effects when they occur and to review several nonlinear approaches.

No text on polymer science and viscoelasticity is complete without a discussion of time-dependent failure and just as with other structural materials, failure must be defined. In this chapter, only failure by a creep to yield or a creep to rupture (separation) will be considered. We will address both the mechanisms of deformation that often precede these types of failures as well as modeling to describe this behavior. The primary focus will be on one-dimensional models but many of the models discussed have been or can be extended to three-dimensions. The procedures to be discussed are not new and are relatively easy to use by the design engineer to make estimates of the time for either yielding or rupture to occur. While no discussion of either viscoelastic fracture mechanics or fatigue crack growth will be given these are very important topics and the reader is referred to Knauss (1973, 2003) for the former and to Kinloch and Young (1983) for the latter for an in-depth discussion of these topics. Fracture based approaches for prediction of time to failure work best when a crack of a known size exists. The same is true for fatigue as a relation between crack growth rates and time to failure can be established. Other approaches provided by damage mechanics (Krajcinovic 1983) and viscoplasticity (Lubliner 1990) provide a more rational but highly mathematical approach to damage and/or failure evolution for three-dimensional stress states and are perhaps best suited for numerical procedures such as the finite element method. Here we restrict ourselves to simpler, analytic approaches to introduce the fundamental issues.