Stress-Strain Behavior of Plastics Materials

Unfortunately, Hooke s Law does not accurately enough reflect the stress-strain behavior of plastics parts and is a poor guide to good successful design. Assuming that plastics obey Hookean based deformation relationships is a practical guarantee of failure of the part. What will be developed in this chapter is a similar type of basic relationship that describes the behavior of plastics when subjected to load that can be used to modify the deformation equations and predict the performance of a plastics part. UnUke the materials that have been used which exhibit essentially elastic behavior, plastics require that even the simplest analysis take into account the effects of 

In order to analyze these effects, models which exhibit the same type of response to applied forces as the plastics are used which are capable of mathematical formalization. By using appropriate analogy models, the mathematics will accurately reflect the behavior of the real materials. 

The elements that are used in such an analysis are a spring, which represents elastic response since the deflection is proportional to the applied force, and the dashpot which is an enclosed cylinder and piston combination which allows the fluid filling the cylinder to move from in front of the piston to behind the piston through a controlled orifice. The behavior of the dashpot is controlled by Newton s Law of Fluid Flow for a perfect liquid which states that the resistance to flow is proportional to the rate of flow. 

The retarded elastic response which occurs in plastics materials is best represented as a spring and dashpot acting in parallel. The creep or cold flow which occurs in plastics is represented by a dashpot. Accordingly, the combination which best represents the plastics structure would be a spring and dashpot parallel combination in series with a dashpot. The basic elements and the combinations are shown in Fig. 2-1. 

One of the results of the viscoelastic response of polymers is to vary the relationship between stress and strain depending on the rate of stress application. The standard test for many materials to determine 

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The basic understanding of stress-strain behavior of plastic materials is of utmost importance to design engineers. One such typical stress-strain (load-deformation) diagram is illustrated in Figure 2-1. For a better understanding of the stress-strain curve, it is necessary to define a few basic terms that are associated with the stress-strain diagram. 

O. H. Varga, Stress-Strain Behavior of Elastic Materials, Wiley-Interscience, New York, 1966. A. Peterlin, Plastic Deformation of Polymers, Marcel Dekker, New York, 1971. 

As mentioned earlier, there have been many attempts to develop mathematical models that would accurately represent the nonlinear stress-strain behavior of viscoelastic materials. This section will review a few of these but it is appropriate to note that those discussed are not all inclusive. For example, numerical approaches are most often the method of choice for all nonlinear problems involving viscoelastic materials but these are beyond the scope of this text. In addition, this chapter does not include circumstances of nonlinear behavior involving gross yielding such as the Luder s bands seen in polycarbonate in Fig. 3.7. An effort is made in Chapter 11 to discuss such cases in connection with viscoelastic-plasticity and/or viscoplasticity effects. The nonlinear models discussed here are restricted to a subset of small strain approaches, with an emphasis on the single integral approach developed by Schapery. 

Brittleness Brittle materials exhibit tensile stress-strain behavior different from that illustrated in Fig. 2-13. Specimens of such materials fracture without appreciable material yielding. Thus, the tensile stress-strain curves of brittle materials often show relatively little deviation from the initial linearity, relatively low strain at failure, and no point of zero slope. Different materials may exhibit significantly different tensile stress-strain behavior when exposed to different factors such as the same temperature and strain rate or at different temperatures. Tensile stress-strain data obtained per ASTM for several plastics at room temperature are shown in Table 2-3. 

The stress-relaxation behavior of a material is normally determined in either the tensile or the flexural mode. In these experiments, a material specimen is rapidly elongated or compressed to produce a specified strain level and the load exerted by the specimen on the test apparatus is measured as a function of time. Specimens of certain plastics may fail during tensile or flexural stress-relaxation experiments. 

The stress-strain behavior of ceramic polycrystals is substantially different from single crystals. The same dislocation processes proceed within the individual grains but these must be constrained by the deformation of the adjacent grains. This constraint increases the difficulty of plastic deformation in polycrystals compared to the respective single crystals. As seen in Chapter 2, a general strain must involve six components, but only five will be independent at constant volume (e,=constant). This implies that a material must have at least five independent slip systems before it can undergo an arbitrary strain. A slip system is independent if the same strain cannot be obtained from a combination of slip on other systems. The lack of a sufficient number of independent slip systems is the reason why ceramics that are ductile when stressed in certain orientations as single crystals are often brittle as polycrystals. This scarcity of slip systems also leads to the formation of stress concentrations and subsequent crack formation. Various mechanisms have been postulated for crack nucleation by the pile-up of dislocations, as shown in Fig. 6.24. In these examples, the dislocation pile-up at a boundary or slip-band intersection leads to a stress concentration that is sufficient to nucleate a crack. 

Both the elastomeric and the plastic castor oil/polystyrene SINs exhibit significant toughening/ " " The synthesis of these materials is discussed in Sect on 5.5.2. Figure shows the stress-strain behavior of COPEN,... 

Special attention is required when selecting the correct indenter tip. Sharp indenters such as the Berkovich tip indenter have been used by most researchers to measure the hardness and Young s modulus. However, the assumption of the transition from elastic to plastic behavior of the material is not permissible with a sharp-tipped indenter because these indenters create a nominally constant plastic strain impression. With a spherical tip, on the other hand, the depth of penetration increases as the contact stress increases therefore, the response of the elastic to plastic transition and the contact stress—strain property of a material can be determined (He and Swain, 2007). 

Figure 11.4 Stress-strain behavior of three types of polymeric materials. Young s modulus of brittle plastics is often close to 3 x 10° Pa, o/e.

Qi J H and Boyce M C (2004) Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials, J Mech Phys Solids 52 2187-2205. Drozdov A A and Dorfmann A (2003) A micro-mechanical model for the response of filled elastomers at finite strains, Int J Plasticity 19 1037-1067. 

Table 2-1 lists the characteristic features of stress-strain curves as they relate to the polymer properties (7). In some applications it is important for a designer to know the stress-strain behavior of a particular plastic material in both tension and compression. At relatively lower strains, the tensile and compressive stress-strain curves are almost identical. Therefore, at low strain, compressive modulus is equal to tensile modulus. However, at a higher strain level, the compressive stress is significantly higher than the corresponding tensile stress. This effect is illustrated in Figure 2-5. 

In Chapter I the basic concepts of testing are discussed along with the purpose of specifications and standards. Also discussed is the basic specification format and classification system. The subsequent chapters deal with the testing of five basic properties mechanical, thermal, electrical, weathering, and optical properties of plastics. The chapter on mechanical properties discusses in detail the basic stress-strain behavior of the plastic materials so that a clear understanding of testing procedures is obtained. Chapter 7 on... 

Figure 9 illustrates the radial brick joint compression-only behavior. In this example the stress-strain behavior of the refractory material was assumed to remain totally elastic. As shown, a portion of the joint on the hot face end of the radial brick joint is in compression, and a portion on the cold face end of the brick is separated. The circumferential loading is a maximum at the lining hot face and decreases linearly to zero at an internal location of the brick joint. For actual elastic/plastic refractory behavior, the circumferential loading would be nonlinear. The internal location is where the joint begins to separate. This joint behavior can be explained fundamentally by considering the temperature of the various locations in the brick joint compared to the steel shell temperature and the coefficient of thermal expansion of the brick material and the steel shell. 

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