The Behavior of Structures Composed of Composite Materials Part 3 docx

49 In the above, is the displacement and From elementary strength of materials the constant of proportionality between the shear stress and the angle is the shear modulus in the plane. From the theory of elasticity From Equation (2.10), or Hence, Similarly, 50 Thus, all components have now been related to mechanical properties, and it is seen that to characterize a three dimensional orthotropic body, nine physical quantities are needed (that is and and using Equation (2.17). However, because of (2.17) only six separate tests are needed to obtain the nine physical quantities. The standardized tests used to obtain these anistropic elastic constants are given in ASTM standards, and are described in a text by Carlsson and Pipes [5]. For convenience, the compliance matrix is given explicitly as: 2.4 Methods to Obtain Composite Elastic Properties from Fiber and Matrix Properties There are several sets of equations for obtaining the composite elastic properties from those of the fiber and matrix materials. These include those of Halpin and Tsai [6], Hashin [7], and Christensen [8]. In 1980, Hahn [9] codified certain results for fibers of circular cross section which are randomly distributed in a plane normal to the unidirectionally oriented fibers. For that case the composite is macroscopically, transversely isotropic, that is and where in the parentheses the quantity could be E, G, or hence, the elastic properties involve only five independent constants, namely and For several of the elastic constants, Hahn states that they all have the same functional form: 51 where for the elastic constant P, the and are given in Table 2.2 below, and where and are the volume fractions of the fibers and matrix respectively (and whose sum equals unity): The expressions for and are called the Rule of Mixtures. In the above is the plane strain bulk modulus, and Also, the are given as follows: The shear modulus of the matrix material, if isotropic, is given by The transverse moduli of the composite, are found from the following equation: where The equations above have been written in general for composites reinforced with anisotropic fibers such as some graphite and aramid (Kevlar) fibers. If the fibers are isotropic, the fiber properties involve and where In that case also becomes 52 Hahn notes that for most polymeric matrix structural composites, If that is the case then the parameters are approximately: Finally, noting that for most epoxies, then and Also, the Poisson’s ratio, can be written as where is the fiber Poisson’s ratio and for see Table 2.2. The above equations along with Equation (2.17) provide the engineer with the wherewithal to estimate the elastic constants for a composite material if the constituent properties and volume fractions are known. In a few instances only the weight fraction of the fiber, is known. In that case the volume fraction is obtained from the following equation, where is the weight fraction of the matrix, and and are the respective densities: For determining the composite elastic constants for short fiber composites, hybrid composites, textile composites, and very flexible composites, Chou [10] provides a comprehensive treatment. 53 2.5 Thermal and Hygrothermal Considerations In the previous two sections, the elastic relations developed pertain only to an anisotropic elastic body at one temperature, that temperature being the "stress free" temperature, i.e. the temperature at which the body is considered to be free of stress if it is under no mechanical static or dynamic loadings. However, in both metallic and composite structures changes in temperature are commonplace both during fabrication and during structural usage. Changes in temperature result in two effects that are very important. First, most materials expand when heated and contract when cooled, and in most cases this expansion is proportional to the temperature change. If, for instance, one had a long thin bar of a given material then with change in temperature, the ratio of the change in length of the bar, to the original length, L, is related to the temperature of the bar, T, as shown in Figure 2.5. Mathematically, this can be written as where is the coefficient of thermal expansion i.e., the proportionality constant between the "thermal" strain and the change in temperature, from some reference temperature at which there are no thermal stresses or thermal strains. The second major effect of temperature change relates to stiffness and strength. Most materials become softer, more ductile, and weaker as they are heated. Typical plots of ultimate strength, yield stress and modulus of elasticity as functions of temperature are shown in Figure 2.6, In performing a stress analysis, determining the natural frequencies, or finding the buckling load of a heated or cooled structure one must use the strengths and the moduli of elasticity of the material at the temperature at which the structure is expected to perform. In an orthotropic material, such as a composite, there can be up to three different coefficients of thermal expansion, and three different thermal strains, one in each of the orthogonal directions comprising the orthotropic material [Equation (2.28) would then have subscripts of 1, 2 and 3 on both the strains and the coefficients of thermal expansion]. Notice that, for the primary material axes, all thermal effects are dilatational only; there are no thermal effects in shear. 54 Some recent general articles and monographs on thermomechanical effects on composite material structures include those by Tauchert [11], Argyris and Tenek [12], Turvey and Marshall [13], Noor and Burton [14] and Huang and Tauchert [15]. During the mid-1970’s another physical phenomenon associated with polymer matrix composites was recognized as important. It was found that the combination of high temperature and high humidity caused a doubly deleterious effect on the structural performance of these composites. Engineers and material scientists became very concerned about these effects, and considerable research effort was expended in studying this new phenomenon. Conferences [16] were held which discussed the problem, and both short range and long range research plans were proposed. The twofold problem involves the fact that the combination of high temperature and high humidity results in the entrapment of moisture in the polymer matrix, with attendant weight increase and more importantly, a swelling of the matrix. It was realized [17] that the ingestion of moisture varied linearly with the swelling so that in fact 55 where is the increase from zero moisture measured in percentage weight increase, and is the coefficient of hygrothermal expansion, analogous to the coefficient of thermal expansion, depicted in Equation (2.28). This analogy is a very important one because one can see that the hygrothermal effects are entirely analogous mathematically to the thermal effect. Therefore, if one has the solutions to a thermoelastic problem, merely substituting for or adding it to the terms provides the hygrothermal solution. The test methods to obtain values of the coefficient of hygrothermal expansion are given in [18]. The second effect (i.e. the reduction of strength and stiffness) is also similar to the thermal effect. This is shown qualitatively in Figure 2.7. Dry polymers have properties that are usually rather constant until a particular temperature is reached, traditionally called by polymer chemists the "glass transition temperature," above which both strength and stiffness deteriorate rapidly. If the same polymer is saturated with moisture, not only are the mechanical properties degraded at any one temperature but the glass transition temperature for that polymer is significantly lower. As a quantitative example, Figure 2.8 clearly shows the diminution in tensile and shear strength due to a long term hygrothermal environment. Short time tensile and shear tests were performed on random mat glass/polyester resin specimens. It is clearly seen that there is a significant reduction in tensile strength, and a 29.3% and a 37.1% reduction in ultimate shear strength of these materials over a day soak period. If these effects are not accounted for in design analysis, catastrophic failures can and have occurred. 56 Thus, for modern polymer matrix composites one must include not only the thermal effects but also the hygrothermal effects or the structure can be considerably under designed, resulting in potential failure. Thus, to deal with the real world of polymer composites, Equation (2.12) must be modified to read where in each equation j = 1 - 6. Two types of equations are shown above because in the primary materials system of axes ( i,j = 1, 2, , 6) both thermal and hygrothermal effects are dilatational only, that is, they cause an expansion or contraction, but do not affect the shear stresses or strains. This is important to remember. Although the thermal and moisture effects are analogous, they have significantly different time scales. For a structure subjected to a change in temperature that would require minutes or at most hours to come to equilibrium at the new temperature, the same structure would require weeks or months to come to moisture equilibrium (saturation) if that dry structure were placed in a 95-100% relative humidity environment. Figure 2.9 illustrates the point, as an example. A 1/4" thick random mat glass polyester matrix material requires 49 days of soak time at 188°F and 95% relative humidity to become saturated. 57 Recently, Woldesenbet [19] soaked a large number of IM7/8551-7 graphite epoxy unidirectional test pieces, some in room temperature water to saturation. For the 1/4" diameter by 3/8" long cylinders soaked at room temperature the time required to reach saturation was 55 weeks. Other test pieces were soaked at an elevated temperature to reduce soak time. For additional reading on this subject, see Shen and Springer [20]. 2.6 Time-Temperature Effects on Composite Materials In addition to the effects of temperature and moisture on the short time properties discussed above, if a structure is maintained under a constant load for a period of time, then creep and viscoelastic effects can become very important in the design and analysis of that structure. The subject of creep is discussed in numerous materials science and strength of materials texts and will not be described here in detail. Creep and viscoelasticity can become significant in any material above certain temperatures, but can be particularly important in polymer matrix materials whose operating temperatures must be kept below maximum temperatures of 250°F, 350°F, or in some cases 600°F for short periods of time, dependent upon the specific polymer material. See Christensen [21]. From a structural mechanics point of view, almost all of the viscoelastic effects occur in the polymer matrix, while little or no creep occurs in the fibers. Thus, the study of creep in the polymeric materials, which comprise the matrix, provides the data necessary to study creep in composites. Jurf and Vinson [22] experimentally studied the effects of temperature and moisture (hygrothermal effects) on various epoxy materials (FM 73M and FM 300M adhesives). They established that at least for some epoxy materials it is possible to construct a master creep curve using a temperature shift factor, and established the fact that a moisture shift factor can also be employed. The importance of this is that by these experimentally determined temperature and moisture 58 shift factors, for the shear modulus of the epoxy, the results of short time creep tests can be used for a multitude of time/temperature/moisture combinations over the lifetime and environment of a structure comprised of that material. Wilson [23,24] studied the effects of viscoelasticity on the buckling of columns and rectangular plates and found that significant reductions of the buckling loads can occur. Wilson found that for the materials he studied, the buckling load diminished over the first 400 hours, then stabilized at a constant value. However that value may be a small fraction of the elastic buckling load if the composite properties in the load direction were matrix dominated properties (described later in the text). Wilson also established that for the problems studied it was quite satisfactory to bypass the complexities of a full-scale viscoelastic analysis using the Correspondence Principle and Laplace transformations. The use of the appropriate short time stiffness properties of the composite experimentally determined with specimens that have been held at the temperature and until the time for which the structural calculations are being made. 2.7 High Strain Rate Effects on Material Properties Another consideration in the analysis of all composite material structures is the effect of high strain rate on the strength and stiffness properties of the materials used. Most materials have significantly different strengths, moduli, and strains to failure at high strain rates compared to static values. However most of the major finite element codes such as those which involve elements using hours of computer time to describe underwater and other explosion effects on structures, still utilize static material properties. High strain rate properties of materials are sorely needed. Some dynamic properties have been found, and test techniques established. For more information see Lindholm [25], Daniel, La Bedz, and Liber [26], Nicholas [27], Zukas [28], and Sierakowski [29,30], Rajapakse and Vinson [31], and Abrate [32,33]. Vinson and his collegues have found through testing over thirty various composite materials over the range of strain rates tested up to 1600/sec, that in comparing high strain rate values to static values, the yield stresses can increase by a factor up to 3.6, the yield strains can change by factors of 3.1, strains to failure can change by factors up to 4.7, moduli of elasticity can change by factors up to 2.4, elastic strain energy densities can change by factors up to 6, while strain energy denstities to failure can change by factors up to 8.1. Thus the use of static material properties to analyze and design structures subjected to impact, explosions, crashes, or other dynamic loads should be carefully reviewed. [...]... elsewhere [2, 3, 4, 6, 7] The effects of transverse shear deformation, shown through the inclusion of the and relations shown in Equations (2 .32 ) and (2 .34 ), must be included in composite materials, because in the fiber direction the composite has many of the mechanical properties of the fiber itself (strong and stiff) while in the thickness direction the fibers are basically ineffective and the shear... Laminae of Composite Materials Almost all practical composite material structures are thin in the thickness direction because the superior material properties of composites permit the use of thin walled structures Many polymeric matrix composites are made in the form of a uniaxial set of fibers surrounded by a polymeric matrix in the form of a tape several inches wide termed as a "prepreg." The basic... Composite Materials and Their Use in Structures Applied Science Publishers, London 3 Vinson, J.R (19 93) The Behavior of Shells Composed of Isotropic and Composite Materials, Kluwer Academic Publisher, Dordrecht, The Netherlands 4 Shames, I.H (1975) Introduction to Solid Mechanics New York: Prentice-Hall, Inc 5 Carlsson, L.A and Pipes, R.B (1996) Experimental Characterization of Advanced Composite Materials, ... comprising the equations of elasticity will be considered: the strain-displacement relations, the equilibrium equations and the compatibility equations Consider a laminate composed of N laminae For the lamina of the laminate, Equation (2. 43) can be written as: 67 where all of the above matrices must have the subscript k due to the material and its orientation for each particular lamina with respect to the. .. be considered further Note that without the hygrothermal terms, the strain-curvature matrix at the right in Equation (2. 53) would suffice for the entire laminate independent of orientation, because the displacements, and strains are continuous over the thickness of the laminate In that case the subscript k on that matrix would not be needed However, even though there is continuity of the mid-surface... dominated by the weaker matrix material Similarly, because quite often the matrix material has much higher coefficients of thermal and hygrothermal expansion and thickening and thinning of the lamina cannot be ignored in some cases Hence, without undue derivation, the Equations (2 .32 ) through (2 .34 ) are modified to be: 61 where Please note the introduction of the factor of as noted in the strain expressions,... (1976) On the Hygrothermal Response of Laminated Composite Systems, Journal of Composite Materials, April, pp 130 148 18 ASTM Standard D5229-92 Test Method for Moisture Absorption Properties and Equilibrium Conditioning of Polymer Matrix Composite Materials 19 Woldesenbet, E (1995) High Strain Rate Properties of Composites, Ph.D Dissertation, Department of Mechanical Engineering, University of Delaware,... strains in either coordinate system 2.9 Laminate Analysis In the previous section the generalized constitutive equations for one lamina of a composite material were formulated Many structures of composite materials including sandwich structures are composed of numerous laminae, which are bonded and/or cured together In fact, over and above the superior properties in strength and stiffness that composites... and shell structures these quantities vary in both the x and y directions 71 On the plate shown in Figure 2. 13, the positive directions of all the stress resultants and stress couples are shown, consistent with the definitions of the quantities given in Equation (2.54) For a laminated plate, the stress components can be integrated across each lamina, but must then be added together across the laminae... accurate calculations one can use the simpler classical form of where the are defined in Equation (2.44), but one can use the of Equation (2.41) instead of Equation (2.40) for consistency with the simpler expressions above One interesting variation of the above classical quantities of Equation (2. 43) resulted when Tsai and Pagano [37 ] rewrote many of the quantities in terms of material invariants and trigonometric . 49 In the above, is the displacement and From elementary strength of materials the constant of proportionality between the shear stress and the angle is the shear modulus in the plane. From the theory. shown through the inclusion of the and relations shown in Equations (2 .32 ) and (2 .34 ), must be included in composite materials, because in the fiber direction the composite has many of the mechanical. material then with change in temperature, the ratio of the change in length of the bar, to the original length, L, is related to the temperature of the bar, T, as shown in Figure 2.5. Mathematically,