The Behavior of Structures Composed of Composite Materials Part 3 docx

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

In the above, is the displacement and From elementarystrength of materials the constant of proportionality between the shear stress and theangle is the shear modulus in the plane.

From the theory of elasticity

From Equation (2.10), or

Hence,

Similarly,

Thus, all components have now been related to mechanical properties, and it is seenthat to characterize a three dimensional orthotropic body, nine physical quantities are

Equation (2.17) However, because of (2.17) only six separate tests are needed to obtainthe nine physical quantities The standardized tests used to obtain these anistropic elasticconstants 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 propertiesfrom 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 ofcircular cross section which are randomly distributed in a plane normal to theunidirectionally oriented fibers For that case the composite is macroscopically,

parentheses the quantity could be E, G, or hence, the elastic properties involve onlyfive independent constants, namely and

For several of the elastic constants, Hahn states that they all have the samefunctional form:

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

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 casealso becomes

Hahn notes that for most polymeric matrix structural composites, Ifthat 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 thewherewithal to estimate the elastic constants for a composite material if the constituentproperties and volume fractions are known In a few instances only the weight fraction ofthe fiber, is known In that case the volume fraction is obtained from the followingequation, where is the weight fraction of the matrix, and and are the respectivedensities:

For determining the composite elastic constants for short fiber composites, hybridcomposites, textile composites, and very flexible composites, Chou [10] provides acomprehensive treatment

2.5 Thermal and Hygrothermal Considerations

In the previous two sections, the elastic relations developed pertain only to ananisotropic 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 arecommonplace both during fabrication and during structural usage Changes intemperature result in two effects that are very important First, most materials expandwhen 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 materialthen 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 betweenthe "thermal" strain and the change in temperature, from some referencetemperature 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 areshown 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 strengthsand the moduli of elasticity of the material at the temperature at which the structure isexpected to perform

In an orthotropic material, such as a composite, there can be up to three differentcoefficients of thermal expansion, and three different thermal strains, one in each of theorthogonal directions comprising the orthotropic material [Equation (2.28) would thenhave subscripts of 1, 2 and 3 on both the strains and the coefficients of thermalexpansion] Notice that, for the primary material axes, all thermal effects are dilatationalonly; there are no thermal effects in shear

Some recent general articles and monographs on thermomechanical effects oncomposite 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 polymermatrix composites was recognized as important It was found that the combination ofhigh temperature and high humidity caused a doubly deleterious effect on the structuralperformance of these composites Engineers and material scientists became veryconcerned about these effects, and considerable research effort was expended in studyingthis new phenomenon Conferences [16] were held which discussed the problem, andboth short range and long range research plans were proposed The twofold probleminvolves the fact that the combination of high temperature and high humidity results inthe entrapment of moisture in the polymer matrix, with attendant weight increaseand more importantly, a swelling of the matrix It was realized [17] that the ingestion ofmoisture varied linearly with the swelling so that in fact

where is the increase from zero moisture measured in percentage weight increase,and is the coefficient of hygrothermal expansion, analogous to the coefficient ofthermal expansion, depicted in Equation (2.28) This analogy is a very important onebecause 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 hygrothermalsolution The test methods to obtain values of the coefficient of hygrothermal expansionare given in [18]

The second effect (i.e the reduction of strength and stiffness) is also similar to thethermal effect This is shown qualitatively in Figure 2.7 Dry polymers have propertiesthat are usually rather constant until a particular temperature is reached, traditionallycalled by polymer chemists the "glass transition temperature," above which both strengthand stiffness deteriorate rapidly If the same polymer is saturated with moisture, not onlyare the mechanical properties degraded at any one temperature but the glass transitiontemperature for that polymer is significantly lower

As a quantitative example, Figure 2.8 clearly shows the diminution in tensile andshear strength due to a long term hygrothermal environment Short time tensile and sheartests were performed on random mat glass/polyester resin specimens It is clearly seenthat 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 effectsare not accounted for in design analysis, catastrophic failures can and have occurred

Thus, for modern polymer matrix composites one must include not only thethermal effects but also the hygrothermal effects or the structure can be considerablyunder designed, resulting in potential failure.

Thus, to deal with the real world of polymer composites, Equation (2.12) must bemodified 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 significantlydifferent time scales For a structure subjected to a change in temperature that wouldrequire minutes or at most hours to come to equilibrium at the new temperature, the samestructure would require weeks or months to come to moisture equilibrium (saturation) ifthat dry structure were placed in a 95-100% relative humidity environment Figure 2.9illustrates the point, as an example A 1/4" thick random mat glass polyester matrixmaterial requires 49 days of soak time at 188°F and 95% relative humidity to becomesaturated

Recently, Woldesenbet [19] soaked a large number of IM7/8551-7 graphite epoxyunidirectional 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 reachsaturation was 55 weeks Other test pieces were soaked at an elevated temperature toreduce 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 propertiesdiscussed 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 andstrength of materials texts and will not be described here in detail

Creep and viscoelasticity can become significant in any material above certaintemperatures, but can be particularly important in polymer matrix materials whoseoperating 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 polymermaterial See Christensen [21]

From a structural mechanics point of view, almost all of the viscoelastic effectsoccur 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 datanecessary to study creep in composites Jurf and Vinson [22] experimentally studied theeffects of temperature and moisture (hygrothermal effects) on various epoxy materials(FM 73M and FM 300M adhesives) They established that at least for some epoxymaterials 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 Theimportance of this is that by these experimentally determined temperature and moisture

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 andenvironment 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 thatsignificant reductions of the buckling loads can occur Wilson found that for thematerials he studied, the buckling load diminished over the first 400 hours, thenstabilized at a constant value However that value may be a small fraction of the elasticbuckling load if the composite properties in the load direction were matrix dominatedproperties (described later in the text) Wilson also established that for the problemsstudied it was quite satisfactory to bypass the complexities of a full-scale viscoelasticanalysis using the Correspondence Principle and Laplace transformations The use of theappropriate short time stiffness properties of the composite experimentally determinedwith specimens that have been held at the temperature and until the time for which thestructural calculations are being made

2.7 High Strain Rate Effects on Material Properties

Another consideration in the analysis of all composite material structures is theeffect 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 highstrain rates compared to static values However most of the major finite element codessuch as those which involve elements using hours of computer time todescribe underwater and other explosion effects on structures, still utilize static materialproperties High strain rate properties of materials are sorely needed Some dynamicproperties have been found, and test techniques established For more information seeLindholm [25], Daniel, La Bedz, and Liber [26], Nicholas [27], Zukas [28], andSierakowski [29,30], Rajapakse and Vinson [31], and Abrate [32,33]

Vinson and his collegues have found through testing over thirty variouscomposite materials over the range of strain rates tested up to 1600/sec, that in comparinghigh 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 densitiescan change by factors up to 6, while strain energy denstities to failure can change byfactors up to 8.1 Thus the use of static material properties to analyze and designstructures subjected to impact, explosions, crashes, or other dynamic loads should becarefully reviewed

2.8 Laminae of Composite Materials

Almost all practical composite material structures are thin in the thicknessdirection because the superior material properties of composites permit the use of thinwalled structures Many polymeric matrix composites are made in the form of a uniaxialset of fibers surrounded by a polymeric matrix in the form of a tape several inches widetermed as a "prepreg." The basic element in most long fiber composite structures is alamina of fiber plus matrix, all fibers oriented in one direction, made by laying theprepreg tape of a certain length side by side In the next section, 2.9, the stacking ofvarious laminae to form a superior structure termed a laminate will be discussed Inmodern manufacturing methods, such as many liquid injection molding techniques, thefibers are placed in the mold as a “preform” In that case the analyst must decide whetherthe molded composite can best be modeled as one lamina or a laminate Also if acomposite lamina has a thermal gradient across the thickness such that the materialproperties vary significantly from one surface to the other, then the analyst could modelthe composite as a laminate with differing material properties in each lamina

To describe this, consider a small element of a lamina of constant thickness h,wherein the principal material axes are labeled 1 and 2, that is the 1 direction is parallel tothe fibers, the 2 direction is normal to them, and consider that the beam, plate or shellgeometric axes are x and y as depicted in Figure 2.10 For the material axes 1 and 2, the 1axis is always in the direction involving the stiffer and stronger material properties,compared to the 2 direction

The element shown in Figure 2.10 has the stresses shown in the positive

directions consistent with references [2,3,34-36] If one performs a force equilibriumstudy to relate and to and it is exactly analogous to the Mohr'scircle analysis in basic strength of materials with the result that, in matrix form,

and where here and is defined positive as shown in Figure 2.10,and the subscripts CL refer to the classical two-dimensional case only, that is, in the 1-2

plane or the x-y plane only.

Analogously, a strain relationship also follows for the classical isothermal case

However, these classical two-dimensional relationships must be modified to treat

a composite material to include thermal effects, hygrothermal effects, and the effects oftransverse shear deformation treated in detail elsewhere [2, 3, 4, 6, 7] The effects oftransverse shear deformation, shown through the inclusion of the andrelations 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 thefiber itself (strong and stiff) while in the thickness direction the fibers are basicallyineffective and the shear properties are dominated by the weaker matrix material.Similarly, because quite often the matrix material has much higher coefficients of thermaland hygrothermal expansion and thickening and thinning of the lamina cannot beignored in some cases Hence, without undue derivation, the Equations (2.32) through(2.34) are modified to be:

Please note the introduction of the factor of as noted in the strain expressions, because

of the way and are defined in Equation (2.5)

For completeness, the reverse transformations are given

Again, please note that the transformations can be made only with tensor strains.

Hence, from Equation (2.5), it is necessary to divide and by two

If one systematically uses these expressions, and utilizes Hooke’s Law relatingstress and strain, and includes the thermal and hygrothermal effects, one can produce thefollowing overall general equations for a lamina of a fiber reinforced composite material

in terms of the principal material directions (1, 2, 3); see Equations (2.8) - (2.21)

In the above, the quantities are used for the stiffness matrix quantities obtaineddirectly from Equation (2.8) through (2.21) One should also remember that

and hence the coefficients of "two"appearing with the tensor shear strains and above Using the notation ofSloan [36], the stiffness matrix quantities can be written as follows:

* can be found by replacing by in [T].

Incidentally in the above expressions, if the lamina is transversely isotropic, i.e.has the same properties in both the 2 and 3 directions, then

with resulting simplification

For preliminary calculations in design or where great accuracy is not needed,simpler forms [2, 3] for some of the expressions in Equation (2.40) can be used, as shownbelow, with little loss in numerical accuracy:

If these simpler forms are used then one would use the classical form of theconstitutive relations instead of Equation (2.39), neglecting transverse shear deformationand transverse normal stress, i.e., letting and equal zero, thus obtaining

where one should remember also that hence the appearance of the factor of twobefore As stated above for many cases it is sufficient to use Equations (2.41) and(2.42) rather than Equations (2.39) and (2.40) for faster and easier calculation

When the structural axes, x, y and z, are not aligned with the principle materialsaxes, 1, 2, 3, as described in Figure 2.10, then a coordinate transformation is necessary