## High performance

As reviewed throughout this book the high performance materials are engineering plastics such as polycarbonate, nylon, acetal, and reinforced plastic (RP). Data on these plasties are provided throughout this book. In this section information on RPs is presented since they can provide a special form of high performance material that provides a designer with different innovative latitudes of performances than usually reviewed in textbooks.

### Reinforced Plastic

They are strong, usually inert materials bound into a plastic to improve its properties such as strength, stiffness/modulus of elasticity, impact resistance, reduce dimensional shrinkage, etc. (Figs 2.2, 2.23, & 2.24). They include fiber and other forms of material. There are inorganic and organic fibers that have the usual diameters ranging from about one to over 100 micrometers. Properties differ for the different types, diameters, shapes, and lengths. To be effective, reinforcement must form a strong adhesive bond with the plastic; for certain reinforcements special cleaning, sizing, etc. treatments are used to improve bonds. A microscopic view of an RP reveals groups of fibers surrounded by the matrix.

In general adding reinforcing fibers significantly increases mechanical properties. Particulate fillers of various types usually increase the modulus, plasticizers generally decrease the modulus but enhance flexibility, and so on. These reinforced plastics (RPs) can also be called composites. However the name composites literally identifies thousands of different combinations with very few that include the use of plastics. In using the term composites when plastics are involved the more appropriate term is plastic composite.

Figure 2.?3 RPs tensile S-S data (Courtesy of Plastics FALLO)

05 200,000

600,000

05 200,000

600,000

STRAIN, INCHES /INCH

STRAIN, INCHES /INCH

Fig u re 2 24 Properties of RPs and other materials (Courtesy of Plastics FALLO)

SPECIFIC STRENGTH

SPECIFIC MODULUS

CARBON/ EPOXY

GLASS/ EPOXY

WOOD

ALUMINUM

### STEEL

Types of reinforcements include fibers of glass, carbon, graphite, boron, nylon, polyethylene, polypropylene, cotton, sisal, asbestos, metals, whiskers, etc. Other types and forms of reinforcements include bamboo, burlap, carbon black, platelet forms (includes mica, glass, and aluminum), fabric, and hemp. There are whiskers that are metallic or nonmetallic single crystals (micrometer size diameters) of ultrahigh strength and modulus. Their extremely high performances (high modulus of elasticity, high melting points, resistance to oxidation, low weights, etc.) are attributed to their near perfect crystal structure, chemically pure nature, and fine diameters that minimize defects. They exhibit a much higher resistance to fracture (toughness) than other types of reinforcing fibers (Chapter 1).

The advanced RP (ARP) refers to a plastic matrix reinforced with very high strength, high modulus fibers that include carbon, graphite, aramid, boron, and S-glass. They can be at least 50 times stronger and 25 to 150 times stiffer than the matrix. ARPs can have a low density (1 to 3 g/cm3), high strength (3 to 7 GPa) and high modulus (60 to 600 GPa).

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 meaningful values.

### Orientation of Reinforcement

RPs behavior is dominated by the arrangement and the interaction of the stiff, strong reinforcing fibers with the less stiff, weaker plastic matrix. The fiber arrangement determines the behavior of RPs where a major advantage is that directional properties can be maximized. Arrangements include the use of woven (with different weaves) and nonwoven (with different lengths and forms) fabrics.

Design theories of combining actions of plastics and reinforcement arrangements have been developed and used successfully. Theories are available to predict overall behavior based on the properdes of fiber and matrix. In a practical design approach, the behavior can use the original approach analogous to that used in wood for centuries where individual fiber properties are neglected; only the gross properties, measured at various directions relative to the grain, are considered. This was the initial design evaluation approach used during the 1940s.

Orientation Terms

Orientation terms of RP directional properties include the following:

Anisotropic construction RP properties are different in different directions along the laminate flat plane.

Balanced construction RP in which properties are symmetrical along the laminate flat plane.

Bidirectional construction RP with the fibers oriented in various directions in the plane of the laminate usually identifying a cross laminate with the direction 90° apart.

Heterogeneous construction RP material's composition varies from point to point in a heterogeneous mass.

Homogeneous construction Uniform RP.

Isotropic construction RPs having uniform properties in all directions along the laminate flat plane.

Nonisotropic construction RP does not have uniform properties in all directions.

Orthotropic construction RP having mutually perpendicular planes of elastic symmetry along the laminate flat plane.

Unidirectional, construction Refers to fibers that are oriented in the same direcdon (parallel alignment) such as filament-winding, pultrusion, unidirecdonal fabric laminate, and tape.

RPs can be constructed from a single layer or built up from muldple layers using fiber preforms, nonwoven fabrics, and woven fabrics. In many products woven fabrics are very practical since they drape better over 3-D molds than constructions that contain predominandy straight fibers. However they include kinks where fibers cross. Kinks produce repetitive variations and induce local stresses in the direction of reinforcement with some sacrifice in properties. Regardless, extensive use of fabrics is made based on their advantages.

The glass content of a part has a direct influence on its mechanical properties where the more glass results in more strength. This relates to the ability to pack the reinforcement. Fiber content can be measured in percent by weight of the fiber portion (wt%) or percent by volume (vol%). (Fig. 2.25) When content is only in percent, it usually refers to wt%. Depending on how glass fibers are arranged content can range from 65 to 95.6 wt% or up to 90.8 vol%. When one-half of the strands are placed at right angles to each half, glass loadings range from 55 to 88.8 wt% or up to 78.5 vol% (Fig. 2.26).

### Basic Design Theory

In designing RPs, certain important assumptions are made so that two materials act together and the stretching, compression, twisting of fibers and of plastics under load is the same; that is, the strains in fiber and plastic are equal. Another assumption is that the RP is elastic, that is, strains are direcdy proportional to the stress applied, and when a load is removed the deformation disappears. In engineering terms, the material obeys Hooke's Law. This assumption is a close approximation to the actual behavior in direct stress below the proportional limit, particularly in tension, where the fibers carry essentially all the stress. The assumption is possibly less valid in shear where the plastic carries a substantial portion of the stress.

In this analysis it is assumed that all the glass fibers are straight; however, it is unlikely that this is true, particularly with fabrics. In practice, the load is increased with fibers not necessarily failing at the same time. Values of a number of elastic constants must be known in addition to strength properties of the resins, fibers, and combinations. In this analysis, arbitrary values are used that are low for elastic constants and strength values. Any values can be used; here the theory is illustrated.

Weight to volume relation example Fiber arrangement for filament wound fabricated products influences properties

Weight to volume relation example Fiber arrangement for filament wound fabricated products influences properties

0.04

0 20 40 60 80 100 Per cent glass by weight or volume c &

0.05

0.06

0.04

0 20 40 60 80 100 Per cent glass by weight or volume

Any material, when stressed, stretches or is otherwise deformed. If the plastic and fiber are firmly bonded together, the deformation is the same. Since the fiber is more unyielding, a higher stress is developed in the glass than the plastic. If the stress-strain relationships of fiber and plastic are known, the stresses developed in each for a given strain can be computed and their combined action determined. Fig. 2.27 stressstrain (S-S) diagrams provide the basis for this analysis; it provides related data such as strengths and modulus.

These S-S diagrams may be applied to investigate a rod in which half of the volume is glass and the other half is plastic. If the fibers are parallel to the axis of the rod, at any cross-section, half of the total is fiber with half plastic. If the rod is stretched 0.5%, the S-S diagrams show that the glass is stressed to 50,000 psi (345 MPa), resin B at 7,500 psi (52 MPa), and resin C at 2,500 psi (17 MPa). If the rod has a total cross-section of Vi in2, the glass is 'A in2. The total load on the glass is 'A x 50,000 or 12,500 lb. Similarly resin B is 1,875 lb and resin C is 625 lb.

The load required to stretch the rod made of resin B becomes the sum of glass and resin load or 14,375 lb. With resin C the load is 13,125 lb.

The foregoing can be put into the form of an equation:

Resin C

14,375 lb 28,750 psi

13,125 lb 26.250 psi

Average values of modulus of elasticity of the entire cross-section may be computed by dividing c by the strain. The strain is 0.5%, therefore the two average values of E of the rod, incorporating resins B and C, are 5.75 x 106 psi and 5.35 x 106- psi, respectively.

For a cross-section made up of a number of different materials, Eq. (215) may be generalized to:

in which c is the tensile strength and Aj the cross-sectional area of any component of the cross-section. This equation can be still further generalized to include tension, compression, and shear:

in which S, is the strength property of the cross-sectional area Aj, and S is the mean strength property over the entire cross-section A.

Similar to finding the overall modulus of a cross-section, the equation becomes:

in which E is the overall modulus of elasticity, A the total cross-section, and E; the modulus of elasticity corresponding to the partial cross-sectional area A;. For shear modulus G the equation becomes:

Fiber Strength Theory

The deformation and strength of filamentary structures subjected to combined loading can be theoretically predicted using experimentally-determined intrinsic stiffnesses and strength of the individual constituent layers. In order to have an integrated material and structure design, the gross properties as functions of the micromechanical parameters represent an important issue on the continuing and expanding use of RPs. It has been established, both in theory and experiment, that four principal elastic moduli and three principal strengths govern the deformation and strength of unidirectional fiber RPs. With the aid of a yield condition, the initial failure of filamentary structures can be predicted. After the initial failure, the structure may carry additional loads. An analysis of a partially failed or degraded structure can be used to predict the ultimate deformation and strength.

With an understanding of the gross behavior of a filamentary structure, a proper assessment of the mechanical and geometric properties of the constituent materials is possible. In particular, the use of fiber strength, the binding resin matrix, and the interface may be placed in a perspective based on the results of a mathematical analysis. They provide accurate guidelines for the design of RPs.

A better understanding exists of the elastic stiffness of filamentary materials than of the strengths. The generalized Hooke's law is usually accepted as the governing equation of the linear elastic deformation of RP materials. The simultaneous or sequential modes of deformation and fracture are difficult to describe from the phenomenological standpoint. In general, a strength theory on one criterion will not be sufficient to cover the entire range of failure modes of RP. In addition, fabrication variables and test methods are also known to introduce uncertainties in strength data that makes the verification of theories more difficult.

A macroscopic theory of strength is based on a phenomenological approach. No dircct reference to the mode of deformation and fracture is made. Essentially, this approach employs the mathematical theories of elasticity and tries to establish a yield or failure criterion. Among the most popular strength theories are those based on maximum stress, maximum strain, and maximum work. The maximum stress theory states that, relative to the material symmetry axes x-y, failure of the RP will occur if one of three ultimate strengths is reached. There are three inequalities, as follows:

With negative normal stress components, compressive strengths designated by X and Y must be used:

ar<X

Shear strength S has no directional property and it retains the same value for bodi positive and negative shear stress components.

The maximum strain theory is similar to the maximum stress theory. Associated with each strain component, relative to the material symmetry axes, e^ ey> or e„ there is an ultimate strain or an arbitrary proportional limit, Xe> Ye> or Se, respectively. The maximum strain theory can be expressed in terms of the following inequalities:

ey<Ve ex<Se

Where ^ and are negative, use the following inequalities:

eY<re

The maximum work theory in plane stress takes the following form:

If ov and <3y are negative, compressive strengths X' and Y' should be used in Eq. 2-34, respectively.

In the following reviews, the tensile and compressive strengths of unidirectional and laminated RPs, based on the three theories, is computed and compared with available data obtained from glass fiber-epoxy RPs. The uniaxial strength of unidirectional RPs with fiber orientation 6 can be determined according to the maximum theory. Strength is determined by the magnitude of each stress component according to Eqs. 2-24, 2-25, and 2-26 or Eqs. 2-27 and 2-28. As fiber orientation varies from 0° to 90°, it is only necessary to calculate the variation of the stress components as a function of 6. This is done by using the usual transformation equations of a second rank tensor, thus:

where av, av, as are the stress components relative to the material symmetry axes, i.e., ax is the normal stress along the fibers, oy, transverse to the fibers, a„ the shear stress; c^ = uniaxial stress along to the test specimen. Angle 0is measured between the 1-axis and the fiber axis. By combining Eqs. 2-35, 2-36, and 2-37 with 2-24, 2-25, and 2-26, the uniaxial strength is determined by:

o, < XIcos2 6 < V/sin2 9 <S/(sin 0 cos2 0)

The maximum strain theory can be determined by assuming that the material is linearly elastic up to the ultimate failure. The ultimate strains in Eqs. 2-29, 2-30, and 2-31 as well as 2-32 and 2-33 can be related directly to the strengths as follows:

Se = 5/G (2-43) The usual stress-strain relations of orthotropic materials is:

Substituting Eq. 2-35, 2-36, and 2-37 into 2-44, 2-45, and 2-46 results in,

p— (cos2 9- v12 sin2 0)ot tn e¥ = (sin2 0 - v21 cos2 0)o, t22

Finally, substituting Eqs. 2-47, 2-48, and 2-49 and 2-41, 2-42, and 2-43 into Eqs. 2-29, 2-30, and 2-31, and after rearranging, one obtains the uniaxial strength based on the maximum theory:

The maximum work theory can be obtained direcdy by substituting Eq. 2-35, 2-36, and 2-37 into Eq. 2-34:

Determining the strength of laminated RPs is no more difficult conceptually than determining the strength of unidirectional RPs. It is only necessary to determine the stress and strain components that exist in each constituent layer. Strength theories can then be applied to ascertain which layer of the laminated composite has failed. Stress and strain data is obtained for E-glass-epoxy, and cross-ply and angle-ply RPs. Under uniaxial loading, only N, is the nonzero stress resultant and when temperature effect is neglected, the calculations become:

where A' and B' matrices are the in-plane and coupling matrices of a laminated anisotropic composite.

The stress and strain components can be computed from Eqs. 2-54 and 2-55. They can then be substituted into the strength theories, from which the maximum M, the uniaxial stress resultant can be determined. Uniaxial tensile strengths of unidirectional and laminated composites made of E-glass-epoxy systems are obtained. Also, uniaxial axial-com-pressive strengths are obtained. The three strength theories can be applied to the glass-epoxy RP by using the following material coefficients:

The maximum stress theory is shown as solid lines in Fig. 2.28. On the right-hand side of the figure is the uniaxial strength of directional RPs with fiber orientation 0from 0° to 90°; on the left-hand side, laminated RPs with helical angle a from 0° to 90°. Both tensile and compressive loadings are shown. The tensile data are the solid circles and the com-

. sin40

. sin40

I : Maximum stress theory

pressive are squares. Tensile data are obtained from dog-bone specimens. Compressive data are from specimens with uniform rectangular cross-sections.

Figure 2.29 shows the comparison between the maximum strain theory and the same experimental data shown in Fig. 2.28. The formats are similar. Fig. 2.28 shows a comparison between the maximum work theory of the same experimental data as shown in Figs 2.29 and 2.30.

Based on a Tsai review, it shows that the maximum work theory is more accurate than the maximum stress and strain theories. The maximum work theory encompasses the following additional features.

1. There is a continuous variation, rather than segmented variation, of the strength as a function of either the fiber orientation G or helical angle a.

2. There is a continuous decrease as the angles 0 and a deviate from 0°. There is no rise in axial strength, as indicated by the maximum stress and strain theories.

3. The uniaxial strength is plotted on a logarithmic scale and an error of a factor of 2 exists in the strength prediction of the maximum stress and strain theories in the range of 30°.

4. A fundamental difference between the maximum work and the other theories lies in the question of interaction among the failure modes. The maximum stress and strain theories assume that there is no interaction among the three failure modes (axial, transverse, and shear failures).

The angular dependence does not vanish, that should not be the case for an isotropic material.

### Fiber Geometry on Strength

Various investigators have developed mathematical means for determining the efficiency of glass-fiber RPs. In order to analyze the effect of fiber geometry on strength, the fundamental mechanics of RP theory is reviewed. Relationships have been derived to relate the load distribution in an RP to the properties of the individual materials. The derivations are based on the following: (1) stress is proportional to the strain in both materials; (2) resin-fiber bond is efficient, so that resin and fiber are strained an equal amount under load; (3) fibers are straight, continuous, and aligned with the axis of the applied load; and (4) material components are isotropic and homogeneous. The nomenclature used is as follows:

Fc h is

5c Sf

Cf em

Area of comosite, inches2

Total fiber area, inches2

Area of fiber in load direction, inches2

Area of matrix, inches2

Fiber diameter, inches

Modulus of elasticity of composite, psi (tension)

Modulus of elasticity of fiber, psi (tension)

Modulus of elasticity of matrix, psi (tension)

Base strength of the fiber, psi

Base strength of the matrix, psi

Theoretical composite strength, psi

Height of shear plane, inches

Length of shear plane and required overlap of fibers

Load of composite, pounds

Load on fiber, pounds

Load on matrix, pounds

Unit stress in composite, psi

Unit stress in fiber, psi

Unit stress in matrix, psi

Unit strain of composite, inches/inches

Unit strain of fiber, inches/inches

Unit strain of matrix, inches/inches

The derivations are as follows: s = £e

where E is the proportionality constant or the modulus of elasticity. For RPs

and the stress in the fiber and matrix is sf = Efif (2-62)

Thus Eq. 2-62 and 2-63 can be written as sf = EfCc (2-65)

Since the load is equal to unit stress times area, the load on the fiber is:

and the load on the matrix is:

The load applied to the composite, P„is resisted by the resisting loads in the fiber and matrix; therefore the following equation exists:

The ratio of the load carried by the fibers to the applied load is

Pi = Pf (2-70) Pc Pf+ Pm and substituting for ?f and Pm

Equation 2-71 can be further simplified by assuming the composite to have an area of one square inch. Thus:

Equation 2-71 can now be written as:

The ratio of fiber stress to composite stress can be determined by dividing the fiber and composite loads by their respective area, thus if Sr and since Ac = 1

Pf/Af PJAC

It can be concluded from equation 2-73 that the percentage of the applied load carried by the fiber is a function of the relative moduli of matrix and fiber and also a function of the area fiber resisting the applied load. The same statement is true for the ratio of the stress and the fibers to the stress in the composite. By equation 2-75, it is determined that the stress in the fiber increases as E,„/Ef decreases and Af decreases.

Continuous fibers, such as those in filament winding, cross laminates, and cloth laminates, can transmit the applied load or stress from the point of application to the reaction via a continuous load padi. If the fibers are not continuous between the load and the reaction, the matrix must transfer the load from one fiber to the next at the points of discontinuity. Fiber continuity also affects the type of failure of the composite.

With continuous fibers, it can be assumed that the failure will ultimately occur by fracture of the fibers. Discontinuous fibers, on the other hand, can have three other types of failures: (1) fracture of the resin at a weak net section; (2) shear failure in the matrix at the points of discontinuity of the fiber; and (3) failure of the bond between the fibers and the matrix.

The theoretical composite strength is defined as the sum of the strengths of the fiber and matrix materials. This can be written as

(where Aft and Am are part of a unit area) when the composite is assumed to have a unit area. The composite efficiency is the ratio of the composite strength as tested to the simple composite theoretical strength expressed in percent. Thus, test strength of the composite (2-78)

composite efficiency =

theoretical composite strength x 100

The effective fiber stress can be determined from the load in the fiber and the fiber area. The percentage of the applied load that is carried by the fiber is dependent on E^Ef and Af. This load divided by the fiber area is the effective fiber stress. Thus,

## Post a comment