INTRODUCTION
The word “composite” in composite material signifies that two or more materials are combined on a macroscopic scale to form a useful material. The key is the macroscopic examination of material different materials can be combined on a microscopic scale, such as in alloying, but the resulting material is macroscopically homogenous. The advantage of composites is that they usually exhibit the best qualities of there constituents and often some qualities that neither constituent possesses. The properties that can be improved by forming a composite material include. 1. strength 2. stiffness . corrosion resistance 4. wear resistance 5. attractiveness 6. weight 7. fatigue life 8. temperature dependant behavior 9. thermal insulation 10. acoustical insulation Naturally not all of the above properties are improved at the same time nor is there usually any requirement to do so.
CLASSIFICATION OF COMPOSITE MATERIALS
There are three commonly accepted types of composite materials 1 fibrous composite which consist of fibers in a matrix 2 laminated composites which consist of layer of various materials 3 particulate composites which are composite of particles in matrix 1
FIBROUS COMPOSITES
Long fibers in various forms are inherently much stiffer and stronger than the same material in bulk form. For example ordinary plate glass fractures at stresses of only a few thousand PSI yet glass fibers have strengths of 400000 to 700000 PSI in commercially available forms and about 1000000 PSI in laboratory prepared forms. Obviously, then the geometry of a fiber is somehow crucial to the evaluation of its strength and must be considered in structural applications. More properly, the paradox of a fiber having different properties from the bulk form is due to the more perfect structure of a fiber.
The crystals are aligned in the fiber along the fibers axis. Moreover, there are fewer internal defects in fibers than in bulk material. For example, in materials that have dislocations, the fiber form has fewer dislocations than the bulk form.
PROPERTIES OF FIBERS
A fiber is characterized geometrically not only by its very high length to diameter ratio by its near crystal sized diameter. Strength and stiffnesses of a few selected fiber materials are shown in the following table |Fiber or wire |Density,? |Tensile strength S|S/? |Tensile stiffness,E |E/? | |Lb/in3 | |105 in. |106 lb/in2 |107 in. | | | |Lb/in2 | | | | |Aluminum |. 097 |90 |9 |10. 6 |11 | |Titanium |. 170 |280 |16 |16. 7 |10 | |Steel |. 282 |600 |21 |30 |11 | |E-glass |. 92 |500 |54 |10. 5 |11 | |S-glass |. 090 |700 |78 |12. 5 |14 | |Carbon |. 051 |250 |49 |27 |53 | |Beryllium |. 067 |250 |37 |44 |66 | |Boron |. 093 |500 |54 |60 |65 | |Graphite |. 51 |250 |49 |37 |72 | PROPERTIES OF MATRICES Naturally , fibers are of little use unless they are bounded together to take the form of structural elements which can take loads . the binder materials is usually called the matrix . the purpose of the matrix is manifold : support, protection ,stress transfer, etc. Typically, the matrix is considerably lower density ,stiffness and strength than the fibers . However ,the combination of matrix and fiber can have very high strength and stiffness ,yet still have low density .
LAMINATED COMPOSITES
Laminated composites consist of layer of atleast two different material that are bounded together. Lamination is used to combine the best aspect of the constituent layers in order to achieve a more use full material . the properties that can be emphasized by lamination are strength ,stiffness, low weight, corrosion resistance, wear resistance, beauty or attractiveness, thermal insulation, acoustical insulation etc. Laminated fibrous composites are a hybrid class of composites involving both fibrous composites and lamination techniques. A more common name is laminated fiber-reinforced composites.
Here, layers of fibrous reinforced material are built up with the fiber direction of each layer typically oriented in different directions to give different strengths and stiffnesses in the various directions. Thus, the strength and stiffnesses of the laminated reinforced composite can be tailored to the specific design requirements of the structural element being built. Example of laminated fibrous reinforced composites include Polaris missile cases, fiberglass boat hulls, aircraft wings panels and body section, tennis rackets, golf club shafts etc.
PARTICULATE COMPOSITES
Particulate composites consist of particles of one or more materials suspended in a matrix of another material. The particles can be either metallic or nonmetallic as can the matrix.
MECHANICAL BEHAVIOR OF COMPOSITE MATERIALS
Composites have many characteristics that are different from more conventional engineering materials. Some characteristics are merely modification of conventional behavior; others are totally new and require new analytical and experimental procedures. Most common engineering materials are homogeneous and isotropic: A homogeneous body has uniform properties throughout, i. . , the properties are not function of position in the body. An isotropic body has material properties that are the same in every direction at a point in a body, i. e. , the properties are not function of orientation at a point in the body. Bodies with temperature dependent isotropic material properties are not homogenous when subjected to a temperature gradient, but still are isotropic. In contrast, composite materials are often inhomogeneous (or heterogeneous – the two terms will be used interchangeably) and nonisotropic (orthotropic or, more generally, anisotropic):
An inhomogeneous body has nonuniform properties over the body, i. e. , the properties are the function of position in the body. An orthotropic body has material properties that are different in three mutually perpendicular directions at a point in the body and, further, have three mutually perpendicular planes of symmetry. Thus, the properties are a function of orientation at a point in the body. An anisotropic body has material properties that are different in all the directions at a point in the body.
There are no planes of material property symmetry. Again, the properties are a function of orientation at a point in the body. Some composite materials have very simple forms of in homogeneity. For example, laminated safety glass has three layers each of which is homogeneous and isotropic: thus, the in homogeneity of composite is a step function in the direction perpendicular to the plane of the glass. Also, some particulate composites are inhomogeneous, yet isotropic although some are anisotropic. Other composite materials are typically more complex.
Because of inherent heterogeneous nature of composite materials, they are conveniently studied from two points of view: micromechanics and macromechanics: Micromechanics is the study of composite material behavior wherein the interaction of the constituent materials is examined on a microscopic scale. Macromechanics is the study of composite material behavior wherein the material is presumed homogeneous and the effects of constituent materials are detected only as averaged apparent properties of the composites.
Use of both the concept of macromechanics and micromechanics allow the tailoring of composite material to meet a particular structural requirement with little waste of material capability. The ability to tailor a composite material to its job is one of the most significant advantages of a composite over an ordinary material. Tailoring of a composite material yields only the stiffness and strength required in a given direction. In contrast, an isotropic material is, by definition, constrained to have excess strength and stiffness in any direction other than that of the largest requirement.
The inherent isotropy (most often only orthotropy) of composite materials leads to mechanical behavior characteristics that are quite different those of conventional isotropic materials. The behavior of isotropic, orthotropic, and anisotropic materials under loading of normal stress and shear stress is shown in the fig. and discussed in the following paragraphs. For isotropic materials, normal stresses cause extension in the direction of applied stress and contraction in the perpendicular direction. Also, shear stress causes only shearing deformation.
For orthotropic materials, like isotropic materials, normal stress in a principle material direction (along one of the intersection of three orthogonal planes of material symmetry) results in the extension in the direction of the applied stress and contraction perpendicular to the stress. However, due to different properties in the two principal directions, the contraction can be either more or less than the contraction of similarly loaded isotropic material with the same elastic modulus in the direction of the load.
Shear stress causes shearing deformation, but the magnitude of the deformation is independent of the various Young’s moduli and the Poisson’s ratio. That is, the shear modulus of an orthotropic material is unlike isotropic material, not dependent on other material properties. For anisotropic materials, application of normal stress leads not only to extension in the direction of the stress and contraction perpendicular to it, but to shearing deformation. Conversely, shearing stress causes extension and contraction in addition to distortion of shearing deformation.
This coupling between both loading modes and both deformation modes is also characteristics of orthotropic materials subjected to normal stress in nonprincipal material direction. For example, cloth is an orthotropic material composed of two sets of inter woven fibers at right angle to each other. If cloth is subjected to a stress at 45? to a fiber direction, stretching and distortion occur.
LAMINA
A lamina is a flat (sometimes curved as in a shell) arrangement of unidirectional fibers or woven fibers in a matrix.
Two typical laminae are shown in figure along with there principal material axis which are parallel and perpendicular to the fiber directions. (diagrams)
LAMINATES
A laminate is a stack of laminae with various orientations of principal material directions in the laminae as in figure. The layers of a laminate are usually bound together by the same matrix material that is used in laminae. A major purpose of lamination is to tailor the directional dependence of strength and stiffness of a material to match the loading environment of the structural element. Laminates are uniquely suited to this bjective since the principal material directions of each layer can be oriented according to need. For example 6 layers of a 10 layer laminate could be oriented in one direction and the other 4 at 90? to that direction the resulting laminate than has a strength and extensional stiffness roughly 50% higher in one direction than the other.
BASIC TERMINOLOGY A COORDINATE AXES
Consider a thin lamina in which fibers are positioned parallel to each other in a matrix as shown in figure. In order to describe the elastic properties, we first define two right handed coordinate systems, namely 1-2-z system and x-y-z system.
Both 1-2 and x-y axis are in the plane of the lamina and the z axis is normal to this plane. In the 1-2-z system, axis 1 is along the fiber length and represents the longitudinal direction for the lamina and axis 2 is normal to the fiber length and represents the transverse direction for the lamina. Together they constitute principal material directions in the plane of the lamina. In the x-y-z system, x and y axes represents the loading directions. The angle between the positive x axis and the 1 axis is called the fiber orientation angle and is represented by ?.
NOTATION
Fibers and matrix properties are denoted by subscripts f and m, respectively. Lamina properties such as tensile modulus, poisons ratio and shear modulus are denoted by two subscripts. The first subscript represents the loading direction and the second subscript represents the direction in which the particular property is measured . for example ? 12 represents the ratio of strain in direction 2 to the applied strain in direction 1 and ? 21 represents the ratio of strain in direction 1 to the applied strain in direction 2. Stresses and strain are also denoted with double subscripts.
The first of these subscripts represents the direction of the out ward normal to the plane in which the stress component acts. The second subscript represents the direction of the stress component . FIG: Normal stress and shear stress component. ?yy ? zz ?yx ?yz ?xy ?xx ? zy ? xx ?zx X ?yy Z C
TRANSFORMATION OF COORDINATES
Consider the behavior of various tensors under the transformation of coordinates in fig. where a rotation about the z-axis is made. That is, the x,y,z coordinates are transformed to the x’,y’,z’ coordinates where the z direction coincides with the z’ direction.
The direction cosines for this transformation are cos? sin? 0 [? ij] = [T] = -sin? cos? 0 0 0 1 Where ? ij is the cosine of angle between the ith direction in the x’, y’ , z’ system and the jth direction in the x,y,z system, that is, for all transformations ? ij = cos (xi ‘,xj) Thus the transformation of coordinates can be written in index notation as X1’ = ? 11×1+? 12×2+? 13×3 X2’ = ? 21×1+? 22×2+? 23×3 X2’ = ? 31×1+? 32×2+? 33×3 Or in matrix form as X1’ ? 11 ? 12 ? 13 x1 X2’ = ? 21 ? 22 ? 3 x2 X2’ ? 31 ? 32 ? 33 x3 This type of transformation will be used to assist in the definition of various orders of tensors. Each will be defined on the basis of the type of transformation is satisfies.
CONTRACTED NOTATION
Contracted notation is a rearrangement of terms such that the number of indices is reduced although their range increases. For second order tensor, the number of indices is reduced from 2 to 1 and the range increased from 3 to 9. The stress and strains, for example, are contracted as in Table A-1. ?i = Cij ? j ?j = Sij ? i bviously, the number of free indices no longer denotes the order of the tensor. Also the range on the indices no longer denotes the number of spatial dimensions. If the stress and strain tensors are symmetric, then ? ij = ? ji ?ij = ? ji and, therefore, the number of independent stresses and strains is reduced to six. This type of symmetry leads to a reduction of the number of independent components of cij and sij from 81 to 36 in space. The cij and sij can further be shown to be symmetric, that is, cij = cji sij = sji whereupon the number of independent components of cij and sij is further reduced from 36 to 21 in space.
The stiffness matrix is then ( the compliance matrix is similar ): Table A-1: contraction of stresses and strain | | |STRESSES | |Tensor notation |Contracted notation | |? 11 | ? 1 | | |? 2 | |? 22 |? 3 | |? 33 |? 4 | |? 23 = ? 23 |? 5 | |? 31= ? 31 |? 6 | |? 12 = ? 12 |? 7 | ? 32 = ? 32 |? 8 | |? 13 = ? 13 |? 9 | |? 21 = ? 21 | | | | |STRAINS | |Tensor notation |Contracted notation | |? 11 |? 1 | |? 22 |? 2 | |? 33 |? 3 | |? 23 = ? 23 |? 4 | |? 31 = ? 31 |? 5 | |? 12 = ? 12 |? 6 | ? 32 = ? 32 |? 7 | |? 13 = ? 13 |? 8 | |? 21 = ? 21 |? 9 | c11 c12 c13 c14 c15 c16 c21 c22 c23 c24 c25 c26 [c] = c31 c32 c33 c34 c35 c3 c41 c42 c43 c44 c45 c46 c51 c52 c53 c54 c55 c56 c61 c62 c63 c64 c65 c66
INTRODUCTION TO CLASSICAL LAMINATION THEORY
The lamination theory is useful in calculating stresses and strains in each lamina of a thin laminated structure.
Beginning with the stiffness matrix of each lamina, the step by step procedure in lamination theory includes 1 calculation of stiffness matrices for the laminate 2 calculation of midplane strains and curvatures for the laminate due to given set of applied forces and moments 3 calculation of in plane strains ? xx , ? yy and ? xy for each lamina 4 calculation of in plane stresses ? xx , ? yy and ? xy in each lamina ASSUMPTIONS 1 The laminate is thin and wide (width is greater than thickness) 2. The perfect interlaminar bonds exist between various laminas 3.
The strain distribution in the thickness direction is linear 4. All laminas are microscopically homogenous and behave in a linearly elastic manner
STRAIN AND STRESS VARIATION IN A LAMINATE
Knowledge of the variation of strain and stress through the laminate thickness is essential to the definition of the extensional and bending stiffnesses of a laminate. Accordingly, if the laminate is, a line a line originally straight and perpendicular to the middle surface of the laminate is assumed to remain straight and perpendicular to the middle surface when the laminate is extended and bends.
Requiring to the normal to the middle surface to remain straight and normal under deformation is equivalent to ignoring the shearing strains in planes perpendicular to the middle surface, that is ? xz = ? yz = 0 where z is the direction of the normal to the middle surface in the figure. In addition , the normals are presumed to have constant length so that the strain perpendicular to the middle surface is ignored as well that is ? z = 0. The foregoing collection of assumptions of the behavior of single layer that represents the laminate constitutes the familiar kirchoff hypothesis for plates and kirchoff – love for shells.
FIG : Geometry of the deformation in the x-y plane U0 AA B x B W0 zc C C ? ? z D D Zc ? The implication of the kirchoff or the kirchoff love hypothesis on the laminate displacements u, v and w in the x, y ,z directions are derived by use of laminate cross section in the x-z plane shown in figure. The displacement in x direction of point b from the undeformed to the deformed middle surface is u0. since line ABCD remains straight under deformation of laminate, uc = u0 – zc ? …………………… eqn 1 but since , under deformations, line ABCD further remains perpendicular to the middle surface, ? is the slope of the laminate middle surface in the x direction that is, ? = ? w0/? x………………………. eqn 2 Then the displacement u at any point z through the laminate thickess is U = u0 – z ? w0/? x………………………. eqn 3 By similar reasoning the displacement v in the y direction is , v = v0 – z? w0/? y……………………. eqn 4 The laminate strain has been reduced to ? x, ? y and ? xy by the virtue of kirchoff love hypothesis that is ? z = ? xz = ? yz = 0 . For small strains, the remaining strains are defined in terms of displacement as ? = ? u/? x ?y = ? v/? y……………………. eqn 5 ?xy = ? u/? y+ ? v/? x Thus, for the derived displacements u and v in equations eqns 3 and 4 the strains are ?x = ? u0/? x – z ? 2we/? x2 …………………………… eqn 6 ?y = ? v0/? y – z ? 2w0/? y2 ?xy = ? u0/? y + ? v0/? x – 2z ? 2w0/? x? y……………………eqn7 ?x? 0x kx………………….. eqn 8 ?y = ? 0y + z ky ?xy? 0xy kxy Where the middle surface strains are ?0x? u0/? x………………………………. eqn 9 ? 0y =? v0/? y ?0xy? u0/? y + ? v0/? y And the middle surface curvatures are kx? 2w0/? x2 ky= _? 2w0/? y2eqn 10 kxy2 ? 2w0/? x? y
The last term in equation 10 is the twist curvature of the middle surface. Thus, the kirchoffs or kirchoff- love hypothesis has been readily verified to imply a linear variation of strain through the laminate thickness. Because of the strain displacement relation in equation 6 the foregoing strain analysis is only valid for plates. For shells the ? y term in equation 6 must be supplemented by w0/r where r is the radius of circular cylindrical shell , other shells have more complicated strain displacement relations. The stress strain relations in principle material coordinate for a lamina of an orthotropic material under plane stress are 1Q11 Q12 0? 1 ?2 =Q12Q220? 2eqn 11 ?120 0Q66? 12 The reduced stiffnesses Qij, are defined in terms of engineering constants as follows Q11 = E1/(1- ? 12 ? 21) Q12 = ? 12 E2/(1- ? 12 ? 21) = ? 21E2/(1- ? 22 ? 21)eqn12 Q22 = E2/(1- ? 12 ? 21) Q66 = G12 In any other coordinate system in plane of lamina the stresses are ?xQ’11 Q’12 Q’16 ? x ?y =Q’12Q’22 Q’26? yeqn13 ?xyQ’16 Q’26 Q’66? xy where the transformed reduced stiffnesses , Q’ij, are given in terms of reduced stiffnesses as follows Q’11 = Q11cos4? + 2(Q12 + 2Q66) sin2 ? cos2 ? + Q22 sin4 ? Q’12 = (Q11 + Q22 – 4Q66)sin2 ? cos2 ? Q12 (sin4 ? + cos4 ? ) Q’22 = Q11 sin4 ? + 2(Q12 + 2Q66 ) sin2 ? cos2 ? + Q22 cos4 ? ………… eqn Q’16 = (Q11-Q12 -2Q66) sin? cos3? + (Q11-Q12 -2Q66)sin 3 ? cos? Q’26 = (Q11-Q12 -2Q66) sin3 ? cos? + (Q11-Q12 -2Q66)sin ? cos3 ? Q’66 = (Q11+Q22 -2Q12 – Q12 – 2Q66) sin2 ? cos2? + Q66 (sin4? + cos4 ? ) The stress strain relations in arbitrary coordinates , eqn13 are useful in the definition of the laminate stiffnesses because of the arbitrary orientation of the constituent laminae . both eqn 12 and 13 can be thought of stress – strain relations for the kth layer of multilayered laminate thus equation 13 can be written as (? k = (Q’)k (? )k. ………………………………….. eqn15 By substitution of the strain variation through the thickness, eqn 8 in the stress strain relation, eqn 15 the stresses in the kth layer can be expresses in terms of laminate middle surface strains and curvatures as Fig. 2. Hypothetical variation of strain and stress through the laminate thickness. 1 2 3 Z 4
LAMINATE VARIATION OF CHARACTERISTIC VARIATION STRAIN MODULIE OF STRESS
?xQ’11 Q’12 Q’16 ? 0xkx ?y =Q’12Q’22 Q’26? y+ zkyeqn 16 ?xyQ’16 Q’26 Q’66? 0xykz k k since the Q’ij can be different for each layer of the laminate the stress variation through the laminate thickness is not necessarily linear , even thought the strain variation is linear instead typical strain and stress variations are shown in above figure.
RESULTANT LAMINATE FORCES AND MOMENTS
The resultant forces and moments acting on a laminate are obtained by integration of stresses in each layer or lamina through the laminate thickness, for example Nx = t/2 ? – t/2 ? x dz ……………………………. eqn17 Mx = t/2 ? t/2 ? x zdz Actually, Nx is a force per unit length (width) of the cross section of the laminate as shown in the figure. Similarly Mx is the moment per unit length as shown in the figure. The entire collection of force and moments resultant for an N layered laminate is depicted in the following figures and is defined as Fig. 3. In plane forces on a flat lamina Fig. 4 Moments on flat lamina x MX NX NXY
NYX Y NY Nx ? x ? x Ny = t/2 ? – t/2 ? y dz = N? k=1 zk? zk-1 ? y dz …… eqn18 Nxy ? xy ? xy k k And Mx ? x ? x My = t/2 ? – t/2 ? y zdz = N? k=1 zk? zk- ? y zdz ……. eqn19 Mxy ? xy ? xy Where zk and zk-1 are defined in the following figure 5. note that z0 = -t/2 .this force and moment resultants do not depend on z after integration but are functions of x and y . he coordinates in the plane of laminate middle surface . the integration indicated in the eqn 18 and eqn 19 can be rearranged to take advantage of the fact that the stiffness matrix for a laminate is constant within the lamina. Thus the stiffness matrix goes outside the equation over each layer but is within the summation of force and moment resultant over each layer when the laminas stress strain relations eqn16 are substituted Fig 5: Geometry of an n layered laminate. 1 t/2 2 z0 z1z2 middle surface
K Z Zk-1 Zk t ZN-1 ZN N Layer number …………………………………………eqn. 20 Nx Q’11 Q’12 Q’16 ? 0x kx Ny = N? k=1 Q’12 Q’22 Q’26 zk? zk-1 ? 0y dz + zk? zk-1 ky zdz Nxy Q’16 Q’26 Q’66 ? 0xy kxy …………………………………………. eqn21 Mx Q’11 Q’12 Q’16 ? 0x kx My = N? k=1 Q’12 Q’22 Q’26 zk? zk-1 ? 0y zdz + zk? zk-1 ky z2dz Mxy Q’16 Q’26 Q’66 ? 0xy kxy However, we should now recall that ? 0x, ? 0y, ? 0xy. kx,ky and kxy are not functions of z but are middle surface values so can be removed from under the summation sign. Thus eqn 20 and eqn 21. can be written as, Nx A11 A12 A16 ? x B11 B12 B16 kx Ny = A12 A22 A26 ? 0y + B21 B22 B26 ky ………… eqn22 Nxy A16 A26 A66 ? 0xy B16 B26 B66 kxy Mx B11 B12 B16 ? 0x D11 D12 D16 kx My = B12 B22 B26 ? 0y + D21 D22 D26 ky ………eqn23 Mxy B16 B26 B66 ? 0xy D16 D26 D66 kxy Where Aij = N? k=1 (Q’ij)k (zk – zk-1)
Bij = ? N? k=1 (Q’ij)k (z2k – z2k-1) ……………………eqn24 Dij = 1/3 N? k=1 (Q’ij)k (z3k – z3k-1) In eqn 22, eqn23 and eqn24 , the Aij are called extensional stiffnesses, the Bij are called coupling stiffnesses and Dij are called bending stiffnesses. The presence of Bij implies coupling between bending and extensional of a laminates. Thus, it is impossible to pull on a laminate that has Bij terms without at the same time bending and/or twisting the laminate. That is, an extensional force results in not only extensional deformation, but twisting and/or bending of a laminate.
Also, such a laminate cannot be subjected to monomer without at the same time suffering extension of the middle surface.
SPECIAL CASES OF LAMINATE STIFFNESSES CASE
SINGLE LAYERED CONFIGURATIONS
Type 1 : single isotropic layer For a single isotropic layer with material properties , E and ? and thickness t the laminate stiffnesses of eqn 24 reduced to A11 = Et/1- ? 2 = A D11 = Et3/12(1- ? 2) = D A12 = ? AD12 = ? D A22 = A Bij = 0 D22 = Deqn25 A16= 0 D16=0 A26 = 0 D26 = 0 A 66 = Et/2(1+ ? ) = (1- ? /2) A D66 = Et3/24(1+ ? = (1- ? /2)D Whereupon the resultant forces are dependant only on the in surface strains of the laminate middle surfaces and the resultant moments are dependant on the curvatures of the middle surfaces. Nx A ? A 0 ? 0x Ny = ? A A 0 ? 0y Nxy 0 0 (1-? /2)A ? 0xy …………………….. eqn26 Mx D D 0 kx My = D D 0 ky …………………. qn27 Mxy D 0 (1-? /2)D kxy Thus there is no coupling between bending and extension of a single isotropic layer. also note that D = At2/12…………………….. eqn28 Type 2 : single specially orthotropic layer . For a single specially orthotropic layer of thickness t and lamina stiffnesses Qij given by equation 12 , the laminate stiffnesses are A11 = Q11tD11 = Q11t3/12 A12 = Q12tD12 = Q12t3/12 A22 = Q22t Bij = 0 D22 = Q22t3/12………………. eqn29 A16= 0 D16=0 A26 = 0 D26 = 0 A 66 = Q66t D66 = Q26t3/12
Whereupon as with a single isotropic layer the resultant forces depend only on the in surface strains and the resultant moments depend only on the curvatures. Nx A11 A12 0 ? 0x Ny = A12 A22 0 ? 0y ……………eqn30 Nxy 0 0 A66 ? 0xy Mx D11 D12 0 kx My = D21 D22 0 ky …….. ………eqn31 Mxy 0 0 D66 kxy
Type 3: single generally orthotropic layer For a single generally orthotropic layer of thickness t and lamina stiffnesses Qij given by eqn 14 the laminate stiffness are Aij = Q’ijt Bij = 0 Dij = Q’ijt3/12 Again there is no coupling between bending and extension so the force and moment resultants are Nx A11 A12 A16 ? 0x Ny = A12 A22 A26 ? 0y ………………. eqn32 Nxy A16 A26 A66 ? 0xy Mx D11 D12 D16 kx
My = D12 D22 D26 ky …… ………eqn33 Mxy D16 D26 D66 kxy Note in contrast to both anisotropic layer and specially orthotropic layer that extensional forces depend upon shearing strain as well as extensional strain. Type 4 : Single anisotropic layer The only difference in appearance between a single generally orthotropic layer and anisotropic layer is that the latter has lamina stiffnesses, Qij , given implicitly by eqn following eqn34. nd whereas the generally orthotropic layer has stiffnesses, Qij given by eqn. 14 ?1Q11 Q12 Q16? 1 ?2 =Q12Q22 Q26? 2………………eqn 34 ?12Q16 Q26Q66? 12 Aij = Qij t Bij = 0Dij = Qij t3 / 12 And the force and moment resultants are given by eqn32 and eqn33. CASE
SYMMETRIC LAMINATES
For laminates that are symmetric in both geometric and material properties about the middle surface, the general stiffness eqn24, simplify considerably. In particular, because of the symmetry of the (Qij)k and the thicknesses tk , all the coupling stiffnesses, that is the Bij can be shown to zero.
The elimination of coupling between bending and extension has two important practical ramification. First, such laminate are usually much easier to analyze the laminates with coupling. Second, symmetric laminates do not have a tendency to twist from the inevitable thermally induced contractions that occurs during cooling following curing process. Consequently symmetric laminates are commonly used unless special circumstances require an unsymmetric laminate. For example, part of the function of a laminate may be to serve as a heat shield, but the heat comes from only one side , thus, an unsymmtric laminate to be used.
The force and moment resultants for a symmetric laminate are Nx A11 A12 A16 ? 0x Ny = A12 A22 A26 ? 0y ……………eqn35 Nxy A16 A26 A66 ? 0xy Mx D11 D12 D16 kx My = D12 D22 D26 ky ……………eqn36 Mxy D16 D26 D66 kxy Special cases of symmetric laminates. Type 1 : Symmetric laminates with multiple isotropic layers
If multiple isotropic layers of various thicknesses are arranged symmetrically about a middle surface from both a geometric and a material property standpoint, the resulting laminate does not exhibit coupling between bending and extension. A simple example of a symmetric laminate with three isotropic layers is shown in fig. 1. a more complicated example of a symmetric laminate with six isotropic layer of a different elastic properties and thicknesses is given in table1. The extensional and bending stiffnesses for the general case are calculated from eqn24 wherein for the Kth layer, (Q’11)k = (Q’22)k = (Ek / 1- ? k) (Q’16)k = (Q’26)k = 0 ……………….. eqn37 (Q’12)k = ? kEk / 1- ? 2 (Q’66)k = Ek / 2(1- ? 2k) The force and moments resultants take the form Nx A11 A12 0 ? 0x Ny = A12 A22 0 ? 0y ……………….. eqn38 Nxy 0 0 A66 ? 0xy E1, ? 1,t E2, ? 2,t y FIG1: Exploded view of three layered symmetri symmetric laminate with isotropic layer. X E1, ? ,t Table1: Symmetric laminate with six multiple isotropic later |Layer |Material properties |Thickness | |1 |E1, ? 1 |T | |2 |E2, ? 2 |2t | |3 |E3, ? 3 |3t | |4 |E3, ? |3t | |5 |E2, ? 2 |2t | |6 |E1, ? 1 |T | Mx D11 D12 0 kx My = D12 D22 0 ky ………. ………eqn39 Mxy 0 0 D66 kxy Wherein for isotropic layers, A11= A22 and D11 = D22 because of the first condition of eqn37.
The specific form of the Aij and Dij can be somewhat involved as can easily be verified by the use of some simple examples. Type 2: Symmetric laminate with multiple specially orthotropic layers Because of the analytical complications involving the stiffnesses A16, A26, D16 and D26, a laminate is desired that does not have these stiffnesses. Laminates can be made with orthotropic layers that have principle material directions aligned with the laminate axes. If the thicknesses, locations and material properties of the laminae are symmetric about middle surface of the laminate, there is no coupling between bending and extension.
A general example is shown in table2. the extensional and bending stiffnesses are calculated from eqn24, wherein the Kth layer. (Q’11)k = Ek1 / (1- ? k12 ? k21 ) (Q’16)k = 0 (Q’21)k = ? k12Ek1 / (1- ? k12 ? k21 ) (Q’26)k = 0 ………………eqn40 (Q’22)k = Ek2 / (1- ? k12 ? k21 ) (Q’66)k = Gk12 Because (Q’16 )k and (Q’26 )k are zero, the stiffness A16 , A26, D16 and D26 vanish. Also, the stiffnesses Bij are zero because of symmetry.
This type of laminate could therefore be called a specially orthotropic laminate in analogy to a specially orthotropic laminate in analogy to a specially orthotropic lamina. The force and moment resultant take the form of eqs. 38 , 39 respectively. Y FIG: Exploded view of three layered regular Symmetric cross-ply laminate. X A very common special case of symmetric laminates with multiple specially orthotropic layer occurs when the laminae are all of the same thickness and material properties, but have their major principle material directions alternating at 0 ° and 90 ° to the laminate axes, for example,0 °/90 °/0 °. uch laminates are called regular symmetric cross-ply laminates. A simple example of a regular symmetric cross-ply laminate with three layers of equal thickness and properties is shown in fig. 2. The fiber direction of each lamina are schematically indicated by the use of light lines in fig. 2. The laminate must have an odd number of layers to satisfy the symmetry requirement by which coupling between bending and extension is eliminated. Cross-ply laminates with an even number layers are obviously not symmetric. The logic to establish the various stiffnesses will be traced to illustrate the simple procedures.
First, consider the extensional stiffnesses Aij = N? k=1 (Q’ij)k (zk – zk-1) …………………………………eqn41 The Aij are the sum of the product of the individual laminae Q’ij and the laminae thicknesses. Thus the only way to obtain a zero individual Aij are all Q’ij to be zero or for some Q’ij to be negative and some positive so that there products with there respective thicknesses sum to zero. For the expressions for the transformed lamina stiffnesses Q’ij in equation 14. apparently Q’11, Q’12,Q’22 and Q’66 are positive definite since all trigonometric functions appear to even powers.
Thus A11,A22,A12 and A16 are positive definite since the thicknesses are always positive. however Q’16 and Q’26 are zero for lamina orientations of 0 and 90 to the laminate axis. Thus A16 and A26 are zero for laminates of orthotropic laminae oriented at either 0 or 90 to the laminater axis. Second consider the coupling stiffnesses Bij = 1/2 N? k=1 (Q’ij)k (z2k – z2k-1) …………………………………. eqn42 If the cross ply laminate is symmetric about the middle surface then the Bij all vanish as can easily be shown. Finally consider the bending stiffnesses Dij = 1/3 N? k=1 (Q’ij)k (z3k – z3k-1) ………………………………….. qn43 The Dij are sum of product of the individual laminae Q’ij and the term( z3k – z3k-1). Since Q’11,Q’12,Q’22 and Q’66 are positive definite and the geometric term is positive definite then D11,D12,D22 and D66 are positive definite. Also Q’16 and Q’26 are zero for lamina principal material property orientations of 0 and 90 to the laminate coordinate axis. Thus D16 and D26 are zero. Type 3 symmetric laminate with multiple generally orthotropic layers A laminate of multiple generally orthotropic layers that are symmetrically disposed about the middle surface exhibits no coupling between bending and extension that is the Bij are zero.
Therefore the force and moment resultant are represented by the equation 38 and 39 respectively. There all the Aij and Dij are required because of coupling between normal forces and shearing strain, shearing force and normal stress, normal moments and twist and twisting moment and normal curvatures. Such coupling is evidenced by the A16, A26, D16,D26 stiffnesses. A special subclass of this class of symmetric laminates is the regular symmetric angle ply laminate. Such laminates have orthotropic lamina of equal thicknesses.
The adjacent laminae have opposite sign of the angle of orientation of the principle material properties with respect to the laminate axis. For example +? /- ? /+ ? . thus for symmetry there must be an odd number of layers. A simple example of a three layered regular symmetric angle ply laminate is shown in figure. FIG: Exploded view of a three layered regular symmetric angle ply laminate. Angle +? Y Angle – ? x angle + ? The aforementioned coupling that involves A16,A26,D16 and D26 takes on a special form of symmetric angle ply laminates.
Those stiffnesses can be shown to be the largest when N = 3 and decreases in proportion to 1/N as N increase. Actually in the expressions for the extensional and bending stiffnesses A16 and D16 . A16 = N? k=1 (Q’16)k (zk – zk-1) ………………………eqn44 D16= 1/3 N? k=1 (Q’16)k (z3k – z3k-1) …… …………………eqn45 Obviously A16 and D16 are sums of there alternating signs since (Q’16) + ? = -(Q’16)- ?. …………………………eqn46 Thus for many layered angle ply laminates the values of A16, A26, D16 and D26 can be quite small when compared to the other Aij and Dij respectively.
When the always present advantage of zero Bij because of symmetry is considered in addition to the low A16 , A26, D16 and D26 many layered symmetric angle ply laminates can offer significant, practically advantageous simplifications over some more general laminates. In addition symmetric angle ply laminates offer more shear stiffness than do the similar cross ply laminates so are used more often. However the knowledge of effect of A16, A26, D16 and D26 on the individual class of problems being considered by designer is essential ecause even a small A16 or D16 might cause significantly different results from cases in which those stiffnesses are exactly zero. Only in the situation where A16 , A26 , D16 and D26 are exactly zero can they be ignored without further thought or analysis .
ANTISYMMETRIC LAMINATES
The symmetry of a laminate about the middle surface is often desirable to avoid coupling between bending and extension. However many physical applications of composites require non symmetric laminates to achieve design requirements. For example coupling is necessary feature to make jet turbine fan blades with pre twist.
As a further example if the shear stiffness of a laminate made of laminae with unidirectional fibers must be increased one way to achieve this requirement is to position layers at some angle to the laminate axis. To stay within weight and cost requirements an even number of such layers may be necessary at orientations that alternate from layer to layer. Example + ? /- ? /+ ? /- ?. Therefore symmetry about the middle surface is destroyed and the behavior characteristics of the laminate can be substantially changed from the symmetric case.
Although the example laminate is not symmetric it is anti symmetric about the surface and certain stiffness simplifications are possible. The general class of anti symmetric laminates must have an even number of layers if adjacent laminae have alternating sign of the principal material property directions with respect to the laminate axis. In addition each pair of laminae must have the same thickness. The only exceptions to the above stipulation occur when the angle of orientation is 0 or 90, then an odd number of layers fits the definition if the central layer is either 0 or 90 layer. the central layer is figuratively split and regarded as two layers of the same orientation ). The stiffnesses of an antisymmetric laminate of isotropic laminae do not simplify from those presented in equations 20 and 21. however as a consequence of anti symmetry of material properties of generally orthotropic laminae but symmetry of there thicknesses the extensional coupling A16 A16 = N? k=1 (Q’16)k (zk – zk-1) Is easily seen to be zero since (Q’16) + ? = -(Q’16)- ? And layer symmetric about the middle surface have equal thickness and hence the same value of geometric term multiplying (Q’16)k. imilarly A26 is zero as is the bending twist coupling stiffness D16 D16= 1/3 N? k=1 (Q’16)k (z3k – z3k-1) Since again equation 46 holds and the geometric term multiplying ( Q’16 ) is the same for two layers symmetric about the middle surface. The preceding reasoning applies also for D26. The coupling stiffnesses Dij vary for different classes of anti symmetric laminate of generally orthotropic laminae and no general representation other than in the following force and moment resultants. Nx A11 A12 0 ? x B11 B12 B16 kx Ny = A12 A22 0 ? 0y + B21 B22 B26 ky ………… eqn47 Nxy 0 0 A66 ? 0xy B16 B26 B66 kxy Mx B11 B12 B16 ? 0x D11 D12 0 kx My = B12 B22 B26 ? 0y + D21 D22 0 ky ………eqn48 Mxy B16 B26 B66 ? 0xy 0 0 D66 kxy Type 1: Antisymmetric cross ply laminate
Antisymmetric cross play laminate consist of an even number of orthotropic laminae laid on each other with principal material directions alternation at 0 and 90 to the laminate axis as in the simple example of following figure. Nx A11 A12 0 ? 0 B11 0 0 kx Ny = A12 A22 0 ? 0y + 0 -B11 0 ky ……. eqn49 Nxy 0 0 A66 ? 0xy 0 0 0 kxy Mx B11 0 0 ? 0x D11 D12 0 x My = 0 -B11 0 ? 0y + D21 D22 0 ky ………eqn50 Mxy 0 0 0 ? 0xy 0 0 D66 kxy Diagram : Exploded view of two layered regular antisymmetric laminate. Y X A regular antisymmetric cross ply laminate is defined to have laminae of all equal thickness and is common because of simplicity of fabrication. As the number of layer increases the coupling stiffness B11 can be shown to approach zero. Type 2: Antisymmetric angle ply laminates.
An antisymmetric angle ply laminate has laminae oriented at +? degrees to the laminate coordinate axis on one side of the middle surface and corresponding equal thicknesslaminae oriented at –? degrees on the other side. A simple example of an antisymmetric laminate is shown in figure. A regular antisymmetric angle ply laminate has laminae all of the same thickness for ease of fabrication. This class of laminates can be further restricted to have a single value of ? as opposed to several orientations. The force and moment resultant for an anti symmetric angle ply laminate are as follows
Nx A11 A12 0 ? 0 0 0 B16 kx Ny = A12 A22 0 ? 0y + 0 0 B26 ky …. eqn51 Nxy 0 0 A66 ? 0xy B16 B26 0 kxy Mx 0 0 B16 ? 0x D11 D12 0 kx My = 0 B11 B26 ? 0y + D21 D22 0 ky ….. …eqn52 Mxy B16 B26 0 ? xy 0 0 D66 kxy FIG: Exploded view of two layered antisymmetric angle-ply laminate. angle ? Y y The coupling stiffnesses B16 and B26 can be shown to go zero as the number of layers in the laminate increases for a fixed laminate thickness. CASE 4. NON SYMMETRIC LAMINATES For general case of multiple isotropic layers of thickness tk and material properties Ek, and ? k the extensional, coupling and bending stiffnesses are given by eqn24. (Q’11)k = (Q’22)k = (Ek / 1- ? 2k) (Q’16)k = (Q’26)k = 0 ……………….. eqn53 (Q’12)k = ? kEk / 1- ? (Q’66)k = Ek / 2(1- ? 2k) In some cases no special reduction of stiffness is possible when tk is arbitrary. That is coupling between bending and extension can be oriented by unsymmetric arrangement about the middle surface of isotropic layers with different material properties and possibly different thicknesses. Thus coupling between of bending and extension is not a manifestation of material orthotropy but rather of laminate heterogeneity that is a combination of both geometric and material properties. The force and moment resultants are Nx A11 A12 0 ? B11 B12 0 kx Ny = A12 A22 0 ? 0y + B12 B11 0 ky …eqn53 Nxy 0 0 A66 ? 0xy 0 0 B66 kxy Mx B11 B12 0 ? 0x D11 D12 0 kx My = B12 B11 0 ? 0y + D21 D22 0 ky ….. …eqn54 Mxy 0 0 B66 ? xy 0 0 D66 kxy Non symmetric laminate with multiple specially orthotropic layers can be shown to have force and moment resultants in equation 53 and 54. But with different A22, B22, and D22 from A11, B11, and D11 respectively. That is there is no shear coupling term and therefore the solution of problem with this kind of lamination is about as easy as isotropic layers. Non symmetric laminates with multiple generally orthotropic layers or with multiple anisotropic layers have force and moment resultants no simpler than eqn 21 and 23.
All stiffnesses are present. Hence configurations with either of those two laminae are much more difficult to analyze than configuration with either multiple isotropic layers or multiple specially orthotropic layers. SUMMARY Single layer laminates or individual laminae with a reference surface at the middle surface do not exhibit coupling between bending and extension. With any other reference surface there is indeed such coupling. Multi layered laminates in general, develop coupling between bending and extension.
The coupling is influenced by geometrical as well as the material property characteristics of laminates. There are however combinations of geometrical and material property characteristics for which there is no coupling between bending and extension. Those special cases have been reviewed in this section along with other special cases . all the special cases find important applications and should be well understood , not from the collection of special cases that the elastic symmetry of the laminae is not necessarily maintained in the laminate. The symmetry can be increased, decreased or remain the same.
Moreover the symmetries of 3 stiffness matrices A,B and D need not be the same. The basic concept of coupling between bending and extension must be understood because there are many applications of the composite materials where neglect of coupling can be catastrophic. These coupling is the key to the correct analysis of eccentrically stiffened plates and shells. For example card and jones showed that longitudinal stiffners are placed on the outside of an axially loaded circular cylindrical shell the bucking load is twice the value when the same stiffners are on the inside of shell.
Previously the coupling between the stiffner and the shell had been ignored. The manner of describing a laminate by use of individual layer thcikeese, principal material property orientations and overall stacking sequence could be quite involved. However fortunately all pertinent parameters are represented in a simple concise fashion by use of following stacking sequence terminology. for regular (equal thickness layers) laminates, a listing of the layers and there orientations suffices for example (00 /900/450). Note that the principal material direction orientations need be given.
Many different laminates could be made with the same layers. For example (900/00/450). For irregular laminates a notation of layer thicknesses must be appended to the previous notation. For example ( 00@t/900@2t/450@3t). Finally for symmetric laminates , the simplest representation of , for example (0°/90°/45°/45°/90°/0°) is ( 0°/90°/45°) symmetric. [pic]
MECHANICS OF MATERIALS APPROACH TO STIFFNESS
The key features of the mechanics of materials approach is that certain simplifying assumptions are made regarding the mechanical behavior of a composite material.
The most prominent assumption is that the strains in the fiber direction of a unidirectional fibrous composite are the same in the fiber as in the matrix as shown in fig. since the strain in both the matrix and fiber are the same, then it is obvious that sections normal to the 1-axis that were plans before being stressed remain plan after stressing. The foregoing is a prominent assumption in the usual mechanics of materials approaches such as in beam, plate and shell theories. We shall derive on that basis, the mechanics of materials expressions for the apparent orthotropic moduli of unidirectionally reinforced fibrous composites materials. Determination of E1 The first modulus to be determined is that of the composite in the 1-direction that is in the fiber direction. ?1 = ? L / L ……………………………. (eq. 1) Where ? 1 applies for both the fiber and the matrix according to the basic assumption. Then, if both constituent materials behave elastically, the stresses are ? f = Ef? 1 ……………………………(eq. 2) ? m = Em? 1 2 variation of E1 with fiber volume MATRIX E1 ? FIBER 1 Em MATRIX L ? L Representative volume element Loaded in dir-1 Vf*100 The average stress ? 1 acts on a c/s areas A, ? f acts on the c/s area of the fiber area Af and ? m acts on the c/s area of the matrix Am. Thus the resultant force on the element of composite material is P = ? 1 A = ? fAf + ? mAm ……………………….. eq. 3) By substitution of eq. 2 in eq. 3 and recognition that ?1=E1? 1 …………………………(eq. 4) apparently E1 = EfAf/A + EmAm/A …………………………. (eq. 5) But the volume fraction of fibers and matrix can be written as Vf = Af/A Vm = Am/A ………………………….. (eq. 6) Thus , E1 = EfVf + EmVm ………………………….. (eq. 7) This is known as the rule of mixtures expression for the apparent Young’s modulus in the direction of the fibers. The rule of mixtures is graphically depicted in above fig.
The rule of mixture represents a simple linear variation of apparent Young’s modulus E1 from Em to Ef as Vf goes from 0 to 1. 2 Determination of E2 The apparent Young’s modulus, E2 in the direction transverse to the fiber is considered next. In the mechanics of material approach, the same transverse stress,? 2 , is assumed to be applied to both the fiber and the matrix as in fig. The strain in the fiber and in the matrix is, therefore, ? f = ? 2/Ef ?m = ? 2/Em 2 ?
MATRIX W FIBER
1 MATRIX Representative volume element loaded in 2 direction
The transverse direction over which, on the average, ? f acts is approximately VfW whereas ? m acts on VmW. Thus, the total transverse deformation is ? 2 = Vf? f + Vm? m Which becomes, upon substitution of above eq. ?2 = Vf ? 2/Ef + Vm ? 2/Em But ?2 = E2? 2 = E2 (Vf ? 2/Ef + Vm ? 2/Em) Whereupon E2= EfEm(VmEf+VfEm) Which is mechanics of materials expression for the apparent young’s modulus in the direction transverse to the fiber. 3 Determination of ? 12 The so called poison’s ratio, ? 12, can be obtained by use of an approach similar to the analysis for E1. The major poison’s ratio is defined as ? 2=- ? 2/ ? 1 ……………………….. (eq1) For the stress state ? 1= ? and all other stresses are zero. The deformations are depicted in fig. 2 ?w/2 MATRIX W FIBER? 1 MATRIX 1 L ? L Fig. Representative volume element loaded in 1-direction The transverse deformation ? w is ?w =-W? 2 = W? 12? 1 ……………………(eq2) But ?w = ? mw + ? fw……………………(eq3) In the manner of analysis for the transverse young’s modulus E2, the deformation ? mw and ? fw are approximately ? w =WVm? m? m……………………(eq4) ?fw = WVf? f? f Thus upon combination of eq. 2, eq. 3, and eq. 4 division by ? 1W yields ? 12=Vm? m + Vf? f Which is a rule of mixtures for the major poison’s ratio. 4 Determination of G12 The in plane shear modulus of a lamina, G12 is determined in the mechanics of material approach by assuming that the shearing stresses on the fiber and the matrix are same. The loading is shown in fig1. by virtue of the basic assumption, ? m =? /Gm………………………………(eq1) ?f = ? /Gf The nonlinear shear stress-strain behavior typical of fiber-reinforced composites is ignored, i. e. he behavior is regarded as linear. 2 MATRIX ? 12 ? FIBER 1 W MATRIX ?m Fig1 Representative volume element loaded in shear On microscopic scale, the deformation are shown in fig.. the total shearing deformation is defined as ? =? W…………………(eq. 2) And is made up of, approximately, ?m = VmW? m …………………………….. eq. 3) ? f = VfW? f Then since ? =? m +? f , division by W yields ? = Vm? m + Vf? f ……………………………(eq. 4) Or upon substitution of eq. 1 and realization that ? = ? /G12 ………………….. (eq. 5) Eq. 4 can be written as ?/G12 = Vm ? /Gm + Vf ? /Gf ……………………………. (eq. 6) Finally G12 = GmGf / (VmGf+VfGm) ………………….. (eq. 7) STRESS-STRAIN RELATION FOR PLANE STRESS IN AN ORTHOTROPIC MATERIAL For a lamina in the plane 1-2 as shown on fig. , a plane stress state is defined by setting ? 3 = 0 ? 3 = 0 ? 31 = 0 in the three dimensional stress-strain relations for anisotropic, orthotropic, transversely, or isotropic materials. For orthotropic material, such a procedure results in implied strains of ? 3 = s13? 1 + s23? 2 ?23 = 0 ? 31 = 0 moreover, the strain-stress relations reduced to ?1 s11 s12 0 ? 1 ? 2 = s12 s22 0 ? 2 ? 12 0 0 s66 ? 12 [pic] FIG. Unidirectionally reinforced lamina. S11 = 1/E1 S12 = -? 12/E1 = -? 21/E2
S22 = 1/E2 S66 = 1/G12 Note that in order to determine ? 3 in above eq. , ? 13 and ? 23 must be known in addition to those engg. constants shown in above eq. The stress-strain relation in above eq. can be inverted to obtain the stree-strain relations: ?1 Q11 Q12 0 ? 1 ? 2 = Q12 Q22 0 ? 2 ? 12 0 0 Q33 ? 12 where the Qij , the so called reduced stiffnesses, are Q11 = S22/ (S11S22-S212) Q12 = – S12/ (S11S22-S212) Q22 = S11/ (S11S22-S212) Q66 = 1/S66
In terms of engineering constants, Q11 = E1/(1- ? 12 ? 21) Q12 = ? 12 E2/ (1- ? 12 ? 21) = ? 21E2/ (1- ? 22 ? 21) Q22 = E2/(1- ? 12 ? 21) Q66 = G12 The preceding relations and stress-strain relations are the basis for the stiffness and stress analysis of an individual lamina subjected to forces in its own plane. The relations are therefore indispensable in the analysis of laminates.
STRESS-STRAIN RELATIONS FOR A LAMINA OF ARBITRARY ORIENTATION
The principle direction of orthotropy often do not coincide with coordinate directions that are geometrically natural to the solution of the problem.
For example, consider the helically wound fiberglass-reinforced circular cylindrical shell in following fig. There, the coordinate natural to the solution of the shell problem are the shell coordinates x,y,z whereas , the principle material coordinate are x’,y’,z’. The wrap angle is defined by cos(y’ y) =cos? : also z’ = z. other example include laminated plates with different laminae at different orientations. Thus a relation is needed between the stress and strain in the principle material direction and those in the body coordinates. Then method of transforming stress-strain relations from one coordinate system to another is also needed.
At this point we recall from elementary mechanics of materials the transformation equations for expressing stresses in x-y coordinate system in terms of stresses in a 1-2 coordinate system. ?x cos2? sin2? -2sin? cos?? 1 ?y =sin2? cos2? 2sin? cos?? 2 ……………eqn ? xy sin? cos? -sin? cos? cos2? – sin2?? 12 where the ? is the angle from the x-axis to the 1-axis . note especially that the transformation has nothing to do with the material properties but is merely a rotation of stresses. Similarly the strain transformation equations are ?x cos2? sin2? -2sin? cos? ?1 ?y =sin2? cos2? 2sin? cos? ?2 ………….. eqn. ?xy/2 sin? cos? sin? cos? cos2? – sin2? ?12/2 A so called specially orthotropic lamina is one whose principle material axes are aligned with the natural body axes for the problem, for example, FIG: Helically wound fiber-reinforced circular cylindrical shell Z Z’ X Y Y’ X’ Y 1 2 X ?x ? 1 Q11Q120 ? 1 ?y = ? 2 = Q12Q220 ? 2 ?xy ? 12 0 0Q66 ? 12 The sress-strain relation in x-y coordinate are ?x ? x Q’11 Q’12 Q’16 ? x ? y = [Q’] ? y = Q’12 Q’22 Q’26 ? ? xy ? xy Q’16 Q’
