Microstructure Lectures

1.0 Introduction:

Figure 1. Microstructures of grains and lamina

Macroscopic mechanical behavior can be influenced by structures at the microscopic scales of 10^-3m. For example elastic properties of a polycrystalline material are controlled by the orientation of individual grains sizes of 10 ^-3m, i.e. random orientation leads to isotropic macroscopic elastic properties. For composite laminates macroscopic properties of laminated composites are influenced by the elastic properties and fiber orientation in each laminate layer ("lamina"). Typical lamina thicknesses are 10^-3m and hence are considered microscopic structures. Cumulative damage events, i.e. ply cracks, at the lamina-microscopic scale can also influence mechanical behavior and failure at the macroscopic laminate scale. Relationships between the microscopic and macroscopic scales embraces a wide variety of other topics and examples. It is not the intention of this section to provide a comprehensive overview of these micro/macro models but rather to introduce the students to examples of micro/macro models that is representative of the research conducted by Dr. Ron Kriz while working with computers at NIST (formerly NBS) from 1980 to 1990 and with computers at NCSA (National Center for Supercomputing Applications) from 1990 to 2000. During this time period high performance computing was rapidly evolving. CDC, Thinking Machine CM2, Cray, and SGI computers were all used to study the mechanical behavior of a variety of problems related to damage in laminated composites and wave propagation in highly anisotropic crystals and fiber-reinforced composites.

Throughout this section we provide links to examples where the reader can work with interactive computer programs, written by Dr. Kriz, as they relate to the theory developed in the sections below. Hence our objective is to create an interactive section that compliments the theory and allows the student to experiment with the various micro/macro models as they are introduced.

2.0 Cracks: Atomistic 10^-9m - at Interfaces 10^-3m - in a Continuum 10⁰m

Cracks span the scales. The mechanical behavior of materials with cracks can be modeled to include: 1) the influence of individual atoms ("nano"), 2) the presence of interfacial layers separating different microstructures ("micro"), or 3) the idealized continuum model where atomic and interfacial effects are neglected ("macro"). Regardless of the scale, models make simplifying approximations. Understanding these approximations is key to understanding how some models bridge the scales. Therefore we will highlight these approximations below as we develope the theory and also when we develope the mathematical model for each of the scales. The images on the home page exemplify these three scales.

Figure 2. Schematic of Mode-I cracks spanning the length scales

3.0 Cracks Near Interfaces Between Dissimilar Isotropic Materials

Numerous models for interfacial cracks exist. In the most general case we assume a crack arrested at an interface separating two dissimilar anisotropic materials and then simplify the model to dissimilar isotropic materials. Such cracks are common in fiber-reinforced composite laminates. In Figure 3 we show a schematic and an acetate replica micro-graph of a 90 degree ply crack arrested at a 0/90 degree ply interface.

4.1 Introduction to Mechanical Behavior of Anisotropic Materials

In the early part of the 20th century most engineering materials were isotropic. In the later part of the 20th century, 1970s -> 1990s, the aerospace industry created the demand for new, high-strength, high stiffness and light-weight fiber-reinforced composite materials. Because these materials are also highly anisotropic, designers could optimize their designs, i.e. orient the direction of high stiffness and strength in the load direction. Other materials used in "Smart" or "Intelligent" structures, such as piezioelectric polymers are also highly anisotropic. Because the properties of these new materials could be tailored, designers, could for the first time, design the material for the application instead of designing an application around fixed material properties. Anisotropic materials are becoming the material of choice in a variety of engineering applications as we enter the 21st century. Studying the mechanical behavior of these highly anisotropic materials under static and dynamic loads at all scales is relevant when considering current engineering practice.

In the early part of the 20th century anisotropy was more a topic of scientific research then a property used in engineering design. Nye's text on "Physical Properties of Crystal", Ref.[6], gives an excellent introduction to anisotropy in crystals from a material scientist perspective. Lekhnitski's text on "Theory of Elasticity of an Anisotropic Body", Ref.[7], began to address issues of anisotropy in engineering design. In this section we will give a very brief introduction to anisotropy for a continuum (macroscale) and then apply what we learned to anisotropic laminates at the microscale. In section 10.0 we will also show how anisotropy can significantly control dynamic mechanical behavior in a continuum (macroscale).

Mechanical behavior of anisotropic materials is characterized in the most general notation as a fourth order stiffness tensor, C_ijkl, which is contracted into a six by six stiffness matrix, C_α,β, relating six components, σ_α, of a symmetric stress tensor to the six components, e_β, of the symmetric strain tensor which are written here as column vectors.

(4)

Although this contracted notation shown above is convenient and will be used below to model mechanical behavior of anisotropic laminates at the microscale, a more complete description of elastic anisotropy will require that we return to the more general form where the stiffness matrix that relates strains to stress is written as a fourth order stiffness tensor C_ijkl, where the stresses, σ_ij, and Lagrangian strains, l_ij, must then be written as second order tensors. We will show that this more general tensoral form will allow us to create unique three dimensional geometric representation surfaces for each unique anisotropic elastic symmetry. A more general tensor equation relating second order stresses and second order strains is more fully developed in any standard text on continuum mechanics, see Ref.[8].

(5)

First we will show how stress can be written in the form of a second order tensor. The proof that stress is a second order tensor because it transforms as a second order tensor is given in Ref.[8].

Figure 7 shows how all possible components of stress acting as forces on a small differential element can be organized into a matrix format which can also be reduced to index notation where i and j are called "free" indices that are assigned values of 1, 2, or 3. All possible combinations of the i and j indices yield the desired three by three matrix.

Figure 7. Stress tensor components shown on a differential element

Writing all six components of strain into a similar matrix is not as straight forward as it was for stress where each stress components could be simply seen as a force acting on the surface of a cube. Strain implies displacement which requires much more development, see Ref.[8] Chapter 3, "Analysis of Deformation in a Continuum". Suffice it to say that strain like stress has 9 components which can be identified with the same indices which identifies a displacement direction or shearing in a plane associated with each strain component. From a Lagrangian viewpoint, for large displacement gradients, the nonlinear strain displacement relationships are shown below.

(6)

If we assume only small displacement gradients then the nonlinear terms can be neglected and the upper case "L" is reduced to the lower case "l" which is used to denote the linear lagrangian strain tensor, l_ij.

(7)

With strain and stress defined as second order tensors, it is not difficult to derive the desired linear anisotropic relationship, equation (5), from the first law of thermodynamics. One form of the first law of thermodynamics is shown below.

(8)

where u is the specific internal energy per unit mass, ρ is the density, e_ij is the Eulerian strain associated with Eulerian displacement gradients, c is the radiative heat transfer per unit time per unit mass, and b_i is the heat transferred through a unit surface per unit time. Note d/dt is the total or comoving derivative. For a complete derivation see Ref.[8], Chapter 4, section 4.5.

If we assume no heat transfer by radiation or conduction and we assume that only small displacement gradients result in lagrangian strains, l_ij, equation (8) reduces to a form where u is now called the strain energy per unit mass since the only mechanism available for storing internal energy in the continuum is through deformation.

(9)

Next let's assume density is constant, or at least a function of strains and that stresses are related to strains by some unknown function (to be determined). Hence strain energy can be a function of stress or strain. For the purpose of discussion here we assume strain energy can be written as some function of strain.

(10)

(11)

Substituting equation (11) into equation (9) yields.

(12)

If this relationship is true for any arbitrary nonzero dl_ij, then the relationship in the parenthesis must vanish which yields an equation that shows the relationship between stress, strain and strain energy per unit mass.

(13)

Recall that for equation (10) we assumed some undefined functional form for strain energy as a function of strain. One possible function would be to expand the strain energy into a power series whose coefficients like strain are tensors but constants with indices that sum ("contraction") with the indices of the strain tensors such that each term in the series is a scalar component of the strain energy. The first term α is an arbitrary energy reference state, so we can set it to zero. The second term is a zero strain reference state. The third term is the linear elastic term and the fourth term is the nonlinear, but elastic, term. Higher order terms can be added here as needed to model the desired constitutive stress-strain relationship.

(14)

Substituting this function for strain energy into equation (13) yields the most general form of the anisotropic elastic constitutive law in tensor format.

(15)

NOTE: 1) it is perhaps more evident at this point that the coefficient β_ij represent the residual stresses when the strains, l_ij, go to zero, and 2) the coefficient δ_ijklmn is the nonlinear elastic term that gives rise to nonlinear stress-strain diagrams not associated with dissipative constitutive models, which we also set to zero. With these simplifications equation (15) reduces to the desired linear-elastic equation.

(16)

Where C_ijkl are the terms that relate components of strain to the components of stress per unit mass. A similar expression without density can be derived where strain energy per unit volume is introduced into the derivation. Interestingly to remove density is not as straight forward as one might think, see Ref.[8].

(17)

Lets briefly look at an arbitrary geometric transformation of equation (17). For a full development of this type of transformation see Ref.[8]. It can be shown that both stress and strain are second order tensors because they transform as second order tensors and C_ijkl transforms as a fourth order tensor.

(18)

(19)

(20)

---

Transformation Law of a vector

Figure 8. Vector drawn with respect to two different Cartesian axes

Introduce the direction cosines between the two sets of axes:

(a₁₁, a₁₂, a₁₃) = direction cosines of the y^'₁-axis with respect to the (y₁,y₂,y₃)-axes.

(a₂₁, a₂₂, a₂₃) = direction cosines of the y^'₂-axis with respect to the (y₁,y₂,y₃)-axes.

(a₃₁, a₃₂, a₃₃) = direction cosines of the y^'₃-axis with respect to the (y₁,y₂,y₃)-axes.

Figure 9. Direction Cosine Matrix With a Simple Rotation About the y₃ axis.

Hence, the relation between the vector A^' and A can be summarized as follows.

(21)

Note that the indices on A^' can be replace as i=1,2,3 hence the three equations in equation (21) reduces to one equation with a "free" index "i".

(22)

Also note that when the number in each term in equation (22) are the same (1 1 in the first term, + 2 2 in the second term, and + 3 3 in the third term) we introduce a new idea of a "summation" notation where any such summation can be notationally simplified and replaced with repeated indices in one term. Hence we can now use this summation index to reduce equation (22).

(23)

---

The same direction cosine matrix, a_ij, can also be used in the transformation of stress, strain, and stiffness, in equations (18), (19), and (20) which when substituted into equation (17) reveals that the same equation exists in the arbitrary prime coordinate system, equation (24). The fact that equations (17) and (24) are identical demonstrates that equation (17) is invariant to any ARBITRARY transformation, hence equation (17) is a "tensor equation". The keyword here is arbitrary.

(24)

NOTE ON ARIBITRARINESS: Before we move on to understanding the different types of anisotropies, there is one very important idea that needs to be introduced about equations (17) and (24) which is that tensor equations by definition exhibits mathematical "invariance" to any ARBITRARY transformation. Although the two ideas of invariance and arbitrariness are mathematical ideas, these same ideas of invariance to an arbitrary transformation is also the same fundamental idea, indeed a requirement, of a physical law. Hence invariant arbitrary transformations are essential properties for both physical laws and tensor equations. Are all tensor equations by definition physical laws, is a question that remains unanswered but an interesting topic for discussion.

Equations (17) and (24) are the form of Hooke's Law we need to continue the discussion of anisotropic elasticity by Nye and also to show some interesting "glyphs" (geometric representations) of the fourth order stiffness tensor, C_ijkl .

Now that we have introduced elastic anisotropy both as a fourth order tensor and a six-by-six matrix with reduced notation, the reader can now understand the comprehensive treatment of elastic anisotropy given by Nye, Ref.[6], in chapter 8, "Elasticity. Fourth-Rank Tensors", pp. 131-149, where both stiffnesses and compliances are described in terms of matrix and tensor notation. Our objective here is to summarize and highlight the different elastic anisotropies commonly encountered by todays materials engineers. Figure 10 summarizes the various elastic anisotropies in matrix notation.

KEY TO NOTATION
TRICLINIC (21)
MONOCLINIC (13)
ORTHORHOMBIC (9)                       CUBIC (3)
(7)          TETRAGONAL          (6)
(7)             TRIGONAL             (6)
HEXAGONAL (5)                          ISOTROPIC (2)

Figure 10. Various anisotropies for S_ij and C_ij.

As shown in Figure 10, the different elastic anisotropies are related to the number of independent terms in the elastic property matrix where symbols represent either stiffnesses or compliances. Each of these unique matrices can be related to crystal symmetry classes but here we will discuss just a few of the major classes and symmetry conditions. Triclinic is the most general with 21 independent components. Starting with Triclinic the number of independent terms can be reduced using symmetry planes and invariant rotations. For example if we could imagine ourselves standing inside a crystal and observed that the crystal structure looked identical if we looked either in the +z or -z direction. With this observation we would conclude that symmetry exists about the x-y (1-2) plane and the transformation matrix, a_ij, would be the identity matrix but with the term a₃₃ set equal to -1, see Figure 11. Using this transformation matrix in equation (17) components of the stress tensor are compared in the primed and unprimed coordinates.

Figure 11. Sample calculation of transformation for symmetry about the x-y (1-2) plane.
NOTE: transformation of stress and strain requires the use of tensor transformations
eqns. (18) and (19) to get the correct sign.

This comparison determines which matrix terms must vanish. For example in Figure 11 comparing σ₁ with σ₁^' requires C₁₄ and C₁₅ vanish. Similar comparisons with the other stress components requires,

(25)

Hence this reduces the Triclinic with 21 independent terms to Monoclinic with 13 independent terms. Figure 10 shows two different types of Monoclinic matrices. If we continue with symmetry in the x-z and y-z plane additional terms must vanish and Monoclinic reduces to Orthotropic with 9 independent terms. Arbitrary rotation about the z axis (3-direction) assuming elastic properties are invariant reduces Orthorhombic to Hexagonal with five independent terms. Of course for isotropic there are only two independent terms or components of the stiffness or compliance fourth order tensor.

Elastic symmetries commonly encountered when working with fiber-reinforced materials commonly used in lightweight aircraft and aerospace structures, are called "orthotropic" which is the same as orthorhombic and "transversely isotropic" which is equivalent to hexagonal.

We continue the development of anisotropic mechanical behavior into two sections. First we focus on engineering elastic properties and represent anisotropy in terms of engineering properties: E, G, and ν, and second we focus on all components of the fourth order stiffness tensor. In both cases we ephasize the use of geometry to study the anisotropy.

Another linear constitutive relationship, similar to equation (24), exists for compliances, S_ijkl, that relate stresses with strains.

(26)

Compliances like stiffnesses can be considered fourth order tensors because they transform as fourth order tensors.

(27)

Hence the relationship between compliances and stiffness can be shown as an inverse relationship. This inverse relationship is shown here in reduced contracted notation. NOTE: although these matrices are inversely related, S_ij and C_ij are not tensors quantities because they do not transform as second order tensors.

(28)

Compliances, unlike stiffnesses, have a simple relationship with engineering elastic properties such as Young's modulus, shear modulus, and Poisson's ratio. For example the compliance along the z axis (3-direction), S₃₃ is defined as the reciprocal of the Young's modulus in the same z-direction.

(29)

Similarly expressions for shear modulus and Poisson's ratio in the y-z (2-3) plane can be simply derived from terms in the compliance matrix.

,	(30), (31)

It is now possible to show how these three engineering elastic properties vary as a function of rotation from the z axes (3-direction) into the x-y (1-2) plane by using the general fourth order compliance tensor transformation of equation (27) where the direction cosine terms, a_ij, are expanded into sines and cosines of the rotation angle, θ, starting at zero along the z axis and rotating 90^o into the x-y plane. This transformation although simply done as a fourth order transformation, becomes a tedious task in reduced contracted notation. After much algebra the final equations are simplified, see Ref.[9].

(32)

(33)

(34)

For all isotropic engineering materials equations (32), (33), and (34) will, by definition, reveal no variation in properties as we rotate from the z axis into the x-y plane, hence circles in a two-dimensional (2-D) plane or spheres in a three-dimensional (3-D) space are visual representations (manifestations) of isotropic elastic properties. Unidirectional fiber-reinforced composite materials, on the other hand, exhibit highly ANisotropic behavior which is exhibited as significant deviations from the isotropic spherical geometry.

Material	C33	C11	C12	C13	C44	ν32	ν12
Epoxy-matrix (Kriz-Stinchcomb, 1979)	8.63	8.63	4.73	4.73	1.95	0.345	0.345
Carbon-fiber (Kriz-Stinchcomb, 1979)	235	20.0	9.98	6.45	24.0	0.279	0.490
Carbon-fiber (Dean-Truner, 1973)	240	20.4	9.40	10.5	24.0	0.35	0.497
Carbon-fiber (Smith, 1972)	221	19.4	6.60	5.80	20.3	0.230	0.320
Composite (60% Fiber) (Kriz-Stinchcomb), 1979)	144	13.6	7.0	5.47	6.01	0.284	0.497

Material	Poisson's Ratio(ν)    Modulus: Bulk(K), Shear(G), Young's(E) Subscripts: Matrix(m), Fiber-Axis(L), Transverse-Plane(T)
Isotropic: two independent properties->	Em	νm
Epoxy-matrix (Kriz-Stinchcomb, 1979)	5.35	0.345
Hexagonal: five independent properties->	EL	ET	GLT	GTT	KTT
Carbon-fiber (Kriz-Stinchcomb, 1979)	232	15.0	24.0	5.02	15.0

For example if we take elastic properties predicted in Ref.[10], see Table 1., for unidirectional carbon-fiber/epoxy composites and substitute these properties into equations (32), (33), and (34), both spherical and nonspherical geometries are predicted, see Figures 12, 13, and 14. In these figures the elastic properties are also shown as a function of fiber volume fraction where 0.0 fiber volume fraction yields pure isotropic epoxy properties, shown as circles, and 1.0 fiber volume yields pure anisotropic fiber properties and hence the geometry significantly deviates from the circular geometry. Even low fiber volume fractions yield significant anisotropy.

4.2 Introduction to Mechanical Behavior of Laminates (Laminate Plate Theory): Analytic Model

Figure 25. Lamina-Laminate Definitions

In the previous section, 4.1 Introduction to Mechanical Behavior of Anisotropic Materials, mechanical behavior of individual anisotropic layers, "lamina", of the "laminated" plate can now be used to model the mechanical behavior of the laminate. The purpose of this section is to introduce the reader to the basic ideas of laminated plate theory so that the reader gains an understanding of the relationship between the two scales. To gain a working knowledge of laminated plate theory, we encourage students to take ESM4044 MECHANICS OF COMPOSITE MATERIALS and ESM6104 MECHANICS OF COMPOSITE STRENGTH AND LIFE . With a working knowledge of laminate plate theory, designers can taylor the elastic properties and orientation of each lamina, hence optimize their design by controlling the properties and orientations of each lamina. At the end of this section we will outline several examples on how the mechanical response of a laminated plate can be controlled ("designed") by lamina properties, but first we briefly outline the derivation of the analytic model needed to understand how properties at the microscale control mechanical behavior at the macroscale.

Analytic Model:

Consider an orthorhombic lamina in plane stress and let the 1-2 axes shown in Figure 25 be aligned with the fiber axis. In plane stress we assume the lamina is very thin and the stress through the lamina thickness is very small.

Figure 25. Lamina axes defined

(44)

The elements of [Q] are related to the engineering constants E,G, and ν as follows

(45)

(46)

Transformation of lamina stress and strain from the 1-2 axes to the rotated x-y axes is shown here in matrix notation. This transformation is different from the transformation for all components of the second order tensor because here we have only three components of stress and strain because of the plane-stress condition.

(47)

where

(48)

The lamina stress-strain relationship in the rotated x-y coordinates

(49)

In reduced matrix notation

(50)

where

(51)

The inverse stress-stain relationship relates stresses(known) to strains(unknown) with the compliance, [S], matrix

(52)

where

(53)

Hence there is an inverse relationship between the [S] and [Q] matrices.

(54)

The equations above apply to each individual laminate layer ("lamina"). Next we stack each lamina into a laminate where we assume each lamina, denoted with a subscript k, is bonded ("glued") to the adjacent lamina. The composite structure is referred to as a laminated plate, see Figure 26.

Figure 26. Lamina-Laminate Configuration Geometry Defined

The laminated plate shown in Figure 27 is loaded, in the Z-direction. Deflections are assumed small and the Kirchoff thin plate approximations can be used here to derive relationships between in-plane and out-of-plane displacements, curvatures, twists, and strains.

Figure 27. Thin Plate Deflections

From Figure 27 it is not difficult to see how in-plane u_o and v_o displacements are augmented by out-of-plane (Z-direction) displacements and rotations.

(55)

From equation (7) the strain-displacement relationship are rewritten here for components of displacement, u and v, confined to the x-y plane ("in-plane").

(56)

From the expression of radius of curvature, ρ, we derive expressions for curvature, k_x and k_y assuming small angular deflections.

(57)

A similar expression for twist, k_xy, can be derived from Figure 28 where twist is defined as an "angular change" per "unit length".

Figure 28. Exaggerated Thin Plate Deflections and Twists Defined

The coefficient of 2 is introduced by convention for engineering shear strain.

(58)

Combining equations (57) and (58), into (56) yields

(59)

Substituting equation (59) into equation (49) yields a more general stress-strain relationship for the k-th lamina which includes out-of-plane displacement implicitly in the curvature terms.

(60)

With stresses defined in terms of the Z-coordinate, through the laminate thickness, we integrate these expressions to get the laminate resultant in-plane loads (N_x, N_y, N_xy), and moments (M_x, M_y, M_xy).

Figure 29. Laminate Plate In-Plane Loads and Moments Defined

(61)

Substitute equation (60) into equation (61) and simplify.

(62)

Equation (62) can be simplified

(63)

where

(64)

Figure 30 shows the relative position of each k-th lamina used by equation (61) to integrate through the thickness to calculate the laminate loads, (N_x, N_y, N_xy), and moments (M_x, M_y, M_xy).

Figure 30. Lamina Positions Defined in the Laminate for Integration of Resultant Loads

Substituting the [A], [B], and [D] matrices defined in equation (64) into equation (63) results in the final relationship that defines the mechanical behavior of laminated fiber-reinforced plate. Equation (63) shows the relationship between known in-plane strains ( ε^o_x, ε^o_y, γ^o_xy ) and curvatures ( k_x, k_y, k_xy ) and unknown in-plane loads and moments.

(65)

And the inverse relates known in-plane loads to unknown in-plane strains and curvatures.

(66)

The [A], [B], and [D] matrices for the laminate are similar to the [Q] matrix for the individual lamina. If you followed the basic ideas in this derivation it is not difficult to understand how microscale lamina properties control the macroscale laminate properties. In deed we now have a predictable way to control, and hence design, the mechanical behavior of a laminate at the macroscale.

Like the lamina at the microscale, anisotropy of the laminate has a significant effect on the mechanical behavior of the laminate. This can best be observed by inspection of equations (65) and (66). If the individual lamina are stacked and glued together such that the [B] matrix is zero then the mechanical behavior exhibits a decoupling of moments and in-plane loads. For example curvatures only affect moments and inplane strains only affect inplane load. On the other hand it is not difficult to see that if [B] is not zero then there is coupling between curvatures and inplane loads, and coupling also exists between inplane strains and moments. This is rather peculiar. For example if we apply an inplane strain to a specially configured laminate then the laminate can bend more then it would extend ("stretch"). Similarly if we apply an inplane load the laminate will twist. The later is actually used to an advantage by designers of aircraft structures where aircraft wings, that are constructed by fiber-reinforced laminates, are designed to twist into a predetermined aerodynamic angle-of-attack as the wing bends and extends. This design methodology is called PASSIVE aero-elastic tailoring. With "smart"-"intelligent" structures it is possible to insert active piezoelectric layers into the laminate stacking sequence and ACTIVELY control the mechanical behavior of the laminate. The degree of coupling is controlled by the [A], [B], and [D] matrices.

In this section a few simple examples are presented that illustrate how simple laminate configurations can provide predictable macroscopic mechanical behavior by inspection of equations (65) and (66).

Case I: Symmetric Laminates, [B]=0

By inspection when [B] is zero, inplane loads and moments decouple in equations (65) and (66).
Inplane loads {N} only produce inplane strains {ε} and inplane moments {M} only produce
curvature {k}

Case II: Specially Orthotropic Symmetric laminates, A₁₆=0 and A₂₆=0

Bi-directional laminates consist only of 0/90 degree lamina. It is not difficult to show Q₁₆ and Q₂₆
are zero. Hence A₁₆ and A₂₆ are also zero and consequently the inplane shear strain {γ} decouples
from the inplane normal strains. Therefore shearing loads can only produce shearing strains and
inplane normal loads can only produce inplane strains

Case III: Specially Orthotropic Bending Behavior, D₁₆ & D₂₆ -> 0 , no. of lamina -> large

For a large number of symmetric pairs, +θ at +h and -θ at -h, D₁₆ and D₂₆ go to zero. Hence
moments {M_x} & {M_y} and torque {M_xy} decouple so that bending caused by moments do not
introduce twisting and torque that causes twisting does not introduce bending.

Accept for a few simple cases outlined above, equations (65) and (66) are sufficiently complex where the designer can not deduce a solution by observation. These equations must be solved numerically, hence the designer must make educated guesses and experiment by submitting multiple jobs and hopefully converge on a particular design. To control the mechanical behavior of the laminate requires attention on controlling the [A], [B], and [C] matrices. For complex stacking sequences this becomes difficult. Many references on designing laminated structures exist. As an introduction we suggest the reader look at Refs.[19,20]. To that end we provide an interactive tool below that will allow the reader to experiment and design the desired laminate mechanical behavior.

5.0 Laminate Singularities Caused by Anisotropy: "Free-Edge Problem": FEM Model

Mechanical behavior of thick laminated plates can be accurately predicted by simple laminated plate theory, but the stresses within these thick laminates are not accurately predicted by classical laminated plate theory. For thick laminates stress singularities result in controlling the sequence of failure events that lead to the ultimate fracture of the laminate. Figure 31 shows a typical sequence of damage events that lead to ultimate fracture at the macroscale.

Figure 31. Typical sequence of damage events leading to laminate fracture

To accurately predict stress singularities in thick laminates we introduce the laminate "Free-Edge" problem. In this section we provide:

6.0 Laminate Singularities Caused by Ply Cracks: FEM Model

After the formation of a ply crack, see Figure 25, new singularities occur near the ply crack tip. The growth of this crack can continue into the adjacent ply, see Figures 1, 3, 4, and 5, or debond along the interface. Below we provide models that demonstrate how anisotropy and residual stress can augment singularities near ply cracks and influence how ply cracks grow and control laminate fracture strength. In this section we provide:

7.0 Cracks Near Interfaces Between Dissimilar Anisotropic Materials

In previous sections we studied stresses near laminate stress free edges and ply cracks that exist at an interface between two dissimilar anisotropic materials. These results were predicted by a Finite Element Model (FEM) that did not include singularities. Instead stresses gradients near the free edge and ply crack tips were approximated with a high density FEM mesh near the singurality. In this section we introduce an analytic model that predicts these singularites: 7.1) at a bimaterial interface intersecting a stress free edge, see Figure 30, and, 7.2) at a ply crack tip intersecting an interface seperating two dissimilar anisotropic media, see Figure 31. Both models are solved using Stroh's method, Ref.[24]. These singularites can be introduced into enriched FEMs that included a variety of different boundary conditions (BCs) without using high density FEM meshes. With these enriched FEMs, stress intensity factors can be solved for as unknowns along with the nodal displacements for each unique geometry and BCs. Section 8 will outline how to create a finite element model with elements "enriched" with these singularities.

For both models, shown in Figs. 30 and 31, it is important to note that the singularity coincides with the origin of the coordinates. Hence each model has a unique coordinate system. Since the objective here is to reproduce published results of these two models the same coordinate system established by Ting et al. will be used. This choice will require rederivation of an arbitray rotation in the 1-2 plane for orthorhombic symmetry outlined in section 4.1. The derivations below will follow the notation and procedures outlined by Ting et al, Refs. [25] and [26]. The notes below will expand on the solution introduced by Ting et al. by developing the numerical approach to solve for these singularities.

7.1 Stress free edge singularities:

T.C.T. Ting and S.C. Chou, Ref.[25], used Stroh's method to solve for the singularity where the interface seperates two dissimilar anisotropic regions and terminates at a stress free edge, see Figure 30. Here we only expand on the development of the numerical solution and use the same notation in Ref.[25] except the exponent on the radius, r, is changed from, δ, to -k similar to the exponent in Ref.[26].

8.0 Cracks in Homogeneous Isotropic Materials

Outline how enriched finite elements are used to augment existing nonsingular finite elements with singularities that exist at crack tips Ref [Benzley]. Use this enriched FEM to model crack tip singularities in a homogeneous isotropic material and compare numerical results with analytic results for Mode-I and Mode-II.

9.0 Stress Concentrations in Homogeneous Anisotropic Plates

Predict stress gradient near holes in an anisotropic and isotropic plate and compare results with Mizra & Olson Ref [1980].

10.0 Wave Propagation in Homogeneous Isotropic/Anisotropic Materials

Introduction to 1-D isotropic wave propagation: Analytic and Numerical Models

1. Kriz, R.D., "Influence of Ply Cracks on Fracture Strength of Graphite/Epoxy Laminates at 76K", Effects of Defects in Composite Materials, ASTM STP 836, American Society for Testing and Materials, pp. 250-265, 1984.

2. Broeck, D., Elementary Engineering Fracture Mechanics, Kluwer Academic Publishers Group, 1982.

3. Eftis, J., Subramonian, N., and Liebowitz, H., "Crack Border Stress and Displacement Equations Revisited," Engineering Fracture Mechanics, Vol. 9, pp. 189-210, 1977.

4. Williams, M. L., "The Stresses Around a Fault of Crack in Dissimilar Media," Bulletin of the Seismological Society of America, Vol. 49, No. 2, pp. 199-204, April, 1959

5. Zak, A. R., and Williams, M. L., "Crack Point Stress Singularities at a Bi-Material Interface," GALCIT SM 62-1, California Institute of Technology, January, 1962.

6. Nye, J. F., Physical Properties of Crystals, Oxford University Press, London, 1957.

7. Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Body, Holden-Day, San Francisco, 1963.

8. Frederick, D. and Chang, T. S., Continuum Mechanics, Scientific Publishers, Inc. Boston, 1972.

9. Kriz, R.D. and Ledbetter, H.M., "Elastic Representation Surfaces of Unidirectional Graphite/Epoxy Composites," Recent Advances in Composites in the United States and Japan, ASTM STP 864, J.R. Vinson and M. Taya, Eds. American Society for Testing and Materials, Philadelphia, pp. 661-675, 1985.

10. Kriz, R.D. and Stinchcomb, W.S., "Elastic Moduli of Transversely Isotropic Fibers and Their Composites," Experimental Mechanics, Vol. 19, pp. 41-49, 1979.

11. Golfein, S., "Investigation of Structure of Graphite FIbers with the Goal of Improving Their Mechanical Properties," Internal Report AD-A013 524, Army Mobility Equipment Research and Development Center, Fort Belvoir, Virginia, 1974.

12. Barnet, F.R. and Norr, M.K., "A Three-dimensional Structural Model for a High Modulus Pan-based Carbon Fibre," Composites, pp. 93-99, 1976.

13. Ledbetter, H.M. and Kriz, R.D., "Elastic Wave Surfaces in Solids," Phys. Stat. Sol. (b) 114, p. 475, 1982.

14. Musgrave, M.J.P., "On an Elastodynamic Classification of Orthorhombic Media," Proc. R. Soc. Lond. A374, p. 401, 1981.

15. Kriz, R.D. and Ledbetter, M.M., "Elastic-wave Surfaces in Anisotropic Media," Rheology of Anisotropic Materials, C. Huet, D. Fourgoin, S. Richmond, Eds. CR 19 Coll. GFR Paris Nov. 1984 Editions CEPADUES, Toulouse, pp. 79-92, 1986.

16. Kriz, R.D. and Heyliger, P.R., "Finite Element Model of Stress Wave Topology in Unidirectional Graphite/Epoxy: Wave Velocities and Flux Deviations,"Review of Progress in Quantitative Nondestructive Evaluation, Vol.8A, Plenum Publishing Corp., pg 141-148, 1989.

17. Kriz, R.D. and Gary J.M., "Numerical Simulation and Visualization Models of Stress Wave propagation in Graphite/Epoxy Composites,"Review of Progress in Quantitative Nondestructive Evaluation, Vol.9, Plenum Publishing Corp., pg 125-132, 1990.

18. Vandenbossche, B. and Kriz, R.D.,and Oshima, T." Stress Wave Displacement Polarizations and Attenuation in Unidirectional Composites: Theory and Experiment, "J. Research In Nondestructive Evaluation, Vol. 8, No. 2, pp. 101-123, 1996.

19. Jones, B., Mechanics of Composite Materials, McGraw Hill, 1975.

20. Ashton, J.E. and Whitney, J.M., "Theory of Laminated Plates", Technomic Publishing Co., Inc., Stamford Conn., 1969.

21. Kriz, R.D. and Stinchcomb, W.W., "Effects of Moisture, Residual Thermal Curing Stresses, and Mechanical Load on the Damage Development in Quasi-Isotropic Laminates," Damage in Composite Materials, ASTM STP 775, American Society for Testing and Materials, 1982, pp.63-80.

22. Kriz, R.D., "Edge Stresses in Woven Laminates at Low Temperatures," Composite Materials: Fatique and Fracture, Second Volume, ASTM STP 1012, Paul A Lagace, Ed., American Society for Testing and Materials, 1989, pp. 150-161.

23. R. B. Pipes and N.J. Pagano, "Interlaminar Stresses in Composite Laminates Under Uniform Axial Extension", J. Composite Materials, Vol. 4, pp.538-548, 1970.

24. Stroh, A.N., "Steady State Problems in Anisotropic Elasticity,", J. Mathematics and Physics, Vol. 41, pp. 77-103, 1962.

25. Ting, T.C.T and Chou, S.C., "Stress Singularities in Laminated Composites," Fracture of Composite Materials, eds. G.C. Sih and V.P. Tamuzs, 1982, Martinus Nijhoff, Proceedins of the 2nd USA-USSR Symposium, Lehigh Univ., Bethlehem, PA, March 9-12, 1981, pp. 265-277, 1982.

26. Ting, T.C.T. and Hoang, P.H., "Singularities at the Tip of a Crack Normal to the Interface of an Anisotropic Layered Composite," Int. J. Solids and Structures, Vol. 20, No. 5, pp. 439-454, 1984.

27. Tewary, V. K. and Kriz, R. D., "Generalized Plane strain Analysis of Bimaterial Composite a Free Surface Normal to the Interface", Journal of Material Research, Vol. 13, pp. 2609-2622, 1991.

28. Tewary, V.K. and R. D. Kriz, "Effect of a Free Surface on Stress Distribution in a Bimaterial Composite", NIST Special Publication 802, National Institute of Standards and Technology, U.S. Gov. Printing Office, Washington D.C., March 1991.

29. Benzley, S.E., "Representation of Singularities with Isoparametric Finite Elements," Int. J. Numerical Methods in Engineering, Vol. 8, pp. 537-545, 1974.

30. Mirza, F.A. and Olson, M.D., "The Mixed Finite Element Method in Plane Elasticity," Int. J. Numerical Methods in Engineering, Vol. 15, pp. 273-289, 1980.

31. Heyliger, P.R. and Kriz, R.D., "Stress Intensity Factors by Enriched Mixed Finite Elements," Int. J. Numerical Methods in Engineering, Vol. 28, 1461-1473, 1989.