1.16 Multilayer Models for Composite and Sandwich Structures

1.16 Multilayer Models for Composite and Sandwich Structures Serge Abrate, Southern Illinois University, Carbondale, IL, United States Marco Di Sciuva, Politecnico di Torino, Torino, Italy r 2018 Elsevier Ltd. All rights reserved.

1.16.1 1.16.2 1.16.2.1 1.16.2.2 1.16.3 1.16.3.1 1.16.3.2 1.16.3.3 1.16.4 1.16.4.1 1.16.4.1.1 1.16.4.1.2 1.16.4.2 1.16.4.2.1 1.16.4.2.2 1.16.4.2.3 1.16.4.2.4 1.16.4.3 1.16.4.3.1 1.16.4.3.2 1.16.5 1.16.6 1.16.6.1 1.16.6.1.1 1.16.6.1.2 1.16.6.1.3 1.16.6.2 1.16.6.2.1 1.16.6.2.2 1.16.6.2.3 1.16.6.3 1.16.6.3.1 1.16.6.3.2 1.16.6.3.3 1.16.6.3.4 1.16.6.3.5 1.16.7 1.16.7.1 1.16.7.2 1.16.7.3 1.16.7.4 1.16.7.4.1 1.16.7.4.2 1.16.8 References

Introduction The Need for Discrete Layer Theories ESL Theories Transverse Stress Continuity and the Zigzag Effect Layerwise Theories Original Layerwise Theories Generalized Layerwise Theories Layerwise Mixed Formulation Zigzag Theories ESL Theories Enriched by MZZF Polynomial type ESL theories with MZZF Non-polynomial theories with MZZF Zigzag Plate Theories With Polynomial Expansions at the Ply Level Piecewise linear theories with transverse stress continuity Third order polynomial expansions at the ply level Refined first order zigzag theories Refined higher order zigzag theories Non-Polynomial Zigzag Theories Sine zigzag theories Hyperbolic zigzag theories Sublaminate Approach Models for Multilayer Structures With Interfacial Damage Imperfect Interface Models Linear spring layer models Nonlinear spring layer models Viscoelastic interface models Models With Multiple Delaminations Layerwise theories with multiple delaminations Multiple delaminations in {1,0} zigzag theory Multiple delaminations in {3,0} zigzag theory Models Including Imperfect Interfaces Elasticity solutions for laminates with imperfect interfaces ESL theories with imperfect interfaces Murakami’s theory Layerwise theories with imperfect interfaces Zigzag theories with imperfect interfaces Sandwich Structures Elasticity Solutions for Three Layer Structures ESL Theories for Sandwich Structures Theories Based on Different Kinematic Assumptions for the Facings and the Core and Displacement Continuity at the Interfaces Mixed Formulation for Sandwich Structures HSAPT Other mixed formulations for sandwich structures Conclusions

Comprehensive Composite Materials II, Volume 1

doi:10.1016/B978-0-12-803581-8.09885-4

399 400 400 401 402 402 403 403 404 404 404 404 404 405 405 408 409 409 409 409 410 410 411 411 411 412 412 412 412 412 412 413 413 414 414 414 414 414 415 415 415 415 416 416 417

399

400

1.16.1

Multilayer Models for Composite and Sandwich Structures

Introduction

Many equivalent single layer (ESL) theories that have been developed over the years. In spite all these developments, these ESL theories do provide satisfactory results for some multilayer structures. This chapter discusses the need for discrete layer models and presents a comprehensive view of the current state of the art. With the axiomatic approach, the formulation of ESL theory starts with an assumed displacement field that is continuous through the thickness. Accordingly, the strains are also continuous through the thickness. For laminated structures with different elastic properties and/or different fiber orientations for adjacent plies, continuous strain distributions result in discontinuity in the transverse shear stresses sxz, syz and the transverse normal stress szz at the interface. Such discontinuities are not physically possible so, when these stresses need to be determined accurately, an improved description of the kinematics of the deformation is necessary. Predictions of the onset and growth of delamination are based on criteria in which accurate values for sxz, syz, and szz are needed. Sandwich structures with thin facesheets and a much thicker core that experiences significant shear deformations and, sometimes, compression in the transverse direction is one example of structures that may not be modeled accurately by ESL theories. ESL theories also assume perfect bonding between adjacent layers. This may not be always the case for laminated composite plates because of resin rich regions between plies that are there as an unintended consequence of the manufacturing process. An added layer of matrix material can also be inserted between plies in order to improve impact resistance. Delaminations are another type of imperfect bonding. In some applications, several layers of wood are joined using mechanical fasteners (nails, screws) or adhesives. Laminated glass used for windshields, windows and other architectural applications generally consist of two or more layers of glass bonded by a softer, thin, transparent polymeric layer. Those are also examples of multilayered structures with imperfect interfaces. This chapter gives an overview of theories for multilayered structures for which the deformations of individual layers are significantly different and/or discontinuities at the interface cannot be ignored.

1.16.2

The Need for Discrete Layer Theories

This section starts by recalling the basic formulation of ESL theories and why they do not provide satisfactory results for some multilayer structures. It also defines two types of discrete layer models: layerwise models and zigzag models.

1.16.2.1

ESL Theories

Typically, ESL theories start with a {m,n} power series expansion of the form u¼

m X

zi ui ;

v¼

i¼0

m X

zi vi ;

w¼

i¼0

n X

ð1Þ

zj wj

j¼0

Using such an expansion results in what is called an unconstrained theory because no extra assumption is made to reduce the number of variables which is 2m þ n þ 3. While it is expected that increasing the number of terms will increase the accuracy of the results, the process will become computationally expensive. The transverse shear strains are given by exz ¼

m X

i zi1 ui þ

i¼1

n X

zj wj;x ;

eyz ¼

j¼0

m X

i zi1 vi þ

i¼0

n X

zj wj;y

ð2Þ

j¼0

Some studies include various {m,n} combinations. In consistent approximations, n¼m  1 so that for each zj term in the transverse strain approximation there is a uj þ 1 term and a wj,x term (or a vj þ 1 term and a wj,y term). Most theories adopt the transverse inextensibility assumption (ezz ¼ w,z ¼ 0) so wi ¼ 0 for i¼ Z1. The transverse shear strains are given by exz ¼

m X

i zi1 ui þ wo;x ;

i¼1

eyz ¼

m X

i zi1 vi þ wo;y

ð3Þ

i¼0

In the Kirchhoff–Love theory, transverse shear deformations are neglected (exz ¼ eyz ¼ 0) so ui ¼ vi ¼ 0 for iZ2, u1 ¼  wo,x, v1 ¼  wo,y. The displacement field for that theory is u ¼ uo  zwo;x

v ¼ vo  zwo;y

w ¼ wo

ð4Þ

This can be stated as: line segments normal to the reference surface remain straight and normal to the reference surface after deformation and does not suffer variation of their length. In the first order shear deformation theory (FSDT), transverse shear stresses are assumed to be constant through the thickness. Therefore, ui ¼vi ¼ 0 for iZ2 and u1 and v1 are variables independent of wo so the kinematics of the deformation are described by u ¼ uo þ zu1

v ¼ vo þ zv1

w ¼ wo

ð5Þ

Multilayer Models for Composite and Sandwich Structures

401

With this theory, line segments normal to the reference surface remain straight but not necessarily normal to the reference surface after deformation. Usually, shear stresses vanish on the top and bottom surfaces. To enforce that requirement, a higher order expansion of exz and eyz has to be considered beyond the constant term exz ¼ u1 þ wo;x þ 2zu2 þ 3z2 u3 ;

eyz ¼ v1 þ wo;y þ 2zv2 þ 3z2 v3

ð6Þ

Keeping only the first order terms is not sufficient and the second order terms are needed  to enforce the zero shear strain conditions on the top and bottom surfaces. Then we find that u2 ¼ 0 and u3 ¼  3h42 u1 þ wo;x . The displacement field becomes u ¼ uo þ zu1  and can be written as 

4z2 3h2



 4z3  u1 þ wo;x ; 3h2

  u ¼ uo  zwo;x þ f ðzÞ u1 þ wo;x ;

v ¼ vo þ zv1 

 4z3  u1 þ wo;y ; 3h2

w ¼ wo

  v ¼ vo  zwo;y þ f ðzÞ v1 þ wo;y ;

w ¼ wo

ð7Þ

ð8Þ

for this theory which is usually referred to as Reddy’s third order theory (RTSDT). where f ðzÞ ¼ z 1  Many ESL theories can be written in that form (Eq. (8)). Table 1 gives several examples of functions f(z) used in the literature. There are only three variables to be determined in the classical plate theory: uo, vo, wo. For all the other theories with displacement fields given by Eq. 8, there are five variables: uo, vo, wo, u1, and v1. This will be an important consideration when considering the use of discrete layer theories.

1.16.2.2

Transverse Stress Continuity and the Zigzag Effect

With ESL theories the displacement field is continuous through the thickness and so will the three transverse strains exz, eyz, and exz. With different elastic properties for the layers above and below the interface, the transverse stresses sxz, syz, and szz are discontinuous. Basic equilibrium conditions indicate that those three stress components should be continuous at each interface. Stress continuity requires strain discontinuities at the interfaces which manifests itself in changes in slopes of the axial displacements (Fig. 1). This is often called the zigzag effect. Accounting for it requires separate approximations for each layer enforcing displacement continuity at each interface. Table 1 Examples of shear shape functions f(z) in equivalent single layer (ESL) theories Theory

f(z)

Kirchhoff–Love Mindlin–Reissner Reddy’s third order1 Sine (Touratier)2

0 z  4z 2 z 1  3h 2

Inverse trigonometric (Thai et al.3) Hyperbolic (Soldatos4) Hyperbolic (Mahi et al.5)

h tan (2z/h)  z h sinh (z/h)  z cosh (1/2)  3 h 4 z 2 2 tanh ð2z =h Þ  h2

Fig. 1 Zigzag effect at the interface between layers k and k þ 1.

h pz p sin h 1

3cosh ð1Þ

402

Multilayer Models for Composite and Sandwich Structures

Many discrete layer theories with the same kinematic assumption for every layer have been proposed. When only displacement continuity conditions are enforced at ply interfaces they are called layerwise theories and we will see that the total number of variables increase with the number of plies. Using a combination of global and local variables and satisfying both stress and displacement continuity conditions leads to theories with a fixed number of variables that are called zigzag theories. Many discrete layer theories assume that all plies can be described by the same displacement field. For example, every ply behaves as a Mindlin–Reissner FSDT plate, as a {3,0} or a {3,2} plate. Implied in the use of a single type kinematic field for all layers is the assumption that there are no major difference in thickness and/or deformation of the plies. In some cases, the deformation of the various plies is drastically different so it cannot be assumed that a single type of kinematic field can be used for all layers. Sandwich structures have a core that is much thicker than the facesheets and is much more flexible in shear. In some cases, it also deforms in the transverse direction. Therefore, many ad hoc theories have been developed using different approximations for different layers.

1.16.3

Layerwise Theories

The term layerwise usually implies the use of a discrete layer theory with different kinematic fields for each layer. Those kinematic assumptions are usually of the same type. The most common one is that the displacement is assumed to vary linearly through the thickness of the ply. Displacements are taken to be continuous at the interfaces but continuity of the transverse stresses is not enforced. In this section we distinguish three types of layerwise theories. In the original formulation, the primary variables are the interface displacements including the top and bottom surface. A variant of that original formulation is when the displacements of the midsurface and the rotation of the normals in each layer are taken as primary variables (Fig. 2). In generalized layerwise theories, the primary variables are not specific displacements or rotations. Instead, kinematic fields are assumed and the variables are unknown functions to be determined. The original and the generalized theories are displacements based. These theories are based entirely on displacement approximations. Layerwise mixed formulations are based on approximations for the displacements and for the transverse stresses in each ply.

1.16.3.1

Original Layerwise Theories

In the original layerwise theories, the primary variables are the displacements at the ply interfaces including the top and bottom surfaces.6–8 The displacements at an arbitrary point through the thickness are obtained by interpolation of the primary variables. For example, Fig. 2(a) shows four axial displacements for a sandwich beam. Linear interpolation is used to determine the displacement at any other location. In general, for laminate with N  1 layers, there are N displacement variables for each coordinate direction and the displacement components at one point can be written as uðx; y; z; t Þ ¼

N X

Uk ðx; y; t ÞCk ðzÞ;

vðx; y; z; t Þ ¼

k¼1

N X

Vk ðx; y; t ÞCk ðzÞ;

wðx; y; z; t Þ ¼

k¼1

N X

Wk ðx; y; t ÞCk ðzÞ

ð9Þ

k¼1

where the functions Ck(z) are piecewise linear interpolation functions (e.g.,9–11) or Lagrange interpolation functions.12 An equivalent approach is to use as primary variables for the in-plane displacements the axial displacement of the mid-surface and the rotation of each layer (Fig. 2(b)).13–26 Similar descriptions of the deformation can be found in Ref. [27]. Application to sandwich plates can be found in Ref. [13]. The kinematics of the deformation illustrated in Fig. 2(b) can be described analytically by expressing the displacement field for an arbitrary layer k as ðkÞ

uðkÞ ¼ uðokÞ þ zk u1

(a)

ðkÞ

vðkÞ ¼ voðkÞ þ zk v1

ðkÞ

wðkÞ ¼ wðokÞ þ zk w1

ð10Þ

(b)

Fig. 2 Layerwise models of a sandwich beam: (a) axial displacements at each interface are taken as primary variables and (b) primary variables: axial displacement at neutral axis and rotation of the normal in each layer.

Multilayer Models for Composite and Sandwich Structures ðkÞ

ðkÞ

403

ðkÞ

where uo , vo , and wo are displacements of points on the midplane of the layer, zk is the position relative to the midplane of layer ðkÞ ðkÞ ðkÞ k, u1 , v1 represent the rotation of the normal to that midplane, and w1 the uniform elongation of the same normal. Eq. (10) describes the kinematics of a {1,1} layerwise theory in the sense that for each ply all three displacements are approximated by a polynomial of order 1. For an N-ply laminate there is a total of 6N variables. Displacement continuity conditions at the N  1 interfaces result in 3(N  1) equations so the total number of equations is reduced to 3(N þ 1) variables. Often it is assumed that the plate is inextensible in the transverse direction (e.g.,19,25) so w(k) ¼ w and there are 4N þ 1 variables initially and with 2(N  1) continuity equations, the total number of variables is 2N þ 3. Whether transverse inextensibility is invoked or not, the number of unknowns increases linearly with the number of layers. For example, for a three-layered plate, the number of variables is 2  3 þ 3¼9 compare to three for the classical plate theory or five for the FSDT. Laminated composite structures often have a large number of plies so, for example, for a 48 ply laminate there is a total of 99 variables with this type of theory which becomes impractical. Analyses based on this displacement field (Eq. (10)) can be found in Refs. [28–30]. In the layerwise theories discussed so far the displacement continuity equations at ply interfaces are used to reduce the number of displacement variables. At the interface between layers k and k þ 1, these conditions can be written as h i ðkþ1Þ ðkÞ ðkþ1Þ ðkÞ 〚u〛  uo  hkþ1 u1 þ hk u1 2 k ¼ uo ð11Þ h i ðkþ1Þ ðkÞ ðkþ1Þ ðkÞ  vo  hkþ1 v1 þ hk v1 2 〚v〛 k ¼ vo Mau31 introduced the interface shear stresses as Lagrange multipliers (lx)k and (ly)k and added the term n1 h X k¼1

i   ðlx Þk 〚u〛k þ ly k 〚v〛k

ð12Þ

to the potential energy functional and derived the equations of equilibrium of the plate that included the interlaminar shear stresses.

1.16.3.2 Fares

32

Generalized Layerwise Theories

added linear and quadratic local terms to the transverse displacement to obtain the displacement field ðkÞ

uðkÞ ¼ uðokÞ þ zk u1

ðkÞ

vðkÞ ¼ voðkÞ þ zk v1

ðkÞ

ð kÞ

wðkÞ ¼ wo þ zk w1 þ z2k wk

ð13Þ

Initially, Eq. (13) uses 6N þ 1 displacement variables for an N-ply laminate. Enforcing three displacement continuity conditions at the N  1 interfaces reduces the number of variables to 3N þ 4. Plagianakos and Saravanos33 introduced a {3,0} layerwise theory with uk ¼ U k Ck1 þ U kþ1 Ck1 þ akx Ck3 þ lkx Ck4 vðkÞ ¼ V k Ck1 þ V kþ1 Ck1 þ aky Ck3 þ lky Ck4 ;

wðkÞ ¼ wo

ð14Þ

where the functions Cki are interpolation functions, Uk and Vk denote the inplane displacements of the interfaces, akx ; aky ; lkx , and lky are amplitudes of quadratic and cubic displacement variations through ply k that vanish on the interfaces. Higher-order LW formulations fulfilling the continuity requirements of displacements as well as the transverse shear and normal stresses at the interface between two consecutive layers have been proposed by Cho et al.34 and Lee et al.35 ({3,2}-order layerwise theories); in the same line, Refs. [36–39] proposed a {3,3}-order layerwise theories. In Wu and Hsu40 LW higher order theory, the displacement and traction continuity conditions at the interface between layers and the traction conditions at the outer surfaces were imposed as the constraint conditions, and introduced into the potential energy functional by the Lagrange multiplier method. Evaluations of the layerwise theories and comparisons of their predictions with those from other types of theories (ESL or zigzag theories) or elasticity solutions can be found in the references cited above and in Refs. [41–44].

1.16.3.3

Layerwise Mixed Formulation

45,46 A layerwise mixed assumes that in each layer the displacements u ¼ (uk, vk, wk)T and the transverse stresses  T formulation k k k sn ¼ sxz ; syz ; szz are used as primary variables in order to a priori satisfy continuity requirements at the interfaces. In addition,  T the inplane strains ep ¼ ekxx ; ekyy ; ekxy are also used as primary variables in order to have only first order derivatives in the formulation. The three equations of equilibrium are written in terms of ep and rn. The three transverse stresses rn are written in terms of the displacements u using the stress–strain relations and the three strains ep are related to the displacements u using the strain–displacements relations. Writing these nine equations so that the right hand side is equal to zero, the left hand side is not uniformily equal to zero in an approximate solution. The left hand side of these equations are called residuals and in this formulation the function to be minimized is the sum of the squares of the nine residuals.

404

1.16.4

Multilayer Models for Composite and Sandwich Structures

Zigzag Theories

Under the phrase “zigzag theories,” this section regroups theories for analyzing multilayer structures with different kinematics for each layer but with a fixed number of variables that is independent of the number of plies. Section 1.16.4.1 presents a general approach in which an ESL theory is enriched by a zigzag function that introduces strain discontinuities at the interfaces. Section 1.16.4.2 describes theories in which a polynomial displacement is assumed for each layer, and the continuity of displacements and transverse stresses is used to reduce the total number of variables to that of the corresponding ESL theory regardless of the number of plies. The number of variables can be reduced further by requiring that shear stresses vanish on the top and bottom surfaces of the structure.

1.16.4.1

ESL Theories Enriched by MZZF

In 1986, Murakami47 introduced a piecewise linear zigzag function in order to model the so called zigzag effect in laminated plates with a periodic layup. This function, being independent of the stratification and the mechanical characteristics of the material of the various layers constituting the laminate, can be added to any component of the displacement field of any ESL theory to easily introduce the zigzag effect, as will be shown below. It should be noted that the Murakami zigzag functions (MZZF) does not allow to fulfill the conditions of continuity of the transverse shear stresses at the interfaces. Applications to displacement based polynomial or non-polynomial theories, mixed formulations with both displacement and stress variables are considered. Complicating effects such as interlayer slip can also be included. A deep investigation of the strength and weakness of this approach has been given by Gherlone48 and Iurlaro et al.49

1.16.4.1.1

Polynomial type ESL theories with MZZF

The FSDT, adding a term with a MZZF results in the following displacement field for an arbitrary layer k uk ¼ uo þ zu1 þ ð1Þk ζk uM

vk ¼ vo þ zv1 þ ð1Þk ζk vM ;

w ¼ wo

ð15Þ

where ζk ¼2zk/hk is a nondimensional quantity varying linearly from  1 at the bottom of the ply to 1 at the top. hk is the thickness of the ply and zk is the distance from the midplane of the ply uM and vM are two additional variables that are functions of x, y, and t. They give the amplitudes of the MZZFs. MZZFs can be added to any theory based on a polynomial expansion of the displacements.48,50–69 In the original work of Murakami,47 uk ¼ uo þ zu1 þ ð1Þk ζk uM

vk ¼ vo þ zv1 þ ð1Þk ζk vM ;

w ¼ wo þ zw1 þ ð1Þk ζk wM

ð16Þ

70

Displacements can also be expanded in a series of Legendre polynomials. Demasi63 used a mixed variational approach in which the displacements are expanded in a polynomial series and enriched by MZZF as discussed above. In addition, the three transverse stresses are expanded in terms of Legendre polynomials (no MZZF).

1.16.4.1.2

Non-polynomial theories with MZZF

In Ref. [71], laminated structures are analyzed using a trigonometric ESL theory with an added MZZF. The displacement field for a beam is given by   ð17Þ uk ¼ uo  zwo;x þ f ðzÞ oz þ wo;x þ ð1Þk ζk uM w ¼ wo where oz is the rotation of the cross section due to shear deformation and f ðzÞ ¼

h pz sin p h

ð18Þ

For sinusoidal transverse shear theories, allowing for a quadratic evolution of the transverse displacement in the thickness-wise direction, the displacement field of a plate are taken as,72,73 pz uk ¼ uo þ zu1 þ sin u2 þ ð1Þk ζk uM h pz ð19Þ vk ¼ vo þ vu1 þ sin v2 þ ð1Þk ζk vM h w ¼ wo þ zw1 þ z2 w2 pz An hyperbolic shear deformation theory with MZZFs where sin pz h is replaced by sinh h is used in Refs. [73,74].

1.16.4.2

Zigzag Plate Theories With Polynomial Expansions at the Ply Level

This subsection considers theories in which kinematic assumptions are made at the ply level and the number of variables is reduced using conditions at ply interfaces and on the top and bottom surfaces. In the development of a new theory one should consider to be made is the order of the approximation, and how many variables vary from ply to ply (local variables) and how many are global variables. For example, a {3,3} polynomial displacement approximation for each ply will result 12N variables for an N ply laminate. Requiring that three displacements and three transverse shear stresses be continuous at each interface produces

Multilayer Models for Composite and Sandwich Structures

405

6(N  1) equations that can be used to reduce the number of variables to 6N þ 6. This number increases with the number of plies. Here there are 6 continuity conditions per interface which means that we cannot have more than 6 local variables per ply. Of the 12 variables in that theory, 6 can be selected to be global variables and the other six will be local variables. The initial number of variables is now 6N þ 6 subtracting the 6(N  1) conditions results in 12 variables for this problem. This illustrates the fact that in order to have a number of variables that is independent of the number of plies, the number of local variables cannot exceed the number of compatibility conditions at each interface. Another decision is the choice of the reference surface to use. Some authors use the bottom surface while most use the midsurface. Several other practical decisions have been made by different authors. The following present types of zigzag theories that have seen significant development and use in the literature.

1.16.4.2.1

Piecewise linear theories with transverse stress continuity

Modeling each ply as a Reissner–Mindlin plate, the displacement field for layer k is uk ¼ uko þ zk uk1 ;

v ¼ vok þ zk v1k ;

w ¼ wo

ð20Þ

where the transverse displacement is constant through the thickness according to the usual transverse incompressibility assumption. For an N ply laminate this will result in 4N þ 1 variables. The continuity conditions for the 2 inplane displacements and the two transverse shear stresses at the interfaces can be used to reduce the number of unknowns by 4(N  1). Therefore, the number of displacement variables is reduced to 5, like in the Reissner–Mindlin plate theory regardless of the number of plies. With this displacement field, the transverse shear stresses cannot be made to vanish on the top and bottom surfaces. Enforcing such a requirement leads to neglecting shear deformation through the thickness of the plate. In effect, it is constant along the thickness,75–77 thus enforcing such a requirement leads to neglecting shear deformation through the thickness of the plate. In applying the continuity conditions to reduce the number of equations to five, different options are available regarding the choice of the reference surface and the displacement variables to be used. The following will describe theories where either the bottom surface or the midsurface are taken as reference and different ways of describing the rotation of the plies are adopted. Di Sciuva,75–78 who first named this class of theories “zigzag theories” uses the bottom surface as reference surface and the following displacement field uk ¼ uo þ zu1 þ

k1 X

ðz  zi Þji Hi

i¼1

vk ¼ vo þ zv1 þ

k1 X

ðz  zi Þci Hi

i¼1

wk ¼ wo

ð21Þ

where uo, vo, and wo are the displacements at the bottom surface of the laminate, u1, v1 the rotations of the normals in the bottom ply (first ply), ji ¼ ai(u1 þ wo,x) and ci ¼ bi(v1 þ wo,y) the so-called zigzag functions, ai and bi known constants only depending on the transverse shear mechanical properties of the constituent layers. Hi are Heaviside unit functions. This theory is known in the literature as first-order shear deformation zigzag theory (FSDZZT). Icardi79 also uses the bottom surface as the reference surface. The displacement field for a beam is taken as uk ¼ u1 þ wk ¼ w1 þ

k1 X i¼1 k1 X i¼1

ðz  zi Þui2 Hi ðz  zi Þwi2 Hi þ

k1 X i¼1

ðz  zi Þ2 wi3 Hi

ð22Þ

where, as before, Hi are Heaviside functions. Here ui2 represents the rotation of the normal in ply i whereas in Eq. (21) there is a global rotation u1 and a relative rotation ji. The same approach is used for plates in Ref. [80].

1.16.4.2.2

Third order polynomial expansions at the ply level

The {3,2} third order polynomial expansion for each layer k uk ¼ uko þ zuk1 þ z2 uk2 þ z3 uk3 ; wk ¼ wko þ zwk1 þ z2 wk2

vk ¼ vok þ zv1k þ z2 v2k þ z3 v3k

ð23Þ

Results in 11 displacement variables per ply, at perfectly bonded interfaces there are 3 displacement continuity equations and 3 transverse stress continuity equations. Requiring the transverse shear stresses to vanish on the top and bottom surface results in 4 additional equations. For perfect bonding between plies this kinematic approach results in 11N  6(N  1)  4¼ 5N þ 2 variables for an N ply laminate. The number of ply-dependent variables should be reduced in order to have a fixed number of variables independently of the number of plies. With n ply-dependent variables and 11  n global variables, the displacement field in Eq. (23) results in a total number of variables Nt ¼Nn þ (11  n)  6(N  1)  4¼ (n  6)N þ 13  n. The number of ply-dependent variables must be equal to six and then Nt ¼7.

406

Multilayer Models for Composite and Sandwich Structures

If the assumption is made that the plate is inextensible in the transverse direction, i.e., if {3,0} third order polynomial expansion there are only two displacement and two stress compatibility conditions at each interface. The total number of displacement variables is Nt ¼ 4Nn þ 5  4(N  1)  4¼ 5. Like in the FSDT, the theory is based on three displacements uo, vo, wo and two rotations cx, cy. The {3,0}-order zig–zag plate theory (TSDZZT) developed by Di Sciuva77 falls into this class of zigzag plate theory. The displacement field is uk ¼ uo þ f ðzÞu1 þ

k1 X

ðz  zi Þji Hi

i¼1

vk ¼ vo þ f ðzÞv1 þ

k1 X

ðz  zi Þci Hi

i¼1

wk ¼ wo  with f ðzÞ ¼ z 1 

4 2 z 3h2

ð24Þ



Note that this third-order zig–zag theory has the same number of independent kinematic variables and the same global kinematics as the Reddy’s third-order ESL theory (see, Section 1.16.2, Eqs. (7) and (8)). In Ref. [81] Cho and Parmerter develop a theory quite similar to that proposed by Di Sciuva, choosing as the reference surface the midplane of the plate and splitting the summation into two summations, one extended to the layers below the reference surface and the other extended to layers above the reference surface. Refs. [82–86] used four ply-dependent variables and five global variables in the displacement field uk ¼ uko  zwo;x þ zuk1 þ z2 u2 þ z3 u3 vk ¼ vok  zwo;y þ zv1k þ z2 v3 þ z3 v3 ;

w ¼ wo

ð25Þ

Since the assumption is made that the plate is inextensible in the transverse direction, there are only two displacement and two stress compatibility conditions at each interface. The total number of displacement variables is Nt ¼ 4Nn þ 5  4(N  1)  4¼5. Like in the FSDT, the theory is based on three displacements uo, vo, wo and two rotations cx, cy. This approach (Eq. (25)) was also used to study the boundary layer effect cross-ply laminated plates with two opposite edges on simple supports.87 Dumir et al.88 the inplane displacements follow Eqs. (25a,b) but the transverse displacement w has a significant variation across the thickness due to large contribution of thermal strain. A piecewise quadratic function accounting for thermal effects is added to wo in this case. In Di Sciuva89,90 the displacements are written in terms of global displacements and two local displacements uk ¼ uG þ ukL ;

vk ¼ vG þ vLk ;

wk ¼ wG

ð26Þ

Lrx ðzÞurG ðx; yÞ

ð27Þ

where (this is the classical series expansion used in the ESL theory) uG ¼

R X r¼0

Lrx ðzÞurG ðx; yÞ;

vG ¼

R X r¼0

Lrx ðzÞ are any set of linearly independent functions, at least continuous with their first derivatives with respect to z, and the local displacements ukL ¼

N1 X

ji ðx; yÞðz  zi ÞHðz  zi Þ;

i¼1

vLk ¼

N1 X

ci ðx; yÞðz  zi ÞHðz  zi Þ

ð28Þ

i¼1

With this approach, in an N-ply laminate, there are 2(N  1) local variables ji and ci, and 2(R þ 1) þ 1 global variables for a total of 2(N þ R) þ 1 initial variables. Using the transverse shear continuity equations Nt ¼ 2(N þ R) þ 1  2(N  1)¼2R þ 3. Requiring that the shear stresses vanish on the top and bottom surfaces reduces the number of variables to Nt ¼ 2R  1. For example, for R¼3, i.e., a {3,0}-third order polynomial theory, Nt ¼ 5. Here, as before, the zigzag functions ji and ci are determined by satisfying the contact condition on the transverse shearing stresses at the interfaces. As before, the result is that the zigzag functions can be expressed in terms of the same generalized displacements of the corresponding ESL theory and the transverse shear mechanical properties of the constituent layers. A {3,0}-third order polynomial theory based on the previous approach has been developed in Refs. [77,91–94] among many others. The global in-plane displacements are expanded in a cubic power series, uG ¼ uo þ zu1 þ z2 u2 þ z3 u3 ;

vG ¼ vo þ zv1 þ z2 v2 þ z3 v3

ð29Þ

A {3,0}-order zigzag theory slightly different from a formal point of view of the previous one, is presented in Refs. [93–98]. In this zigzag theory the zigzag behavior is obtained by specifying the slopes above and below the interface. The laminate has nu plies above the reference surface and nl plies below (Fig. 3).

Multilayer Models for Composite and Sandwich Structures

407

Fig. 3 Lamination layup and in-plane displacement with jumps in slope.

The displacement field uk ¼ uo þ vk ¼ vo þ

nX u 1

l 1     nX     Skx z  zuk H z  zuk þ Txs z  zls H z  zls þ z2 u2 þ z3 u3

k¼0

s¼0

nX u 1

nX l 1

    Sky z  zuk H z  zuk þ

k¼0

s¼0

    Tys s z  zlk H z  zls þ z2 v2 þ z3 v3

ð30Þ

wk ¼ wo has seven global variables uo, uo, vo, wo, u2, u3, v2, v3, and 2(nu þ nl) local variables. The displacement continuity conditions are already satisfied. The transverse shear continuity conditions and the zero shear condition on top and bottom surfaces makes the number of variables independent of the number of layers. A similar approach was used by Cho et al.99,100 to analyze multilayer shells. In Refs. [101–104], with reference to the specific case of sandwich plate with multilayered faces, a {3,2}-third order polynomial theory has been developed. The in-plane displacements u and v are also approximated as in the previous {3,0}-third order polynomial theory. The transverse displacement is assumed to have a constant value wu in the upper facesheet and a constant value wl in the lower facesheet, and be equal to wo on the mid-surface. The transverse displacements in the core are taken as interpolated by w ¼ wu l1 þ wo l2 þ wl l3

ð31Þ

where l1, l2, l3 are Lagrangian interpolation functions. In Ref. [105], in addition to the {3,0} global displacement field (Eq. (25)), two local displacements field are introduced so ^kL ; uk ¼ uG þ ukL þ u

vk ¼ vG þ vkL þ ζk3^vkL ;

wk ¼ wG

ð32Þ

with ukL ¼ ζk uk1 þ ζk2 uk2 ;

vkL ¼ ζk v1k þ ζk2 ;

^kL ¼ ζk3 uk3 ; u

^vkL ¼ ζk3 v3k

ð33Þ

where ζk ¼2(z  zm)/hk, zm is the position of the mid-surface and hk is the thickness of layer k. There are 6N local variables and nine ^kL , and ^vkL we global variables. Matching the ukL variables at each interface gives N  1 equations. Proceeding the same way for vkL , u find that a total of 4(N  1) equations. The total number of equations becomes Nt ¼ 6N þ 9  4(N  1)¼2N þ 13. Another 2(N  1) equations are obtained from shear stress continuity conditions so Nt ¼ 15. The number of variables can be reduced by four by imposing zero shear stresses on the top and bottom surfaces. Then Nt=11. However, in order to avoid having first

408

Multilayer Models for Composite and Sandwich Structures

derivatives as variables, Ref. [105] used 13 variables in order to have Coz instead of C1z continuity requirements in the finite element formulation.

1.16.4.2.3

Refined first order zigzag theories

The FSDZZT developed by Di Sciuva,75–78 and presented in Section 1.16.4.2.1, Eq. (20), has the advantage to retain the same kinematic variables of the Timoshenko beam theory (TBT) and the Mindlin plate theory, but suffers of the same drawbacks of these theories, that is, the transverse shear stresses are constant along the thickness of each layer. This means that enforcing the continuity conditions on the transverse shear stresses leads to a constant shear stresses along the thickness. As a consequence, with the kinematics of the FSDZZT given by Eq. (20), the transverse shear stresses cannot be made to vanish on the top and bottom surfaces. Furthermore, see Ref. [106], at the clamped support, the correct shear force and the average shear stress can be determined only from an equilibrium equation relating the shear force to the derivative of the bending moment, as in Bernoulli–Euler theory, not from the constitutive equations. Moving from these remarks, Tessler et al.106,107 develop a refined zigzag theory where the proposed zigzag function vanishes at the top and bottom surfaces and does not require full shear stress continuity across the laminate thickness. Furthermore, the proposed theory (1) is able to model correctly all boundary conditions including the fully clamped condition, and (2) relies on C0-continuous kinematics for finite element modeling, as do Timoshenko and Mindlin theories. The theory is a natural extension of Timoshenko theory to laminated composites. The refined zigzag theory for multilayer and sandwich beam developed by Tessler et al.106,107 uses the following kinematics uk ¼ uo þ zu1 þ fk ðzÞuR ;

w ¼ wo

ð34Þ

  1 1 1  ζ k ukt þ 1 þ ζ k ukb 2 2

ð35Þ

where fk ðzÞ ¼

The midplane is used as the reference surface and the thickness of the beam is taken to be 2h. uo is the uniform axial displacement, wo is the transverse deflection, and u1is the average bending rotation (Fig. 4). With only these three displacements, Eq. (34) defines the classical kinematics of TBT. The additional kinematic variable, uR, is the amplitude of the zigzag contribution to the axial displacement, fk(z) denotes the zigzag contribution to the axial displacement and is assumed to be a piecewise linear function of the thickness coordinate z to be defined. Then, the refined zigzag theory is a natural refinement of the TBT for multilayered beams; the zigzag term fk(z)uR, added to the expression of the axial displacement, takes into account the cross section distortion. Following Tessler et al. the zigzag function fk(z) is prescribed to vanish on the outer beam surfaces, that is (1) f (  h) ¼f(N)( þ h) ¼ 0 Then, [u3(h)  u1(  h)]/(2h) is the rotation obtained by joining the displacement on the top and bottom surfaces. A typical through-the-thickness pattern of a zigzag function for a three-layer beam is depicted in Fig. 4(b). The derivative of the refined zigzag function, constant in each layer, is given by the following expression in terms of transverse shear ðkÞ elastic stiffness coefficient, Gxz , and the thickness, 2h(k) of the layer k, !1 X hk dfk ðzÞ G  1 ð36Þ ¼ ðkÞ  1 where G ¼ h Gk dz Gxz k ¼ 1;N xz It is interesting to note that the refined zigzag theory and Murakami’s zigzag theory have the same number of independent kinematic variables (four for beams and seven for plates and shells), the difference being in the zigzag functions: while MZZF (  1)kζk in Eq. (25) has no physical meaning, the refined zigzag function defined by Eq. (36) is dependent on the layup and on the ply mechanical characteristic of the multilayer structure. Also to be remarked is the number of independent kinematic variables in both Murakami’s zigzag theory and the refined zigzag theory is greater than that of Di Sciuva’s FSDZZT; specifically, one

Fig. 4 Three-layer laminate: layer notation and geometry (a) and zigzag function (b) in the refined zigzag theory.

Multilayer Models for Composite and Sandwich Structures

409

additional kinematic variable for beams and two for plates and shells. The methodology is extended to plates108,109 and shells.110,111 Tessler112 used Reissner’s mixed variational theorem to formulate a mixed RZT beam theory. The assumed law of variation of the transverse shear stress is based on a closed-form integration procedure of elasticity-theory equilibrium equations. The approach has been extended to the plate by Iurlaro et al.113 where also a mixed formulation based on the Murakami’s polynomial approach for the transverse shear stresses has been developed. Closed-form solutions for the bending of sandwich beam using RZT are given by Gherlone;114 two approaches are used to estimate the transverse shear stresses. C0 finite elements for beams,115–120 plates121–124 and shells110,111 based on the RZT have been also developed. A RZT Bernoulli–Euler beam theory is proposed in Ref. [125].

1.16.4.2.4

Refined higher order zigzag theories

The mixed cubic zigzag theory of Iurlaro et al.126 starts with the displacement approximation uk ¼ uo þ zu1 þ z2 u2 þ z3 u3 þ fkx u3 ; vk ¼ vo þ zv1 þ z2 v2 þ z3 v3 þ fky v3 ; w ¼ Hb wb þ Ht wt þ Ha wa

ð37Þ

where fkx and fky are zigzag functions not to be confused with the MZZFs in Section 1.16.4.1. The polynomials in the expansion of the transverse displacements are 1 1 3 1 1 3 3 3 ð38Þ Hb ¼   z þ 2 z2 ; Ht ¼  þ z þ 2 z2 ; Ha ¼  2 z2 4 2h 4h 4 2h 4h 2 2h Then, wb and wt are the values of the transverse displacement at the bottom and top beam surfaces, respectively, whereas wa is the average transverse displacement, Z 1 þh wa ¼ wdz 2h h There are 13 variables in this displacement field. The same kinematic assumption is used for multilayered beams in Ref. [127]. Iurlaro et al.126–128 developed also a mixed formulation of the refined zigzag theory using Reissner’s mixed variational principle.

1.16.4.3

Non-Polynomial Zigzag Theories

In developing these theories, the approach generally adopted is the global–local approach, where the global contribution is that of the corresponding ESL theory and the linear local contribution allows for the jump in the derivative along the thickness, Di Sciuva.89,90

1.16.4.3.1

Sine zigzag theories

Another approach consists of defining the displacement in reference to a single coordinate system. For example, Roque et al.129,130 used a trigonometric layerwise system where pz uk ¼ uo  zwo;x þ sin yx þ Ak þ zBk h pz vk ¼ vo  zwo;y þ sin yy þ Ck þ zDk h w ¼ wo ð39Þ Eq. (39) are the displacement field for laminated beams in an earlier publication.131 The classical plate theory is recovered when only the first two terms are retained in Eq. (39). Retaining the first three terms gives the trigonometric theory of Touratier2 and others. Continuity of the inplane displacements and of the transverse shear stresses at the interfaces gives four equations relating the constants Ak, Bk, Ck, Dk, for layer k to those of layer k  1. This approach is also used in Ref. [132]. Further refinement of this approach are given by Vidal and Polit in Refs. [133,134] where they develop a sine zigzag theory that allows for transverse normal continuous strains (cubic variation of the transverse displacement). Applications of this approach to bending, vibration, and buckling are given in Karama et al.135 Arya136 and Mantari et al.137 introduce a trigonometric theory with both global and local variables in which both displacement and stress continuity conditions are satisfied at the interfaces.

1.16.4.3.2

Hyperbolic zigzag theories

The word hyperbolic is usually used for theories for which the displacement field include hyperbolic sine and/or hyperbolic cosine functions. In the inverse hyperbolic ESL theory of Grover et al.138, the displacement field is u ¼ uo  zwo;x þ ½g ðzÞ þ Ozbx ;

v ¼ vo  zwo;y þ ½g ðzÞ þ Ozby w ¼ wo

ð40Þ

410

Multilayer Models for Composite and Sandwich Structures

 pffiffiffiffiffiffiffiffiffiffiffiffiffi with g(z) ¼sinh1(pz/h) and O ¼  2r= h r 2 þ 4 where r is a parameter to be optimized. This theory has five variables (uo, vo, wo, bx, by) like the FSDT. The inverse hyperbolic zigzag theory of139–141 is an extension of Grover’s theory with uk ¼ uo  zwo;x þ

nX u 1 i¼1

vk ¼ vo  zwo;y þ w ¼ wo

nX u 1

nX    l 1

     z  zui Hi z  zui aixu þ z  zlj Hi z  zlj alxl þ g ðzÞ þ Ok z bx



i¼1

j¼1

n    l 1  X

    z  zui Hi z  zui aiyu þ z  zlj Hi z  zlj alyl þ g ðzÞ þ Ok z by

ð41Þ

j¼1

An inverse trigonometric zigzag theory is obtained using g(z)¼cot1(pz/h) and O ¼  4r=½hð4r 2 þ 1Þ in Eq. (41). Karama et al.142,143 develop an exponential zigzag beam theory with u ¼ uo  zwo;x þ f ðzÞuz ;

w ¼ wo

ð42Þ

where f ðzÞ ¼ g ðzÞ þ



z hðzÞ lm  þ þ ðz  zm ÞHðz  zm Þ 2 2 m¼1 N 1 X

ð43Þ

  hðzÞ ¼  g ðzÞ ¼ z exp 2ðz=hÞ2

This approach was extended to plates144 and it can also be considered to be a hyperbolic zigzag theory.

1.16.5

Sublaminate Approach

In the previous sections we presented the layer-wise approach and the zig–zag approach. Simply stated, we can say that layer-wise approach adopt a layer-wise kinematics, i.e., an independent approximation for the displacements and/or the transverse stresses in each discrete (physical or mathematical) layer through the thickness.6–8 Generally the number of mathematical layers in the laminate is taken to be equal to or less than the number of physical layers. In the zig–zag theories the layer-wise degrees of freedom (DOFs) are eliminated by enforcing transverse stress continuity conditions. This reduces the number of DOFs of freedom to that of the corresponding ESL theory. These latter theories are able to take into account the zigzag effect and the transverse shear deformability giving through-the-thickness continuous distributions of the relative stresses, thus being a good compromise between computational simplicity and accuracy. On the contrary, Layer-wise models are much more accurate and are able to capture the behavior of thick laminates with large layer-wise variations of the engineering constants; their main drawback is the computational complexity, also due to the fact that the number of unknowns depend on the number of layers. The sub-laminate approach, first proposed in Refs. [145,146], can be considered as intermediate between the LW theory and zigzag theory, in that, the number of mathematical layers in the laminate is less than the number of physical layers; this is achieved by grouping plies together in a sub-laminate and using the zigzag approach to model the sub-laminate kinematics. In other words, if N is the number of physical layers of laminates, we can group these layers, for example, in three sub-laminates, each consisting of NS1, NS2, and NS3 layers, respectively, so that N ¼NS1 þ NS2 þ NS3. The resulting mathematical three-layer laminate is then modeled with the LW approach. This allows, in general, an accuracy greater than the zigzag approach with a computational cost lower than the LW approach. In other words, in the sub-laminate approach, the through-the-thickness approximation of the inplane displacement components takes the form of a layer-wise theory in which each layer is really a sub-laminate containing several, even many, physical layers. Within each sub-laminate, a zig–zag through-the-thickness approximation of the in-plane displacement components is taken in which the layer-wise DOFs are eliminated by enforcing continuity of the transverse stresses. Shear traction conditions at the top and bottom of each sub-laminate are also satisfied. The approach allow for including the effect of transverse normal strain. As an example, Cho and Averill147–149 assumed the following displacement field for layer k in the sublaminate (the coordinate system is located at the bottom of a sublaminate) uk ¼ u1 þ zu2 þ

k1 X i¼1

ðz  zi Þui3 ;

vk ¼ v1 þ zv2 þ

k1 X i¼1

ðz  zi Þv3i ;

 z z w ¼ wb 1  þ wt h h

ð44Þ

where u1, v1, and wb are the displacements at the bottom of the sublaminate, wt is the transverse displacement at the top, u2 and v2 are the rotations of the normal at the bottom, u3 and v3 are the rotations of the normal. Further developments, improvements and applications, with FEM formulations, of this approach may be found in Refs. [33,150–157].

1.16.6

Models for Multilayer Structures With Interfacial Damage

Most theories assume perfect bonding between adjacent layers. However, in many applications, there may be some relative motion at the interface between these two layers. It was first recognized that with wooden beams nails could not prevent sliding (slip) at the interfaces. With bonded interfaces, relative motions parallel, and normal to the interfaces are possible. In laminated composite

Multilayer Models for Composite and Sandwich Structures

411

materials, resin-rich layers between plies may experience significant deformations that cannot be neglected. Adding an extra layer of matrix material at certain interfaces (interleaving) has been suggested to reduce impact damage. Debonding between plies (delamination) is one of the failure modes of composite structures. Several types of imperfect interfaces have been considered in the development of mathematical models. Linear relationships between the stresses at the interface and the relative displacements in what are called linear spring layer models (Section 1.16.6.1). Nonlinear relationships between stresses and displacement jumps are discussed in Section 1.16.6.2, viscous or viscoelastic interface models in Section 1.16.6.3. The vast literature dealing with cohesive zone models and its application to laminated composite materials has been reviewed comprehensively.158 Once failure of the interface has been predicted by a failure criterion, many cohesive models are available to predict the extent of the delamination. Cohesive elements are now available in some well-known commercial finite element programs and their use has enabled some researchers to predict the extent of impact damage with remarkable accuracy. The work on cohesive zone models will not be discussed here. The following describes how to model laminated structures with existing imperfections.

1.16.6.1

Imperfect Interface Models

The interface is a layer of negligible thickness connecting two layers. It allows relative motion in the tangential and normal directions. The behavior of the interface can be linear elastic, nonlinear, viscoelastic, or viscous. In this section we give a review of the most commonly adopted models to study the effect of imperfect bonding of two adjacent layers (nonrigid interface). In short, these models define the jumps in the displacement and their time derivative to the transverse shear and out-of-plane normal stresses. It should be emphasized that the same models allow also to study the behavior of the laminates when interface delaminations are present (Section 1.16.6.2). This is obtained by imposing that for delaminated interface the relative compliance coefficients are infinity in Eqs. (46) and (47), and that relative stiffness coefficients are zero in Eqs. (48) and (49). In both cases, the transverse shear and out-of-plane normal stresses are zero.

1.16.6.1.1

Linear spring layer models

The work of Newmark et al.159 is recognized as the first study of composite beams with partial interaction. A model for a two-layer beam is developed. Both layers are modeled as Bernoulli–Euler beams and a linear relationship between the relative interface displacements and the shear stress is used. Murakami160 first used the TBT to model this problem. Jumps in displacements at interface k are defined as the difference between the displacements of the bottom face of layer k þ 1 and the displacements of the top face of layer k ðkþ1Þ

〚u〛k ¼ ub

ðkÞ

 ut ;

ðkþ1Þ

〚v〛k ¼ vb

ðkÞ

 ut ;

ðkþ1Þ

〚w〛k ¼ wb

ðkÞ

 wt

ð45Þ

where the subscripts t and b stand for the layer’s top and bottom surfaces, respectively. With what is called the (linear) spring-layer approach, the inplane displacement jumps are related to the transverse shear stresses 〚u〛k ¼ Rk11 skxz þ Rk12 skyz ;

〚v〛k ¼ Rk21 skxz þ Rk22 skyz

Rk11 ; Rk12 ; Rk21 ; Rk22

ð46Þ 115,116,122,152,161–168

are compliance coefficients of interface k. This approach is used in several studies. In where many cases, the coupling between the effects of the two shear stresses is neglected and Rk12 ¼ Rk21 ¼ 0 as in Refs. [169–189]. In most cases, including Refs. [169–182], it is assumed that there are no displacement jump in the z-direction so 〚w〛k ¼ 0. However, it is also possible to introduce another linear relationship of the form116,183,185,187,190 〚w〛k ¼ Rk33 skzz

ð47Þ

Accounting for relative displacements at the interfaces certainly complicates the analysis and raises the question: when are those effects important. One study189 concluded that, for sandwich structures, the shear modulus of the core should be “high enough to develop the interaction required between the” facesheets and that what constitutes perfect depends on the ratio of the core stiffness to the bonding stiffness. The bonding stiffness has some effect on the plate deflection up to a certain level and beyond that the perfect bonding assumption is acceptable.

1.16.6.1.2

Nonlinear spring layer models

Refs. [191–193] modeled two-layer beams with nonlinear behavior of the interface in shear. FT, the tangential force per unit length, is related to ST, the tangential jump in displacements by FT ¼ K1 ST þ K3 S3T

ð48Þ

Campi and Monetto,194 and Monetto195 used a bilinear interface model illustrated in Fig. 5, These models can be seen as a simplification of the Foschi–Bonac load–slip relationship196 FT ¼ ðPo þ P1 ST Þ½1  expðkST =Po Þ

ð49Þ

412

Multilayer Models for Composite and Sandwich Structures

Fig. 5 Shear force per unit area in Fosci–Bonac model (left), Monetto195 (center), Campi and Monetto194 (right).

1.16.6.1.3

Viscoelastic interface models

Viscoelastic interfaces are modeled using a Kelvin–Voigt model _ k; skxz ¼ Zkox 〚u〛k þ Zk1x 〚u〛

skyz ¼ Zkoy 〚v〛k þ Zk1y 〚_v〛k ;

_ k skzz ¼ Zkoz 〚w〛k þ Zk1z 〚w〛

ð50Þ

in Refs. [197,198]. Viscoelastic effects on the transverse shear stresses are considered using the first two equations in Refs. [199,200] but not on skzz . Hansen and Spies201 used Eq. (50a) for laminated beams. Chen et al.171,202–204 considered viscous interfaces so that Zkox ¼ Zkoy ¼ 0. Bai and Sun205 studied the damping of sandwich beams with viscoelastic core and viscoelastic adhesives between the core and the facesheets. For steady state harmonic loading, the interfaces are modeled using skxz ¼ k uk where k is the frequency-dependent complex shear stiffness parameter of the adhesive layers.

1.16.6.2

Models With Multiple Delaminations

To account for the presence of delaminations, jumps in displacements at the interfaces should be included in the kinematic field by adding the following expressions ukD ¼

k1 X

ui ðx; yÞHðz  zi Þ;

i¼1

vDk ¼

k1 X

vi ðx; yÞHðz  zi Þ;

i¼1

wkD ¼

k1 X

wi ðx; yÞHðz  zi Þ

ð51Þ

i¼1

to the displacements in the x, y, and z-directions.〚u〛k〚 ; v〛k , and〚w〛k represent possible jumps in the displacements due to slipping and opening at delaminated interfaces.

1.16.6.2.1

Layerwise theories with multiple delaminations

Starting with Reddy’s layerwise theory, Barbero and Reddy206 add the terms ukD ; vkD ; and wkD in Eq. (51) to account for delaminations at the interfaces. This approach is also used in Refs. [207–210]. Similar approaches can be found in Ref. [211].

1.16.6.2.2

Multiple delaminations in {1,0} zigzag theory

In Refs. [212–215] the kinematics of the deformation is described using the superposition of (1) first order shear deformation to address the overall response of the entire laminate; (2) layerwise functions to describe the zigzag-like in-plane deformation through the laminate thickness, and to satisfy the interlaminar shear traction continuity requirement; and (3) to model the delamination, the assumed displacement field is supplemented with Heaviside unit step functions, which allows discontinuity in the displacement field (Eq. (51)).

1.16.6.2.3

Multiple delaminations in {3,0} zigzag theory

Refs. [123,216–220] start with a zigzag theory in which the kinematics are described by the superposition of the displacement field of the{3,0} ESL plate theory and terms accounting for zigzag in-plane deformations. To this {3,0} zigzag theory, the displacement jumps ukD ; vkD ; and wkD in Eq. (52) are added to account for eventual delaminations at the N  1 interfaces. A similar {3,0} zigzag theory with delaminations is given in Refs. [161,162]. A global {1,0} first order plate theory with local terms describing the zigzag effect and terms to account for delaminations (Eq. (52)) is given in Refs. [213,214]. For imperfect bonding, the interface displacements can be modeled using imaginary springs.124

1.16.6.3

Models Including Imperfect Interfaces

With imperfect interfaces, there are jumps in displacements between the two adjacent layers that have to be taken into account in the displacement field. In addition, since this relative motion is neither completely prevented nor completely free, some strain

Multilayer Models for Composite and Sandwich Structures

413

energy is absorbed during this deformation and has to be included in the total potential energy during the derivation of the equations of motion. The total potential energy can be written as Z N Z 1  T Y 1Z Z X rT edzdO þ t ðrÞ DðrÞ dO  q wdO ð52Þ ¼ 2 O h Or¼1 O where r and e are the stress and strain tensors, t(r) is the vector of surface tractions at interface r, D(r) is the vector with the three displacement discontinuities at that interface, q is the external pressure in the transverse direction and w is the transverse deflection of the reference surface.166 The second integral accounts for the deformation of the interface and it is zero in two cases: (1) perfect bonding and (2) complete delamination. With Eq. (52), the theorem of minimum potential energy can be used to derive the equilibrium equations for static problems. For dynamic problems, the problem is usually formulated using Hamilton’s principle or the Reissner mixed variational principle. In all cases the energy absorbed by the deformation of the interface must be accounted for.

1.16.6.3.1

Elasticity solutions for laminates with imperfect interfaces

Semi-analytical solutions based on the theory of elasticity for laminates with imperfect interfaces are based on the state-space approach developed by Refs. [117,118], that is described in detail in Ref. [119]. Following179 the equations of motion from threedimensional elasticity are written in terms of the three displacements u, v, w and the three transverse stress components sxz, syz, szz. Defining the state vector V¼{szz, u, v, w, sxz, syz,}, the equations of motion can be written as ∂ V ¼ DV ð53Þ ∂z 179 For a simply supported rectangular plates, V can be expressed in terms of sine and cosine where D is a differential operator. functions so that the boundary conditions are satisfied. After substitution in the equations of motion, it is found that V ðkÞ u , the state ðkÞ vector on the upper surface of layer k, is related to V l , the state vector on the lower surface of that ply by ðkÞ

V ðkÞ u ¼ Mk V l

ð54Þ

where Mk is the transfer matrix. For imperfect interfaces, the state vector on the lower surface of layer k þ 1 is related to that on the upper face of layer k by ðkþ1Þ

Vl

¼ Pk V ðkÞ u

ð55Þ

where Pk is a transfer matrix that accounts for the behavior of the interface. Using this transfer matrix approach, the behavior of the laminate is written as ð1Þ



V ðNÞ u ¼ TV l



where T ¼ ∏1j ¼ N Mj Pj1 is the 171,178,180,197,199,203,204,223–226

ð56Þ 120,221,222

global transfer matrix. This approach has been used to study beams, plates, and cylindrical panels170,172,176,179,198,200,202 with simply supported boundary conditions. Interfaces were modeled as spring layers (Eqs. (46) and (47)) in Refs. [170–172,176,178–180,222–225] as viscoelastic layers using the Kelvin–Voigt model (Eq. (50)) in Refs. [120,197–200]. With the Maxwell–Weichert viscoelastic model in Ref. [221] the time-dependent shear modulus of the interface G(t) is written as Gðt Þ ¼ G1 þ

n X

Gj et=yj

ð57Þ

j¼1

where G1 is the long term modulus, Gj and yj are the relaxation moduli and the relaxation times. Refs. [171,202–204,226] use this state-space approach for viscous interfaces (Eq. (50) with Zkox ¼ Zkoy ¼ 0). This state-space approach and the results presented in the publications cited in this subsection can serve to validate results obtained from using multilayer plate theory that include interfacial bonding imperfections.

1.16.6.3.2

ESL theories with imperfect interfaces

Murakami160 included interlayer slip in the formulation of a TBT for a two-layer beam. The problem is formulated in a general way so that any interface slip law can be introduced in the equations of motion. Shu169 included bonding imperfections in the Kirchhoff–Love theory using the spring layer approach and derived an analytical solution for laminates with shear slip under cylindrical bending. Shu and Soldatos included interlaminar imperfections modeled as spring layers in the FSDT and developed an analytical solution for plates under cylindrical bending166 and for cylindrical panels under external pressure.167 Many two-layer beams or plates with slipping interfaces have been proposed. The dynamic behavior of two-layer beams227,228 and sandwich beams229–231 with and without interlayer slip is governed by the 6th order equation of motion w;xxxxxx  l2 w;xxxx þ

m m l2 m € ;xx  l2 €¼  w w p þ p;xx B1 Bo B1 Bo

ð58Þ

where p is the distributed load, m is is the mass per unit length, Bo is the sum of the bending rigidities of the individual layers, B1 is the bending rigidity of the beam assuming perfect bonding at the interfaces and

414

Multilayer Models for Composite and Sandwich Structures   2b B1 l2 ¼ κ 2 G 2 h2 D1 Bo

ð59Þ

for sandwich beams, and l2 ¼ k

B1 D1 Bo

ð60Þ

for beams with interlayer shear slip. D1 is the bending rigidity of the top beam or the top facing. Many sandwich beam theories (e.g.,232–235) result in a 6th order equation of motion of this form. _ k Þ is Heuer236 showed that the behavior of beams with Kelvin–Voigt viscoelastic interfaces where skxz ¼ Zkox ð〚u〛k þ u〚u〛 governed by 

   m m € ;xx  l2 € ¼0 w;xxxxxx þ uw_ ;xxxxxx  l2 w;xxxx þ uw_ ;xxxx þ w w Bo B1

ð61Þ

which reduces to Eq. (58) when u-0.

1.16.6.3.3

Murakami’s theory

Toledano and Murakami237 used a mixed formulation in which the displacement of a two-layer plate with interlayer slip are approximated by (See Section 1.16.4.1) 〚u〛=h þ ð1Þk ζk uM uk ¼ uo þ zu1 þ 2z vk ¼ vo þ zv1 þ 2z 〚v〛=h þ ð1Þk ζk vM ;

w ¼ wo

ð62Þ

where〚u〛and〚v〛are the interlayer slips and are functions of x, y and time. The transverse shear stresses are also interpolated through the thickness.

1.16.6.3.4

Layerwise theories with imperfect interfaces

Kimpara et al.124 use a {3,2} polynomial expansion for each layer (Eq. (23)) with 11 displacement variables per ply. For imperfect bonding, the interface can be modeled using imaginary springs which gives 6(N  1) equations for an N-ply laminate. With this layerwise approach, the number of variables increases with 5N.

1.16.6.3.5

Zigzag theories with imperfect interfaces

A linear spring layer model with displacement jumps ukD ; vkD ; and wkD from Eq. (51) has been added to a {1,0} zigzag theory,115 a {1,1} zigzag theory in Ref. [116] a {3,0} zigzag theory in Refs. [115,122,161,162,164,165,168,238], and a {3,2} zigzag theory in Ref. [163].

1.16.7

Sandwich Structures

Typically sandwich consists of two outside layers separated by a thicker core layer (Fig. 1). For a homogeneous beam with a solid cross section, the linear stress distribution through the thickness indicates that the material near the neutral axis is not used efficiently. Therefore, splitting this material into two layers and separating them by a distance d results in a much higher moment of inertia. This requires that the two facesheets be connected by an intermediate layer called the core whose primary function is to ensure that the whole three-layered structure functions as a beam. Sandwich structures have been studied extensively and this work led to the publication of several books239–243 and literature surveys.41,244–249 The following briefly mentions elasticity solutions and the use of ESL theories for sandwich structures. It then describes displacement-based theories with different kinematic assumptions for the core and the facings followed by theories in which both displacements and stresses are used as primary variables. At this point, it should be remarked that the layer-wise and zigzag theories reviewed in the previous sections are applicable and have been applied to the analysis of sandwich structures. Layer-wise theories have been used in Refs. [14–27]. Di Sciuva’s zigzag theory has been used to analyze the mechanical behavior of sandwich structures in Refs. [250–252]. Brischetto et al.50,51,66 and Demasi63 used MZZF within the framework of Carrera’s Unified Formulation. Neves et al.73 added MZZF to the inplane displacements u and v in the hyperbolic sine theory. Application of the refined zigzag theory to sandwich structures can be found in Refs. [48,61,62,67,80,106–111,113,114,121,126,253,254].

1.16.7.1

Elasticity Solutions for Three Layer Structures

Elasticity solutions have been obtained for special cases of laminated beams and plates. The work of Srinivas255 and Pagano256 are often used for generate solutions that can be used to validate various models. Kardomateas257,258 considered special cases of Pagano’s solution not considered in Ref. [256]. New elasticity solutions were developed for three-layered structures.259–261 Elasticity solutions were developed for sandwich structures with functionally graded materials.262–270

Multilayer Models for Composite and Sandwich Structures 1.16.7.2

415

ESL Theories for Sandwich Structures

ESL theories such as the Kirchhoff–Love (e.g.,239–241) and Mindlin–Reissner plate theories are used for sandwich structures. Examples of refined higher order ESL theories used to model sandwich structures include the FSDT,252,271–273 Reddy’s TSPT,273–278 {1,2} theories,279,280 {3,0},281,282 {3,2} theories,252,281,283–288 the trigonometric theory.132,273,289,290 The hyperbolic sine theory with u ¼ uo þ zu1 þ sinh ðpz=hÞuz ;

v ¼ vo þ zv1 þ sinh ðpz=hÞvz ;

w ¼ wo þ zw1 þ z2 wz

ð71Þ

is used in Refs. [291].

1.16.7.3

Theories Based on Different Kinematic Assumptions for the Facings and the Core and Displacement Continuity at the Interfaces

For sandwich structures, theories can be developed starting with different kinematic assumptions for the facings and the core. Table 2 summarizes the current state of the art and shows that in most cases the facings are modeled using a simple theory (Kirchhoff–Love or FSDT) while much refined descriptions of the core’s behavior are used.

1.16.7.4

Mixed Formulation for Sandwich Structures

While most theories are based on approximate representations of the displacement field, another approach uses one set of variables to describe the displacements of the facings and another set of variables to describe the transverse stresses in the core. With such a mixed formulation, the transverse shear and normal stresses are primary variables and are expected to be predicted more accurately than with a displacement-based formulation in which they will be determined from the strain–displacement relations and the constitutive equations. The following describes the basics of Frostig’s higher order sandwich plate theory (HSAPT) that is widely used in the literature and other mixed formulations for sandwich structures.

1.16.7.4.1

HSAPT

In what has been subsequently called the HSAPT, Frostig et al.361 developed a mixed formulation for the bending of symmetric sandwich beams based on the theorem of minimum total potential energy δðU þ V Þ ¼ 0

ð72Þ

where U is the strain energy and V is the potential energy of the external forces. Assuming that the top and bottom facing are subjected to bending stresses sxx only and that the core is subjected to transverse shear and normal stresses tc and szz Z Z Z Z δU ¼ sxx δexx dVt þ sxx δexx dVb þ tc δgc dVc þ szz δezz dVc ð73Þ Vt

Vb

Vc

Vc

The first variation δV accounts for distributed pressures in the normal and tangential directions and distributed moments on the top and bottom surfaces. Concentrated forces and moments can also be applied on those surfaces. The facings behave as Bernoulli–Euler beams. In 2D elasticity, equilibrium in the x direction is governed by sxx,x þ sxz,z ¼ 0. If the bending stress sxx in the core is negligible, as assumed in Eq. (73), this equation of equilibrium indicates that sxz,z ¼ 0 so the transverse shear stress tc ¼ sxz is constant through Table 2 Theories based on different kinematic assumptions for the facings and the core and displacement continuity at the interfaces Facings

Core

References

Kirchhoff–Love theory

FSDT {1,1} {1,2} {2,1} {2,2} RSDT {3,2} {5,4}

292–303 304–306 307 308–320 321 322–325 326–338 339–343

FSDT {1,0}

FSDT {1,0} {3,0} {3,2} {1,1} {3,2} {3,2}

13–16,344–350 351,352 349,353–354 22 356–359 349,360

{1,1} {3,0} {3,2}

Abbreviations: FSDT, first shear deformation theory; RSDT, reddy’s shear deformation theory.

416

Multilayer Models for Composite and Sandwich Structures

the thickness. Equilibrium in the transverse direction is governed by sxz,x þ szz,z ¼ 0. Then, szz,z is constant through the thickness, szz varies linearly, and, since szz ¼ Eezz ¼ Ewc,z, the transverse displacement in the core (wc) is a quadratic polynomial function of z. The equations of motion are developed in terms of uto ; wto ; ubo ; wbo , the axial and transverse displacements of the facing, and tc the shear stress in the core. In the end, bending is governed by three fourth-order partial differential equations with variables uto ; wto ; ubo ; wbo , and tc. This is a mixed formulation in the sense that it involves both displacements and stress variables. For dynamic cases, the equations of motion are derived using Hamilton’s principle Z t2 δ ðU þ V  T Þdt ¼ 0 ð74Þ t1

where T is the kinetic energy. To study the free vibration of sandwich beams, V¼ 0 and the first variation of the kinetic energy is written as

Z t2 Z L Z L Z Z δT ¼ mt ðu_ ot δu_ ot þ w_ t δw_ t Þdx þ mb ðu_ ob δu_ ob þ w_ b δw_ b Þdx þ rc u_ c δu_ c dVc þ rc w_ c δw_ c dVc dt ð75Þ t1

0

0

Vc

Vc

where mt and mb are the mass per unit length of the upper and lower skins, u_ ot w_ t u_ ob w_ b are the velocities of the top and bottom skins in the horizontal and vertical directions, u_ c and w_ c are the velocities in the core.362 This approach was extended to sandwich plates.363 The facings are assumed to behave as Kirchhoff–Love plates and eight equations of motion are derived in terms of six displacement variables uto ; vot ; wto ; ubo ; vob ; wbo and the two transverse shear strains in the core tx and ty. This model was extended further to include the dynamic behavior of sandwich plates using Hamilton’s principle.364

1.16.7.4.2

Other mixed formulations for sandwich structures

Desai and coworkers developed a mixed formulation for laminated and sandwich structures in which, for each layer, the displacements and the transverse stresses on the top and bottom surfaces are taken as field variables. For plates,36,365 the formulation starts with a {3,3} power series expansion of the displacements u, v, and w in terms of the transverse coordinate z. These displacements are then written in terms of 12 field variables (three displacements and three transverse stresses on each surface): ut, (sxz)t, wt, (szz)t, vt, (syz)t, ub, (sxz)b, wb, (szz)b, vb(syz)b where the subscripts t and b describe the top and bottom surfaces of the layer. For beams,366 six variables are selected: the two displacements u and w and the transverse shear stress sxz on each surface (ut, (sxz)t, wt, ub,(sxz)b, wb). Within the framework of the refined zigzag theory, mixed formulation for sandwich beam and plates have been proposed in Refs. [112,126,127].

1.16.8

Conclusions

The aim of the chapter is to give an overview of the state-of-art of multilayer models for composite and sandwich structures. To this end, we started briefly recalling the ESL formulation, which uses a single type of kinematic field for all layers, thus reducing the laminate to a single layer. We noted that implicit in ESL formulation is the assumption that the displacements and stresses are smooth functions of the thickness-wise co-ordinate. Generally, this is the case when the mechanical properties vary little through from one layer to the adjacent one. In some cases, the deformation of the various plies is drastically different so it cannot be assumed that a single type of kinematic field can be used for all layers. Such is the case of fiber reinforced laminated composite structures and sandwich structures, which generally have a core that is much thicker than the facesheets, is much more flexible in shear and, in some cases, it also deforms in the transverse direction. Therefore, the analysis of these multilayer structures generally need ad hoc theories, that is, theories that allow to take into account the jump, often strong, in the distribution of mechanical properties along the thickness at the interface of two adjacent layers. The approaches generally used to reduce the 3D problem of elasticity to a 2D one are axiomatic in nature, i.e., they a priori assume the law of the distribution of the displacements (displacement-based formulation) in conjunction with the principle of virtual work or the law of the distribution of the displacements and transverse stresses (mixed formulation) in conjunction with the Reissner’s mixed variational principle. The first formulation is the one that is most commonly adopted. Generally speaking, the first is more appealing from the point of view of computational cost and also of the formulation of finite elements, while the second one is more expensive but more accurate in predicting the state of the stress. Within each of the two previous formulations (displacement-based formulation and mixed formulation), the specific models for multilayered composite laminates and sandwich structures can be grouped into two main classes: layer-wise theories and zig–zag theories. Although we did not go into details of the various theories, the bibliographic survey allows us to draw some generally accepted conclusions. The first is that the layer-wise theories are more accurate than the zigzag theories, but suffer from the fact that the number of kinematic parameters (displacement-based theories) or the number of kinematic and stress parameters (theories based on mixed formulations) to be calculated increases with the number of the layers. Therefore, for laminates with 10 of layers (as usually happens in practical applications) they become very expensive from a point of view of the computational cost. This aspect is also reflected in the formulation of finite elements.

Multilayer Models for Composite and Sandwich Structures

417

So, the conclusion could be that zig–zag models are less expensive than layer-wise models and in many instances their computational accuracy is comparable to that of layer-wise models. Within the zig–zag models, two models are widely used in literature, that first proposed by Di Sciuva and subsequently amended in the so-called refined zigzag theory and that based on the Murakami’s zigzag function. The first model has the advantage of using zigzag functions that take into account the geometric and mechanical characteristics of the constituent layers, while the zigzag functions of Murakami are dependent only on the thickness of the layers. Generally, the first approach provides more accurate estimates, especially when applied to non-periodic (non-repetitive) laminations, while the second can be readily extended to all of the ESL models. In any case, the numerical results quoted in the literature show that refined zig–zag theory gives results equal, when not better, than that given by Murakami’s zigzag function. Both models have also been used for the study of damaged interfaces and delaminations and for the formulation of finite elements.

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