Appendix: Differential Operators

The differential operators that appear throughout the various derivations in this book are sum-marized in this appendix, which also examines the characteristics of some of these operators.

Note that although additional versions of the operators exist, the following is confined to those that are relevant to the discussions in this book. Due to the desire to put the operators in a con-sistent sequence, their order of appearance here does not always match the one used throughout the derivation (mainly in Chapter 3).

In this section we denote the polar coordinates(ρ,θ) as (r,θ), respectively.

3.6.1 Generalized Laplace’s Operators

Two versions of generalized Laplace’s operators are of interest.

The generalized Laplace’s operator in the x y-plane for Cartesian anisotropy is defined by

∇⁽²⁾₁ = b44

∂²

∂x²− 2b45

∂²

∂x∂y+ b55

∂²

∂y². (3.194)

The following Laplace’s operators have been developed in S.3.3.1 (see (3.97)), and may be obtained from (3.194) by replacing bi jwith ai j, namely,

∇⁽²⁾₃ = a44 ∂²

∂x²− 2a45 ∂²

∂x∂y+ a55 ∂²

∂y², (3.195)

and its “orthotropic” version form (i.e., a45= 0)

∇⁽²⁾₄ = a44

∂²

∂x²+ a55

∂²

∂y². (3.196)

Recall that for MON13z material, and clearly for orthotropic material, the b44, b45, b55 set of coefficients is identical to the a44, a45, a55set, respectively, see (3.28). Therefore, in such cases,∇⁽²⁾₃ =∇⁽²⁾₁ and their orthotropic versions coincide.

The isotropic version of the operator (3.194) is given by

∇⁽²⁾₀ =2(1 +ν)

E ∇⁽²⁾, (3.197)

where∇⁽²⁾is the simplest Laplace’s operator, namely,

∇⁽²⁾= ∂²

∂x²+ ∂²

∂y². (3.198)

The isotropic version of operator (3.196) is identical to the one presented in (3.197).

In polar coordinates(r,θ) and for cylindrical anisotropy, see S.7.5.2, (7.152), Laplace’s operator becomes

which for cylindrical orthotropic material takes the form, see S.7.5.3.2,

∇⁽²⁾₆ = a44(∂²

In the isotropic case, Laplace’s operator of (3.199) may be written as (3.197) while∇⁽²⁾, the simplest Laplace’s operator, is written in polar coordinates as

∇⁽²⁾= ∂²

The operators documented in (3.194–3.196) are all written in Cartesian coordinates and for Cartesian anisotropy. On the other hand, the operators documented in (3.199–3.201) are all written in polar coordinates and for cylindrical anisotropy. Therefore, employing the coor-dinate transformations of S.7.5.1 can not be used to obtain∇⁽²⁾₅ from∇⁽²⁾₁ or∇⁽²⁾₆ from∇⁽²⁾₂ and vice versa. Such a transformation is allowed only in the isotropic case. In other words, by simple coordinate transformation, (3.198) can be transformed into (3.201) and vice versa, and therefore, these operators are denoted identically as∇⁽²⁾.

3.6.2 Biharmonic Operators

We shall first list the various biharmonic operators used in the book. As shown by (3.31), for Cartesian anisotropy, the biharmonic operator for plane-strain analysis is

∇⁽⁴⁾₁ = b22 The orthotropic version of this operator is given as

∇⁽⁴⁾₂ = b22

The isotropic version of the biharmonic operator is given in (3.204) as

∇⁽⁴⁾₀₁ =1−ν²

E ∇⁽⁴⁾, (3.204)

where∇⁽⁴⁾is the simplest biharmonic operator,

∇⁽⁴⁾=∇⁽²⁾·∇⁽²⁾= ∂⁴

∂x⁴+ 2 ∂⁴

∂x²∂y²+ ∂⁴

∂y⁴. (3.205)

The biharmonic differential operator for x y-plane-stress analysis given in (3.45) is obtained and its orthotropic version is given by

∇⁽⁴⁾₄ = a22

The isotropic version of (3.206) is

∇⁽⁴⁾₀₂ = 1

E∇⁽⁴⁾, (3.208)

where∇⁽⁴⁾appears in (3.205).

Finally, for bending of a thin plate, the following operator has been found, see S.3.5.1.2 and (3.164): For cylindrical anisotropy, the biharmonic differential operator for plane-strain in polar co-ordinates(r,θ) is derived in S.7.5.3.1, as

∇⁽⁴⁾₅ = b22

The orthotropic version of this operator is

∇⁽⁴⁾₆ = b22 ∂⁴ The isotropic version of the biharmonic operator of (3.210) is given by (3.204), where the simplest biharmonic operator∇⁽⁴⁾in polar coordinates is written as

∇⁽⁴⁾=∇⁽²⁾·∇⁽²⁾= (∂²

or, by opening brackets and differentiating, as

∇⁽⁴⁾= ∂⁴ Analogously to the discussion in S.3.6.1, some clarification is required here. The opera-tors documented in (3.202–3.208) are all written in Cartesian coordinates and for Cartesian anisotropy. On the other hand, the operators documented in (3.210–3.213) are all written in polar coordinates and for cylindrical anisotropy. Therefore, employing the coordinate trans-formations of S.7.5.1 can not be used to obtain∇⁽⁴⁾₅ from∇⁽⁴⁾₁ or∇⁽⁴⁾₆ from∇⁽⁴⁾₂ and vice versa.

Such a transformation is allowed only in the isotropic case, namely, (3.205) can be transformed into (3.212) and vice versa, and therefore, these operators are denoted identically as∇⁽⁴⁾.

3.6.3 Third-Order and Sixth-Order Differential Operators

The third-order differential operator ∇⁽³⁾₁ that appears for GEN21 materials but vanishes for MON13z (and simpler) materials, is given in S.3.4 for Cartesian anisotropy as

∇⁽³⁾₁ = −b24 We also define an additional sixth-order operator∇⁽⁶⁾₁ that may be written for Cartesian anisotropy using the∇⁽²⁾₁ ,∇⁽³⁾₁ and∇⁽⁴⁾₁ operators as

∇⁽⁶⁾₁ =∇⁽⁴⁾₁ ·∇⁽²⁾₁ −∇⁽³⁾₁ ·∇⁽³⁾₁ . (3.215) No use of these operators with the ai j coefficients appears in this book. The above operators are presented in (Lekhnitskii, 1981) for polar coordinates.

3.6.4 Generalized Normal Derivative Operators

The generalized normal derivative operators presented here are typically used along with the generalized Laplace’s operators presented in S.3.6.1. For Cartesian anisotropy, the first opera-tor is which in the orthotropic case becomes

Dⁿ₂= a44

∂

∂xcos(¯n,x) + a55

∂

∂ycos(¯n,y). (3.217)

In the isotropic case the operator of (3.216) is simplified to Dⁿ₀=2(1 +ν)

E d

dn, (3.218)

where _dn^d is the (geometrical) normal derivative, namely, d

dn≡ ∂

∂xcos(¯n,x) + ∂

∂ycos(¯n,y). (3.219)

No use of these operators with the bi jcoefficients appears in this book.

In polar coordinates(r,θ) and for cylindrical anisotropy, see S.7.5.2, (7.152), the generalized normal derivative operator becomes while for orthotropic cylindrical anisotropy we obtain, see S.7.5.3.2,

Dⁿ₄= a44

In the isotropic case, we use (3.218) where _dn^d becomes again the (geometrical) normal deriva-tive in the r,θ-plane and takes the form

d

By employing the coordinate transformations of S.7.5.1 one may obtain Dⁿ₃from Dⁿ₁and Dⁿ₄ from Dⁿ₂and vice versa only by substitutingθ= 0. Yet, for the isotropic case, (3.219) can be directly transformed into (3.222).

3.6.5 Ellipticity of the Differential Operators

In this section, we shall discuss the characteristic polynomials piα, α∈ {a,b}, i = 2,3,4,6 corresponding to ai jand bi jversions of the operators defined by i= 2 in (3.194), (3.195), i = 4 in (3.202), (3.206), i= 3 in (3.214) and i = 6 in (3.215), namely

p2α=α55µ²− 2α45µ+α44, (3.223a)

p_3α=α15µ³− (α14+α56)µ²+ (α25+α46)µ −α24, (3.223b) p_4α=α11µ⁴− 2α16µ³+ (2α12+α66)µ²− 2α26µ+α22, (3.223c)

p6α= p4αp2α− p²_3α. (3.223d)

Note that for the sake of completeness, we have documented here the polynomial p3aand p6a

although no corresponding operator has been used.

To show that the differential operators (3.223a,c,d) have elliptical properties, i.e. their char-acteristic polynomials cannot have a real root, we write the strain energy, W , see (2.27), as

W=1

2

∑

⁶i=1σiεi=1

2

∑

⁶i, j=1ai jσiσj, (3.224)

and exploit the fact that W attains only positive values, see S.2.10. We treat each characteristic polynomial separately.

As an example, for the polynomial p2a, by assigning the following values to the stress com-ponents: σx= 0,σy= 0,σz= 0, τyz= −1, τxz= µ, τxy= 0, one obtains an expression for the energy that coincides with (3.223a) since 2W = p2a(µ), and therefore, one may conclude that p2a(µ) > 0, µ ∈ R.

Similarly, for the remaining operators of even degree (as odd-degree polynomials will always have a real root) we select the following five sets, where only the non-vanishing components are given:

For p_2b:σz= 1

a₃₃(a34− a35µ), τyz= −1, τxz= µ. (3.225a)

For p4a:σx= µ², σy= 1, τxy= −µ. (3.225b)

For p4b:σx= µ², σy= 1, σz= − 1

a33(a13µ²+ a23− a36µ), τxy= −µ. (3.225c) For p_6a:σx= µ², σy= 1, τyz= −λa, τxz=λaµ, τxy= −µ. (3.225d) For p6b:σx= µ², σy= 1, σz= − 1

a33(a13µ²+ a23− a36µ− a34λb+ a35λbµ),

τyz= −λb, τxz=λbµ, τxy= −µ. (3.225e)

In the above, we have used the notationλa= −^p_p^3a_2a, λb= −^p_p^3b_2b. Thus, it has been shown that p2b(µ) > 0, p4a(µ) > 0, p4b(µ) > 0, p6a(µ) > 0, p6b(µ) > 0 for µ ∈ R. Note that the polyno-mials in (3.223a,c,d) that contain bi jwere rewritten with ai j. P.3.12 illustrates the positiveness of polynomials as described in this section.

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