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The polar method in plane anisotropy

Università di Pisa Facoltà di Ingegneria 10 luglio 2007. The polar method in plane anisotropy. P. Vannucci UVSQ - Université de Versailles et Saint-Quentin-en-Yvelines. Foreword.

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The polar method in plane anisotropy

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  1. Università di Pisa Facoltà di Ingegneria 10 luglio 2007 The polar method in plane anisotropy P. Vannucci UVSQ - Université de Versailles et Saint-Quentin-en-Yvelines

  2. Foreword • This presentation concerns a series of researches made during the last years, firstly at ISAT, University of Burgundy and presently at the University of Versailles, by G. Verchery, A. Vincenti, E. Valot and myself. • These researches are based upon an original idea of G. Verchery, the so-called polar method, for the description of plane anisotropy. • The polar method is just an alternative way to the Cartesian one to represent a tensor: in the polar method the whole set of independent invariants are found and they are used to effectively represent the tensor. • In a sense, all these researches turn around a basic question: is there an alternative, effective and advantageous way to represent plane anisotropy? We only tried to answer to this question…

  3. Content • How to describe anisotropy? • The polar method • The mechanical meaning of the polar parameters • Some other results concerning the polar method • Conclusions and perspectives

  4. How to describe anisotropy? • Anisotropy is the dependence of some property upon the direction; roughly speaking, that property is not invariant with the direction. • Mathematically, this implies that any orientation-based description of anisotropy is not invariant, i.e. it is based upon some quantities which are not intrinsically representative of the property, being frame dependent: it is the case of the Cartesian representation. • For instance, let us consider the case of elasticity: an isotropic material is described, in any frame, by only two quantities, the Young’s modulus and the Poisson’s ratio, or alternatively the Lamé’s moduli, or C1111 and C1122 in the stiffness tensor or even two other independent combinations of these quantities. • Any of these is an intrinsic quantity describing the elastic behaviour of the material; in a sense, it belongs to the material, and serves to its identification.

  5. How to describe anisotropy? • Actually, once you know any couple of these numbers, you know everything about the elastic behaviour of the material, in particular you do not need to specify the frame. • But when you want to make the same with an anisotropic material, for instance a fiber reinforced layer, the situation is much more complicated. • First of all, you need more quantities (how much??? a nice question…), and, more important, these quantities can be frame-dependent, according to the mathematical description of the property you choose. • As a matter of fact, the tensor components and most of their combinations, like engineer constants, are no more invariants. • So, they describe the given property in a given frame and only in this one; when you need the description in another frame, you must use a change-of-frame law (tensor rank dependent and normally rather cumbersome).

  6. How to describe anisotropy? • For instance, a fiber reinforced layer is orthotropic and described in its material axes by 4 quantities: E1, E2, n12 and G12 or alternatively by the stiffness matrix: • Nevertheless, none of these quantities is intrinsic: a change of frame transforms these quantities in a rather cumbersome way and the skyline of the stiffness matrix changes: with a Cartesian representation in a general frame, nobody can say in a glance if a layer is orthotropic!

  7. How to describe anisotropy? • In fact, an orthotropic layer in a general frame looks like this:

  8. How to describe anisotropy? • To remark that in a general frame it is apparent that 6, of which 4 independent, quantities are needed to describe an orthotropic, just like a totally anisotropic, layer. • In a sense, the Cartesian representation is frame dependent and for this reason it is effective whenever frame dependent results are needed (stresses, strains, displacements). • Nevertheless, in some problems the use of an intrinsic representation can be more effective than the Cartesian one. • Another reason to use an alternative representation of the tensor components is to have simpler formulae for the frame rotation; this can be of great importance in inverse problems, where the direction is one of the design variables: it is the case of laminate design.

  9. How to describe anisotropy? • In other words, alternative tensor representations for anisotropy are interesting per se and for applications. • In the past, different authors have proposed not-Cartesian tensor representations, mainly in elasticity: in the field of composite mechanics, and hence in two-dimensional elasticity, the most known attempt is that of Tsai and Pagano (1968), who proposed the following formulae:

  10. How to describe anisotropy? • Here, Uiare the Tsai and Pagano parameters given by: • This representation is not very effective (indeed, it just simplifies a little the rotation formulae) for two reasons: the parameters Uiare not tensor invariants and they do not have a physical meaning. • In addition, 7 parameters are introduced for describing a 6-parameters property.

  11. How to describe anisotropy? • So, the ideal should be the following: to have an intrinsic representation of an anisotropic property, i.e. making use only of tensor invariants and of a sufficient number of direction parameters fixing the frame, and in addition the tensor invariants should be chosen in such a way that they represent some physical property, if possible linked to the kind of anisotropy of the material. • This is just what has been done in 2D elasticity by Professor G. Verchery in 1979, who has introduced what he called the polar method.

  12. The polar method • The origin of the polar method is the complex variable representation of plane quantities, a classical approach in mathematical physics (Klein 1896, Michell 1902, Kolosov 1909, Muskhelishvili 1933, Green et Zerna 1954). • Actually, Verchery introduces a new complex transformation, more effective than that of Green and Zerna, and develops the tensor representation of symmetric tensors of the 2nd rank and of elasticity tensors. • The Verchery’s transformation: for a vector x= (x1, x2) the contravariant components are given by • Some algebraic, a little bit complicate, manipulations give the transformation of symmetric 2nd rank tensors and of elasticity tensors. The results are the following:

  13. The polar method • Second rank symmetric tensors: • Elasticity tensors:

  14. The polar method • This transformation has a peculiar and fundamental point: it allows to find the whole set of tensor invariants and so to express the Cartesian components of the tensor by only its invariants plus one parameter (actually, an angle) that fixes the frame. • In the case of 2nd order symmetric tensor the result is very well known: it is the Mohr’s representation: • T, R and F are the polar components of T. Indeed, T and R are invariants, and they represent respectively the centre and the radius of the Mohr’s circle (the spherical part and the deviator of the tensor), while F is an angle fixing the frame (it gives the direction of the principal components, and depends on the frame where the Cartesian components are known).

  15. The polar method • More interesting is the case of elasticity tensors: • Conversely,

  16. The polar method • T0, T1, R0, R1, 0 and 1 are the polar components of T, the last two being angles and the remaining ones moduli. • It can be shown that T0, T1, R0, R1 and the angular difference0-1 are five independent invariants of T, which can be represented by these intrinsic quantities. The choice of 0 or 1 fixes the frame (normally, 1 =0). • Actually, they describe the elastic properties of a layer, in a different, but intrinsic, way from the usual Cartesian one. • So, now the questions are: • is this representation interesting, i.e. useful? • have the polar components a mechanical meaning? • what are the bounds of polar components? • The answer to the first question is crucial: I will try to convince you that in some problems, but indeed not everytime, the polar representation can be advantageous. • But, to start…

  17. The polar method • ...what happens for the components in a rotated frame? • The result is rather interesting: it is sufficient to subtract the rotation angle from the polar angles! • This property reveals to be rather useful in laminate design.

  18. The polar method • The polar components of the inverse tensor S=T-1 can be given as function of T0, T1, R0, R1, 0 and 1 :

  19. The polar method • The technical moduli can also be easily calculated, for a given angle θ:

  20. The mechanical meaning of the polar parameters • A glance at the equations expressing the Cartesian components in a general frame as functions of the polar parameters; fixing F1=0 it is shows immediately that T0 and T1 represent the isotropic part of T, while R0, R1 and 0 the anisotropic part. • The necessary and sufficient condition for plane isotropy is then R0=R1=0. • More generally, the mechanical meaning of the polar components concerns material symmetries and strain energy decomposition.

  21. The mechanical meaning of the polar parameters • Let us first consider the link between elastic symmetries and polar components. • Actually, the polar method has allowed the first intrinsic definition of plane orthotropy, and the discovery of a special type of orthotropy. • In fact, there are 3 alternative invariant sufficient conditions of plane orthotropy: • general orthotropy (Vong & Verchery, 1986): • square symmetry (Verchery, 1979): • R0 orthotropy (Vannucci, 2002): • In brackets, the same condition expressed through the Tsai and Pagano parameters: the comparison shows the effectiveness of the polar method.

  22. The mechanical meaning of the polar parameters • The case of general orthotropy is stated by • This means that a layer is generally orthotropic if the harmonics describing the elastic properties are shifted of a multiple of 45°. • If we pose F1=0 (i.e. the frame is fixed so as the x axis is the strong axis) the Cartesian components look like

  23. K K =0 =0 K K = 1 = 1 K K =1 =1 K K = 0 = 0 q q ( ( ° ° ) ) The mechanical meaning of the polar parameters • To be remarked that there are two different types of general orthotropy, K= 0 or 1, for the same invariants T0, T1, R0 and R1 . In other words, two different orthotropic materials share the same values of T0, T1, R0 and R1: The directional and Cartesian diagram of the Young’s modulus E() for two orthotropic materials sharing the same values of T0, T1, R0 and R1 but with different K (T0 = 1.3, T1= 0.8, R0=0.7, R1=0.3).

  24. K=0 K=1 K=0 K=1 q (°) The directional and Cartesian diagram of the Poisson's coefficient nxy() for two orthotropic materials sharing the same values of T0, T1, R0 and R1 but with different K (T0 = 1.3, T1= 0.8, R0=0.7, R1=0.3). The mechanical meaning of the polar parameters

  25. K=0 K=0 K=1 q (°) The directional and Cartesian diagram of Gxy() for two orthotropic materials sharing the same values of T0, T1, R0 and R1 but with different K (T0 = 1.3, T1= 0.8, R0=0.7, R1=0.3). K=1 The mechanical meaning of the polar parameters • K=0 corresponds to what Pedersen, 1993, called a low shear modulus orthotropy (U3>0) while the case K=1 corresponds to the case of high shear modulus orthotropy (U3<0) . • It can be shown that materials with K=1 show some peculiar properties, especially for the design with respect to stiffness properties of laminates.

  26. The mechanical meaning of the polar parameters • The value of K influences also the kind of variation of the component Txxxx(q), hence of the normal stiffness. Three are the possible cases, depending upon the value of K and of the parameter r=R0/R1; if once again we choose F1=0, then: • The case of the Young's modulus can be treated in the same way, but using the compliance parameters and remembering that Ex(q)=1/Sxxxx(q). • K=0 or K=1 and r<1: Txxxx(0) is the highest value and Txxxx(p/2) is the lowest; • K=0 and r≥1 Txxxx(0) is the absolute and Txxxx(p/2) the relative maximum; the minimum is at the angle W, with • K=1 and r≥1 Txxxx(0) is the relative and Txxxx(p/2) the absolute minimum; the maximum is at the angle W.

  27. E(q) K=0 Gxy(q) K=0 K=1 K=1 The directional diagrams of the Young’s modulus E() and of Gxy()for two square-symmetric materials sharing the same values of T0, T1, R0 but with different K (T0 = 1.3, T1= 0.8, R0=0.7). The mechanical meaning of the polar parameters • The case of orthotropy stated by R1=0 corresponds to the so-called square-symmetry. It is the planar equivalent of the cubic syngony. • Though it is still an orthotropic case, algebraically it is different from the previous one. In fact, unlike the previous case, stated by a third-order invariant, this case is ruled by a second-order invariant.

  28. The mechanical meaning of the polar parameters • Now, the change from K=0 to K=1 corresponds simply to a rotation of the frame through an angle of 45°. • In fact, the Cartesian components in this case are • The Cartesian conditions for square-symmetry are easily found to be

  29. The mechanical meaning of the polar parameters • R0 orthotropy is a special case of orthotropy. Like the previous case of square–symmetry, it is ruled by a second order invariant. • The components of an elasticity R0 orthotropic tensor T behave like those of a 2nd rank symmetric tensor. If 1= 0, then: • Txxyy and Txyxy are isotropic, while Txxxy and Tyyxy are identical .

  30. The mechanical meaning of the polar parameters • To be remarked that R0=0 does not imply the same in compliance: • Nevertheless, the independent constants are still 3 also in compliance, as • So, there are also materials r0-orthotropic in compliance but not in stiffness.

  31. The mechanical meaning of the polar parameters • The Cartesian conditions corresponding to R0-orthotropy can be easily found: • Once again, two kinds of variations of the Young's modulus E(q) are possible: i. T02R1: E(0) is the highest and E(/2) the lowest of E(q) ; ii. T0<2R1: E(0) is a local minimum and E(/2) the absolute minimum; the absolute maximum is attained for the orientation • Two examples are shown in the next figures.

  32. E(q) Gxy(q) nxy() The directional diagrams of E(), Gxy() and nxy() for a R0-orthotropic material (T0 = 1.3, T1= 0.8, R1=0.3). The mechanical meaning of the polar parameters • First case: T02R1.

  33. The mechanical meaning of the polar parameters • Second case: T0<2R1 Gxy(q) E(q) nxy() The directional diagrams of E(), Gxy() and nxy() for a R0-orthotropic material (T0 = 1.3, T1= 0.8, R1=0.7).

  34. The mechanical meaning of the polar parameters • A question arise: how R0-orthotropic materials can be obtained? • If we consider a lamina reinforced by fibers disposed in equal amount along two directions, the polar condition to get R0-orthotropy is the following one • So, the solution is • This is only a sufficient condition for R0-orthotropy. • Nevertheless, it is the only possible solution for a lamina reinforced by fibers disposed along only two directions. • In fact, if l is the relative volume fraction, the only solution is get for l=1 and a=d1-d2=p/4, see the figure.

  35. The mechanical meaning of the polar parameters • Some remarks about plane orthotropy: we have seen that: • there is not a unique kind of plane orthotropy, algebraically speaking; • the three kinds of plane orthotropy are distinguished by different tensor conditions: • two of them, the R0 and R1 orthotropy are stated by second-order invariants, the third, the general orthotropy, is stated by a third-order invariant; • in addition, two general orthotropic materials can share the same polar moduli but a different value of the angle F0; so, two are, in principle, the general orthotropic materials sharing the same invariant moduli, and they have qualitative different properties; • R0-orthotropy does not correspond to r0-orthotropy and vice-versa; • R1-orthotropy corresponds to the cubic syngony in 3 dimensions; • the existence of R0-orthropy in 3 dimensions has been recently proved by S. Forte (2005); • in some senses, R0 and R1 orthotropy are stronger forms of orthotropy than the general one: they are conserved in some cases of homogenization.

  36. The mechanical meaning of the polar parameters • A last consideration about orthotropy. • Usually, it is said that plane orthotropy depends upon 4 constants, and so that 4 independent measures are needed to experimentally characterize such a case. • But, in the stiffness matrix, 6 are the components to be assigned; two of them are zero if the material is orthotropic and if it is described in its material axes: these are two data, numerically equivalent to two zero components. • When we look at the same problem in terms of polar invariants, we find that, once the frame fixed (usually by posing F1=0): • general orthotropy is characterized by 5 invariants: T0, T1, R0, R1 and K; • R0 and R1 orthotropy are characterized by 3 invariants: T0, T1 and R0 or R1. • So, it is not so immediate to say how much constants determine plane orthotropy !

  37. The mechanical meaning of the polar parameters • Let us now consider the energetic meaning of the polar components. • If then • It can be shown that • T1 is directly responsible of WS, T0, R0 and 0 of WD while R1 couples the two parts.

  38. The mechanical meaning of the polar parameters • And now, let us look at the bounds on the elastic polar components. • It can be shown that the following two conditions are necessary and sufficient to ensure the positive definiteness of W: • For orthotropic materials: SurfaceS= • surface in 3D; • b) level curves of S • and existence domain for the two cases of orthotropy.

  39. Some other results concerning the polar method • The polar method has been applied to the study of some problems, mainly concerning laminates but not only. • The problem of assessing the influence of orientation errors on some elastic properties of laminates has been addressed. • For instance, in the case of extension-bending coupling, the degree of coupling has been introduced to describe the average consequences of angle errors. • It has been shown that, once again, the ratio is the material parameter that describes the importance of an orientation error.

  40. Some other results concerning the polar method • In particular, if there is only one orientation error e, the expression of b is: • l is a parameter depending upon the number of layers n: • The maximum of b is obtained when the error e is located in correspondence of one of the outer layers and when its value is

  41. b/l r e Some other results concerning the polar method • In both cases, the maximum of b is 2l. The results are summarized in the figures below, that show a bifurcation-like behaviour with respect to the parameter control r, accounting for the anisotropy of the material.

  42. Some other results concerning the polar method • The previous figure show that the less sensitive laminates to orientation errors are those composed by R0-orthotropic materials (r=0), while the most sensitive are those composed by R1-orthotropic materials (r), i.e. those whose layers are reinforced by balanced fabrics. • A similar analysis has been conducted also for the property of quasi-homogeneity, and numerically also for a random vector of orientation errors for each layer, with similar results. • Another result concerns the classical equation for linear buckling of anisotropic plates

  43. Some other results concerning the polar method • If one expresses not only the bending stiffness tensor with polar parameters, but also 4w, which is a completely index-symmetric tensor, described by 4 polar invariants, indicated with lower case letters, one gets a simpler equation: • At the first member, there are only 3 terms, not 5, accounting for the isotropic and R0- and R1-anisotropic parts of D and 4w respectively. • At the second member there are only 2 terms, not 3, accounting for the spherical and deviatoric parts of N (capital letters) and of 2w (lower case letters). • This equation is still to be used….

  44. Some other results concerning the polar method • Others results concerning the mechanics of laminates: • a new kind of tests for angle-ply laminates ; • an experimental study of the strength of in-plane isotropic laminates; • a linear theory of laminates composed by intrinsically coupled layers; • an analytical study of the thermal and piezoelectric expansion coefficients for composite laminates. • The polar method has been recently extended also to the description of other plane properties than elasticity, namely of the tensors describing the phason and the coupled phonon-phason behaviours of quasi-crystals (Lauretti and Vannucci, 2006) and of piezoelectric tensors (Vannucci, 2007).

  45. Conclusions and perspectives • The polar method is only an alternative method to describe plane properties based upon the use of tensor invariants. • It reveals to be effective in the qualitative description of some properties, because the polar invariants have a mechanical meaning. • In addition, it is rather useful in some optimal design problems of laminates. • At present, some studies are considering the role played by polar invariants in the minimization of the elastic energy for some structural optimization problems in anisotropic elasticity. • In the future, it should be interesting to apply the same method to the qualitative study of other mechanical properties, like for instance plasticity, damage and strength criteria in anisotropic layers. Thank you very much for your attention.

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