Nathalie Boudeau

Jean-Claude Gelin

Salima Salhi

Computational prediction of the localized necking in sheet forming based on microstructural material aspects

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The analysis of the formability in sheet metal forming processes requires to account to different parameters as strain paths, initial and induced anisotropy, material rate sensitivity. The paper investigates the role of initial and induced anisotropy for various strain paths using crystalline computations. Usually, sheet metal used in Ž. stamping is obtained by cold rolling that has effect on their internal morphological structure grain shape and Ž. wx their crystallographic texture grain orientation 10,34 . These effects modify the material macroscopic properties of the sheet. In fact, if the material is initially isotropic, it becomes anisotropic due to the cold rolling process. This initial anisotropy influences both forming behavior and the deformation stage at which necking wx occurs. Different approaches that allow to predict the necking phenomenon, such as the two zones model 29 , wx wx the bifurcation theory 37 , the stability of the weak velocity form associated to equilibrium equations 13 , or wx the perturbation technique 16 , are usually applied with phenomenological constitutive laws including isotropic wx or orthotropic materials. The orthotropy is usually described with the quadratic 21 or non-quadratic yield wx wx function 5,27 . These models need assumptions to take into account the rotation of the orthotropic axes 15 . On 1 N. Boudeau, J.C. Gelin, S. Salhi Laboratoire de Mécanique Appliquée R. Chaléat, UMR CNRS 6604-Université de Franche-Comte, 24 Chemin de l'Epitaphe, 25030 Besançon Cedex, France Ž the other hand, the quadratic Hill's model does not very well represent the f.c.c. materials like aluminium for . wx example and the materials with low Lankford coefficient 25 . The non-quadratic model gives better results, but its real exponent is a purely numerical one and is difficult to be estimated.

wx In recent years, considerable efforts 6,26,48 , have been made in order to incorporate the plastic anisotropy Ž. that results particularly from crystallographic texture into constitutive models of the deformation of polycrystalline solids. Moreover, lots of studies are concerned to understand the texture evolution under complex strain wx paths 3,4,7, 40,43 . In this paper, we adopt a micromechanical approach to study the problem of localized necking in sheet metal stamping. The constitutive model used in this approach is based on crystalline plasticity where the plastic deformation is due to crystallographic slip. Such model with a microscopic hardening law allows to include the initial anisotropy and the induced one resulting from plastic deformation. The purpose of Ž. this work is to investigate the effect of initial and induced anisotropy on the Forming Limit Diagrams FLD and to find a relation between the characteristics of the pole figures and the necking phenomenon. The FLD represent in the principal strain space the domain over which localized necking appears for different strain paths, so it characterizes the straining ability of the sheet for stamping. Related studies are conducted in Refs. wx 11,32,47 where the influence of the induced texture and the effect of grain shape on the FLD were investigated. However, strain paths used in these studies were linear, and the effects of sequential strain paths on the texture evolution were ignored.

Locally the sheet metal is supposed to be a polycrystal described by 200 grains, where each grain is assumed Ž. to be a single crystal with a given crystal structure, b.c.c. or f.c.c. see Fig. 1 . The behavior of each crystal is wx w x given by the rate dependent model proposed by 35 . Taylor's assumption 44,45 is then employed to obtain the behavior of the aggregate. This model is incorporated in a linear perturbation technique for sheet necking wx analysis 16 . The main purpose of the perturbation method is to assume a certain additional inhomogeneity in the strain state, and to see under which condition the material will allow such a perturbation to develop, and to study the development of this perturbation, particularly whether its size will increase or not. As this method is wx questionable for the case of a rate-dependent material analysis 25 , firstly it is important to remark that the Ž. strain rate sensitivity used in the present paper is very small of the order of 0.005 . Secondly, and from the wx experience of the first author 8 , this method leads to more severe previsions at necking than the well known wx MK model 29 . Furthermore the linear stability analysis based on the study of the equilibrium state for given stresses, strains and strain rates can be locally applied directly for the treatment of finite elements results wx avoiding the integration along the strain path 9 . In that paper we focus our study on the effect of crystallographic texture and strain paths on the onset of strain localization.

In the first section, we present the microscopic constitutive laws used. The strain path is imposed so that it induces a simple form to the velocity gradient tensor. The second section is devoted to the adaptation of the linear perturbation technique to these constitutive laws. The third section concerns the application of the method to build FLD numerically. Some results will be presented and discussed. Numerous experimental observations will be found numerically and will give more explanations on the necking phenomenon for sequential strain paths with induced anisotropy.

Kinematic and constitutive behavior

The kinematic description of crystal plastic deformation in finite strains is described by a material shear flow along well defined slip systems a which are characterized by their slip direction s Ž a . and their slip normal plane m Ž a . . When a stress tensor is applied to the single crystal, slip systems glide and cause a microscopic shearing. wx To describe this feature, one can use the theory of thermally activated dislocation glide 46 . However, for cold deformation, the stress dependence on the slip rate can be reasonably approximated by the power law proposed wx by Ref. 24 . This power law takes the following form:

1rm Ž a . t Ž a .Ž a . g s g sign t 1 Ž. Ž . ˙0 Ž a . t c
where g is a reference strain rate and the exponent 1rm characterizes the material rate sensitivity. t Ž a . is the ˙0 c critical shear stress that depends on the physical variables and t Ž a . is the resolved shear stress on slip system a , which is given by

t Ž a . s P Ž a . :t 2 Ž. Ž a .
Ž. where the symbol : stands for the double contracted product of tensors. The tensor P introduced in Eq. 2 is defined as:

S a Ž a .Ž a . P s s m m 3 Ž . Ž.
where s Ž a . is the unit slip direction and m Ž a . the unit slip plane normal characterizing the slip system a. The superscript S denotes the symmetric part of the tensor. ŽŽ . . Ž. Ž a . < Ž a . < Since the exponent of the power law Eq. 1 is very large 200 , g is quite small, unless t is close to Ža . Ž.

< Ž a . < Ž a .
or nearly greater than t . Thus when using Eq. 1 , the activation condition t s t can be taken into

c c
account by filtering out the most inactive slip systems and keeping t Ž a . close to t Ž a . . In this description, we do c Ž. not limit the number of the slip systems in each crystal. Indeed, the value of the exponent in the Eq. 1 is so Ž.

< Ž a . < Ž a .

large 1rm s 200 that all the slip systems a where t t are automatically eliminated.

c

The description of the behavior is completed by a law that describes strain hardening in polycrystal on the basis of slip in single crystals. During polycrystal deformation, experimental studies show that more than one slip system are activated simultaneously, and that the slip in each slip system has an effect on the hardening of all systems. This is taken into account by adopting an evolution equation of the critical shear stresses in the form wx given by 22,23 :

Ž a . ab Ž a . t s h g 4 Ž. ˙Ý c a
where h ab are the hardening moduli. These are assumed to be given by the form ab a b h s h g q q 1 y q d 5 Ž. Ž . Ž .

< a < ab where g s Ý g is the sum of the slip strains on all slip systems and d is the Kronecker symbol.

a Ž. Ž. The parameter q characterizes the hardening behavior. A form of h g in Eq. 5 appropriate to Al-Cu wx alloys is 12

h g 0 2 h g s h sech h 6 Ž. Ž. 0 ž / t y t S0
where h is the initial hardening rate, t is the critical resolved shear stress and t is the saturation strength.

S

Under the stress tensor applied, the single crystal will induce a strain rate tensor, which is decomposed into wx the elastic and plastic part. Following Ref. [START_REF] Peirce | Material rate dependence and localized deformation in crystalline solids[END_REF] , the plastic part of the strain rate tensor is:

p Ž a .Ž a . p Ž a .Ž a . D s g P , V s g W 7 Ž. ˙ÝÝ a a
where D p and V p are the symmetric and skew-symmetric parts of the strain rate tensor. We assume throughout Ž.

Ž a .

this paper that sums like in Eq. 7 are extended over all active slip systems, by simply setting g s 0 ẇhenever the slip system a is not active. Finally, the tensor W Ž a . is given by the following relation:

A Ž a .Ž a .Ž a . W s s m m 8 Ž . Ž.
where the superscript A denotes the skewsymmetric part of the tensor.

The linear perturbation analysis

The localized necking phenomenon is a local instability problem associated to local equilibrium equations wx and material constitutive equations for a given strain and stress state. The solution proposed by Refs. 17,18,31 is employed in conjunction with the micro mechanical constitutive model described above to build forming limit diagrams and to study the effects of anisotropy on the localized necking problem.

Concerning the linear perturbation analysis relatively to other methods to predict the onset of necking, the wx comparisons between computations and experiments conducted by the first author 8 reveal the following Ž. conclusions. For a mid carbon steel SDDK steel the agreement is very good between the results given by the wx Ž. linear stability analysis and by the experiments 33 in the range from uniaxial tension r sy0.5 to Ž. equibiaxial tension r s 1 . In that case the MK model fails to give satisfactory results in the sense that the wx initial defect cannot be chosen in a reliable manner. Other comparisons related in Ref. 11 lead to the same conclusions. The linear stability analysis and the MK model are in good agreement for the cases corresponding to the overall strain ratios using a phenomenological approach. In the case of a polycrystalline approach, the limit strain corresponding to both analyses are similar in the range where r -0 and the MK model appears less accurate than the linear stability analysis. Ž Then, the local equilibrium of a material point is governed locally by the following equations in plane stress . state :

Ž. -the hardening law for each slip system N equations :

s 1rm Ž a . t Ž a .Ž a . g s g sign t , Ž. ˙0 Ž a . t c Ž.
-the plastic flow law 3 equations : Ž.

-the strain rate compatibility 1 equation :

D q D s 2 D .
11 ,22 22,11 12,12 Ž p . Moreover, the assumption of a rigid plastic behavior is done D s D . The elasticity is then entirely neglected, which seems justified since the material reaches large strains and is fully plastically loaded. The set of equations becomes in a formal form the non-linear system: Let u be a solution of Eq. 9 . To study the stability of the equilibrium state u ,a perturbation d u of the solution at equilibrium is introduced:

Mu s 0 9 Ž. Ž. 1 N s 4 T where u s t , t , t , D , D , D , D , g ,. . . ,g .
u s u 0 q d u 10 Ž.
where the applied perturbation has the following form:

0 d u s d u exp h t exp i j x P n 11 Ž. Ž . Ž. Ž .
x is the current position of the point where an instability can expand along a direction orthogonal to n with ™ ™ 0 Ž. n s cos c P e q sin c P e see Fig. 2 . d u represents the amplitude of the perturbation, h is the temporal part 12 of the instability and j is the spatial one.

The perturbation of the hardening law and the behavior law gives after linearization the following Ž. expressions see Appendix A for details :

g 0Ž a . Ža .Ž a . dg s dt 12 Ž. ˙0Ža . mt N s Ž a .Ž a . d D s P dg 13 Ž. Ý as1
For the compatibility condition, we obtain the following relation: sin 2 cdD q cos 2 cdD y 2 cos c sin cdD s 01 4 Ž.

11 22 12
The perturbation and linearization of the equilibrium equations give:

s cos c q s sin cd ´q cos cds q sin cds s 0, Ž .

11 12 33 11 12
s cos c q s sin cd ´q cos cds q sin cds s 01 5 Ž . Ž .

12 22 33 12 22

We can also show that:

dg Ž a . s hdg Ž a . , d D s hd´16 Ž. ˙33 33 Ž.
The perturbed solution must satisfy Eq. 9 :

Mu 0 , h , cd u 0 s 0 17 Ž.
Ž .

The instability occurs when u becomes different than u 0 . This is satisfied when there is a non-trivial solution Ž. for Eq. 17 , that means: det Mu 0 , h , c s 0 18 Ž.

Ž .

After some straightforward calculations, the expression of the matrix M takes the following form:

dt b A0 h I g 00 0E yhB d D Mu, h , cd u s Ž . ij C h 0 Ž a .
Ž. Ž. Due to the form of M, Eq. 18 gives a quite simple relation that must be satisfied by the unknown quantity h:

h N s D h q E s 0 19 Ž.
Ž .

NN
Ý N g h k dt k ks1 ij wxwx h s where i , j g 1, 2 = 1, 2 20 Ž. Ž . ij Nk g Ý dt ks 1 ij Ž.
k

The use of Eq. 20 requires the knowledge of dt which can be determined from the set of equations given ij Ž. by Eq. 17 : This set of equations is over-determined and the non-independent equations are suppressed, that allows to determine the vector direction fixing its norm arbitrarily. To compute its norm, the maximal work principle is applied. Then a power quantity p is defined as follows: Ž.

i BA I 0 0 dt s i C h 0 Ž.
2 wx Ž. Finally, since h ) 0, ;I g 1, 2 h is the principal value of the matrix h , there is instability at the

II ij
polycrystalline scale that corresponds to a localized necking appearance.

Numerical results

In this part, some applications of the proposed approach are presented. The influence of the strain path upon the formability of the sheet using different types of FLDs under sequential strain paths is investigated. The relation between the sheet formability and strain paths history was already shown in numerous experimental wx works 2,20,39,41 . So, the results obtained from our simulations are compared with experimental ones published in the literature in order to validate the proposed approach. We examine the effects of initial material anisotropy and crystal structure on the formability of the material for several strain paths. We are especially interested by the evolution of the anisotropy which accompany the localized necking.

Theoretical FLDs are determined for proportional simple or sequential strains paths, i.e. the ratio of the major to minor strain is constant at any stage of deformation: d ´II r ss const. 24

Ž.

d ´I

In stamping of sheets metal, numerous experimental works have shown that this ratio can vary from uniaxial Ž. Ž . tension r sy0.5 to balanced biaxial expansion r s 1. strain paths. The material is initially isotropic and we present the evolution of crystallographic texture when the localized necking is detected. Material parameters used in our simulations are reported in Table 1, and wx correspond to an aluminium alloy with 2.8% of copper 28 that exhibits a f.c.c. crystal structure.

FLDs are built in a two dimensional strain diagram for different imposed loading paths and recording the critical strain at which failure occurs or at which a given strain gradient appears.

Isotropic material

Ž.

As a first application of the proposed method an initially isotropic material Fig. 3 is considered. For this material, the effects of strain path changes on the forming limit diagrams are examined and the anisotropy of the material using crystallographic texture is characterized. For a selected number of grains, the crystal plasticity model is employed to compute their rotations and re-orientations from the strain path imposed. In this paper, the initial and updated textures are described in the form of the pole figures.

Ž. First, FLD are built for direct strain paths Fig. 4 . Their shape and level are analogous to those obtained by the macroscopic phenomenological approach. The strain level at necking is found higher for an uniaxial tension test than one obtained for a balanced biaxial expansion test. This characteristic is in good agreement with Ž. FLD obtained with a sequential strain path are also presented Figs. 4 and6 and their main characteristics are found. Premature instabilities are observed for strain paths consisting of prior balanced biaxial prestrain Ž. followed by uniaxial tension since prestrain reaches 10% Fig. 4, curve c . Conversely, uniaxial prestrain Ž followed by balanced biaxial stretching considerably increases limit strains at the onset of necking Fig. 4, curve . wx b . That fact is in good agreement with experimental observations 20,19,38 and several analyses of the necking wx phenomenon 8,29,37 . This behavior was also explained by microscopic observations that have revealed the wx strong stability of the microstructure generated by an equibiaxial expansion strain mode 38 .

The lowest FLD under sequential strain path is obtained when the second stage of deformation consists of Ž. plane strain Fig. 6 , and it is therefore possible to predict the severity of a plane strain second mode deformation in stamping.

Ž.

Ž. Fig. 9. FLDs obtained by linear strain paths for initial isotropic curve a and anisotropic materials curve b .

Ž.

Ž.

The observation of the pole figures Figs. 5 and7 shows a slight concentration of poles in the middle of the figure that proves the existence of an induced anisotropy due to plastic deformation along strain paths. This concentration is more pronounced when one of the strain paths is a balanced biaxial expansion.

Anisotropic material

Ž Sheets for stamping usually obtained by cold rolling present a particular texture called rolling texture Fig. Ž initial anisotropy. The classical macroscopic phenomenological studies assume that the anisotropy and then the . texture does not develop during the forming process.

The FLDs shown in Fig. 9 reveal an improvement of the sheet formability in case of an initial anisotropic Ž. wx material especially in thinning mode r -0 . This result is in agreement with Refs. 1,30 , pointing out that the anisotropy of the sheet metal modifies the strain ratio and that an important value of the anisotropy ratio has the effect to decrease the thickness strain compared to the other components for strain paths corresponding to drawing.

The observation of the pole figures at necking reveals a dependence of the evolution of the texture relatively Ž. Ž . to strain paths Fig. 10 . For r s 0 the initial rolling texture is reinforced Fig. 10a and for r s 1 the texture Ž. corresponding to necking Fig. 10b is largely different from the initial one. These observations reveal the existence of an induced anisotropy whose degree depends strongly on the initial anisotropy degree. The pole figures also show that their evolutions depend on the strain paths.

Ž. FLDs for sequential strain paths have been built Figs. 11 and 13 and the characteristics indicated for isotropic materials are found again. Uniaxial tension followed by a balanced biaxial expansion leads to an Ž. Ž . Fig. 13. FLDs of an initially anisotropic material for direct curve a and sequential curves b and c strain paths. important improvement of the sheet formability. A second straining mode in plane strain is critical for sheet stamping and leads to premature necking. Ž. The observation of the pole figure Fig. 12 shows that a sequential strain path corresponding to balanced Ž. biaxial expansion-uniaxial stretching curve c, Fig. 11 decreases the forming limit level in comparison to the Ž. 4 Ž. inverse sequence curve b, Fig. 11 . The associated 100 pole figures Fig. 12 corresponding to the balanced biaxial expansion-uniaxial stretching sequence reveal a more pronounced texture than in the case of the reverse wx sequence. These observations are in agreement with experimental ones 1,38 where the authors observe that a 4 reinforcement of the pole 100 decreases the forming limit level.

Influence of the crystallographic texture

One of the main interest of the microscopic based approach is that it includes the crystal structure in the constitutive laws. The crystal nature is given by the slip systems which can be activated by the external boundary conditions. These slip systems are different and more or less numerous, depending on the crystal structure. We propose here to study two different crystal structures: f.c.c. and b.c.c. The set of material parameters for the b.c.c. material studied is given in Table 2. These crystal structures are different in the Ž. description of the slip systems Tables 3 and4 .

The FLD obtained for these two different structures are given in Fig. 14. The strain level at necking is higher for the b.c.c. structure than for the f.c.c. one. The curve shapes are quite identical in the thinning domain, but very different for the expansion modes. We have shown here the importance of the crystal structure on the necking phenomenon as it was expressed wx by Keeler in the discussion following the presentation of Hutchinson and Neale's paper 25 and the results wx presented are in good agreement with those presented in Ref. 6 .

Conclusions

An original approach of the localized necking problem using a microscopic modelling for the constitutive laws is proposed. This new approach better reflects physical features such as crystallographic slip, work hardening and lattice rotation during deformation and takes into account their effects in the localized necking problem. We use the constitutive behavior resulting from texture evolution and microscopic hardening to characterize the sheet formability. The theoretical analysis developed in this work can predict the plastic behavior of sheet metal under sequential strain paths. In this analysis, the effect of the strain path coupled with a modelling of initial or induced anisotropy is shown to be the most important parameter controlling instability in multiple stage forming operations. In particular, it is shown, with the help of different combinations of two stages linear strain paths, how premature instability can occur due to the strain path imposed to the material. The Ž. main characteristics of the FLD have been found: i an uniaxial tension followed by balanced biaxial expansion Ž. improves the sheet formability considerably, ii an equibiaxial expansion followed by an uniaxial tension is Ž. quite insensitive until prestrain attains 10% for which the formability diminishes, iii an initial anisotropy improves the sheet formability, especially in the thinning domain.

In the case of an isotropic material, we have not observed a strong evolution of the texture. However, we have proved the existence of a slight induced anisotropy. In case of an initially anisotropic material, we have observed an important evolution of texture by examination of the pole figures. We have demonstrated that the induced anisotropy degree depends strongly on the initial anisotropy degree, and on the strain paths and the plastic strain level, which is consistent with experiment trends.

We have also shown the importance of the crystal structure on the phenomenon that proves again the large interest to use the microscopic based approach to study the necking phenomenon.

Finally, the model developed in this work is shown to be a powerful tool to understand and predict the onset of localized necking of sheet metal under simple and sequential strain paths. Ž. The hardening law has the expression given in Eq. 1 . We will distinguish here the two cases corresponding to the positivity or not of t Ž a .

Notations

Ø case No. 1: t Ž a . -0:

The hardening law can be written as:

1rm Ž a . t Ž a . g syg y ˙0 Ž a . ž / t c
The perturbed equation is:

1rm 1rm 0Ž a .Ž a . 0Ž a .Ž a . t q dt t dt 0Ž a .Ž a .
g q dg syg ys y g y Finally, we find the following relation:

1 q ˙˙˙0 0 Ž a .Ž a . 0Ž a . ž / ž / ž
g 0Ž a . dt Ž a .

Ža

. dg s ˙0Ža .

m t Ø case No. 2: t Ž a . ) 0 The hardening law can be written as:

1rm Ž a . t Ž a . g s g ˙0 Ž a . ž / t c
The perturbated equation is:

1rm 1rm 1rm 0Ž a .Ž a . 0Ž a .Ž a . t q dt t dt 0Ž a .Ž a . g q dg s g s g 1 q ˙˙0 0 Ž a .Ž a . 0Ž a . ž / ž / ž / tt t cc
The linearization of the above relation leads to the following equation:

1rm 0Ž a .Ž a .Ž a . t 1 dt 1 dt 0Ž a .Ž a . 0Ž a . g q dg f g 1 qs g 1 q ˙˙0 Ž a . 0Ž a . 0Ž a . ž / ž / ž / mm tt t c
Finally, we find the following relation:

g 0Ž a . dt Ž a . Ža . dg s ˙0Ža . t
Finally, the perturbation of the hardening law gives the following relation:

g 0Ž a . dt Ž a .

Appendix B

T is the Kirchhoff stress tensor at the polycrystal scale and t is the Kirchhoff stress tensor at the grain scale. The relation between T and t is given by the Taylor's assumption as following:

Ý t g g T s N g
N represents the number of grain constituting the polycrystal.

g

The perturbation of the above equation gives the following relation:

Ý dt g g d T s N g
To find a relation between the temporal perturbation part at polycrystal scale and the one at grain scale, we perform a temporal derivation:

Fig. 1 .

 1 Fig. 1. The micro-macro approach.

Fig. 2 .

 2 Fig. 2. The different frames.

  incompressibility condition 1 equation : trace D p s 0, Ž.

  corresponds to the number of slip systems and ga g 1, ..., N is the shear on each slip system

d D 4

 4 Fig. 3. 111 pole figure of an initial isotropic material.

a

  The perturbed quantity p has to be extremal which gives:Edp k b t b q g b ḃ s 0 m dt sy22From Eq. 22 it is possible to compute the amplitude of the perturbation on the stress tensor:m y 1 dts P t 23

  . FLDs for a direct curve a or sequential curves b and c strain paths applied to an initially isotropic material. Sequences of uniaxial Ž.Ž. tension followed by balanced biaxial expansion curve b and the inverse one curve c .Ž.Ž . These different values of r are used in the simulation to represent simple r or sequential r and r 11 2

  4 Ž Fig. 5. 111 pole figures corresponding to deformation at necking for an uniaxial tension-balanced biaxial expansion sequences strain path .Ž . 1 or a balanced biaxial expansion-uniaxial tension sequences strain paths 2 and 3 in relation with Fig. 4.

Fig. 6 .

 6 Fig.6. Uniaxial or biaxial prestrain up to different deformation levels and subsequent straining at fixed strain ratio for an initially isotropic material.

  wx w x experimental observations 19,38 . That fact has been explained by Ref. 38 introducing the mean free path of mobile dislocations. In the study proposed in this paper, the stress dependency on the slip rate is taken into ŽŽ . . wx account using a power law Eq. 1 . It is shown in Ref. 14 that a physical law based on density of mobile Ž. dislocations can be used for expressing g in Eq. 1 as ˙0 g s N P b P n 25 Ž. ˙0 with N the density of mobile dislocations, b the magnitude of the Burgers vector and n the critical dislocation 0 Ž Ž a .Ž a . . velocity with n s n when t s t . Thus, one can introduce in the approach proposed in this paper a 0 c 4 Ž Fig. 7. 111 pole figures corresponding to deformation at localized necking for uniaxial tension-plane strain sequences strain paths 1 and .Ž . 2 and balanced biaxial expansion-plane strain sequences strain path 3 in relation with Fig. 6. Fig. 8. Pole figures for an initial anisotropic material corresponding to a rolling texture: a 111 poles figure, b 100 poles figure.

  wx physical description 36,42 of the shear rate at the level of the slip system in terms of density of mobile dislocations to model many experimental observations.

Fig. 10 .

 10 Pole figures at localized necking in relation with Fig. 9. For r s 0 the rolling texture is reinforced a . For r s1 the resulting Ž. texture b is largely different from the initial one.

  .

Ž

  8 . The pole figures related in Fig.8and corresponding to a rolling texture have been obtained by a plane strain compression test that simulates relatively well the rolling process. This rolling texture is characterized by an Ž. Ž . Fig. 11. FLDs of an initially anisotropic material for direct curve a and sequential curves b and c strain paths. Uniaxial tension-balanced Ž. Ž. biaxial expansion sequences curve b and the inverse ones curve c . 4 Fig. 12. 100 pole figure at localized necking in relation with Fig. 12 for uniaxial tension-balanced biaxial expansion sequences strain path .Ž . 1 and the inverse ones strain path 2 .

Fig. 14 .

 14 Influence of crystal structures. FLD for f.c.c. material curve a and b.c.c. material curve b .

  Perturbation of the plastic constitutiÕe lawŽ.The plastic flow law is given in Eq. 7 . The perturbed one is then:00 Ž a .Ž a .Ž a . 0 Ž a .Ž a . D q d D s g q dg P s D q P dg ˙˙Ž . ÝÝ a aFinally:Ž a .Ž a . d D s P dg Ý a wx The other equations are classical and we can find the results of the perturbed equations in Ref. 16 .

  ´ẃ x 14 I.L. Dillamore, J.G. Roberts, A.C. Bush, Occurrence of shear bands in heavily rolled cubic metals, Metal Science, February, 1979, pp. 73-77. wx 15 A. Dogui, F. Sidoroff, Quelques remarques sur la plasticite anisotrope en grandes deformations, C.R. Acad. Sc., Paris, T 299, Serie II, ´´Ń o. 18, 1984. wx 16 D. Dudzinski, A. Molinari, Modelisation et prevision des instabilites plastiques en emboutissage, in: Physique et Mecanique de la Mise ´´´é n Forme des Metaux, Presses du CNRS, 1990, pp. 444-460. ẃx Ž.Ž . 17 C. Fressengeas, A. Molinari, Inertia and thermal effects on the localization of plastic flow, Acta Metall. 33 3 1985 387-396. wx Ž.Ž . 18 C. Fressengeas, A. Molinari, Instability and localization of plastic flow in shear at high strain rates, J. Mech. Phys. Solids 35 2 1987 185-211. wx Ž. 19 A.K. Ghosh, Plastic flow properties in relation to localized necking in sheets, in: D.P. Koistinen, N.M. Wang Eds. , Mechanics of Sheet Metal Forming, Plenum Press, NY-London, 1978, pp. 287-312. wx Ž. 20 J. Gronostajski, A. Dolny, T. Sobis, Proc. 12th IDDRG Congress, International Deep Drawing Research Group WGI , ed. by the Associazione Italiana Metallurgia, Italy, 1982, 39 pp. wx 21 R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, London, 1950. wx Ž. 22 R. Hill, J.R. Rice, J. Mech. Phys. Solids 20 1972 401. wx Ž. Ž . 23 R. Hill, K.S. Havner, Perspectives in the mechanics of elastoplastic crystals, J. Mech. Phys. Solids 30 1r2 1982 5-22. wx Ž. 24 J.W. Hutchinson, Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. R. Soc. Lond. A 348 1976 101-127. wx Ž. 25 J.W. Hutchinson, K.W. Neale, Sheet necking -III. Strain-rate effects, in: D.P. Koistinen, N.M. Wang Eds. , Mechanics of Sheet Metal Forming, Plenum Press, NY-London, 1978, pp. 269-285. wx 26 U.F. Kocks, M.G. Stout, G.R. Canova, 114th. TMS-AIME meetings, New York, 1985. wx 27 J. Lian, F. Barlat, B. Baudelet, Plastic behavior and stretchability of sheet metals. Part II: Effect of yield surface shape on sheet Ž. forming limit, Int. J. Plasticity 5 1989 131-147. wx 28 A.M. Maniatty, P.R. Dawson, Y.-S. Lee, A time integration algorithm for elasto-viscoplastic cubic crystals applied to modelling Ž. polycrystalline deformation, Int. J. Num. Meth. Eng. 35 1992 1565-1588. wx 29 Z. Marciniak, K. Kuczynski, T. Pokora, Influence of the plastic properties of a material on the forming limit diagram for sheet metal in Ž. tension, Int. J. Mech. Sci. 15 1973 789-805. wx Ž. 30 Z. Marciniak, J. Duncan, in: E. Arnold Ed. , Mechanics of Sheet Metal Forming, Publ. 1992. wx Ž.Ž . 31 A. Molinari, Instabilite thermoviscoplastique en cisaillement simple, J. Meca. Theor. Appl. 4 5 1985 659-684. ´ẃ x 32 K.W. Neale, E. Van Der Giessen, Crystal plasticity based computations of localized necking instabilities in rolled sheets, 3th World Ž. Congress on Computational Mechanics, WCCM III, Chiba Japan , Vol. 1, D13-1, 1994, pp. 580-581.

  solving of Eq. 19 gives the temporal perturbation on the grain scale h . To know if the equilibrium state is stable or not at the polycrystal scale, it is necessary to analyze the temporal perturbation h Ž. of the polycrystal. Using Taylor's hypothesis see Appendix B , we find the following relation between h and h

ss

Coefficients D and E depend on the current equilibrium state and have been determined by recurrent NN ss relations that allow to compute them for any number of slip systems.

Ž.

k For each grain k, k :

Table 1

 1

	Ž. Material parameter of an aluminium alloy with 2.8% of copper f.c.c. used in the crystalline model from Ref. 28 Ž wx .
	Ž. Reference slip rate identical for all the slip systems	g ˙0	y3 10	Ž . Eq. 13
	Ž a . Ž. MPa cc Initial hardening t tt s60.84 0	Ž. Eq. 11
	Coefficient of the hardening law	h	8.9t 00	Ž. Eq. 11
	Yield stress at saturation	1.8t t s0	Ž. Eq. 16
	Strain rate sensitivity exponent	m	0.02	Ž. Eq. 13
	Latent hardening coefficient	q		1.0	Ž. Eq. 15

Table 2

 2 Material parameters of a b.c.c. material

	Ž. Reference slip rate identical for all the slip systems	g ˙0	y3 10	Ž . Eq. 13
	Ž a . Ž. MPa cc Initial hardening t tt s60.84 0	Ž. Eq. 11
	Coefficient of the hardening law	h	8.9t 00	Ž. Eq. 11
	Yield stress at saturation	1.8t t s0	Ž. Eq. 16
	Strain rate sensitivity exponent	m	0.02	Ž. Eq. 13
	Latent hardening coefficient	q		1.1	Ž. Eq. 15

Table 3

 3 List of slip systems for b.c.c. crystal a s1r 2, a s1r 3, a s1r 6

	'' Ž. List of slip systems for f.c.c. crystal where a s1r 2 and a s1r 3 12
	Systems	Ž a . sm	Ž a .
	1	a 11 ya	0		aa 22	a	2
	2	ya 1	0	aaa 122	a	2
	30	a 1122 yaaa	a	2
	4	a 1	0	a 122 yaa	a	2
	5	ya 11 ya	0	yaa 22	a	2
	60	a 1122 ya yaa	a	2
	7	ya 1	0	aa 122 yaa	2
	80	ya 1122 yaa yaa	2
	9	aa0 11		a	yaa 22	2
	10	yaa0 11	ya	yaa 22	2
	11	a 1	0	a 122 ya yaa	2
	12	0	ya 1122 ya ya yaa	2

Ž.

  /

	tt cc	t
	The linearization of the relation above leads to the following equation:
	1rm 0Ž a .Ž a .Ž a . 1 dt 1 dt 0Ž a . 1 qs g 1 q Ž a . 0Ž a . 0Ž a . ž / ž ž / mm t q dg fyg y 0Ž a .Ž a . ˙˙˙0 g tt t c	/

  recherche, MRT -Departement Materiaux, Fascicule 2, Influence de la microstructure sur la deformabilite a froid des produits ´´Ž . plats, Institut de Recherches de la Siderurgie Franc ¸aise IRSID , Aout, 1987, pp. 1-100. ´ŵx Ž. 2 R. Arrieux, C. Bedrin, M. Boivin, Proceedings of the 12th IDDRG Congress, International Deep Drawing Research Group WGI , ed. by the Associazione Italiana Metallurgia, Italy, 1982, p. 61. wx 3 B. Bacroix, P. Genevois, C. Teodosiu, Plastic anisotropy in low carbon steels subjected to simple shear with strain path changes, Eur. J. Bacroix, H.U. Zaiqian, Texture evolution induced by strain path changes in low carbon steel sheets, Metall. Mater. Trans. 26A Ž. 1995 601-613. wx 5 F. Barlat, Endommagement, anisotropie et courbes limites de formage, Ph.D. Thesis, INP-Grenoble, France, 1984. wx Ž. 6 F. Barlat, Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals, Mater. Sci. Eng. 91 1987 55-72. wx 7 N. Boudeau, J.C. Gelin, S. Salhi, Effects of mechanical anisotropy on material instabilities in sheet metal forming, Mater. Instab. Ž. Theory Applic. ASME AMD 183 vol. 50 1994 303-314. wx 8 N. Boudeau, Prediction des instabilites elastoplastiques locales. Application a l'emboutissage, Ph.D. Thesis, University of Besancon, ´´F rance, 1995. wx 9 N. Boudeau, J.C. Gelin, Postprocessing finite element results and prediction of the localized necking in sheet metal forming, J. Mater. Ž. Ž . Proc. Technol. 60 1-4 1996 211-216. wx 10 F. Boutin, Aspects metallurgiques du laminage froid, in: Mise en Forme des Metaux et Alliages, Editions du CNRS, 1975, pp. Canova, U.F. Kocks, C. Fressengeas, D. Dudzinski, Ph. Lequeu, G. Sornberger, in: J.S. Kallend, G. Gottstein Eds. , Textures Ž. and Sheet Forming, 8th Int. Conf. Textures of Materials ICOTOM 8 , The Metallurgical Society, 1988, pp. 1019-1030. wx Ž. 12 Y.M. Chang, R.J. Asaro, An experimental study of shear localization in aluminium-copper single crystals, Acta Metall. 29 1981 241. wx Ž.Ž . 13 J.P. Cordebois, P. Ladeveze, Sur la prevision des courbes limites d'emboutissage, J. Meca. Theor. Appl. 5 3 1986 341-370.

	Ž.Ž Mech. ArSolids 13 5 1994 661-675. .
	wx 4 B. ´267-281.
	wx 11 G.R.		Ž.
	dd d T s hdT and d t d t	gg dt s hd t g
	Finally, we obtain the following relation:
	h s	Ý hd t g gg

g Ý dt g ´´ˆà nnee de


Milan Kvicera

Jonathan Israel

Fernando Perez-Fontan

Pavel Pechac

Sensitivity Analysis of Multiple Scattering Theory Applied to Tree Canopies at Microwave Frequencies

Keywords: Multiple scattering theory, radio-wave propagation, scattering, vegetation

A sensitivity analysis of the multiple scattering theory applied to propagation in tree canopies is presented. This analysis aims at addressing the influence of input parameters on the canopy's equivalent scattering amplitude as well as the overall coherent and incoherent scattered fields. The main input parameters of interest were identified as frequency, number densities and dielectric parameters of branches and leaves, polarization, and a canopy shape. A number of graphical results are provided enabling to identify the key parameters of corresponding radio-wave propagation models based on the multiple scattering theory.

I. INTRODUCTION

T HE MULTIPLE scattering theory (MST), as developed in [START_REF] Foldy | The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers[END_REF]- [START_REF] Twersky | Multiple scattering of electromagnetic waves by arbitrary configurations[END_REF], allows us to obtain the coherent and incoherent fields scattered by an ensemble of randomly oriented and positioned particles where their density is low and their sizes are smaller than their separations [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF]. Alongside the radiative transfer theory [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF], in the last few decades, it has been extensively used in the area of remote sensing, e.g., [START_REF] Ulabyandd | Microwave Radar and Radiometric Remote Sensing[END_REF] for the case of scattering from a layer of vegetation.

In [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] Karam | Leaf-shape effects in electromagnetic wave scattering from vegetation[END_REF], the scattering characteristics for a thin dielectric cylinder representing a tree branch and a flat cylinder (disc) representing a leaf have been introduced. The scattering amplitude tensors of these individual tree canopy elements can then be utilized to obtain the equivalent scattering amplitude, to be introduced shortly, of the whole canopy.

It should be noted that the area of application for remote sensing lies mainly in the monostatic scattering case [START_REF] Ulabyandd | Microwave Radar and Radiometric Remote Sensing[END_REF]. However, MST is not restricted to this case and enables arbitrary bistatic configurations, which is the case of communication and M. Kvicera and P. Pechac are with the Department of Electromagnetic Field, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague 6, Czech Republic (e-mail: kvicemil@fel.cvut.cz; pechac@fel.cvut.cz).

J. Israel is with ONERA-The French Aerospace Lab, 31055 Toulouse, France (e-mail: Jonathan.Israel@onera.fr).

F. Perez-Fontan is with the University of Vigo, 36200 Vigo, Spain (e-mail: fpfontan@tsc.uvigo.es). navigation scenarios in the presence of vegetation. In fact, the current version of ITU-R Rec. P.833 [START_REF] Itu | Attenuation in vegetation[END_REF] dealing with propagation in the presence of vegetation has incorporated MST.

The applicability of MST to such scenarios has been validated in [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] Chee | Radiowave propagation prediction in vegetated residential environments[END_REF]. However, although several simplifying assumptions can be made as in [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF] and [START_REF] Torrico | Modeling tree effects on path loss in a residential environment[END_REF], the underlying calculations depending on a number of input parameters are still very complex. As a consequence, it is not possible to address the dependence of the obtained results on the input parameters analytically in a straightforward way and, rather, a numerical approach should be applied. Although such analysis has not been published so far, it is crucial for addressing the validity range of the propagation models based on MST, such as [START_REF] Itu | Attenuation in vegetation[END_REF] or [START_REF] Israel | Fading and scattering due to trees in L to Ka band propagation simulations[END_REF]. It should be noted that performing such an analysis is very demanding from a computation-time point of view and, thus, could be of interest to a reader wanting to implement such models.

II. MULTIPLE SCATTERING THEORY

Based on [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] Karam | Leaf-shape effects in electromagnetic wave scattering from vegetation[END_REF] and [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF], we provide here the main set of equations utilized in MST. We consider a plane wave with amplitude and polarization incident in the direction on a tree canopy (Fig. 1) [START_REF] Foldy | The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers[END_REF] where is the free-space wavenumber. The coherent field at point inside the canopy is given by [START_REF] Lax | Multiple scattering of waves[END_REF] where is the effective propagation constant and is the distance through the canopy to a point along direction .Note that is in general complex and is equal to [START_REF] Twersky | Multiple scattering of electromagnetic waves by arbitrary configurations[END_REF] where represents the canopy's equivalent scattering amplitude per unit volume in the scattering direction .T h e canopy's specific attenuation, in dB/m, is then [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF] (4) The field scattered from points inside the canopy to a point outside the canopy can be written in terms of its coherent part as [START_REF] Ulabyandd | Microwave Radar and Radiometric Remote Sensing[END_REF] and incoherent part as (6)

TABLE I SIZE OF BRANCHES AND LEAVES

where denotes the statistical average and is an operator given by [START_REF] Karam | Electromagneticwave scattering from some vegetation samples[END_REF] with being the distance through the canopy from a point to the point along the scattering direction .T h e coherent part is a complex electric field while the incoherent part represents power (with no phase information).

The canopy is assumed to consist of various branch types of different sizes (divided into five categories in [START_REF] Itu | Attenuation in vegetation[END_REF] and [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF]; see Table I) and leaves with relative complex permittivity of and , respectively. The orientations of the various types of branches and of the leaves are random and can be described by means of two rotation angles in azimuth and elevation, and , with respect to the tree reference coordinate frame as in [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF]. Following [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF], we have assumed a uniform distribution, , for each single branch or leaf . The possible range of orientation angles for the larger branches tends to be limited and centered about the vertical direction, while the smaller ones and the leaves tend to be uniformly distributed over all spatial angles.

The mean scattering amplitude is [START_REF] Karam | Leaf-shape effects in electromagnetic wave scattering from vegetation[END_REF] with being the scattering amplitude tensor for the branches or leaves [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] Karam | Leaf-shape effects in electromagnetic wave scattering from vegetation[END_REF]. The various types of branches and leaves are also characterized by their respective number densities . For the overall canopy, we can write ( 9)

For a single branch or leaf, the mean scattering cross section can then be written as [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF] and the equivalent scattering cross section per unit volume as [START_REF] Cheffena | Land mobile satellite channel simulator along roadside trees[END_REF] In Section III, we will refer to a tree configuration studied in [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF] with the same branch and leaf sizes and number densities. We may also note that following [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] Karam | Leaf-shape effects in electromagnetic wave scattering from vegetation[END_REF], the branches need to have lengths sufficiently greater than their radii, and the thickness of the leaves needs to be electrically thin.

III. SENSITIVITY ANALYSIS

From the various input parameters and their unlimited possible combinations, we have selected a representative set that enables us to provide a thorough sensitivity analysis of the behavior of MST. As mentioned above, we consider a scenario similar to the one studied in [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF], i.e., a vertically polarized plane wave is incident in the -direction on the canopy, and the scattering characteristics are investigated in the horizontal -plane for scattering angles from 0 to 180 at the distance of 50 m from the canopy center. In the analysis, ( 8) and [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF] were evaluated by means of Monte Carlo integration. We should also note that, following [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] Karam | Leaf-shape effects in electromagnetic wave scattering from vegetation[END_REF] and [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF], an infinite sum of Bessel and Hankel functions shall be utilized to obtain the scattering amplitude tensor of a single branch while [START_REF] Itu | Attenuation in vegetation[END_REF] suggests that a maximum order of 20 is sufficient. However, even higher orders such as 50 may be required at frequencies above 10 GHz, despite the resulting increase in computation time.

A. Frequency

Keeping the dielectric parameters fixed and scanning in frequency, we obtain higher scattering amplitudes as the frequency increases for each individual canopy element. This increase is very significant, and we observe that MST calculations are strongly frequency-dependent; see for example Fig. 2, which compares the scattering amplitudes for branches of category one and leaves at 5 and 10 GHz together with the resulting overall equivalent scattering amplitudes .

B. Dielectric Parameters

We studied the impact of the relative permittivity and conductivity as well. It was found that at frequencies below 10 GHz, a precise characterization is not too significant; an increase/decrease of and of individual branches and leaves by 25% from their original values ( and , and , and , and ) did not significantly influence .InFig.3, this is shown at 2 GHz for the case of the increase only as the other cases had the same negligible impact.

However, at higher frequencies, the change of influences mainly the scattering amplitude of leaves, while the change in influences the resulting specific attenuation of the canopy.

C. Number Densities

Number densities of branches and leaves are given in Table I [9], [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF] up to a precision of three decimals. However, these values will actually vary from tree to tree, even if they belong to the same class. As the number densities influence in ( 9), they have a direct and significant impact on the overall coherent and incoherent fields calculated according to ( 5) and ( 6), regardless of the frequency. Fig. 3 shows for the case where all the number densities were decreased/increased by 25%. It should also be noted that at frequencies above 2 GHz, the scattering amplitude tensor of leaves increases and starts to play a major role in due to their corresponding high number density.

D. Canopy Volume Integration

Unlike in [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF], the volume integration in ( 5) and ( 6) can be performed numerically, thus avoiding simplifications. Considering the coherent field calculations, the incremental volume used in ( 5) and ( 6) needs to be sufficiently small to provide reliable phase information. Thus, wassettobeacube of side , which, for large canopies and high frequencies, results in unwieldy computational demands. However, it was noticed that by increasing this side up to its limit, where the undersampling effects occur, the phase information in the forward scatter direction is still preserved and inaccuracy appears only towards the backscatter direction, as shown in Fig. 4. This allowed us to shorten the computation period by using of side at 2 GHz and at frequencies above 10 GHz. In this way, we were still able to well represent the coherent fields w.r.t. shadowing effects behind a tree canopy described in Section III-G. In contrast, it should be noted that the incoherent intensity calculations not significantly influenced by utilizing the coarser canopy segmentations as they do not include the phase information.

E. Canopy Shape

References [START_REF] Itu | Attenuation in vegetation[END_REF] and [START_REF] De | A tree-scattering model for improved propagation prediction in urban microcells[END_REF] suggest a cylindrical canopy. Other, more realistic shapes for some types of trees, closer for example to a sphere or an ellipsoid, can be considered.

We compared canopies approximately similar to a cylindrical one of radius 5 m and height 8 m. The comparison included a spherical canopy with radius of 5 m and an ellipsoid with and semi-axes of 5 m and semi-axis of 4 m. It is obvious from Fig. 4 that the differences in the resulting fields are not very significant at 2 GHz. A similar situation was also observed at a higher frequency, 10 GHz.

In addition, no noticeable change was observed when the canopy dimensions were increased by 10%, suggesting that such precision does not play a key role.

However, it should be noted that the shadowing effects behind a tree canopy addressed in Section III-G must correspond to the canopy shape. Therefore, considering for example scenarios with inclined and directions, the canopy shape may be of greater importance than in the scenario studied in this letter.

F. Polarization

So far, only vertical polarization has been assumed. However, [START_REF] Karam | Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation[END_REF]- [START_REF] Karam | Leaf-shape effects in electromagnetic wave scattering from vegetation[END_REF] provide equations for the horizontal polarization case as well. The corresponding mean scattering cross sections for the branches of categories one and four and leaves at 2 GHz are compared in Fig. 5 for vertical and horizontal polarizations together with the resulting . We can observe that the general levels for all branch types and the leaves remain similar when the polarization is switched, especially in the forward scatter direction.

Furthermore, the specific attenuation determined from the imaginary part of by ( 4) was slightly lower (0.81 dB/m) than for the vertical polarization case (0.97 dB/m).

G. Shadowing Effects

When added to the direct field, the coherent scattered field provides a diffraction pattern in the shadow region behind the canopy [START_REF] De | Measurement and modeling of radiowave propagation in urban microcells[END_REF]. For this case, the shape of the canopy may be of importance as, in general, mainly the contributions above and below the canopy are different for spherical or cylindrical shapes. Fig. 6 shows the differences obtained at 2 and 10 GHz for cylindrical, spherical, and ellipsoidal canopies with the same dimensions as in Section III-E. Here, we can observe an overall similar trend with slightly different diffraction patterns depending on the canopy shape. The absolute values correspond to the obtained specific attenuation ( dB/m, dB/m) multiplied by the propagation distance through the canopy, which is about 10 m. The geometric shadow region behind the canopy is about 6 wide, which is in good agreement with the pattern in Fig. 6.

IV. SUMMARY

The results enabling to address the validity of corresponding propagation models based on the multiple scattering theory were presented. We have shown that apart from polarization, there are two other main input parameters of interest: frequency and number densities of branches and leaves. Although the frequency dependence can be considered to be self-explanatory, the direct influence of the number densities, often required to a precision of up to three decimals, may not be that evident. However, it was this parameter that determined the dominant role of the leaves throughout the presented analysis. Although one selected scenario was studied, the findings have a general applicability as they are based on the principal MST equations.

Fig. 1 .

 1 Fig. 1. Tree canopy scattering configuration.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Scattering amplitudes for branches of category one and leaves at the frequencies of 5 and 10 GHz together with the resulting equivalent scattering amplitude.

Fig. 4 .

 4 Fig. 4. Coherent and incoherent scattered fields at the frequency of 2 GHz for three different types of a tree canopy: cylindrical (radius of 5 m, height 8 m), spherical (radius of 5 m), and ellipsoidal ( and semi-axes of 5 m, semi-axis of 4 m). The effect of undersampling is shown by the dash-dottted line.

Fig. 5 .

 5 Fig. 5. Mean scattering cross sections for the leaves and branches of categories one and four at the frequency of 2 GHz for the horizontal and vertical polarizations.

Fig. 6 .

 6 Fig. 6. Shadowing behind the canopy caused by the coherently scattered field at the frequencies of 2 and 10 GHz for the three different canopy shapes with dimensions as in Fig. 4.