Polygonization: Theory and comparison with experiments

International Journal of Engineering Science 59 (2012) 211–218

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Polygonization: Theory and comparison with experiments K.C. Le a,⇑, B.D. Nguyen b a b

Lehrstuhl für Allgemeine Mechanik, Ruhr-Universität Bochum, D-44780 Bochum, Germany Computational Engineering, Vietnamese-German University, Ho Chi Minh City, Viet Nam

a r t i c l e

i n f o

Article history: Received 15 August 2011 Accepted 21 November 2011 Available online 6 April 2012 Keywords: Single crystal Bending Tilt boundaries Dislocations Polygon

a b s t r a c t Within continuum dislocation theory (CDT) the energy functional of a bent beam, made of a single crystal having only one active slip system, is asymptotically analyzed. By relaxing the smooth minimizer of this energy functional, we construct a sequence of piecewise smooth displacements and piecewise constant plastic distortions reducing the energy and exhibiting polygonization. The number of polygons can be estimated from above by equating the surface energy of small angle tilt boundaries to the contribution of the gradient terms from the smooth minimizer in the bulk energy. The comparison with Gilman’s experimental results shows a good agreement. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction When a single crystal beam is plastically bent and then annealed, one often observes its polygonization in the final stage as shown schematically in Fig. 1. Each polygon is a single crystal oriented slightly differently with respect to its neighbors so that the boundaries between them are low angle tilt boundaries. The dislocations align themselves into ordered arrays at these boundaries, and there are practically no dislocations inside the polygons. The experimental observations of polygonization have been reported in the late forties of the last century; see for example (Cahn, 1949; Gilman, 1955). The first attempt of taking into account the dislocations in the plastically bent beam was made by Nye (1953) who expressed the curvature of a beam caused by dislocations in terms of the dislocation density tensor bearing now his name. Read (1957) and Bilby, Gardner, and Smith (1958) have extended this result to the case when the stress due to dislocations does not vanish. However, the qualitative modelling of polygonization based on the continuum dislocation theory (Berdichevsky, 2006a, 2006b; Berdichevsky & Le, 2007; Le & Sembiring, 2008a, 2008b, 2009; Kochmann & Le, 2008; Kochmann & Le, 2009a, 2009b; Kaluza & Le, 2011) was proposed only recently by Le and Nguyen (2011). In that paper the simplest case of polygonization of the single crystal beam with one active slip system parallel to the beam axis was considered. The obtained results confirmed the LEDS (low energy dislocation structures) hypothesis (Kuhlmann-Wilsdorf, 2001). The present paper aims at extending this result to the case of single crystal having one active slip system inclined at some angle to the beam axis and comparing with the experiments reported in Gilman (1955). To match Gilman’s experimental setup, we specify the displacements of one face of the beam rather than applying the bending moment to the ends of the beam. We then consider the exact two-dimensional variational problem of minimizing energy of the bent beam within the continuum dislocation theory. Applying the variational asymptotic procedure, we simplify this energy functional and then find the smooth minimizer in closed analytical form. Based on this smooth solution we then construct a sequence of piecewise smooth displacements and piecewise constant plastic distortions having the same bending moment as that of ⇑ Corresponding author. Tel.: +49 2343226033. E-mail address: [email protected] (K.C. Le). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.03.005

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the smooth minimizer. By including also energy contributions at jumps of the plastic distortion, proposed in accordance with the Read–Shockley formula for the low angle tilt boundaries (Read & Shockley, 1950), we show that these discontinuous functions do reduce the total energy of the bent beam. We give also the estimation of the number of polygons depending on the radius of curvature of the bent beam. 2. Energy of the bent beam Consider a single crystal in form of a beam deforming under the plane strain condition. Let the two-dimensional domain occupied by the undeformed beam be a rectangle of width a and height h, x 2 (0, a), y 2 (0, h), where h  a. A rigid cylinder is used as a jig along which the beam is bent so that the bending axis is fixed relative to the crystal axes. For the bending in this way the displacements at point (x, 0) on the lower face of the beam are specified:

ux ðx; 0Þ ¼ r sinðx=rÞ  x;

uy ðx; 0Þ ¼ r cosðx=rÞ  r;

ð1Þ

where r is the radius of the generating circle. The upper and side boundaries of the beam are free from traction (see Fig. 2). If the radius r is sufficiently small, then edge dislocations may appear causing the plastic deformations. We admit only one active slip system for which the slip planes are inclined at an angle u to the plane y = 0. Under the plane strain condition the only non-zero components of the displacements, ux(x, y) and uy(x, y), yield the nonzero components of the total strain tensor

1 2

exx ¼ ux;x ; eyy ¼ uy;y ; exy ¼ eyx ¼ ðux;y þ uy;x Þ; where the comma in indices denotes the spatial derivative with respect to the corresponding coordinate. For the slip system with the slip planes inclined at the angle uwith respect to the plane y = 0, the incompatible plastic distortion tensor is given by bij = b(x, y)simj, where the unit vector s = (cosu, sinu, 0) denotes the slip direction, while the unit vector m = (sinu, cosu,0) is normal to the slip plane. Thus, the non-zero components of the plastic strain tensor, 1 eðpÞ ij ¼ 2 ðbij þ bji Þ, read

Fig. 1. A piece of polygonized bent beam made of single crystal.

Fig. 2. Single crystal beam bent along a rigid cylinder.

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1 2

1 2

1 2

ðpÞ ðpÞ exx ¼  b sin 2u; eðpÞ b sin 2u; eðpÞ b cos 2u: yy ¼ xy ¼ eyx ¼

Consequently, the non-zero components of the elastic strain tensor,

ðpÞ eðeÞ ij ¼ eij  eij , are

ðeÞ ðeÞ exx ¼ ux;x þ 12 b sin 2u; eyy ¼ uy;y  12 b sin 2u; ðeÞ ðeÞ exy ¼ eyx ¼ 12 ðux;y þ uy;x  b cos 2uÞ:

The distribution of geometrically necessary edge dislocations associated with this active slip system is described by the Nye dislocation density tensor (Nye, 1953), aij = ejklbil,k, whose non-zero components read

axz ¼ b;x cos2 u þ b;y cos u sin u; ayz ¼ b;x cos u sin u þ b;y sin2 u: These are the components of the resultant Burgers’ vector of all edge dislocations whose dislocation lines cut the area perpendicular to the z-axis. Therefore the scalar dislocation density (or the number of dislocations per unit area) equals

q¼

1 b

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðaxz Þ2 þ ðayz Þ2 ¼ jb;x cos u þ b;y sin uj; b

where b is the magnitude of the Burgers vector. Because of the prescribed displacements at y = 0 dislocations cannot reach the lower face of the beam which is in contact with the bending jig in the deformed state, therefore

bðx; 0Þ ¼ 0:

ð2Þ

So, in this model the boundary y = 0 serves as the obstacle leading to the dislocation pile-up. Under the assumptions made the bulk energy density per unit volume of the crystal with dislocations takes a simple form (Berdichevsky, 2006b)

U¼

 2  2 2 2 1  1 1 1  k ux;x þ uy;y þ l ux;x þ b sin 2u þ l uy;y  b sin 2u þ l ux;y þ uy;x  b cos 2u þ lk 2 2 2 2 1  ln ; jb;x cos uþb;y sin uj 1 bq

ð3Þ

s

with k and l the Lamé constants, qs the saturated dislocation density, and k the material constant. The first four terms of (3) describe the elastic energy, the last term is the energy of the dislocation network. Bypassing the whole process of dislocation motion and neglecting the associated dissipation due to this motion, we require that, in the final equilibrium state, the true displacements and plastic distortion minimize the total energy of the beam per unit depth

"  2  2 2 2 1  1 1 1  I¼ k ux;x þ uy;y þ l ux;x þ b sin 2u þ l uy;y  b sin 2u þ l ux;y þ uy;x  b cos 2u 2 2 2 2 0 0 3 1 5 dxdy þlk ln jb cos uþb sin uj 1  ;x bq ;y Z

a

Z

h

ð4Þ

s

among all admissible functions ux(x, y), uy(x, y), and b (x, y) satisfying the kinematic conditions (1) and (2). Note that dislocations may reach the free boundary y = h forming there the steps, so, actually their surface energy must be accounted for. Since this energy contribution is small compared with the bulk energy, we shall neglect it. For small up to moderate dislocation densities the logarithmic term in (4) may be approximated by the formula

ln

1 1

jb;x cos uþb;y sin uj bqs

ffi

jb;x cos u þ b;y sin uj 1 ðb;x cos u þ b;y sin uÞ2 þ : 2 bqs ðbqs Þ2

We shall use further only this approximation. 3. Variational-asymptotic analysis Functional (4) contains a small parameter h and can therefore be simplified in the limit h ? 0. The simplification is based on the variational asymptotic method (Berdichevsky, 2009; Le, 1999). For this purpose it is convenient to non-dimensionalize our variational problem. Introducing the following dimensionless variables and quantities

x ¼ bqs x;

 ¼ bqs y; y

 ¼ bqs a; a

 ¼ bq h; h s

we rewrite the energy functional in the form

r ¼ bqs r; u  x ¼ bqs ux ;

 y ¼ bqs uy ; u

E¼

Iðbqs Þ2

l

;

k

c¼ ; l

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E¼

Z

Z

a

0

"  2  2  1  1 1 c ux;x þ uy;y 2 þ ux;x þ b sin 2u þ uy;y  b sin 2u 2 2 2

h

0

 2 2 1 1  dxdy: þ ux;y þ uy;x  b cos 2u þ kjb;x cos u þ b;y sin uj þ k b;x cos u þ b;y sin u 2 2

ð5Þ

As we shall deal further only with the dimensionless quantities, the bars over them will be dropped for short. In order to fix the domain of the transverse coordinate in the passage to the limit h ? 0, we introduce the dimensionless coordinate f = y/h, f 2 (0, 1). Now h enters the action functional explicitly through the formulas

ui;y ¼

1 ui;f ; h

b;y ¼

1 b : h ;f

Keeping the asymptotically principal terms in functional (5) we have at the first step of the variational-asymptotic procedure

E0 ¼ h

Z

Z

a

0

1



1

2h

0

cðuy;f Þ2 þ 2

 1 k 1 2 2 2 dxdf: ðu Þ þ ðu Þ þ sin u j þ kðb sin u Þ jb y;f x;f ;f 2 2 2 h ;f h 2h 2h 1

Since functional E0 is positive definite, its minimum is zero and is achieved at

ux;f ¼ uy;f ¼ b;f ¼ 0: Taking the boundary conditions (1) and (2) into account, we obtain

ux ¼ r sinðx=rÞ  x;

uy ¼ r cosðx=rÞ  r;

b ¼ 0:

ð6Þ

Thus, the displacements do not depend on f and the plastic distortion is identically zero at the first step. At the second step we look for the minimizer in the form

ux ¼ r sinðx=rÞ  x þ u0x ðx; fÞ; u0x ðx; fÞ; u0y ðx; fÞ,

uy ¼ r cosðx=rÞ  r þ u0y ðx; fÞ;

b ¼ b0 ðx; fÞ:

ð7Þ

0

Now, functions and b (x, f) must vanish at f = 0. Substituting these formulas into functional (5) and keeping the asymptotically principal terms containing u0x ; u0y , and b0 , we get

"  2  2  2 1 x 1 x 1 1 1 E1 ¼ h c cos  1 þ u0y;f þ cos  1 þ b0 sin 2u þ u0y;f  b0 sin 2u 2 r h r 2 h 2 0 0 #  2

2 1 1 x k 1 ux;f  sin  b0 cos 2u þ jb0;f sin uj þ 2 k b0;f sin u þ dxdf: 2 h r h 2h Z

Z

a

1

We first fix b0 and minimize the functional with respect to u0x and u0y . Varying E1 with respect to u0x and u0y and taking into account that the variations of u0x and u0y are arbitrary for f in (0, 1) and at f = 1 we obtain

1 0 x u  sin  b0 cos 2u ¼ 0; h x;f r    x 1 1 1 c cos  1 þ u0y;f þ 2 u0y;f  b0 sin 2u ¼ 0: r h h 2

ð8Þ

These equations can be used to find u0x and u0y once b0 is known. Substituting (8) into the energy functional and returning to the original variable y, we reduces it to the functional depending on b0 only

E1 ¼

Z

a

Z

0

h

0

"  # 2

2 x 1 0 1 0 0 dxdy; j cos  1 þ b sin 2u þ kjb;y sin uj þ k b;y sin u r 2 2

ð9Þ

where j is given by

j¼

1 ; 1m

with m being the Poisson’s ratio. 4. Smooth minimizer Since the boundary y = h is traction free and may adsorb dislocation, there should be a dislocation-free zone near this boundary for the crystal to be in equilibrium. This leads to the following Ansatz for the minimizer of (9)

b0 ðx; yÞ ¼



b1 ðx; yÞ for y 2 ð0; lðxÞ; b0 ðxÞ

for y 2 ðlðxÞ; hÞ;

ð10Þ

where l(x) is an unknown function of x, 0 6 l(x) 6 h, and b1(x, l) = b0(x) at y = l(x). We have to find b1(x, y) and the functions b0(x) and l(x). Functional (9) becomes

K.C. Le, B.D. Nguyen / International Journal of Engineering Science 59 (2012) 211–218

"  # 2 x 1 1 j cos  1 þ b1 sin 2u þ kjb1;y sin uj þ kb1;y sin uÞ2 dy r 2 2 0 0 )  2 x 1 þj cos  1 þ b0 ðxÞ sin 2u ðh  lðxÞÞ dx: r 2

E1 ¼

Z

a

(Z

215

lðxÞ

ð11Þ

Varying this functional with respect to b1(x, y), l(x), and b0(x), we obtain the following equation



x r



1 2

j cos  1 þ b1 sin 2u sin 2u  kb1;yy sin2 u ¼ 0

ð12Þ

for y 2 (0, l(x)), subject to the boundary conditions

b1 ðx; 0Þ ¼ 0;

b1;y ðx; lðxÞÞ ¼ 0;

b1 ðx; lðxÞÞ ¼ b0 ðxÞ;

ð13Þ

and

  x 1 k signðb1;y Þ sin u þ j cos  1 þ b0 ðxÞ sin 2u sin 2uðh  lðxÞÞ ¼ 0: r 2

ð14Þ

Condition (13)2, obtained from the variation of l(x), guarantees the continuity of the dislocation density, while condition (14) is obtained from the variation of b0(x), where sign(b1,y) is the limiting value as y approaches l(x) from below. For simplicity we consider the case when sign(b1,y) = 1. The solution to (12) and (13) reads

b1 ðx; yÞ ¼ b1p ð1  cosh vy þ tanh vlðxÞ sinh vyÞ;

ð15Þ

where

b1p ¼ 

2 x cos  1 ; sin 2u r

v¼

rffiffiffiffiffiffiffi 2j cos u: k

The continuity condition for the plastic distortion implies that

 b0 ðxÞ ¼ b1p 1 

 

 1 2 x 1 ¼ : cos  1 1  cosh vlðxÞ sin 2u r cosh vlðxÞ

ð16Þ

Substituting this solution into the condition (14) we obtain the following transcendental equation to determine l(x)

   

1 x 1 k sin u þ j cos  1 þ b1p 1  sin 2uÞ sin 2uðh  lðxÞÞ ¼ 0: r 2 cosh vlðxÞ

ð17Þ

It is easy to show that the real root l(x) 2 (0,h) of Eq. (17) exists only for x > x⁄, where

 x ¼ r arccos 1 

 k : 2jh cos u

For x < x⁄ we must put l(x) = 0, i.e. in the interval (0, x⁄) there is no dislocation at all. Setting x⁄ = a, we get the critical threshold value of r for the dislocation nucleation in the bent beam

rcr ¼

a

: arccos 1  2jh kcos u

Thus, if the radius of the jig r > rcr, then l(x) = 0 and b = 0 everywhere yielding the purely elastic deformation without dislocations. For r < rcr the dislocations are nucleated and pile-up against the lower boundary y = 0 with x > x⁄ forming there the boundary layer. To simulate the minimizer numerically, we choose h = 1, a = 10, r = 5, m = 0.25, k = 104, and the angle u = p /5. The plots of l(x) and of b0(x) are shown in Figs. 3 and 4, respectively. 5. Energy reducing sequence As we know from the experiments, dislocations may climb in the transversal direction during annealing, and then glide along the slip direction and be rearranged as shown in Fig. 1. In the final polygonized relaxed state the dislocations form low angle tilt boundaries between polygons which are perpendicular to the slip direction, while inside the polygons there are no  yÞ dislocations. We want to show that this rearrangement of dislocations correspond to a sequence of piecewise constant bðx; reducing energy of the beam compared with (11). Here and below check is used to denote the polygonized relaxed state after  means the dislocations concentrated at the surface, therefore we ascribe to each jump point the annealing. The jump of b normalized Read–Shockley surface energy (Read & Shockley, 1950)

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l 0.08

0.06

0.04

0.02

2

4

6

8

10

x

8

10

x

Fig. 3. Function l(x).

β0 3.0 2.5 2.0 1.5 1.0 0.5

2

4

6

Fig. 4. Function b0(x).

   ¼ c jsbtj  ln eb ; c sbt  

ð18Þ

jsbtj

b  i Þ ¼ bðx  i þ 0Þ  bðx  i  0Þ denoting the jump of bðxÞ,  with sbtðx c ¼ 4pð1 mÞ, and b⁄ the saturated misorientation angle. For this purpose we divide the interval (x⁄, a) into N equal subintervals. We replace the smooth function b0(x) from for0 ðxÞ shown in Fig. 5 for N = 5 and define bðx;  yÞ as piecewise constant function mula (16) by a piecewise constant function b i in the ith polygon. Concerning the displacements u  x ðx; yÞ and u  y ðx; yÞ we define them according to which is equal to b

 0x ðx; yÞ;  x ðx; yÞ ¼ r sinðx=rÞ  x þ u u

 y ðx; yÞ ¼ r cosðx=rÞ  r þ u  0y ðx; yÞ; u

 yÞ. Since bðxÞ   0x ðx; yÞ and u  0y ðx; yÞ should be found by integrating Eqs. (8), with b0 being replaced by bðx; where u is piecewise  0x ðx; yÞ and u  0y ðx; yÞ are continuous and piecewise linear functions. However, their gradients constant, the displacements u describing the lattice rotation suffers jumps across the boundaries of the polygons. Thus, this sequence of displacements and plastic distortions exhibits polygonization of the bent beam. Due to our choice of the plastic distortion and displacements, it is easy to show that the asymptotically main contributions to the energy of crystal in the final relaxed polygonized state are given by

E¼

Z 0

a

Z 0

h



2

x 1 j cos  1 þ b sin 2u r

2

dxdy þ

N   P  iÞ c sbtðx i¼1

h : cos u

ð19Þ

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β0 3.0 2.5 2.0 1.5 1.0 0.5

2

4

6

8

10

x

0 ðxÞ. Fig. 5. Piecewise constant function b

 and As compared with the similar formula (9) we see that the gradient terms disappear due to the piecewise constant b, instead of them, the surface energy of the low angle tilt boundaries are added. As N becomes large, the contribution of the first term in (19) approaches the corresponding contribution of the smooth minimizer in the state before polygonization apart from the small contributions in the elastic zone near x = 0 and in the boundary layer near the lower face. If the surface energy of the tilt boundaries is less than the contribution of the gradient terms from the smooth minimizer, the constructed  yÞ and displacements u  x ðx; yÞ and u  y ðx; yÞ do reduce energy of the relaxed state. The number of polygons plastic distortion bðx; can be estimated from above by requiring that the increase of the surface energy is less than the reduction in gradient terms giving N   P  iÞ c sbtðx i¼1

h < cos u

Z

a

0

Z

lðxÞ



0

2  1 dxdy: kjb0;y sin uj þ k b0;y sin u 2

ð20Þ

For the rough estimation at large N we may substitute

 ¼ sbt

b0m ; N

b0m ¼ b0 ðaÞ ¼ 



 2 a 1 cos  1 1  sin 2u r cosh vlðaÞ

on the left hand side of (20). The calculation of the integral on the right hand side of (20) can be done by Gauss’ numerical integration according to

100

lnN

80

60

40

20

0

0

2

4

6

Fig. 6. Plot of lnN as function of r.

8

r 10

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Z

a

Z

0

lðxÞ

0

f ðx; yÞ dxdy ¼

  4 lðxÞ P 4 aP a a lðxÞ lðxÞ ni þ ; nj þ wi wj f 2 i¼1 2 j¼1 2 2 2 2

ð21Þ

where f ðx; yÞ ¼ kjb0;y sin uj þ 12 kðb0;y sin uÞ2 ; wi and wj are weighting factors, and ni and nj are coordinate of Gauss’ points. If we take a = 10 mm, h = 1.3 mm, b = 2.68  1010 m, u = p/5, r = 1.17 mm, qs = 1.454  1014 m2, k = 1.56  104, m = 0.25, which correspond to the sizes of the bent beam and the material parameters for zinc, then the estimated average polygon distance (taken as the length of the beam divided by the number of polygons) is equal to around 2.7  107 m which is in good agreement with the experimental result obtained in Gilman (1955). The plot of lnN as a function of r presented in Fig. 6 shows the decrease of the estimated number of polygons with the increasing radius of the generating circle as expected. 6. Conclusion In this paper we have shown that there exists a sequence of piecewise constant plastic distortions reducing the energy of the annealed and relaxed state of the bent beam and exhibiting the polygonization. We mention that the theory developed above does not provide any information about the kinetics of polygonization, which may be quite complicated due to the temperature-dependent dislocation climb and due to the interaction between dislocations and vacancies. Thus, for the kinetics of polygonization the knowledge about dissipation due to the dislocation climb becomes unavoidable. This will be the topics of our future works. References Berdichevsky, V. L. (2009). Variational principles of continuum mechanics. Berlin: Springer. Berdichevsky, V. L. (2006a). Continuum theory of dislocations revisited. Continuum Mechanics and Thermodynamics, 18, 195–222. Berdichevsky, V. L. (2006b). On thermodynamics of crystal plasticity. Scripta Materialia, 54, 711–716. Berdichevsky, V. L., & Le, K. C. (2007). Dislocation nucleation and work hardening in anti-planed constrained shear. Continuum Mechanics and Thermodynamics, 18, 455–467. Bilby, B. A., Gardner, L. R. T., & Smith, E. (1958). The relation between dislocation density and stress. Acta Metallurgica, 6, 29–33. Cahn, R. W. (1949). Recrystallization of single crystals after plastic bending. Journal of the Institute of Metals, 76, 121–143. Gilman, J. J. (1955). Structure and polygonization of bent zinc monocrystals. Acta Metallurgica, 3, 277–288. Kaluza, M., & Le, K. C. (2011). On torsion of a single crystal rod. International Journal of Plasticity, 27, 460–469. Kochmann, D. M., & Le, K. C. (2008). Dislocation pile-ups in bicrystals within continuum dislocation theory. International Journal of Plasticity, 24, 2125–2147. Kochmann, D. M., & Le, K. C. (2009a). Plastic deformation of bicrystals within continuum dislocation theory. Journal of Mathematics and Mechanics of Solids, 14, 540–563. Kochmann, D. M., & Le, K. C. (2009b). A continuum model for initiation and evolution of deformation twinning. Journal of the Mechanics and Physics of Solids, 57, 987–1002. Kuhlmann-Wilsdorf, D. (2001). Q: Dislocation structures – how far from equilibrium? A: Very close indeed. Materials Science and Engineering A, 315, 211–216. Le, K. C. (1999). Vibrations of shells and rods. Berlin: Springer. Le, K. C., & Sembiring, P. (2008). Analytical solution of the plane constrained shear problem for single crystals within continuum dislocation theory. Archive of Applied Mechanics, 78, 587–597. Le, K. C., & Sembiring, P. (2008). Plane constrained shear of single crystal strip with two active slip systems. Journal of the Mechanics and Physics of Solids, 56, 2541–2554. Le, K. C., & Sembiring, P. (2009). Plane constrained uniaxial extension of a single crystal strip. International Journal of Plasticity, 25, 1950–1969. Le, K. C., & Nguyen, Q. S. (2011). Polygonization as low energy dislocation structure. Continuum Mechanics and Thermodynamics, 22, 291–298. Nye, J. F. (1953). Some geometrical relations in dislocated crystals. Acta Metallurgica, 1, 153–162. Read, W. T., & Shockley, W. (1950). Dislocation models of crystal grain boundaries. Physics Review, 78, 275–289. Read, W. T. (1957). Dislocation theory of plastic bending. Acta Metallurgica, 5, 83–88.