Method and computer program product for characterising the bending response of a material

Abstract

Method for characterizing a material (10), characterized in that it comprises the steps of carrying out a bending test and calculating a cross-section moment, M of said material (10) using the following equation: $M=\frac{F.Math.L_m\left({}_1\right)}{2.Math.{\cos }^2\left({}_1\right)}$
where F is the applied bending force, L.sub.m (.sub.1) is the moment arm, and .sub.1 is the bending angle. The expression for the moment, M, fulfils the condition for energy equilibrium:
Fds=2Md.sub.2
when the true bending angle, .sub.2 is: ${}_1-\frac{t.Math.\sin \left({}_1\right)}{L_m}d{}_1.$

Claims

1. A method of characterising a material that comprises the steps of: a. providing a sample of the material simply supported between two parallel die supports, said supports having the same edge shape; b. bending the sample by providing an external force, F, via a bending knife, said force acting in a plane perpendicular to the plane formed by the centres of the die supports and which intersects the material at the centre line between the die supports, said bending knife extending at least the entire length of the sample; c. calculating a cross-section moment, M, of the material using the following equation [1]: $\begin{array}{cc}M=\frac{F.Math.L_m\left({}_1\right)}{2.Math.{\cos }^2\left({}_1\right)}& \left[1\right]\end{array}$ where F is the applied bending force, L.sub.m(.sub.1) is the moment arm, calculated according to the following equation:
L.sub.m(.sub.1)=L.sub.0(R.sub.k+R.sub.d).Math.sin(.sub.1) wherein .sub.1 is calculated using the following equation [0]:
.sub.1=Sin.sup.1([L.sub.0.Math.Q+{square root over (L.sub.0.sup.2+(SQ).sup.2Q.sup.2)}.Math.(SQ)]/[L.sub.0.sup.2+(SQ).sup.2].Math.180/[0] where L.sub.0 is half the die width, R.sub.d is the radius of the die edge, R.sub.k is the radius of the knife, .sub.1 is the bending angle, Q=R.sub.k+R.sub.d+t, t is the sample thickness, and S is a vertical distance through which the bending knife has been displaced.

2. The method of claim 1, wherein the method comprises solving the energy equilibrium expression: $\mathrm{Fds}=2\mathrm{Md}{}_2$ $\mathrm{where}$ ${}_2={}_1-\frac{t.Math.\sin \left({}_1\right)}{L_m\left({}_1\right)}d{}_1$ where t is the thickness of the plate.

3. The method according to claim 1, characterized in that it comprises the step of calculating the flow stress, .sub.1 using the following equation: ${}_1=\frac{2}{B.Math.t^2.Math.{\mathrm{.Math.}}_1}.Math.\frac{d}{d{\mathrm{.Math.}}_1}\left(M.Math.{\mathrm{.Math.}}_1^2\right)$ where the main strain, .sub.1, is calculated from: ${\mathrm{.Math.}}_1={}_2.Math.\frac{t}{L_m\left({}_1\right)}$ where B is the sample length, t is the sample thickness and .sub.2 is the true angle that said material is bent to during said bending test.

4. The method according to claim 1, characterized in that it comprises the step of calculating the flow stress, .sub.1 using the following equation: ${}_1=\frac{2}{B.Math.t^2.Math.{\mathrm{.Math.}}_1}.Math.\frac{d}{d{\mathrm{.Math.}}_1}\left(M.Math.{\mathrm{.Math.}}_1^2\right)$ where the main strain, .sub.1, is calculated from: ${\mathrm{.Math.}}_1=t.Math.\frac{U}{M.Math.L_N\left({}_1,{}_C\right)}.Math.\cos \left({}_C\right)$ where B is the sample length, t is the sample thickness .sub.2 is the true angle that said material is bent to during said bending test, L.sub.N(.sub.1, .sub.C) is the moment arm, calculated as:
L.sub.N(.sub.1,.sub.C)=L.sub.0R.sub.d.Math.sin .sub.1R.sub.k.Math.sin .sub.C U is the energy, and .sub.C is the contact angle between the knife and the material.

5. The method according to claim 1, characterized in that it comprises the step of estimating the Young's modulus, E, of said material by plotting a graph of .sub.2 and said calculated cross-section moment, M and determining the gradient of the elastic part of the moment curve, whereby the gradient is $\left(\frac{2.Math.E^{}.Math.I}{L_m}\right)$ where I is the moment of inertia and where E is the Young's modulus in plain strain and is given by: $E^{}=\frac{E}{\left(1-v^2\right)}$ where is Poisson's ratio.

6. The method according to claim 1, characterized in that it comprises the step of estimating the Young's modulus in plain strain, E, of said material using the following formula: $E^{}=\mathrm{Max}\left(\frac{M^2{L}_m}{2.Math.I.Math.U}\right)$ where I is the moment of inertia, and U is the energy.

7. The method according to claim 1, characterized in that it comprises the step of using said cross-section moment, M of said material to estimate spring-back of said material using the following equations: ${}_{\mathrm{tot}}={}_{\mathrm{Cel}}+{}_{\mathrm{Sel}}+{}_{12}$ ${}_{\mathrm{Cel}}=\frac{M_L.Math.W_C}{E^{}I}=\frac{M_L.Math.\left(R_k+\frac{t}{2}\right).Math.{}_C}{E^{}I}$ ${}_{\mathrm{Sel}}=\frac{M_L.Math.W_m}{2E^{}I}=\frac{M_L}{2E^{}I}.Math.\frac{L_N}{\cos \left({}_1\right)}$ ${}_{12}=\frac{t.Math.\sin {}_1}{L_m}d{}_1$ $M_L=M\frac{L_m}{L_N}$ the approximate length of the flange being tested is: $\frac{L_N}{\cos \left({}_1\right)}$ and the length of material in contact with the knife is: $\left(R_k+\frac{t}{2}\right).Math.{}_C$ where: A .sub.tot is the total amount of spring-back, .sub.Sel is the spring-back of the flange, .sub.Cel is the spring-back related to the material in contact with the knife, M.sub.L is the reduced moment due to the limitation of curvature of the knife, L.sub.N is the moment arm, L.sub.S is the length of the flange, L.sub.e is the length of the material shaped by the knife, R.sub.k is the knife radius, and c is the contact angle between the material and the knife.

8. The method according to claim 1, characterized in that it comprises the step of using said cross-section moment, M of said material to estimate a friction coefficient, , of said material using the equation: $=\frac{M-M_{\mathrm{mtrl}}}{M_{\mathrm{mtrl}}}.Math.\frac{1}{\tan {}_1}$ where M.sub.mtrl is the cross-section moment obtained using friction-free bending test equipment.

9. The method according to claim 1, characterized in that said calculated cross-section moment, M, or the calculated flow stress, .sub.1 or the estimated Young's modulus, E, or the calculated plain strain, .sub.1, is used to optimize a product comprising said material.

10. The method according to claim 1, wherein the sample of material is a plate.

11. The method according to claim 1, wherein the die supports are rollers.

12. The method according to claim 1, wherein the material is a metallic material.

13. The method according to claim 12, characterized in that said metallic material is hot-rolled metallic material.

14. The method according to claim 12, characterized in that said metallic material is a cold-rolled metallic material.

15. The method for characterizing a material according to claim 1, characterized in that it comprises the steps of carrying out a bending test according to the VDA 238-100 standard.

16. The method according to claim 1, characterized in that the method comprises calculating the ratio M/M.sub.e, defined as either: $\frac{M}{M_e}=\frac{3}{\left(\left(\frac{\mathrm{dM}}{d{}_2}/\frac{M}{{}_2}\right)+2\right)}$ $\mathrm{or}$ $\frac{M}{M_e}=\frac{3}{{\left(\left(\frac{\mathrm{dM}}{d{\mathrm{.Math.}}_1}/\frac{M}{{\mathrm{.Math.}}_1}\right)+2\right)}^{\underset{-}{.}}}$

17. The method of claim 16, wherein the ratio M/M.sub.e is calculated for at least two different materials, and the properties of a composite comprising those materials is calculated from the values of the individual materials.

18. A method for determining the conditions under which a material remains stable during bending, said method comprising the method of claim 16 and further characterised by determining the conditions under which the ratio M/M.sub.e remains below 1.5.

19. The method of claim 1, wherein the knife radius, R.sub.k, is less than or equal to the thickness of the material, t.

20. The method of claim 1, wherein the knife radius, R.sub.k, is 0.7 timed the thickness of the material, t, or less.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present disclosure will hereinafter be further explained by means of non-limiting examples with reference to the appended figures where;

(2) FIG. 1 shows a diagram used to determine the bendability of a metallic material using the VDA 238-100 standard test according to the prior art,

(3) FIG. 2 shows the steps of a method according to an embodiment of the present disclosure,

(4) FIG. 3a schematically shows the forces and moment acting on a material during bending in a method according to an embodiment of the disclosure,

(5) FIG. 3b schematically shows the positioning of the sample, rollers and bending knife, and the variables used to describe the various dimensions referred to herein,

(6) FIG. 4 shows a conventional force-curve from VDA 238-100 standard test,

(7) FIG. 5 shows a conventional moment curve calculated using the conventional formula for the moment,

(8) FIG. 6a shows the contact angle .sub.C at the knife that arises when the flange becomes curved during the bending test,

(9) FIG. 6b shows how the contact angle .sub.C develops as the knife moves down and the bend is induced,

(10) FIG. 7 shows the moment curve, M.sub.tot, calculated using the calculated cross-section moment according to the present disclosure,

(11) FIG. 8 shows tensile test data obtained from tests on a 700 MPa steel,

(12) FIG. 9 shows a comparison between the calculated bending force based on 700 MPa steel tensile test data and three bending tests performed on the same material with different stroke lengths,

(13) FIG. 10 schematically illustrates a coil of metallic material,

(14) FIG. 11 shows the theoretical elastic cross-section moment and the maximum cross-section moment at bending,

(15) FIG. 12 shows the force versus knife position measured at bending,

(16) FIG. 13 shows the calculated cross-section moment according to the present disclosure,

(17) FIG. 14a shows the calculated non-dimensional moment M/M.sub.e, plotted against strain calculated using formula [3],

(18) FIG. 14b shows the calculated non-dimensional moment M/M.sub.e, plotted against strain calculated using formula [4b],

(19) FIG. 15a shows the flow stress obtained using the calculated cross-section moment according to the present disclosure, using formula [3] to calculate strain,

(20) FIG. 15b shows the flow stress obtained using the calculated cross-section moment according to the present disclosure, using formula [4b] to calculate strain,

(21) FIG. 16a shows an estimation of Young's modulus obtained using a method according to the present disclosure,

(22) FIG. 16b shows an estimation of Young's modulus obtained using an alternative methodology,

(23) FIG. 17 shows a comparison of the ratio of M/M.sub.e values obtained from tensile tests and bending tests,

(24) FIG. 18 shows the steps of the method for estimating spring back for a free choice of geometrical setup,

(25) FIG. 19 shows the principal for estimation of the shape of curvature of a material,

(26) FIG. 20 shows the distribution of energy within a bend,

(27) FIG. 21 shows the changes in contact and shape angles .sub.C and .sub.S respectively during bending from a particular material being tested,

(28) FIG. 22 shows the force versus the displacement calculated (based on a single reference test) using a method according to the present disclosure, and comparisons with tests performed,

(29) FIG. 23 shows the force vectors representing the bending load and friction force during a bending test,

(30) FIG. 24 shows the moment versus the bending angle for a range of 6 mm thick high strength steels subjected to bending with different levels of friction involved,

(31) FIG. 25 shows the force versus displacement for a range of 6 mm thick high strength steels subjected to bending with different levels of friction involved,

(32) FIG. 26 shows a comparison between measured force achieved from real production in steel bending and force calculated using the methods of the disclosure,

(33) FIG. 27 shows a comparison between the experimentally measured force and calculated angle at F.sub.max, depending on the geometry of the bending setup,

(34) FIG. 28 shows the moment characteristics of the base materials used to form the composite in Example 5,

(35) FIG. 29 shows the moment curves for the various layers in the composite structure of Example 5,

(36) FIG. 30 shows the predicted unit-free moment curve of the composite structure of example 5, in comparison to the actual curve of an equivalent thickness of DX960 base material,

(37) FIG. 31 shows a representation of the difference between angles .sub.1 and .sub.2 with the same incremental increase in S,

(38) FIG. 32 shows a representation of the coordinate X which traverses the bending arm L.sub.N in the curvature vs position plot,

(39) FIG. 33 shows a plot of 1/R vs. X for the steel bent to 30 from Example 6,

(40) FIG. 34 shows the plot of theoretical and measured bending force vs bending angle from Example 6,

(41) FIG. 35 shows the plot of bending moment vs bending angle from Example 6,

(42) FIG. 36 shows the theoretical and measured force vs knife position for the steel bent in Example 7,

(43) FIG. 37 shows the plot of bending moment vs bending angle from Example 7,

(44) FIG. 38 shows the plot of bending moment vs bending angle from Example 8,

(45) FIG. 39 shows the theoretical and measured force vs knife position for the steel bent in Example 8,

(46) FIG. 40 shows the strain vs X-position for the 6 mm materials in Example 8,

(47) FIG. 41 shows the plot of FIG. 40 enlarged to show the kinking feature,

(48) FIG. 42 shows M/M.sub.e vs strain for the 6 mm materials in Example 8,

(49) FIG. 43 shows the flow stress vs strain for the 6 mm materials in Example 8,

(50) FIG. 44 shows a representation of the neutral layer,

(51) FIG. 45 photograph of the microbending apparatus used in Example 9,

(52) FIG. 46 shows a plot of force vs knife position for the material tested in Example 9,

(53) FIG. 47 shows a plot of stress vs strain for the material tested in Example 9, and

(54) FIG. 48 shows a plot of the moment vs bending angle .sub.1 for the material tested in Example 9.

DETAILED DESCRIPTION OF EMBODIMENTS

(55) The following abbreviations are used herein: M=Cross-section (bending) moment M.sub.Max=Maximum bending moment M.sub.L=Reduced moment due to the limitation of curvature due to the knife radius M.sub.mtrl=Cross-section moment characteristics obtained using friction free bending test equipment M.sub.1=Component of cross-section moment M M.sub.2=Component of cross-section moment M M.sub.tot=M.sub.1+M.sub.2 M.sub.e=Elastic cross-section moment F=Applied force F.sub.max=Maximum applied force F.sub.theor=Theoretical force S=Vertical distance through which the bending knife has been displaced S.sub.x=Horizontal movement of the contact point at the knife S.sub.y=Total vertical movement of the contact point, taking account that it moves upwards along the knife surface B=Sample length (length of the bend, or the length of the sample in the dimension parallel to the rollers) t=Sample thickness .sub.1=Bend angle .sub.2=True angle to which the sample (e.g. plate) is bent .sub.12=Difference between .sub.1 and .sub.2 .sub.C=Contact angle between the material and the knife .sub.S=Shape angle (shape of curvature of the flanges) .sub.tot=Total amount of spring-back .sub.Sel=Spring-back of the flange .sub.Cel=Spring-back related to the material in contact with the knife .sub.F max=Bend angle at F.sub.max *=Selected bending angle, fixed when calculating other parameters L.sub.0=Half die width (i.e. half the distance between the centre points of the rollers L.sub.m(.sub.1)=Moment arm at angle .sub.1 (horizontal distance between the tangential contact points) L.sub.N(.sub.1,.sub.C)=Real moment arm (horizontal distance between the real contact points at the knife and die radius, i.e. the distance between where the knife and die contact the sample (e.g. plate) being bent) L.sub.e=Moment arm of the horizontal distance of the neutral layer (i.e. the horizontal distance of the neutral layer between the points of intersection of angle .sub.1 with the neutral layer) W.sub.m=Length of the neutral layer in the plate between the die and point of contact with the knife, corresponding to the length starting from the point of intersection of .sub.1 at the die with the neutral layer to the point of intersection of .sub.C at the knife with the neutral layer (i.e. the estimated length of the neutral layer of the flange between the contact with the knife and diesee FIG. 32) W.sub.C=Length of the neutral layer of the material shaped by the knife (see FIG. 32) dW.sub.m=The increment in the length of the curved material along the neutral layer a=Length of the mid-layer based on moment arm L.sub.e A=Length of the mid-layer based on moment arm L.sub.m X=Coordinate along bending arm L.sub.N b=Peripheral distance on incremental move, as shown in FIG. 31 R=Radius of curvature of the material which is bent R.sub.d=Radius of the die edge (i.e. the radius of the curved edge portion of the die which the sample pivots round during bending). When the dies are rollers, this corresponds to the roller radius R.sub.k=Knife radius R.sub.m=Ultimate strength .sub.1=Flow stress (plain strain) =Effective stress .sub.1=Main strain (plain strain) =Effective strain E=Young's modulus E=Young's modulus in plain strain U=Elastic and plastic energy absorbed by the material during bending U.sub.el=Elastic energy at bending I=Moment of inertia =Friction coefficient of the material .sub.d=Friction between the material and the die edge (roller) radius =Poisson's ratio

(56) By simply supported is meant that each end of the sample (e.g. plate) can freely rotate, and each end support has no bending moment. This is typically achieved by supporting the sample (e.g. plate) with parallel rollers, such that the moment created by the knife when the external force is applied is balanced by the moment created along the centre line where the bending takes place, and no additional bending or force dissipation takes place at the point of contact between the plate and the rollers.

(57) Typically, the sample (e.g. plate) is substantially horizontal when provided on the rollers (or more generally die edges). By substantially horizontal is meant that the sample (e.g. plate) does not move due to gravity when balanced on the rollers prior to bending. In practice, the sample (e.g. plate) will typically be horizontal, though the skilled person would understand that very small variations from horizontal can also be used, providing the force applied by the bending knife is in a plane perpendicular to the plane formed by the centres of the rollers and which intersects the sample (e.g. plate) along the entire length of the centre line between the rollers. In other words, if the sample (e.g. plate) is e.g. 2 degrees from horizontal when the test begins, the bending knife moves (and consequently applies the force) in a direction the same amount (2 degrees) from vertical during the test, such that the bending force is applied perpendicular to the sample's starting position.

(58) The bending force is applied across the entire length of the sample (e.g. plate). This ensures that the sample (e.g. plate) is bent evenly during the test and the force resisting the knife corresponds to the bending moment of the material, rather than internal forces arising due to deformation of the sample (e.g. plate) from incomplete bending. To ensure the bending force is applied across the entire length of the sample (e.g. plate), the length of the bending knife typically is greater than the sample (e.g. plate) length. Typically, the bending knife extends beyond the edge of the sample (e.g. plate) during bending. Due to the end-effects, i.e. not plain-strain condition, the knife will however not be in contact with the material close to the edges. Therefore, the length of the sample should preferably be at least 10 times the thickness, to ensure the main response of plain strain condition.

(59) The sample (e.g. plate) is typically positioned such that cutting burs or fracture surface portions, possibly existing at the edges, are located on the knife side (i.e. on the sample side which will be under compression during bending).

(60) FIG. 1 shows a diagram used to determine the bendability of a metallic material using the VDA 238-100 standard test according to the prior art in which the bendability of the metallic material is determined by measuring the knife position, S at the maximum applied bending force, F.sub.max.

(61) FIG. 2 shows the steps of an exemplary method according to an embodiment of the present disclosure. The method comprises the steps of carrying out a plate bending test according to the VDA 238-100 standard (or a similar friction-free bending), and calculating a cross-section moment, M, of said material using the following equation:

(62) $\begin{array}{cc}M=\frac{F.Math.L_m\left({}_1\right)}{2.Math.{\cos }^2\left({}_1\right)}& \left[1\right]\end{array}$
where F is the applied bending force, L.sub.m(.sub.1) is the moment arm, and .sub.1 is the bending angle. The calculated cross-section moment, M, may be used to predict a real response of the material during bending.

(63) This improved method for characterizing a material was found by studying the energy balance expression:
Fds=2Md.sub.2[5a/5b]
Where F is the applied force, S is the knife position, M is the moment of the material test sample and .sub.2 is the true bending angle.

(64) This expression indicates that there has to be a balance between the energy input during air bending and the energy absorbed by the test sample. Friction between the material and the die edge (roller) radius, .sub.d, is assumed to be negligible.

(65) FIG. 3a shows the forces and moment acting on the material test sample during bending. L.sub.m(.sub.1) is the moment arm that will start with the initial value of L.sub.0 (which is half of the die width) and will decrease during the stroke. The bending angle, .sub.1, is half of the total bending angle (as per VDA test).

(66) FIG. 4 shows a typical force diagram from a VDA 238-100 standard test showing the applied force, F and the vertical displacement of the knife, S.

(67) The vertical displacement of the knife, S can be expressed geometrically as a function of the bending angle .sub.1 as:

(68) $\begin{array}{cc}S\left({}_1\right)=L_0.Math.\tan \left({}_1\right)+\left(R_d+R_k+t\right).Math.\left(1-\frac{1}{\cos \left({}_1\right)}\right)& \left[23\right]\end{array}$

(69) By applying the conventional expression from the literature to calculate the cross-section moment,

(70) $\begin{array}{cc}i.e.=\frac{F.Math.L_m\left({}_1\right)}{2},& \left[24\right]\end{array}$
and also converting the distance S to the corresponding bending angle .sub.1, then the plot of cross-section moment, M, versus the bending angle, .sub.1, will have the form shown in FIG. 5.

(71) It was observed that there is a mismatch between the energy input, F ds [5a], during bending and the internal momentum and its energy, 2Md.sub.1 [5b], i.e. if applying the common expression for the momentum,

(72) $\begin{array}{cc}M=\frac{F.Math.L_m\left({}_1\right)}{2}& \left[24\right]\end{array}$
is used (as shown in FIG. 5). The cross-section moment in bending, referring to the literature, should rather be constant after complete plastification.

(73) It was thereby found that:

(74) $\begin{array}{cc}{}_{}^{}\mathrm{Fds}{}_{}^{}\frac{F.Math.L_m\left({}_1\right)}{2}2d{}_1& \left[26\right]\end{array}$

(75) Reasonably, it must be a relationship between the travel distance of the knife, S, and the bending angle, .sub.1, that gives the correct expression and thereby achieves an energy-balance. By investigating the non-linearity between S and .sub.1, the true relationship between the applied force, F, and the cross-section, M, was derived, as follows.

(76) Taking the first derivative of the geometrical function, equation [23] gives:

(77) 0$\begin{array}{cc}\frac{\mathrm{dS}}{d1}=\frac{L_0-\left(R_k+R_d+t\right).Math.\mathrm{Sin}\left(1\right)}{{\mathrm{Cos}\left(1\right)}^2}=\frac{L_e}{{\mathrm{Cos}\left(1\right)}^2}& \left[27\right]\end{array}$

(78) The following function:
L.sub.e=L.sub.0(R.sub.k+R.sub.d+t).Math.sin(.sub.1)[28]

(79) L.sub.e is almost equal to the moment-arm, L.sub.m, in bending, except for the material thickness, t.

(80) Geometrically, the real moment-arm is (see FIG. 3a):
L.sub.m(.sub.1)=L.sub.0(R.sub.k+R.sub.d).Math.sin(.sub.1)[29]

(81) The energy balance expression, equation[5a/5b], can then be expressed as follows using the derivative, equation [27]:

(82) $\begin{array}{cc}{}_{}^{}F.Math.\mathrm{ds}={}_{}^{}M.Math.2\frac{d{}_2}{d{}_1}.Math.d{}_1={}_{}^{}M.Math.2\frac{d{}_2}{d{}_1}.Math.\frac{d{}_1}{\mathrm{dS}}\mathrm{dS}.Math.F=M.Math.2\frac{d{}_2}{d{}_1}.Math.\frac{d{}_1}{\mathrm{dS}}=M.Math.2\frac{d{}_2}{d{}_1}.Math.\frac{{\mathrm{Cos}\left(1\right)}^2}{L_e}& \left[30\right]\end{array}$

(83) Here a new angle, .sub.2, has been introduced, i.e. the real angle that will be at the bend due to the energy balance, and which is different from the geometrical bending angle, .sub.1, applied, see FIG. 3a. For small bending angles, .sub.1, it is true that the cross-section moment, M, is equal to

(84) $\begin{array}{cc}\frac{F.Math.L_m\left({}_1\right)}{2};& \left[24\right]\end{array}$
called M.sub.1 herein.

(85) Assuming that for large bending angles, the total moment, M.sub.tot, is a sum of M.sub.1 and M.sub.2 where M.sub.2 is an unknown function but is assumed to be a multiple of function M.sub.1, M.sub.tot can be expressed as follows:

(86) $\begin{array}{cc}{M}_{\mathrm{tot}}=M_1+M_{1}f\left({}_1\right)=M_1\left(1+f\left({}_1\right)\right)=\frac{F.Math.\mathrm{Lm}\left(1\right)}{2}\left(1+f\left({}_1\right)\right)& \left[31\right]\end{array}$

(87) In order to balance the energy balance expression, the ratio:

(88) $\begin{array}{cc}\frac{d{}_2}{d{}_1}& \left[32\right]\end{array}$
is assumed to be equal to:

(89) $\begin{array}{cc}\frac{L_e}{L_m}& \left[33\right]\end{array}$
which gives:

(90) $\begin{array}{cc}F=\frac{F.Math.\mathrm{Lm}\left(1\right)}{2}\left(1+f\left({}_1\right)\right).Math.2\frac{L_e\left(1\right)}{L_m\left(1\right)}.Math.\frac{{\mathrm{Cos}\left(1\right)}^2}{L_e\left(1\right)}& \left[34\right]\end{array}$

(91) Hence;
F=F.Math.Cos(.sub.1).sup.2(1+f(.sub.1))[35]

(92) It follows that:

(93) $\begin{array}{cc}f\left({}_1\right)={\mathrm{Tan}\left({}_1\right)}^2,\left(i.e.\frac{1}{{\mathrm{Cos}}^2\left(\right)}=1+{\mathrm{Tan}}^2\left(\right)\right)& \left[36\right]\end{array}$

(94) Finally, the expression for the cross-section moment, M, will then become:

(95) $\begin{array}{cc}M=\frac{F.Math.L_m\left({}_1\right)}{2.Math.{\cos }^2\left({}_2\right)}& \left[1\right]\end{array}$

(96) This correct formulation of the cross-section moment, M, which more accurately predicts the bending behaviour of materials, is even valid for large bending angles (i.e. typically angles greater than typically 6).

(97) FIG. 7 shows the total moment, M.sub.tot as a sum of M.sub.1 and M.sub.2. The common expression, M.sub.1 is valid only for small bending angles (i.e. angles up to about 6). The angle .sub.2 is the true angle that the material is bent to, not the same as the bending angle, .sub.1 applied.

(98) It can be theoretically confirmed that this solution is valid even if the flange becomes curved, i.e. the contact point will occur at the angle of .sub.C instead of .sub.1 (see FIGS. 6a, 6b and 32), as is usually the case during bending.

(99) In more detail, FIG. 6a shows how the contact angle between the material and the bending knife, .sub.C, differs from the bending angle moved by the surface normal of the sample (e.g. plate) at the rollers, .sub.1, due to the curvature of the sample (e.g. plate) at the knife, and due to the radius of the knife itself. FIG. 6b shows how this contact point diverges outwards from the centre point of the bend as the knife moves down (i.e. the vertical displacement dS increases). FIG. 32 shows a clearer representation of how .sub.C relates to the curvature of the sample (e.g. plate).

(100) Thus:

(101) $\begin{array}{cc}M=\frac{d}{d{}_2}\left[\frac{U}{2}\right]=\frac{d}{d{}_1}\left[\frac{F_y}{2}.Math.\frac{{\mathrm{dS}}_y}{d{}_1}d{}_1+\frac{F_x}{2}.Math.\frac{{\mathrm{dS}}_x}{d{}_1}d{}_1\right].Math.\frac{d{}_1}{d{}_2}=& \left[37a\right]\\ =\frac{d}{d{}_1}\left[\frac{F_y}{2}.Math.\frac{{\mathrm{dS}}_y}{d{}_1}d{}_1+\frac{F_y}{2}\tan {}_C.Math.\frac{{\mathrm{dS}}_x}{d{}_1}d{}_1\right].Math.\frac{L_m}{L_e}=& \left[37b\right]\\ =\frac{F_y}{2}.Math.\left[\frac{{\mathrm{dS}}_y}{d{}_1}+\tan {}_C.Math.\frac{{\mathrm{dS}}_x}{d{}_1}\right].Math.\frac{L_m}{L_e}& \left[37c\right]\end{array}$

(102) Where the movements of the contact-point is described by;

(103) 0$\begin{array}{cc}\frac{{\mathrm{dS}}_y}{d{}_1}=\frac{d}{d{}_1}\left[S-R_k.Math.\left(1-\cos {}_C\right)\right]=\frac{L_e}{{\cos }^2{}_1}-R_k\sin {}_C.Math.\frac{d{}_C}{d{}_1}& \left[38\right]\\ \frac{{\mathrm{dS}}_x}{d{}_1}=\frac{d}{d{}_1}\left[R_k.Math.\sin {}_C\right]=R_k\cos {}_C.Math.\frac{d{}_C}{d{}_1}& \left[39\right]\end{array}$

(104) S, is the vertical knife-movement, S.sub.x is the horizontal movement of the contact point at the knife, and S.sub.y is the total vertical movement of the contact-point, taking into account that it moves upwards along the knife surface (see FIGS. 6a, 6b and 32).

(105) Hence;

(106) $\begin{array}{cc}M=\frac{F_y}{2}.Math.\frac{L_e}{{\cos }^2{}_1}.Math.\frac{L_m}{L_e}=\frac{F_y}{2}.Math.\frac{L_m}{{\cos }^2{}_1}& \left[40\right]\end{array}$

(107) The above calculations refer to the bending angles .sub.1 and .sub.2, and it is worthwhile clarifying the distinction between these two parameters.

(108) The bending angle, .sub.1, is the conventional angle used in bending that can be geometrically calculated assuming a strictly straight flange. Using such an assumption, the contact angle at both the knife and die is the same and equal to .sub.1.

(109) The true angle, .sub.2, takes account right length of the moment arm, i.e. L.sub.m, (still considering a straight flange) based on the energy-equilibrium, Fds=2Md. The combination of the true angle, .sub.2, and the correct formulation of the moment arm, L.sub.m, allows a more accurate calculation especially for thicker materials and for large bending angles.

(110) The moment-arm, L.sub.m, is the horizontal distance between the tangential contact-points (knife and die-radius), while the moment arm L.sub.N is the horizontal distance between the actual contact points. The moment arm associated with the neutral layer is referred to as L.sub.e, and is defined in relation to the actual bending angle .sub.2.

(111) These parameters are shown in FIGS. 31 and 32. In FIG. 31, both measures, a, and, A, represent the length of the mid-layer, based on the moment-arms, L.sub.e and L.sub.m, according to following expressions:

(112) $\begin{array}{cc}a=\frac{L_e}{\cos {}_1}<A=\frac{L_m}{\cos {}_1}& \left[41\right]\end{array}$

(113) The difference between, A, and, a, are the end-points of the lines. Thus, a is defined relative to the distance between the points at which the tangential lines intersect a surface of the sample. In contrast, A is defined relative to the vertical line that runs from the tangential point on the die and knife, thus representing the mid-layer and its angular-change, d.sub.2.

(114) However, the vertical displacement, dS, relates directly to the peripheral distance, b via the bending angle .sub.1. The peripheral distance, b, can be expressed both via the distances, a, and, A, but as a function of the different increments in angles d.sub.1 and d.sub.2 respectively, as A>a. In both cases, the vertical displacement, dS, has to be the same.

(115) $\begin{array}{cc}\mathrm{dS}=\frac{b}{\cos {}_1}=\frac{a.Math.d{}_1}{\cos {}_1}=\frac{L_e}{\cos {}_1^2}.Math.d{}_1=\frac{A.Math.d{}_2}{\cos {}_1}=\frac{L_m}{\cos {}_1^2}.Math.d{}_2& \left[42\right]\end{array}$

(116) As the length, A, is larger than, a, the bending angle d.sub.2 will be less than d.sub.1

(117) Hence:

(118) $\begin{array}{cc}d{}_2=\frac{L_e}{L_m}.Math.d{}_1& \left[43\right]\end{array}$

(119) The plain strain (or main strain), .sub.1, can also derived from the amount of energy input in bending, as set out below.

(120) The following expression is well known regarding the relation between the shape radius and the total length of the middle-layer of the curved material.

(121) $\begin{array}{cc}2d=\frac{1}{R}.Math.{\mathrm{dW}}_m& \left[44\right]\end{array}$

(122) Where; d, is the increment in the angle-acting in the interval between current status of bending angle, .sub.2, and the contact angle, .sub.C dW.sub.m, is the increment in length of the material, along the neutral-layer R, is the radius of curvature of the material.

(123) The horizontal component, dL.sub.N, of the length of curvature increment, dS.sub.m, can be expressed as;
dL.sub.N=dW.sub.m.Math.cos .sub.C[45]

(124) The contact-angle;

(125) $\begin{array}{cc}{}_C=\frac{U}{M}-{}_2& \left[46\right]\end{array}$
(in line with the expression for, .sub.C, as earlier demonstratedsee particularly eq. [17])

(126) Hence;

(127) $\begin{array}{cc}{}_{{}_C}^{}2d=2\left({}_2-{}_c\right)=\frac{1}{R}.Math.{\mathrm{dW}}_m=\frac{1}{R}.Math.\frac{{\mathrm{dL}}_N}{\cos {}_C}& \left[47\right]\end{array}$

(128) It also follows that;

(129) $\begin{array}{cc}\frac{1}{R}=2.Math.\frac{d{}_S}{\mathrm{dM}}.Math.\frac{\mathrm{dM}}{{\mathrm{dL}}_N}.Math.\cos {}_C=2.Math.\frac{d{}_S}{{\mathrm{dL}}_N}.Math.\cos {}_C& \left[48\right]\end{array}$

(130) Equation [48] can be derived as follows:

(131) $\begin{array}{cc}{}_{{}_C}^{{}_2}2d=2\left({}_2-{}_c\right)=\frac{1}{R}.Math.{\mathrm{dW}}_m=2.Math.\frac{d{}_S}{{\mathrm{dL}}_N}.Math.\cos {}_C.Math.\frac{{\mathrm{dL}}_N}{\cos {}_C}==2.Math.d{}_S=2{}_S& \left[49\right]\\ \mathrm{hence}{}_2-{}_c={}_S,\left(i.e.{}_2={}_c+{}_S\right)& \left[50\right]\\ \mathrm{Where}:\frac{d{}_S}{\mathrm{dM}}=\frac{1}{M}.Math.\left(2{}_2-{}_S\right)=\frac{U}{M^2}& \left[51\right]\end{array}$
is the derivative expression, of the shape-angle with respect to the moment, M.

(132) These formulae can be derived starting from the formula for the moment, M, equal to;

(133) 0$\begin{array}{cc}M=\frac{F.Math.L_m\left({}_1\right)}{2.Math.{\cos }^2\left({}_1\right)}& \left[1\right]\end{array}$

(134) As follows:

(135) $\begin{array}{cc}\frac{d{}_S}{\mathrm{dM}}=\frac{d}{\mathrm{dM}}\left[\frac{1}{M}.Math.2{}_2\mathrm{dM}\right]=\frac{1}{M}.Math.\left(2{}_2-{}_S\right)& \left[52\right]\\ U=\mathrm{FdS}=2.Math.\mathrm{Md}{}_2=M.Math.\left(2{}_2-{}_S\right)\mathrm{Hence},& \left[53\right]\\ \frac{d{}_S}{\mathrm{dM}}=\frac{U}{M^2}& \left[54\right]\end{array}$

(136) $\frac{\mathrm{dM}}{{\mathrm{dL}}_N},$
is the linear distribution of the moment, M, between the contact points of the knife and the die radius, simply equal to;

(137) $\frac{M}{L_N}.$

(138) It makes the expression to;

(139) $\begin{array}{cc}\frac{1}{R}=2.Math.\frac{U}{M.Math.L_N}.Math.\cos {}_C& \left[55\right]\end{array}$

(140) The expression [55] can be verified as follows:

(141) Elastic energy at bending;

(142) $\begin{array}{cc}{U}_{\mathrm{el}}=\frac{M^2.Math.L}{2.Math.E^{}.Math.I}& \left[56\right]\end{array}$

(143) Where I, is the moment of inertia, E, The young-modulus for plain-strain

(144) At small bending angles and at elastic state, .sub.C=0 and .sub.2<<1, L.sub.NL.sub.m

(145) It makes;

(146) $\begin{array}{cc}\frac{1}{R}=2.Math.\frac{U}{M.Math.L_N}=2.Math.\frac{\left(\frac{M^2.Math.L_m}{2.Math.E^{}.Math.I}\right)}{\left(M.Math.L_N\right)}=\frac{M}{E^{}.Math.I}& \left[57\right]\end{array}$

(147) This is in agreement with the formula known from the literature for elastic bending,

(148) $\begin{array}{cc}M=E^{}.Math.I.Math.\frac{1}{R}& \left[58\right]\end{array}$

(149) At elastic deformation,

(150) $\begin{array}{cc}{}_S={}_2\mathrm{and}{}_2<<1,L_N{L}_m& \left[59\right]\\ U_{\mathrm{el}}=M\left(2{}_2-{}_S\right){}_{2}M& \left[60\right]\\ \frac{1}{R}=2.Math.\frac{U}{M.Math.L_N}=2.Math.\frac{{}_{2}M}{M.Math.L_N}\frac{2{}_2}{L_m}& \left[61\right]\end{array}$

(151) By the expression for the shape of curvature; 1/R, the strains can be calculated, assuming the neutral-layer, i.e. where strain is zero, located at the middle of the cross-section of the material (at t/2).

(152) $\begin{array}{cc}{\mathrm{.Math.}}_1=\frac{t}{2.Math.R}=t.Math.\frac{U}{M.Math.L_N}.Math.\cos {}_C& \left[62\right]\end{array}$

(153) As has been demonstrated before that;

(154) 0$\begin{array}{cc}\frac{\mathrm{dM}}{d{L}_N}=\frac{M}{L_N}& \left[63\right]\end{array}$

(155) This allows the curvature vs position at a coordinate X along the bending arm L.sub.N to be displayed at a given bending angle .sub.2*:

(156) $\begin{array}{cc}X={}_0^{M\left({}_2^{*}\right)}\frac{L_N\left({}_2^{*}\right)}{M\left({}_2^{*}\right)}\mathrm{dM}& \left[64\right]\end{array}$

(157) By the expression for 1/R, the Young-modulus can also be estimated, as shown below.

(158) $\begin{array}{cc}\frac{1}{R_{\mathrm{el}}}=\frac{M}{E^{}.Math.I}& \left[65\right]\\ \frac{1}{R}=2.Math.\frac{U}{M.Math.L_N}.Math.\cos {}_C& \left[66\right]\end{array}$

(159) At elastic state,
.sub.C=0 and .sub.2<<1,L.sub.NL.sub.m[67]

(160) Which gives;

(161) $\begin{array}{cc}2.Math.\frac{U}{M.Math.L_m}.Math.\frac{M}{E^{}.Math.I}& \left[68\right]\end{array}$

(162) Then for plain strain:

(163) $\begin{array}{cc}{E}^{}=\mathrm{Max}\left(\frac{M^2{L}_m}{2.Math.I.Math.U}\right)& \left[69\right]\end{array}$

(164) Or for effective strain:

(165) $\begin{array}{cc}E=\mathrm{Max}\left(\frac{M^2{L}_m}{2.Math.I.Math.U}\right).Math.\left(1-v^2\right)& \left[70\right]\end{array}$

(166) The present disclosure also comprises a carrier containing a computer program code means that, when executed a computer or on at least one processor, causes the computer or at least one processor to carry out the method according to an embodiment of the present disclosure (i.e. whereby the computer program code means may be used to calculate the cross-section moment, M, and/or any of the other properties of the material described herein), wherein the carrier is one of an electronic signal, optical signal, radio signal or computer readable storage medium.

(167) Typical computer readable storage media include electronic memory such as RAM, ROM, flash memory, magnetic tape, CD-ROM, DVD, Blueray disc etc.

(168) The present disclosure further comprises a computer program product comprising software instructions that, when executed in a processor, performs the calculating step of a method according to an embodiment of the present disclosure.

(169) The present disclosure further comprises an apparatus comprising a first module configured to perform the calculating step of a method according to an embodiment of the present disclosure, and optionally a second module configured to perform the calculating step of a method according to a further embodiment of the present disclosure.

(170) For example, the first module may be configured to perform a calculating step to calculate the cross-section moment M, with the optional second module configured to perform a calculating step to calculate a further property of the material, such as the flow stress, main strain etc.

(171) The disclosure further relates to a method in which said calculated cross-section moment, M, or the calculated flow stress, .sub.1, or the estimated Young's modulus, E, or the ratio M/M.sub.e, or the calculated plain strain, .sub.1, or other property calculated using the methods disclosed herein, is used to optimize a product comprising said material.

(172) The dimensionless ratio M/M.sub.e described further in Example 2 is particularly useful, as it shows the point at which a material becomes unstable during bending. Specifically, when M/M.sub.e is below 1.5, the material shows deformation hardening behaviour and is stable during bending. When M/M.sub.e reaches the level of 1.5, the material becomes unstable and by that close to failure.

(173) The disclosure therefore relates to a method for determining the conditions under which M/M.sub.e remains below 1.5 for a given material. With knowledge of these conditions, the skilled person is able to ascertain the suitability of a particular material to a given application. For instance, the skilled person can easily ascertain whether a material is capable of being bent into a desired configuration without (or with minimal risk of) failure, allowing the suitability of the material to be predicted without extensive testing. This method may therefore comprise a further step of utilizing the material as a structural element in a composite product, characterized in that the material is bent under conditions wherein the ratio of M/M.sub.e is below 1.5 during the manufacture of the composite product.

(174) The disclosure also relates to a method for determining the point at which a material becomes unstable during bending, said method comprising determining the point at which the ratio M/M.sub.e becomes 1.5.

(175) The method may also be used to evaluate different materials to determine which materials have bending properties that meet predetermined values necessary for a certain use.

(176) Advantageously, the moment characteristics obtained for different materials may also be super-positioned, allowing the cross-section behaviour of multi-layer materials to be predicted. In this way, the skilled person is able to use the methodology of the disclosure to design new composite materials, and to predict the bending properties of multi-layer materials based on knowledge of the individual layers.

(177) For instance, high strength metallic materials such as high strength steel often have poor bending properties. Adding a layer of more ductile, lower strength material can provide composite materials with improved bending properties. Using the methodology of the disclosure, the skilled person can, without undue experimentation, determine what type of material is required in order to provide the desired bending properties to the high strength material.

(178) Further details of how the moment characteristics for different materials may be super-positioned are provided in Example 5.

(179) The method may also be used to evaluate sample (e.g. plate)s of the same material having different thicknesses, e.g. by studying the ratio of M/M.sub.e.

(180) The following examples implement the methodology of the disclosure to investigate and characterise the properties of various steels during bending.

Example 1

(181) To confirm the correctness of the new expression for the cross-section moment, M, the bending force, F, was calculated using tensile-stress data. The material investigated was: a high strength hot-rolled steel having a tensile strength of >700 MPa a thickness of 2.1 mm. Bending data: Die width L.sub.o=70.5 mm, knife radius R.sub.k=16 mm and roller-radius Rd=25 mm.

(182) The tensile-data is:
()[71]
Converting tensile data to flow stress and plain strain, as:

(183) $\begin{array}{cc}{}_1=\frac{2}{\sqrt{3}}.Math.\stackrel{\_}{}\mathrm{and}{\mathrm{.Math.}}_1=\stackrel{\_}{\mathrm{.Math.}}.Math.\frac{\sqrt{3}}{2}& \left[72\right]\end{array}$
assuming that

(184) $\begin{array}{cc}{}_2{}_1\left(\mathrm{as}{}_2={}_1-\frac{t.Math.\sin {}_1}{L}\mathrm{and}L>>t\right)& \left[73\right]\end{array}$

(185) And furthermore by making the approximation that the relationship between bending angle and strain is:

(186) $\begin{array}{cc}{}_2{\mathrm{.Math.}}_1.Math.\frac{L_m}{t}\mathrm{then};& \left[74\right]\\ L_m\left({}_{2\left(i=0\right)}=0\right)=L_0\mathrm{and}{}_{2\left(i+1\right)}={\mathrm{.Math.}}_{1\left(i+1\right)}.Math.\frac{L_m\left({}_{2\left(i\right)}\right)}{t}& \left[75\right]\end{array}$

(187) The expression for the total moment, M, can then be written as:

(188) $\begin{array}{cc}\begin{array}{c}M=2.Math.B.Math.R^2{}_1.Math.{\mathrm{.Math.}}_1.Math.d\mathrm{.Math.}\\ =\frac{B.Math.t^2}{2.Math.{\mathrm{.Math.}}_1^2}{}_1.Math.{\mathrm{.Math.}}_1.Math.d\mathrm{.Math.}\\ =\frac{B.Math.t^2}{2.Math.{\mathrm{.Math.}}_1^2}\frac{2}{\sqrt{3}}\stackrel{\_}{}.Math.{\mathrm{.Math.}}_1.Math.d{\mathrm{.Math.}}_1\end{array}& \left[76\right]\end{array}$

(189) Combining it with the expression:

(190) 0$\begin{array}{cc}M=\frac{F.Math.L_m\left({}_1\right)}{2.Math.{\cos }^2\left({}_1\right)}\mathrm{and}\mathrm{set}{}_2={}_1& \left[1\right]\end{array}$
then the force, F, becomes;

(191) $\begin{array}{cc}F=\frac{2.Math.{\cos }^2\left({}_2\right)}{L_m\left({}_2\right)}.Math.\frac{B.Math.t^2}{2.Math.{\mathrm{.Math.}}_1^2}\frac{2}{\sqrt{3}}\stackrel{\_}{}.Math.{\mathrm{.Math.}}_1.Math.d{\mathrm{.Math.}}_1& \left[77\right]\end{array}$

(192) The relationship between bending angle, .sub.2, and the knife position, S, is given by:

(193) $\begin{array}{cc}\frac{\mathrm{dS}}{d{}_2}=\frac{\mathrm{dS}}{d{}_1}.Math.\frac{d{}_1}{d{}_2}\frac{L_e}{{\cos }^2\left({}_2\right)}.Math.\frac{L_m}{L_e}=\frac{L_m}{{\cos }^2\left({}_2\right)}\mathrm{Hence}:& \left[78\right]\\ S\frac{L_m\left({}_2\right)}{{\cos }^2\left({}_2\right)}.Math.d{}_2& \left[79\right]\end{array}$

(194) By using the tensile data, shown in FIG. 8, an estimation of the bending force can then be obtained (see FIG. 8), which confirms the correctness of the expression for the cross-section moment, M.

(195) FIG. 9 shows a comparison between the calculated bending force (solid line) based on tensile test data and three individual bending tests performed on the same material but with different stroke length, S. The right-hand sides of the three bending test curves represent the unloading. The bending line was placed along the rolling direction (RD) and the tensile test data was performed perpendicular to the rolling direction (TD).

(196) FIG. 10 schematically illustrates a coil of hot-rolled steel product 10 from which samples may be cut for a bending test. Bending tests may be performed in both the rolling (RD) direction and in a direction transverse to rolling (TD). Additionally, tests can also preferably be performed by turning the samples with rolling-mill side up and down verifying the symmetry of the textures. FIG. 10 shows the bend orientation with respect to a coil of hot-rolled steel product 10.

(197) This example showed that the metallic material 10 has a similar behaviour during a bending test and a tensile test. As a tensile test is an average value of the cross-section properties, compared to bending where the properties are scanned from outer surface and inwards, this case shows that the metallic material 10 behaved uniformly throughout its thickness. Furthermore, FIG. 9 shows that the force drops naturally, and not because of any failures in this case, which illustrates the shortcomings in the VDA 238-100 standard test.

Example 2

(198) In this example a non-dimensional moment (as described in the publication entitled Plastic BendingTheory and Application by T. X. You and L. C. Zhang, ISBN 981022267X) will be exemplified. The non-dimensional moment may be derived by the ratio between the maximum cross-section moment, M.sub.max, and the elastic cross-section moment; M.sub.e. This ratio has two limits; a lower limit that is equal to 1.0 and an upper limit equal to 1.5. The first case is when the material is deformed elastically; the latter case is the state that the material reaches at its absolute maximum moment. Previously, it has not been possible to obtain the material plastification characteristics in between these limits. FIG. 11 shows the equations representing these two limits, i.e. the theoretical elastic cross-section moment, M.sub.e and the maximum cross-section moment, M.sub.max, at bending, and also schematic drawings of the stress distributions in the both cases.

(199) The lower and upper limits of the ratio are as follows, using the two equations shown in FIG. 11:

(200) $\begin{array}{cc}\frac{M_e}{M_e}=1.0& \left[80\right]\\ \frac{M_{m\mathrm{ax}}}{M_e}=\frac{6}{4}=1.5& \left[81\right]\end{array}$

(201) However, to get the entire material response in the whole interval from the elastic state up to the maximum load-carrying capacity, the expression is written as:

(202) $\begin{array}{cc}1\frac{M\left({}_2\right)}{M_e\left({}_2\right)}1.5& \left[82\right]\end{array}$
where M(.sub.2) is the newly disclosed function.

(203) The metallic material 10 that was investigated in this example was a high strength cold-reduced dual phase grade steel having a thickness of 1.43 mm and a tensile strength of >1180 MPa.

(204) FIG. 12 shows the applied force versus the knife position S during bending in a VDA 238-100 standard test. From the bending test, the responses were obtained by measuring the force applied and the position of the knife. The material was tested in both transverse (TD) direction and along the rolling direction (RD). Then the force was transformed to the calculated cross-section moment, M, using the newly disclosed expression, see FIG. 13. The angle .sub.2, was obtained by subtracting the fault angle, .sub.2, (which is important to take into account when calculating the spring back, i.e. over-bending angle) from the .sub.1, bending angle applied, calculated as indicated below:

(205) Using the relationship based on the condition for energy equilibrium:

(206) $\begin{array}{cc}\frac{d{}_2}{d{}_1}=\frac{L_e}{L_m}& \left[83\right]\end{array}$

(207) Then .sub.2, can be obtained from the integral:

(208) $\begin{array}{cc}{}_2=\frac{L_e}{L_m}d{}_1={}_1-\frac{t.Math.\mathrm{Sin}\left({}_1\right)}{L_m\left({}_1\right)}d{}_1={}_1-{}_2& \left[84\right]\end{array}$
where .sub.1 is calculated using equation [23].

(209) The ratio M/M.sub.e was derived herein as:

(210) $\begin{array}{cc}\frac{M}{M_e}=\frac{3}{\left(\left(\frac{dM}{d{}_2}/\frac{M}{{}_2}\right)+2\right)}& \left[85a\right]\end{array}$

(211) Formula [85a] was derived by taking a derivative of the strain as expressed using formula [4a]. As a result, it is less accurate when the material undergoes plastic deformation. An alternative expression for this ratio is as follows:

(212) $\begin{array}{cc}\frac{M}{M_e}=\frac{3}{\left(\left(\frac{dM}{d{\mathrm{.Math.}}_1}/\frac{M}{{\mathrm{.Math.}}_1}\right)+2\right)}& \left[85b\right]\end{array}$

(213) This expression may be solved numerically using [4b] for the strain to give a more accurate result under all bending conditions. A further methodology for calculating the ratio M/M.sub.e is to use the tensile strength, as shown in formula [92].

(214) The expression can easily be verified for the elastic part of deformation, as the derivative

(215) $\frac{\mathrm{dM}}{d{}_2^{}},\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{ratio}\frac{M}{{}_2},i.e.\left(\frac{2E^{}I}{L_m}\right)$
making the ratio equal to 1.0.

(216) When the derivative

(217) 0$\begin{array}{cc}\frac{\mathrm{dM}}{d{}_2}=0& \left[86\right]\end{array}$
then the ratio will become equal to 1.5. This means that when the moment M drops, the material is failing or strain is localized.

(218) The flow stress can also be obtained from the moment derived from equation [76]:

(219) $\begin{array}{cc}{}_1=\frac{2}{B.Math.t^2.Math.{\mathrm{.Math.}}_1}.Math.\frac{d}{d{\mathrm{.Math.}}_1}\left(M.Math.{\mathrm{.Math.}}_1^2\right)& \left[87\right]\end{array}$

(220) Where the main strain, .sub.1 is calculated from:

(221) $\begin{array}{cc}{\mathrm{.Math.}}_1={}_2.Math.\frac{t}{L_m\left({}_1\right)}& \left[3\right]\end{array}$ or alternatively:

(222) $\begin{array}{cc}{\mathrm{.Math.}}_1=t.Math.\frac{U}{M.Math.L_N\left({}_1,{}_C\right)}.Math.\cos \left({}_C\right)& \left[4b\right]\end{array}$

(223) By applying equation [86] in this example, the calculated non-dimensional moment diagram, M/M.sub.e, plotted against main strain will be as shown in FIGS. 14a and 14b (using formula [3] and [4b] respectively).

(224) FIGS. 15a and 15b shows a plot of the flow stresses versus main strain, .sub.1, again calculated using formula [3] and [4b] respectively.

(225) Using the method according to the present disclosure makes it possible to use a material's bending behaviour to estimate the material's Young's modulus, E.

(226) Young's modulus in plain strain, E, is given by:

(227) $\begin{array}{cc}{E}^{}=\frac{E}{\left(1-v^2\right)}& \left[11\right]\end{array}$

(228) For steel, this can be expressed as:

(229) $\begin{array}{cc}{E}^{}\frac{E}{\left(1-{0.3}^2\right)}& \left[12\right]\end{array}$

(230) In this example, Young's modulus was given by:
2.18.Math.10.sup.5 MPa

(231) FIG. 16a shows a graph of main strain, ac versus flow stress, .sub.1, with the Young's modulus being extrapolated therefrom.

(232) Another way of obtaining Young's modulus, E is by determining the gradient of the elastic part of the moment curve (such as that shown in FIG. 13), whereby the gradient is:

(233) $\begin{array}{cc}\left(\frac{2.Math.E.Math.I}{L_m}\right)& \left[10\right]\end{array}$

(234) Preferably, the Young's modulus may be estimated numerically using formula [13]:

(235) $\begin{array}{cc}{E}^{}=\mathrm{Max}\left(\frac{M^2{L}_m}{2.Math.I.Math.U}\right)& \left[13\right]\end{array}$

(236) An example of this methodology is shown in FIG. 16b, which again gives a value of 2.1810.sup.5 MPa.

(237) The relationship between effective stress and strain, flow stress, .sub.1 can be converted using the following expressions, assuming plain strain conditions:

(238) $\begin{array}{cc}\stackrel{\_}{}=\frac{\sqrt{3}}{2}.Math.{}_1\mathrm{and}& \left[88\right]\\ \stackrel{\_}{\mathrm{.Math.}}=\frac{2}{\sqrt{3}}.Math.{\mathrm{.Math.}}_1& \left[89\right]\end{array}$
and converting to true values using:
.sub.tr=.Math.(1+)[90]
and
.sub.tr=.Math.ln(1+)[91]

(239) It is even possible to plot and compare the graph with tensile test data. This will indicate how the hardening behaviour should act if the material's properties are the same from its surface to its centre. If the results of the deformation mechanisms in bending and during pure tension are similar, this is evidence that the material is homogeneous throughout its thickness.

(240) To define the M/M.sub.e ratio from tensile data, the following derived expression is used:

(241) $\begin{array}{cc}\frac{M}{M_e}=\frac{\frac{B.Math.t^2}{2.Math.{\mathrm{.Math.}}_1^2}{}_1.Math.{\mathrm{.Math.}}_1.Math.d{\mathrm{.Math.}}_1}{\frac{B.Math.t^2.Math.{}_1}{6}}=\frac{3}{{}_1.Math.{\mathrm{.Math.}}_1^2}{}_1.Math.{\mathrm{.Math.}}_1.Math.d{\mathrm{.Math.}}_1=\frac{3}{\stackrel{\_}{}.Math.{\stackrel{\_}{\mathrm{.Math.}}}^2}\stackrel{\_}{}.Math.\stackrel{\_}{\mathrm{.Math.}}.Math.d\stackrel{\_}{\mathrm{.Math.}}& \left[92\right]\end{array}$

(242) FIG. 17 shows a comparison between a tensile test and bending tests. In the illustrated case the metallic material seems to harden approximately in similar way, comparing bending and uniform stretching.

(243) According to an embodiment of the disclosure, the method comprises the step of obtaining a cross-section moment, M, of a material and using it to estimate the spring-back for a free choice of set up in bending.

(244) When bending, spring-back is always compensated for by making a certain number of degrees of over-bending to get the final degree of bend. It is difficult to estimate the amount of degrees of over-bending to finally get the desired bend. When handling a material such as high strength steel, it is even more complicated as the spring-back behaviour is higher compared to a material such as mild steel. A thin (3.2 mm) Ultra High Strength Steel was used to investigate the spring-back-effect in four cases of setup for bending. The ultimate strength was approximately 1400-1450 MPa.

(245) The method comprises three steps, see FIG. 18: In the first step the material is tested to determine the material-characteristics in bending, e.g. by performing a VDA 238-100 standard test type of bending, i.e. friction-free bending, obtaining a fully plastified cross-section. In the second step, the moment curve is transformed regarding geometry of a free choice of geometrical setup for a certain case in bending. In the third step these data are used to calculate the spring-back. Even thickness can be converted from the material that has been investigated in the first step. The most accurate result is obtained when using same batch of material in the first and second steps, due to differences in material characteristics.

(246) Material characteristics are obtained by performing the VDA 238-100 standard test, or another type of friction free bending equipment, giving a thumb-print of a current material, by obtaining a moment-curve vs angle diagram. When testing the material characteristics, a narrow die-width is used and a small radius of the knife, approximately 0.7*t for thicker hot-rolled material. The roller radii are friction free, i.e. able to rotate. The maximal bending angle (half bending angle, .sub.1) should not be more than 30-35, eliminating every kind of friction adding a fault energy not connected to the material behaviour.

(247) By using a method according to the present disclosure, a moment-diagram, such as the moment-diagram shown in FIG. 13, can be obtained, based on the measured force vs knife-position, such as the diagram shown in FIG. 4, and the geometry for the trial-setup.

(248) R.sub.d representing the roller radius may for example be 40.0 mm, the knife radius may be 2.0 mm, t (the material thickness) may be 3.2 mm, L.sub.0 the half die-width may be 46 mm and finally, B, the length of the material (i.e. bending length) may be 75 mm.

(249) It was found that if the knife-radius is larger in relation to material thickness and if an increased die-width (compared to the VDA 238-100 standard test) is used, the material between the supports, i.e. the knife and rollers, will be subjected to a curvature, see dashed curve in FIGS. 19 and 20. This means the contact between the knife and the material will not be at the tangent point of a straight line, instead at angle, c, rather than at .sub.1, resulting in a moment arm, L.sub.N, that is longer compared to L.sub.m (see FIG. 19). To be able to estimate the reduced cross-section moment, M.sub.L, the real point of contact has to be defined. Then, the curvature must be obtained. In this Figure, the shape angle, .sub.S, is shown. This angle represents the difference between the contact angle, .sub.C, and the hypothetical contact point of the moment arm at angle .sub.2 (see also FIG. 32). .sub.S therefore represents the difference between the hypothetical contact angle assuming no curvature at the knife, and the observed contact angle at the knife, .sub.C. It is noticed from literature that the shape or curvature of the material (between the contact-points, knife and rollers) is proportional to the complimentary energy, see the shaded area of FIG. 19.

(250) It has been found that by studying the entire distribution of energy within a bend (which is illustrated in FIG. 20), the following expression for the contact-angle, c, can be obtained:

(251) 0$\begin{array}{cc}{}_C={}_2-\frac{2}{M}{}_{}^{}{}_2\mathrm{dM}& \left[93\right]\end{array}$

(252) The contact angle, c, is approximately equal to 0 during elastic deformation, see FIG. 21. This can be shown and confirmed using the integral for the curvature angle, c, putting in the expression for the elastic moment. The contact angle, c, therefore starts to increase at the moment when the bend gets plastified. In FIG. 21, the dotted curve represents the bending angle, the dashed curve represents the true bending angle, the dash-dot curve represents the contact angle between knife and material and finally the solid curve represents the shape angle of the flange.

(253) The expression [29] for the real moment-arm, L.sub.m, may be used when the knife radius is small, i.e. typically 0.7 times the material thickness or less (i.e. R.sub.k0.7t). However, when considering a large knife radius, it is evident that the material will not make contact with the knife at the tangent for a straight line, but at angle, c, shown in FIG. 19. In such a case, the moment arm, L.sub.N, will be:
L.sub.N(.sub.1,.sub.C)=L.sub.0R.sub.d.Math.sin .sub.1R.sub.k.Math.sin .sub.C[94]

(254) It is evident that for large knife-radii, the strain will stop increasing when the material starts to follow the curvature of the knife. At that moment the strain becomes constant and will be limited by the knife-radius, even though the bending angle is increasing. It was found that this level of strain is possible to calculate by applying the contact angle, c, earlier obtained.

(255) For free bending where the knife radius is small compared to material thickness, the bend radius will become free to decrease without any limitation. The cross-section of moment, M, will thereby finally reach its maximum, i.e. fully plastified. If a large knife radius is used, the bending radius will become limited by the knife's geometry, thus the cross-section of moment, M, will be reduced to a certain level, M.sub.L.

(256) It has been assumed that, as the moment is a linearly dependent with respect to the horizontal axis, L (again with reference to FIGS. 19 and 20);

(257) $\begin{array}{cc}\frac{L_m}{L_N}=\frac{M_L}{M}\text{=}\text{>}{M}_L=M.Math.\frac{L_m}{L_N}& \left[95\right]\end{array}$

(258) Where M is the fully, maximal moment that the material can achieve (transformed geometrically from the reference friction free test). M.sub.L is the moment, limited by the knife radius, representing the case to be simulated.

(259) If a small knife radius is used, then the contact point movement is negligible, in relation to the length of the moment arm, resulting in; M.sub.LM. However, if a large knife-radius is used then there will be a difference between the full moment and M.sub.L as they are positioned at two different cross-sections, along the L-axis, hence a difference between L.sub.N and Lm.

(260) The expression for calculating the bending force, F was derived to be:

(261) $\begin{array}{cc}F=\frac{2M{\cos }^2\left({}_1\right)}{L_N}& \left[96a\right]\\ =\frac{2M{\cos }^2\left({}_1\right)}{\left[L_0-R_d.Math.\sin \left({}_1\right)-R_k.Math.\sin {}_C\right]}& \left[96b\right]\\ =\frac{2M{\cos }^2\left({}_1\right)}{\left[L_0-R_d.Math.\sin \left({}_1\right)-R_k.Math.\sin \left[{}_2-\frac{1}{M}{}_{}^{}2{}_2\mathrm{dM}\right]\right]}& \end{array}$
where L.sub.0=the half die-width, R.sub.k=knife radius, R.sub.d=roller radius, .sub.1=bending angle [rad], .sub.2=true bending angle [rad] transformed geometrically from the reference test, M=the full-moment, obtained from the reference test and transformed geometrically.

(262) It is possible to estimate the spring back, .sub.tot, in a very accurate manner using the following equations:

(263) $\begin{array}{cc}{}_{\mathrm{tot}}={}_{C\mathrm{el}}+{}_{S\mathrm{el}}+{}_{12}& \left[14\right]\\ {}_{C\mathrm{el}}=\frac{M_L.Math.W_C}{E^{}I}=\frac{M_L.Math.\left(R_k+\frac{t}{2}\right).Math.{}_C}{E^{}I}& \left[15\right]\\ {}_{S\mathrm{el}}=\frac{M_L.Math.W_m}{2E^{}I}=\frac{M_L}{2E^{}I}.Math.\frac{L_N}{\cos \left({}_1\right)}\mathrm{Where}& \left[16\right]\\ E^{}=\frac{E}{\left(1-v^2\right)}& \left[11\right]\end{array}$
where is Poisson's ratio and E, is the Young's-modulus

(264) For steel, this can be expressed as:

(265) $\begin{array}{cc}{E}^{}\frac{E}{\left(1-{0.3}^2\right)}\mathrm{Furthermore},& \left[12\right]\\ {}_{12}={}_1-{}_2={}_0^{{}_1}\frac{t.Math.\sin {}_1}{L_m}& \left[18\right]\\ M_L=M\frac{L_m}{L_N}& \left[19\right]\end{array}$

(266) The approximate length of the flange being tested is:

(267) $\begin{array}{cc}\frac{L_N}{\cos \left({}_1\right)}& \left[20\right]\end{array}$
and the length (along the neutral layer) of the material in contact with the knife is:

(268) $\begin{array}{cc}{W}_C=\left(R_k+\frac{t}{2}\right).Math.{}_C& \left[21\right]\end{array}$

(269) FIG. 22 shows the measured force versus displacement plus the curve obtained from a friction free bending. The dotted and dashed lines in FIG. 22 represent forces calculated using a method according to the present disclosure and using data from the reference bending test performed similar to the VDA 238-100 standard (i.e. the high load curve in FIG. 22). The solid lines represent actual measured values. It can be seen that using a method according to the present disclosure, substantially the exact bending force can be obtained using data from a reference-test as input. It was found that results obtained from the calculation of spring-back using a method according to the present disclosure were in very good agreement with performed tests.

(270) According to an embodiment of the disclosure the method comprises the step of obtaining a cross-section moment, M of the material by carrying out a friction-free bending test according to the VDA 238-100 standard, or a similar friction free bending-test, and using the cross-section moment, M to estimate a friction coefficient of the material, whereby a friction coefficient can be determined during production.

(271) The bending force and knife position must be measured during the entire bending cycle. If the bending force increases more than what the material is able to absorb in the form of energy (plastic and elastic energy), this has to be due to friction. By studying the cross-section moment behaviour of a material it is thereby possible to isolate the energy-loss related to friction. It is therefore also possible to estimate the friction coefficient of the material. Such a method can thereby be used not only to estimate the friction of coefficient of a material in production, but also to determine coefficients of friction in general, using a dummy material with well-known behaviour as a base for bending, and adding layers of materials whose friction properties are to be investigated.

(272) FIG. 23 shows the force vectors representing the bending load during a bending test. The cross-section moment, M, will make a normal force, F.sub.N against the roller radius, hence a friction force will be developed. The vertical force vector F.sub.y acting and measured during bending is shown in FIG. 23 and corresponds to the bending force.

(273) The friction coefficient, , is calculated using the following equation:

(274) $\begin{array}{cc}=\left[\frac{M-M_{\mathrm{mtrl}}}{M_{\mathrm{mtrl}}}\right].Math.\frac{1}{\tan {}_1}\mathrm{where}& \left[22\right]\\ M_{\mathrm{Measured}}=\frac{F_{y\mathrm{TOT}}.Math.L_m}{2}\frac{1}{{\cos }^2{}_1}& \left[97\right]\end{array}$
and the total force acting vertically is:

(275) $\begin{array}{cc}{F}_{y\mathrm{TOT}}=\frac{M_{\mathrm{mtrl}}}{L_m}{\cos }^2{}_1+\frac{M_{\mathrm{mtrl}}}{L_m}\cos {}_1.Math..Math.\sin {}_1\mathrm{hence};& \left[98\right]\\ =\left[\frac{M_{\mathrm{Measured}}-M_{\mathrm{mtrl}}}{M_{\mathrm{mtrl}}}\right].Math.\frac{1}{\tan {}_1}& \left[22\right]\end{array}$

(276) Where the parameter, M.sub.Measured, is the moment-characteristics obtained from a test where friction is involved. M.sub.mtrl is the reference characteristics of the material, obtained from a friction-free test. However, as the moment characteristics is almost constant after full plastification, this parameter can be set to constant, see thick solid line in FIG. 24.

Example 3

(277) A number of bending tests were performed on hot-rolled high strength steel, 6 mm, with different conditions, i.e. low friction and extremely high, playing without or with different lubricants using same type of material in all cases. In FIG. 25, the force curves are shown. By converting the forces to the cross-section moment, by using the disclosed expression, the influence of friction becomes more obvious, see FIG. 24 and possible to evaluate by the disclosed expression for estimation of the friction coefficient.

Example 4

(278) Comparison has been done between bending tests verifying the disclosed formula.

(279) Within the test-series, different materials, thicknesses and geometrical tooling-setups are used and with ormal conditions for production in bending. In FIG. 26, a good correlation between tests and disclosed formula can be seen. FIG. 27 shows the comparison between the experimentally measured force (F.sub.max) and the calculated angle at F.sub.max. In these data, B/t are between 12 and 67.

(280) Regarding scattering, no friction is assumed in the model. The ultimate strength of the materials bent is not verified.

Example 5Composite Materials

(281) This Example provides a demonstration of how the moment characteristics of composite materials may be calculated based on the characteristics of its component materials. Thus, the properties of a material formed from 5 mm of DX960 (i.e. base layer or the substrate material) and 1 mm skin-layer material made of DX355 (both forms of steel) can be predicted based on the moment characteristics of the individual materials.

(282) Both strain and moment can be transformed, using the following equations:

(283) $\begin{array}{cc}{\mathrm{.Math.}}_{\mathrm{Base}\mathrm{material}}={\mathrm{.Math.}}_{\mathrm{Measured}}.Math.\frac{t_{\mathrm{Base}\mathrm{material}}}{t_{\mathrm{Measured}}}& \left[99\right]\end{array}$

(284) The moment per length unit:

(285) 00$\begin{array}{cc}M/B=\frac{M_{\mathrm{Measured}}}{B}.Math.\frac{{\left(t_{\mathrm{Base}\mathrm{Material}}\right)}^2}{{\left(t_{\mathrm{Measured}}\right)}^2}& \left[100\right]\end{array}$

(286) FIG. 28 shows a plot of the moment characteristics of the base materials, with DX355 being measured at t=4 mm and scaled up to t=6 mm. From FIG. 28, it can be seen that DX355 has much larger deformation-hardening behaviour, which is preferable from a bendability performance point of view.

(287) To calculate the moment contribution from a 1 mm skin layer of DX355 together with 5 mm DX960 (i.e. two skin layers of 0.5 mm DX355 either side of a 5 mm core of DX960 the following calculations are used:

(288) 01$\begin{array}{cc}M/B_{\mathrm{Layer}}=\left[M/B_{\mathrm{Layer}6\mathrm{mm}}-M/B_{\mathrm{Layer}6\mathrm{mm}}.Math.\frac{{\left(t_{\mathrm{measured}}-t_{\mathrm{Layer}}\right)}^2}{{\left(t_{\mathrm{measured}}\right)}^2}\right]==\left[M/B_{\mathrm{Layer}6\mathrm{mm}}.Math.\left(1-\frac{{\left(6.0-1.0\right)}^2}{{\left(6.0\right)}^2}\right)\right]& \left[101\right]\end{array}$

(289) Thus, the moment-characteristics of the full thickness material, t.sub.full, minus the moment-characteristics for the reduced thickness, t.sub.fullt.sub.layer, provides the moment-impact (or contribution) of the skin layers.

(290) The thickness of the substrate (or base material) will in this case be:
t.sub.fullt.sub.layer=6.01.0=5.0 mm

(291) Using the above expressions, this gives:

(292) 02$\frac{M}{B_{\mathrm{Substrate}}}=\frac{M}{B_{\mathrm{Substrate}6\mathrm{mm}}}.Math.\frac{{\left(t_{\mathrm{measured}}-t_{\mathrm{layer}}\right)}^2}{{\left(t_{\mathrm{measured}}\right)}^2}=\frac{M}{B_{\mathrm{Substrate}6\mathrm{mm}}}.Math.\frac{{\left(6.0-1.0\right)}^2}{{\left(6.0\right)}^2}$

(293) The moment-characteristics for the for the skin layer, base layer and composite are shown in FIG. 29.

(294) FIG. 30 shows the unit-free moment curves for the original DX960 material and the predicted properties of the composite material having 5 mm DX960 and two 0.5 mm skin layers of DX355. As can be seen, the composite material is predicted to have unit-free moment of below 1.5 for higher strains, which means that the materials is predicted to be more stable during bending, with less risk of failure.

Example 6Properties of the Material Along the Bend

(295) A rolled 960 Mpa, thickness t=8 mm was bent using a knife radius R.sub.k=4 mm in accordance with a protocol similar to the VDA 238-100 standard. The moment curve is shown in FIG. 35.

(296) FIG. 33 shows a plot of 1/R (i.e. the reciprocal of the radius of the bend) for each point X along the bending length L.sub.N when the true bend angle .sub.2 is 30. The plot shows that at low values of X, i.e. those parts of the plate close to the die, the bending radius is large and approaching infinite at the limit. This means that closer to the contact points with the bending die, the material itself is acting more like a rigid plate (i.e. no bending of the plate is occurring so the plate does not have any curvature). Any deformation is elastic. However, closer to the knife, the bend radius decreases (and 1/R increases). Furthermore, plastic deformation is occurring in this region close to the knife.

(297) The moment vs. bending angle plot is shown in FIG. 35. The Figure shows that the bending moment, M, becomes constant when the material plastifies.

(298) By applying a constant moment, M, using the equations disclosed herein, a theoretical force, F.sub.theor, can be estimated. This theoretical force is shown plotted alongside the actual force in FIG. 34. The results show that the failure point of the steel occurs after the natural force maximum. According to the VDA 238-100 standard, the test should be interrupted at once the bending angle yields F.sub.max, which indicates that the tested steel did not comply with the requirements of the standard test.

Example 7Bend Induced Hardening

(299) In the following example, a hot rolled steel with yield strength of 355 MPa and thickness of 4 mm was subjected to a bending test in line with the VDA 238-100 standard. FIG. 36 shows the plot of bending moment vs knife position, while FIG. 37 shows the plot of theoretical force vs bending angle.

(300) In FIG. 37, it can be seen that a constant moment is not obtained despite the high bending angles. This behaviour is due to hardening of the material at increased bending angles. Despite not being constant, the hardening gives rise to a linear relationship, and a linear regression can be carried out to provide a linear equation for the bending moment (dashed line).

(301) These plots show how the methodology described herein may be used to provide further insight into the bending properties of materials beyond their maximum force, where plastic deformation or bend induced hardening may be occurring.

Example 8Investigating Kinking Behaviour

(302) A hot-rolled 960 MPa steel (approximately 1050 MPa tensile strength) with a thickness of 6 mm was subjected to a bending test. The moment vs. bending angle plot is shown in FIG. 38. FIG. 38 shows a decreasing moment until a discontinuity occurs at failure. The moment is modelled using linear regression to provide a linear equation that this then used to calculate the force, shown plotted versus the knife position in FIG. 39.

(303) The methodology used to plot FIGS. 38 and 39 can therefore be used to investigate whether kinking (i.e. non-uniform curvature of the material) has occurred.

(304) A further way to investigate kinking behaviour is by plotting the plain strain along the length of the bend for a constant bending angle. The area where kinking has occurred will be visible as a discontinuity in the bend, where strain increases stepwise as the length is traversed rather than smoothly.

(305) To demonstrate this, two 6 mm steel sheets were subjected to a bending test. FIGS. 40 and 41 show the plot of plain strain vs. position, while FIGS. 42 and 43 show, respectively M/M.sub.e and the flow stress vs the plain strain. In FIGS. 40 and 41, the material that shows kinking tendencies (solid line) shows a marked increase in the strain when X is around 30.5 mm. This discontinuity in the curve is characteristic of the strain increasing significantly in a short distance, which arises due to the shearing deformations that give rise to the non-uniform curvature. This behaviour also leads to the plots in FIGS. 42 and 43 showing distinctly different profiles from the non-kinking material.

Example 9Microbending

(306) This example shows a small sample prepared from a Gleeble-test specimen. Gleeble specimens are typically too small to make a tensile test samples from, and can only be bent using a microbending apparatus. The bending apparatus used (shown in FIG. 45) was designed to fit the small dimensions of the samples. The dimensions of the bending apparatus were;

(307) TABLE-US-00001 Die-width Knife-radius Die-radius [mm] [mm] [mm] 22 0.4 6

(308) The samples tested had thicknesses from 1-1.5 mm and a width of 10 mm. To aid in bending, the friction was reduced by applying grease between sample and die. The results from the bending tests are shown in FIGS. 46-48. Even though the thickness to width ratio (t/B) was close to the limit where plain strain conditions can be assumed, the results were satisfying.

(309) From the foregoing disclosure, it is evident that the new methodology disclosed herein provides the skilled person with a wide range of options for investigating the bending properties of a material. For ease of understanding, an overview of the steps needed to perform the calculations disclosed herein is provided below: 1) Performing a three point bending test, measuring the force, F, vs the stroke distance, S, by the knife. Friction between the sample and the support shall be minimized preventing consumption of extra energy not related to energy absorbed by the sample itself. 2) Measuring the following parameters relating to the test equipment; Die width, 2.Math.L.sub.0, die radius, R.sub.d, knife radius, R.sub.k, sample length, B and sample thickness, t. 3) Calculate the energy, by making the integral; U=F.Math.dS [5a] 4) Calculate the bending angle, .sub.1, using following formula:

(310) 03$\begin{array}{cc}{}_1={\mathrm{Sin}}^{-1}\left(\frac{\left[L_0.Math.Q+\sqrt{L_0^2+{\left(S-Q\right)}^2-Q^2}.Math.\left(S-Q\right)\right]}{\left[L_0^2+{\left(S-Q\right)}^2\right]}\right).Math.\frac{180}{}& \left[0\right]\end{array}$
where; Q=R.sub.k+R.sub.d+t

(311) Or alternatively by the integral;

(312) 04$\begin{array}{cc}{}_1=\frac{d{}_1}{\mathrm{dS}}\mathrm{dS}=\frac{{\cos }^2{}_1}{L_e}\mathrm{dS}\mathrm{Where};& \left[{27}^{}\right]\\ L_e=L_0-\left(R_k+R_d+t\right).Math.\sin {}_1.& \left[28\right]\end{array}$ 5) Calculate the moment, M, by the formula;

(313) 05$\begin{array}{cc}M=\frac{F.Math.L_m\left({}_1\right)}{2.Math.{\cos }^2\left({}_1\right)}& \left[1\right]\end{array}$
Where; L.sub.m=L.sub.0(R.sub.k+R.sub.d).Math.sin .sub.1 6) Calculate the real bending angle, .sub.2, i.e. in accordance with the energy equilibrium: F.Math.dS=2M.Math.d.sub.2, [5a]/[5b] That is

(314) 06$\begin{array}{cc}{}_2=\frac{L_e}{L_m}d{}_1={}_1-\frac{t.Math.\sin {}_1}{L_m\left({}_1\right)}d{}_1& \left[84\right]\end{array}$ Verify calculated moment, M, and real bending angle, .sub.2, by calculating the energy; 2M.Math.d.sub.2 [5b], and compare it with U (should be equal). 7) The real bending angle consists of two parts; i.e. the shape angle, .sub.S, and the contact angle at the knife; .sub.C; Energy, U, can be expressed in the following way, which allows the calculation of the two angles individually;

(315) 07$U=M.Math.\left(2{}_2-{}_S\right)=>{}_S=2{}_2-\frac{U}{M}$ Hence; .sub.S=.sub.2.sub.C [50] 8) The plain strain, .sub.1, is calculated by following formula:

(316) 08$\begin{array}{cc}{\mathrm{.Math.}}_1=\frac{t}{2}.Math.\frac{1}{R}=\frac{t.Math.U}{M.Math.L_N\left({}_1,{}_C\right)}.Math.\cos {}_C\mathrm{Where};& \left[4a\right]\text{/}\left[4b\right]\\ L_N\left({}_1,{}_C\right)=L_0-R_d.Math.\sin {}_1-R_k.Math.\sin {}_C& \left[94\right]\end{array}$ 9) Then the flow stress, .sub.1, can be estimated by;

(317) 09$\begin{array}{cc}{}_1=\frac{2}{B.Math.t^2.Math.{\mathrm{.Math.}}_1}.Math.\frac{d}{d{\mathrm{.Math.}}_1}\left(M.Math.{\mathrm{.Math.}}_1^2\right)& \left[2\right]\end{array}$ 10) The dimensionless moment,

(318) 0$\frac{M\left({}_2\right)}{M_e\left({}_2\right)},$ has following maximum range of interval;

(319) $\begin{array}{cc}1.0\frac{M\left({}_2\right)}{M_e\left({}_2\right)}1.5,& \left[82\right]\end{array}$ from elastic state to fully plastified cross-section. Calculated as;

(320) $\begin{array}{cc}\frac{3}{\left(\left(\frac{\mathrm{dM}}{d{}_2}/\frac{M}{{}_2}\right)+2\right)}& \left[85a\right]\end{array}$ or alternatively:

(321) $\begin{array}{cc}\frac{M}{M_e}=\frac{3}{\left(\left(\frac{\mathrm{dM}}{d{\mathrm{.Math.}}_1}/\frac{M}{{\mathrm{.Math.}}_1}\right)+2\right)}& \left[85b\right]\end{array}$ 11) Young modulus, E, for plain strain condition can easily be defined by;

(322) $\begin{array}{cc}{E}^{}=\mathrm{Max}\left(\frac{M^2{L}_m}{2.Math.I.Math.U}\right)& \left[13\right]\end{array}$ or alternatively checking the gradient between stress, .sub.1, and the strain, .sub.1, at elastic state. 12) Estimation of maximum load, F.sub.Max, and the bending angel, .sub.F max, where it appears (considering more a less a constant steady state moment, M.sub.max)

(323) $\begin{array}{cc}{F}_{\mathrm{Max}}=\frac{4.Math.M_{\max }}{\left(R_d+R_k\right)}.Math.\sin {}_{F\max }\mathrm{Where};& \left[8a\right]\\ {}_{F\max }={\sin }^{-1}\left[\frac{L_0-\sqrt{L_0^2-{\left(R_d+R_k\right)}^2}}{\left(R_d+R_k\right)}\right]\frac{180}{}& \left[7\right]\end{array}$ The approximate maximum bending moment, M.sub.Max, can be estimated as;

(324) $\begin{array}{cc}{M}_{\mathrm{Max}}=\frac{B.Math.t^2.Math.\left(R_m.Math.\frac{2}{\sqrt{3}}\right)}{4}& \left[9\right]\end{array}$ 13) As the moment, M, is linearly distributed between the knife and the supports, the parameters set out above can then be plotted between these points at any stage of the bending procedure (i.e. .sub.1*). The horizontal X-coordinates (starting at die support and ending at contact point .sub.C is calculated by the expression:

(325) $\begin{array}{cc}X={}_0^{M\left({}_2^{*}\right)}\frac{L_N\left({}_2^{*}\right)}{M\left({}_2^{*}\right)}\mathrm{dM}& \left[64\right]\end{array}$ Where L.sub.N(.sub.1*) is the real moment-arm at the chosen point of bending, and M(.sub.2*) is the moment at that point.

(326) Further modifications of the disclosure within the scope of the claims would be apparent to a skilled person. In particular, the methodology of the disclosure allows the skilled person to investigate the properties of materials such as steel during bending. By comparison to pre-determined threshold values, the skilled person is able to evaluate the suitability of materials such as steel for a particular use using the method of the present disclosure.