136 PAK vol. 58, nr 1/2012 Krzysztof KUKIEŁKA POLITECHNIKA KOSZALIŃSKA, KATEDRA INŻYNIERII PRODUKCJI, ul. Racławicka 15-17, 75-364 Koszalin

Numerical modelling of strain and stress states in quick thread cold rolling process Ph.D. eng. Krzysztof KUKIEŁKA PhD degree in the discipline of machine building and exploitation received in March 2009. Topics of scientific work undertaken by the author relate to problems of modelling and numerical analysis of thread with different outlines in rolling process and treatments as well as burnishing. He works in Department of Production Engineering at the Faculty of Mechanical Engineering in Koszalin University of Technology. e-mail: krzysztof.kukielka@tu.koszalin.pl

Abstract

Numerical modelling of the thread cold rolling process is described in the paper. The solutions of discrete equations of motions and deformations of an object with use of the explicit integration method are presented. The algorithm and application of numerical analysis in ANSYS program were elaborated. This application allows determining the state of strain and stress in thread in which influence of the friction coefficient is taken into account. Exemplary results of numerical analysis of displacement and strain in thread for different conditions of rolling process are presented. Keywords: thread rolling, round thread, mathematical model, numerical analysis.

Modelowanie numeryczne stanów odkształceń i naprężeń w procesie walcowania gwintów o dużym skoku na zimno

Streszczenie

W pracy [7] przedstawiono problematykę modelowania zagadnienia kontaktowego w procesach walcowania gwintów na zimno. W niniejszej pracy przedstawiono model fizyczny i matematyczny procesu walcowania na zimno gwintów o zarysie łukowym. Wyprowadzono równanie ruchu i deformacji obiektu stosując metody wariacyjne i metodę elementów skończonych. Opracowano algorytm i aplikacje do analizy numerycznej w programie ANSYS, przy wykorzystaniu tych modeli. Możliwa jest analiza stanów odkształceń i naprężeń dla przypadku walcowania gwintów o zarysie łukowym. Zamieszczono przykładowe wyniki analizy numerycznej dla różnych warunków realizacji procesu. Wykorzystując przedstawione w niniejszym artykule modele matematyczne procesu walcowania gwintów, algorytmy rozwiązań dyskretnych równań ruchu oraz aplikację w systemie ANSYS można istotnie rozszerzyć i ulepszyć proces projektowania technologii walcowania gwintów, co istotnie przyczyni się do wzrostu jakości gwintu oraz zmniejszenia kosztów wykonania operacji (mniejsze zużycie energii, większa trwałość narzędzi itp.). Słowa kluczowe: walcowanie gwintów, gwint łukowy, model matematyczny, analiza numeryczna. 1. Introduction

One of the most important problems of modern manufacturing techniques is to ensure the proper quality of products while minimising their production costs and increasing the process efficiency. These problems are closely related, and their relative proportions depend on construction precision level, type and quality of used materials, also the applied technology. The adequate quality of the machine parts depends on the type and conditions of finishing during which specified properties in the product surface layer are achieved. These properties have a decisive impact on the reliability and durability of the machines [12, 13].

Over 60% of modern machinery has bolts, pins or threaded holes. Threaded parts are widely used in wood industry [14], mechanical and building engineering [8, 9, 14], medicine and fancy (Fig. 1).

There are two basic ways to create a thread. The first one is shaping the thread by removal of material from the grooves [2, 10] and the second one - by plastic deformation of the material [3÷5, 7÷10, 14]. Selection of the treatment method depends on the dimensions and outline of the thread, the accuracy requirements for its implementation, the material properties and structure of the workpiece, also the production volume.

Fig. 1. Examples of threaded parts: a) in wood industry, b – d) mechanical

engineering, e) medical engineering, f) fancy industry Rys. 1. Przykłady zastosowań gwintów: a) w przemyśle drzewnym, b – d) w budowie

maszyn, e) w technice medycznej, f) przemyśle galanteryjnym

The plastic thread forming consists in giving a thread bar the

outline surface by generating plastic strain. Plastic processing of the thread contains the methods for [1]:

thread rolling on the mill with flat die, thread rolling on the mill with roll die, thread rolling through heads, thread embossing through rollers.

For the purpose of this paper the thread rolling method by the axial head was used. The thread cold rolling process is a doubly (geometrical and physical) nonlinear boundary-initial problem, with nonlinear and moving boundaries. The boundary conditions in the contact zone are not known. Physical nonlinearities, called also as a material ones, result from the nonlinear equations describing the material properties of the processed part. Geometrical nonlinearities caused by a change in the initial part geometry, lead to nonlinear relationships between strain-displacements. Inclusion of these nonlinearities in the analysis of the thread rolling issue is extremely complex, making it necessary to operate increments of particular kinematic quantities (displacement, velocity, acceleration), static (force, force moment, stress) and use of incremental description [6]. An algorithm and application in the ANSYS program for the thread rolling process were elaborated. Numerical analysis allows forecasting the behaviour of the rolled thread during the whole multistage technological process. The most important possibilities of the numerical analysis when applied to the thread rolling is the determination of: the pipe dimensions (mainly nominal and outline diameter), geometry and thread outline during thread rolling and after

relieving elastic, local strain and stress states in the thread, maximum strain – where crack of the thread is possible,

a)

b)

c)

e)

d) f)

PAK vol. 58, nr 1/2012 137

expected rolling force, number and geometry of the rolls, including active roll surface

in the introducing, shaping, calibrating and outing zone, influence of the friction coefficient on the process flow and the

thread quality, state of loads, stresses and strains of the tools, areas of contact, slip and stick.

The incremental mathematical model of the thread rolling process, in the updated Lagrange formulation, contains constitutive equations (model of dynamical yield stress, elastic/visco-plastic stains model, elastic/visco-plastic stress model), the model of contact between tool-workpiece [7], dynamic equation of motion and deformation and boundary-initial conditions.

In the first variational method there was developed the equation of motion and deformation for a typical step time. Then the equation was discretized with the finite element method, given the equations of motion and deformation of a discrete object. There was elaborated the algorithm of numerical analysis in ANSYS in which the discrete equation together with initial and boundary conditions was used . In this work this model was used for strain and stress state analysis for case of the external round thread rolling process. Exemplary results of the numerical analysis are given.

2. Mathematical model of the thread rolling

process

The mathematical model of the thread rolling process is formulated in increments and contains: 1. incremental material model of: yield stress; total strain tensor;

total stress tensor; translation stress tensor, 2. contact model tool-workpieces: contact forces; contact stiffness;

shape and area of the contact area of bodies; contact boundary conditions; the friction condition in contact area of bodies,

3. dynamical equation of motion and deformation, 4. geometrical conditions, 5. initial conditions, 6. boundary conditions.

The general discretized equation of motion and deformation of objects, in the updated Lagrange’s formulation on the typical step time, takes the following form:

}{}{}{Δ}{Δ

}{Δ])[Δ]([}{][}{][

RFFRrKKrΔCrΔM

(1)

where: the global temporary mass matrix [M], the global temporary damping matrix [C], the global temporary stiffness matrix [K] and the temporary internal {F} and external {R} global force vectors are known at time t. However, the increment stiffness matrix [K], the external incremental load vector {R}, the internal incremental forces vector {F}, the incremental vectors of nodal displacement {r}, the nodal velocity }{ rΔ and

the nodal acceleration }{ rΔ of finite element assembly at a typical step time are not known.

The matrix equation (1), which is a system of N second order differential equations, with the constant (on the incremental step) coefficient and the adequate initial }{)}({ 00 rr t ,

}{)}({ 00 rr t and boundary conditions are the incremental

formulation of the dynamical equilibrium deformable solid objects in contact for the thread rolling case. 3. Dynamic Explicit Method solution

Assuming that time increment Δt is very small, it is possible to execute a linearization of equation (1) and using the incremental decomposition we obtain:

.tt }{}{}]{[}]{[ QFrCrM (2)

Then using the central difference method (DEM), in which it is assumed that:

}){}({}{ rrr ttttt

t

2

1 , (3a)

}){}{}({}{ rrrr tttttt

t

21

2 (3b)

and substituting the relations (3a) and (3b) into (2) we obtain:

}{]{[ QrM ~}

~ , (4)

where:

.

tt

~

,t/,t]~

ttttt][

}{][

}{}{}{}{}{

][]/[[

CrMrrQFQ

CMM

2

2

50

2

2

(5)

The integration method requires that the step time Δt is smaller

than the critical value Δtcr, which can be calculated from the mass and stiffness properties of the complete element assemblage:

,/Ttt Ncr π where TN is the smallest period of the finite

element assemblage with N degrees of freedom. 4. Investigation of deformation states

In the case of rolled threads, most of them have incomplete shaped outline, but the full filled outline is a borderline case and it can be assumed that in the thread rolling process is a plane state of material deformation, which is moved parallel to the object axial plane [3, 4, 7]. In order to verify this assumption, experimental research was conducted. In a semi-finished product, namely a grinded shaft, the axial grooves were made and filled with flat steel S235JR (Fig. 2a) and copper M1E (Fig. 2b).

Fig. 2. Samples view for the thread rolling with axial inserts of flat steel

S235JR (a) and copper M1E (b) Rys. 2. Widok próbek do walcowania gwintów z wkładkami osiowymi ze

stali S235JR (a) i miedzi M1E (b)

On the bars the thread was made by axial method on the stand

with angular head type F3, made by FETTE, which contain three rolls. View samples with carried out the thread is shown in Fig. 3.

Fig. 3. View samples with the thread with flat steel S235JR (a) and cooper M1E (b) Rys. 3. Widok próbki z gwintem z wkładką ze stali S235JR (a) i z miedzi M1E (b)

Then carried out cross-sections to the samples axis, in three

planes passing properly through the surface without thread (section I-I), the crest (section II-II) and the root of the thread (section III-III) (Fig. 4). Because the rectilinear plane (1) and (2) of the groove, which are parallel to the axial planes are also parallel after thread rolling, it can be assumed, that during thread rolling exist plane states of displacement and strain. Also in the cross-section lateral plane of the grooves are almost linear. Visible small curvature of the plane at the crest is rather due to discontinuities introduced by the material, than to the spatial state

a)

b)

a)

b)

138 PAK vol. 58, nr 1/2012

of deformation. Therefore, with a sufficient accuracy for engineering practice can be assumed that, during the thread rolling exist spatial state of stress and plane state of strain and displacement in planes parallel to the axial plane of the workpiece.

Fig. 4. View samples with flat steel insert after rolling and scanning pictures

of cross-sectional of the sample at the edge if the groove before rolling (section I-I) and passing through the crest (section II-II) and the root of the thread (section III-III)

Rys. 4. Widok próbki z wkładką stalową po walcowaniu oraz zdjęcia skaningowe przekroju poprzecznego próbki na brzegu rowka przed walcowaniem (przekrój I-I) oraz przechodzących przez wierzchołek (przekrój II-II) i dno gwintu (przekrój III-III)

5. Simulation

The round threads with quick pitch make up the specific group. The screw joint folding from nut and screw is used mainly in the construction of communication tunnels and engineering construction to support ceilings and in earth works for the protection of excavations.

The most commonly used are the threads with a major diameter of d1=31 mm and d1=38 mm, the pitch of P=1256 mm and P=1278 mm respectively, and typical lengths of L=6000 mm (Fig. 5). However, the required length, for a given application, is achieved by shortening the screws or join two or more screw segments by using special threaded sleeve.

The basic problem to design the rolling process of the round thread on pipe is elaborate proper construction of the tool (threading head) and select of the processing condition for provide technical requirements and property of surface layer of the screw, simultaneously increase the tool life and process productivity. Presently this technology isn’t used in Polish industry.

Fig. 5. a) View the screw joint with round profile 3112.56, b) the thread

parameters: major diameter d1, pitch diameter d2, minor diameter d3, the crest filet radius r1, root filet radius r2, height of thread h3

Rys. 5. a) Widok połączenia gwintu o zarysie łukowym 3112.56, b) parametry gwintu: średnica zewnętrzna d1, średnica podziałowa d2, średnica wewnętrzna d3, promień zaokrąglenia wierzchołka gwintu r1, promień zaokrąglenia dna gwintu r2,wysokość gwintu h3

The THREAD ROLLING application in ANSYS system with

APDL (Ansys Parametric Design Language), to parameterize geometric dimensions and material, allowing for easy

modification and change its model parameters was developed and presented in this article. This application fully satisfy advanced requirements and can be successfully used to simulate thread rolling process on cold in real conditions. The main advantages of computer simulation in comparison with other methods of the process analysis are: the model flexibility – is easy to making changes in the

simulated model also filling model with new phenomena, easy to introduce various types of extortion and disturbance

(random) and extreme extortion and disturbance without destroying usually expensive material model,

relatively low cost and short preparation time of simulation, the reliability of simulation results - especially when we can

compare the simulation results with the results obtained from measurements on a real object. The thread rolling process is considered as vertical penetration

of a stamp in rectangular workpiece. Way to reduce this model, eg. the transition from the spatial state to the plane is described in work [3, 4, 7].

Developed in this paper application THREAD ROLLING in ANSYS system enables a complex time analysis of deformation (displacements, strains) and stress states occurring in object folding from a workpiece and a tool. It is possible a complete analysis for following data: any object geometry (eg. shaft, sleeve) and tools (an outline of

the active surface, the number of rings, etc.) any material of workpiece and the tools material as elasto/visco-

plastic model (Young's modulus, non-linear dependence of the properties plastic materials: strain and strain rate, various models of hardening, etc.)

different conditions of friction in the contact area, any horizontal and vertical movement of the tool in time.

Numerical calculations in ANSYS system were realized by the following algorithm: a) data preparation (PREPROCESSOR): defining the workpiece and the tool geometry, nodes defining, defining the material properties of the workpiece and tool, generating the finite element grid (mesh), introducing the contact between tool-workpieces,

b) solving (SOLVER) introducing number of steps and iterations, convergence

conditions, etc. introducing loads, boundary and initial conditions, setting the course of calculations,

c) analysis of solutions and editions of the results (POST-PROCESSOR - General Postproc or TimeHistory Postproc). The first stage of the analysis is determine the geometry of the

object and tool. In carried out investigations plane models is used, assuming that the tool is a rigid or elastic body. In the developed simulation program can make any outline of the active surface of the tool and any condition of process realization. It’s possible to modify this factors during the simulation. It’s possible to observe the stick and slip zones, depending on the tool geometry, friction in the contact zone and working conditions.

The application developed with regard to the method of finite element in ANSYS program provides determination of forces, contact pressure, displacement, strain and stress in each stage of the process.

In the exemplary simulation is shown the possibility of using reduced model to simulate the impact of the flat profile of the working rollers active surface on the thread rolling process realization and quality. The tool is a rigid body E→∞, however the model material as an elsto/visco-plastic body with nonlinear hardening, which was approximated by bilinear model (Fig. 6). The model has discretized by finite element PLANE162 with linear function of the shape. In the contact zone the boundary conditions for displacement are unknown. On the tool displacement on x and y direction was applied with accordance to Tab. 1, this means that the tool in numerical analysis has 18 acts.

I

I II

II

III

III

1 2

h3

d1 d3

PAK vol. 58, nr 1/2012 139

On the bottom of the model translational and rotational degrees of freedom was took off. The calculation was carried out for following properties: material – steel C55, yield stress

18000660924 .iY . )( MPa, where i is intensity of true

strain, Poisson’s coefficient =0.29, Young’s modulus E=210 GPa. Shape coefficient (SC=B/A=1) rectangular finite element is define as a ratio height B to width A of elements (Fig. 6).

model

tooluy=0.8 mm

ux=0 uy=0

Graph Y - i for steel C55 x

y I

J

K L 3

2

1

4

PLANE162

A

BA

BSC

ux=12.56 mm

Fig. 6. Discretized model for the external thread rolling with round outline Rys. 6. Model dyskretny do analizy walcowania gwintów zewnętrznych o zarysie

łukowym

Tab. 1. Boundary condiotion for tool Tab. 1. Warunki brzegowe dla narzędzia

Step Displacement

OX, mm Displacement

OY, mm Step

Displacement OX, mm

Displacement OY, mm

1 -12.56 0.0 10 -50.24 0.0

2 -12.56 -0.8 11 -50.24 -0.8

3 -12.56 0.0 12 -50.24 0.0

4 -25.12 0.0 13 -62.80 0.0

5 -25.12 -0.8 14 -62.80 -0.8

6 -25.12 0.0 15 -62.80 0.0

7 -37.68 0.0 16 -75.36 0.0

8 -37.68 -0.8 17 -75.36 -0.8

9 -37.68 0.0 18 -75.36 0.0

6. Results of numerical simulations

Exemplary results of numerical simulation are presented in Figs 6÷7. Analysing, the intensity distribution of strain and stress, where the impact of successive rings after thread rolling process for various value of friction coefficient is observed.

Fig. 6. The stress intensity maps (MPa) for successive rings and various value of

friction coefficient after the thread rolling process with round outline Rys. 6. Mapy intensywności naprężeń w gwincie o zarysie łukowym dla różnych

pierścieni rolki oraz wartości współczynnika tarcia po procesie walcowania

Fig. 7. The strain intensity maps for successive rings and various values of friction

coefficient after the thread rolling process with round outline Rys. 7. Mapy intensywności odkształceń w gwincie o zarysie łukowym dla różnych

pierścieni rolki oraz wartości współczynnika tarcia po procesie walcowania

7. Conclusion

We can use obtained results from simulation to design the optimization of thread rolling process, reduce design cycles and costs reduction at the production beginning, reduce production costs and increase the product quality. These upper topics are necessary for the proper design and control of the complex process of plastic forming.

The external thread rolling on cold with round profile is a complex process in technological terms. The round thread rolling process on the pipes is a geometrical, physical and thermal non-linear initial and boundary problem. Measurement of a process parameters decide on the technological quality, such as: a displacement zone, a temperature, stress, structural change etc. During the thread rolling process with nowadays technique of a measurement is impossible. About their course, we could conclude on the property of the product after rolling.

An application of modern numerical methods and computing systems allows an analysis of complex physical phenomena occurring in the process under investigation. The application developed in the ANSYS system enables a time analysis of the rolling process with the consideration of the changeability of the rings profile. On the course of physical phenomena in the working zone we can forecast a technological quality of the round thread.

The obtained results from computer simulation of the thread rolling process show that the successive rings has influence on the states of displacements, strains and stresses in the surface layer of the thread, also is one of the factors deciding about the technological and the exploitation quality.

The simulation results can be used to design the round thread rolling process: making a selection of the process condition and kind of the lubrication factor in the aspect of the technological quality of the screw. 8. References [1] Feld M.: Basis of design processes typical machine parts.

Wydawnictwa Naukowo-Techniczne, Warszawa 2000, (in Polish). [2] Feld M.: Mechanical Engineering Technology. Wydawnictwo

Naukowe PWN, Warszawa 2000, (in Polish). [3] Domblesky J.P., Feng F.: A parametric study of process parameters in

external thread rolling. Journal of Materials Processing Technology, Volume 121, ELSEVIER, 2002, pp. 341÷349.

[4] Domblesky J.P., Feng F.: Two-dimensional and three-dimensional finite element models of external thread rolling. Professional Engineering Publishing, Volume 216, Number 4/2002, pp. 507÷517.

[5] Górski E.: Machining. WSiP, Kraków, 1974, (in Polish). [6] Kleiber M.: Finite element method in nonlinear continuum mechanics.

Biblioteka Mechaniki Stosowanej Inst. Podst. Problemów Techniki PAN, PWN, Warszawa – Poznań 1985, s. 300, (in Polish).

[7] Kukiełka K.: Modelling and numerical analysis of the states of deformations and stresses in the surface layer of the trapezoidal and round threads rolled on cold. PhD Thesis, Wydział Mechaniczny Politechniki Koszalińskiej, Koszalin 2009, (in Polish).

[8] Łyczko K.: The tools technology and embossing of the internal threads. Politechnika Częstochowska, Częstochowa, 1999, (in Polish).

[9] Łyczko K.: External thread rolling technology. WNT, Warszawa 2010, (in Polish).

[10] Olszak W.: Machining. WNT, Warszawa, 2008, (in Polish). [11] Patyk R., Kukiełka L.: Modelling and simulations of contacts

phenomena in the surface roughness burnishing in product quality aspects. PAK, vol. 54, no 7/2008, s. 433÷435, (in Polish).

[12] Przybylski W.: Plastic forming. Technology and equipment. WNT, Warszawa 1979, (in Polish).

[13] Przybylski W.: Technology of the plastic forming. WNT, Warszawa 1987, (in Polish).

[14] Żurawski Z., Sikora J., Błużek J.: The thread rolling. WNT, Warszawa, 1962, (in Polish).

_____________________________________________________ otrzymano / received: 09.02.2011 przyjęto do druku / accepted: 01.12.2011 artykuł recenzowany / revised paper

before I II III IV V VI after µ=0

I II III IV V VI after µ=0.2

I II III IV V VI after µ=0.39

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

before I II III IV V VI after µ=0

I II III IV V VI after µ=0.2

I II III IV V VI after µ=0.39

0 120 240 360 480 600 720 840 960 1080