Learning about Stress and Strain Due to Creep for Real

Learning about Stress and Strain Due to Creep for Real World Applications

Kyozo Arimoto1

1. Arimotech Ltd., 4-3-2-701 Habucho, Kishiwada, Osaka 596-0825, Japan, e-mail:

[email protected] (Senior External Education Consultant, Granta Design Ltd., UK)

Abstract The author has an opportunity to give a talk in one-day courses for heat treatment engineers. The aim of this course is to explain about heat treatment simulation based on the finite element method and its applications. An introductory part of the course uses teaching materials to intuitively understand complex phenomena induced from elasticity, thermal expansion and plasticity. These materials, which were reported previously, include a crystal model based on knowledge of both mechanics of materials and materials science. The author has expanded the materials to phenomena including not only elasticity, thermal expansion and plasticity but also creep which occurs generally at high temperature and in a prolonged time. Although the crystal models are useful to understand an essence of the phenomenon, an experiment has been conducted additionally. The devised materials may be meaningful for engineers involved in manufacturing techniques such as casting, plastic working, machining, welding, and design of machines and processes operating at high temperatures. Keywords: Learning, stress, strain, creep, real world applications.

1. Introduction

When students become engineers and face real world applications, some of them think that their skills relating to the stress and strain are not sufficient. The same problem arises when heat treatment engineers use simulation software based on the finite element method to tackle deformation and residual stress generated in heat treated parts. The author has been involved in one-day courses on the problem for them, which are operated by private companies.

In the introductory part of the course, total strain, elastic strain, thermal strain, plastic strain and stress, and their relationships inside the parts are explained using materials created independently, as shown in the previous paper [1]. Some schematic diagrams for the purpose of understanding the generation of stress and strains more intuitively were created by expanding the Boden and Tabor’s crystal model containing 40 atoms [2].

In heat treatment processes, creep phenomenon may occur in parts which are treated for a prolonged time at high temperature in furnace [3]. It is known from materials science that creep phenomenon is caused by dislocation or atomic diffusion. Also, from the viewpoint of mechanics of materials, methods to obtain deformation and stress-strains in structures due to this phenomenon have been developed. However, general textbooks on materials science and mechanics of materials have not described introductory for gaining an intuitive understanding of phenomena coupled with elasticity, thermal expansion and creep.

The author, therefore, extended the materials based on the Boden and Tabor’s crystal model to intuitively understand phenomena with creep additionally. On the other hand, in order to identify actual creep phenomena, a simple bending test of the beam was devised. In the following, after summarizing explanations of creep phenomena in textbooks of mechanics of materials and materials science, teaching materials created using the crystal model and the creep bending test were described. 2. Stress and strains in creep phenomena

Gere’s textbook [4], which inherits a traditional teaching of mechanics of materials, explains uniaxial stress and strains in phenomena of elasticity, plasticity and creep in a loaded round bar. For example, when an elastic round bar with initial length L and section area A is strained by the load P to the length L+δ, elastic strain εE and stress σ are obtained as δ/L and P/A, respectively.

We consider a round bar held at a specified temperature and tensile stress. Figure 1 shows a total strain change during this period. Initially, elastic strain occurs, and then phenomena of primary creep, steady state creep, and tertiary creep progress in order with the passage of time [5-8]. Here, total strain is obtained as a sum of elastic strain εE and creep strain εC as follows:

E C . (1)

The two straight lines shown in Figure 2 depict a relationship between creep rate C and stress σ in a steady state creep condition as a double logarithm graph, which is based on results of above-described creep tests under some stress conditions [5, 6]. From this straight line, the following antilog function is obtained:

Proceedings of 2019 JSEE Annual Conference & AEESEAP Workshop – Sendai September 5th-6th, 2019

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W − 12

C nB . (2)

Dislocation creep (or power-law creep) occurs when the parameter n is approximately 3 to 8. On the other hand, diffusional flow appears in a lower stress region when n is almost 1 [5, 6].

Figure 1 Creep curve Figure 2 Stress dependence of creep rate

The mechanism of the creep phenomenon is explained from a viewpoint of materials science [5, 6]. First,

dislocations obstructed by precipitated particles are considered to climb in a crystal structure at higher temperature as shown in Figure 3. This climb causes dislocations to start gliding, which induces creep deformation. Dislocation gliding is likely to be blocked by precipitated particles again, but after a while, may be started by next climb phenomenon. The mechanism of the climb of dislocations is believed to be due to the fact that atoms present in extra half atomic planes of dislocations are dislodged from the lower end by diffusion. The above is the generation mechanism of dislocation creep.

Figure 3 Climb and glide of dislocation Figure 4 Deformation due to diffisional flow

Diffusional flow is considered to be caused by another mechanism in lower stress [5, 6]. In this type of creep, crystal grain showed by solid lines is deformed into the form depicted by broken lines as shown in Figure 4. This deformation is considered to occur when atoms move by diffusion in crystal grains in the direction of the arrows. At somewhat lower temperature, atoms move by diffusion in grain boundaries. The stress and temperature contribute to the diffusion. 3. Stress-strain and their classification

Stress and strains (total, elastic, thermal, plastic and creep) in structures are induced by force, prescribed displacement, temperature change, etc. The author explains this complicated phenomenon using the schematic diagram shown in Figure 5, which includes the relationship between total strain ε, elastic strain εE, thermal strain εTH, plastic strain εP and creep strain εC:

E TH P C . (3)

In Figure 5, stress and elastic strain , which corresponds to changes in atomic force and interatomic distance, respectively, are linked by a double arrow which describes a linear relationship according to the Hooke’s law. On the other hand, plastic strain and creep strain are

Figure 5 Strains and stresses generated in the loaded structure


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expressed using one-way arrows, because they are based on different kinds of stress-related phenomena due to dislocation gliding and/or atoms diffusion.

It is useful to know about the conditions under which plasticity and/or creep occur. Plastic and creep phenomena are considered to be largely dependent on temperature and stress. Frost and Ashby [9] made a total of 40 deformation mechanism maps for pure metals, alloys and ceramics based on fruits of materials science in order to clarify the conditions.

Figure 6 Example of deformation mechanism map Figure 7 Melting point vs. Young’s modulus

(304 stainless steel, grain size: 50µm) (Chart is created using CES EduPack 2019, Granta Design Limited, Cambridge, UK, 2019)

As an example, Figure 6 shows a reproduced deformation mechanism map for the 304 stainless steel (grain size:

50 μm) [9]. The vertical axis corresponds to shear stress σs normalized by shear modulus G in the test piece. On the other hand, the horizontal axis is homologous temperature T normalized by melting point TM. The unit of each temperature is the absolute temperature.

In the deformation mechanism map shown in Figure 6, areas where plasticity, dislocation creep and diffusional flow occur are shown separately. On the other hand, multiple curves corresponding to different strain rates in the figure show influences of temperature and shear stress on progress of the phenomenon. Besides this, a straight line for the ideal shear strength is drawn in the figure.

Creep strain rate equations used to create deformation mechanism maps includes characteristic data: rigidity, atomic volume, Burger’s vector, grain size, diffusivity (corresponding to lattice diffusion, dislocation core diffusion, and grain boundary diffusion), etc., and experimental parameters related to stress and grain size. An Ashby chart [10] is created by the author to depict the relationship between melting point and Young’s modulus in typical metals as shown in Figure 7 for reference of the data.

4. Stress and strains in crystal models under tests with creep

It is not easy to visualize the occurrence of stress and strains in phenomena coupled with elasticity, thermal expansion, plasticity, and creep. The author created schematic diagrams [1] by applying the Boden and Tabor’s crystal model [2] containing 40 atoms to phenomena with elasticity, thermal expansion, and plasticity, in order to understand the situations more intuitively. This diagram is expanded to describe phenomena including creep in the following.

4.1 Crystal model subjected to creep test

Here, the Boden and Tabor’s crystal model is applied to a creep test. Elastic and creep strains occur as shown in equation (1) during the test, and the sum of these strains gives the total strain. Since creep phenomenon occurs at high temperature in ordinary metals, it needs to consider thermal strain. For the sake of simplicity, the generation of thermal strain is ignored by using a material in which creep occurs at room temperature.

Figure 8 shows changes in force-deformation and stress-strain states in the creep test using the crystal model as a schematic diagram. The elastic deformation state A occurs immediately after applying a tensile force to the initial state O. After holding time, the state B is produced due to creep deformation, and finally state C happens by unloading. In the figure, elongations of the crystal model in states A, B and C are shown as a, b and c, respectively. This amount of elongation also corresponds to total strain in each state, since the original length of the model is assumed to be unit length. The total strain c in the unloaded state C means the creep strain is detected in this state.

Stress and strains in each state are shown in Figure 8. Since the initial state O is unloaded, stress and strains are not generated. In the elastic deformation state A, loaded with a tensile force, the elastic strain εE is a, which corresponds to stress σ: Ea. In the state B after the passage of time, a creep phenomenon occurs, which is symbolically depicted as a glide occurred in the crystal. During this glide, the vertical alignment of atoms changes from 8 to 9 lines.


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Figure 8 Force-deformation and stress-strain changes in crystal model subjected to creep test

The total strain ε in state B is b which includes the creep strain εC: c, which is detected in the unloaded state C.

Then, the elastic strain εE in state B is b–c. The stress σ, Ea at the state A is kept in the state B by ignoring the decrease in the cross section of the model. Therefore, b–c of elastic strain εE is equal to a. Since the unloaded state C is no-load and stress-free, the elastic strain εE is zero, which means the total strain ε contains only the creep strain εC: c.

4.2 Crystal model subjected to temperature cycle test

Stress is generated in a region where thermal expansion is constrained locally from a structure shape, and this stress is relieved by holding at high temperature. In order to understand phenomena caused by such thermal expansion constraints, we consider a crystal model subjected to a temperature cycle test. The relational expression of strains in this case corresponds to the condition where plastic strain is omitted in equation (3) as follows:

E TH C . (4)

Figure 9 Deformation and Stress-strain changes in crystal model subjected to temperature cycle test

As shown in Figure 9, the crystal model is heated from room temperature at the initial state O to a predetermined

temperature in the constrained state A when elastic phenomenon occurs. After holding this condition, the elastic and creep phenomena happen in the state B. Thereafter, returning the model to room temperature produces the initial temperature state C, and releasing the restraint obtains the restraint-free state D. When both ends are constrained, total strain ε in the model is zero. In the state A, the thermal strain εTH: a appears as an elongation as shown in the free expansion state. Using equation (4), the elastic strain εE becomes –a from the thermal strain εTH: a, and E(–a) is obtained as the corresponding stress σ.

In the state B, since the temperature condition in the state A continues, the thermal strain εTH is a, and there is no change in the free thermal expansion state. As time passes under this condition, a creep phenomenon occurs, and as a result, a glide appears in the model as shown in Figure 9. The vertical alignment of the atoms changes from 8 to 7 lines, which results in –d as creep strain εC. Therefore, the elastic strain εE in the state B becomes d–a, corresponding to the stress σ: E (d–a), which means that the absolute value of the compressive stress has decreased.

Since the thermal strain εTH is zero and the creep strain εC remains at –d in the state C when the temperature returns to initial value, the elastic strain εE is d according to the relation of equation (4). Therefore, the stress σ in this state is Ed, which is a tensile stress. Furthermore, when the constraint is removed in the state D, the stress σ is zero since the elastic strain εE is zero. On the other hand, total strain ε is equal to –d of creep strain εC, and as a result the model shrinks. In the above explanation, the temperature dependence in Young’s modulus is ignored due to simplicity.

The stress reduction in restrained strauctures due to creep such as shown in the above case is called as


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relaxation or stress relaxation [6-8]. For example, it occurs at flange tightening bolts as a practical problem.

5. Simple test of creep phenomenon It is meaningful to observe structures in which

creep phenomena occurred. Figure 10 is a part of photographed lead pipes under creep by their own weight on a 75-year-old building in southern New England in the book by Frost and Ashby [9]. They described the creep-induced curvature of these pipes is typical of Victorian lead water piping.

On the other hand, creep deformation conditions of 75 years external drain, in Cambridge, England, and 70 years hot water pipe, in Montgomery, Wales, were studied by them. Their deformation mechanism map of 0.5% antimony lead for a grain size of 50 μm shows that the temperature and stress condition ranges of drain and hot pipe are both in the region of diffusional flow, and are crossed by a creep strain rate line of 10-12/s. It is found that another map for a grain size of 1 mm slows the creep-rate by a factor of 103.

An experiment on creep phenomena in pure bending of lead bars was reported by Tapsell and Johnson [11]. They measured creep deflections in bended lead bars, 7 in height, 2in wide and 15 in long, at room temperature. Time changes of fiber expansion or contraction were measured at each grid, which was drawn at equal intervals (every 1/2 inch) from the neutral axis on both front and back sides.

The author devised a simple four-point bending apparatus to conduct the similar test by Tapsell and Johnson. A lead beam (5 mm square and 500 mm long) marketed for adjusting a horizontal level of stone monuments was installed on two L-shaped hooks with a 300 mm distance as shown in Figure 11 (a). A PET bottle with water (120 g) was connected to loading wires provided at each end of the specimen. The amount of deflection at the center of the bended specimen in the elastic state was about 0.8 mm when loading initially. By holding for 6 days at room temperature, the bending due to creep appeared as shown in Figure 11 (b). The amount of beam deflection at the center was measured as about 2.3 mm.

The deflection produced at the beam center by a pure bending due to creep can be determined by the graphical method described in the work by Lubahn and Felgar [12]. Although the chemical composition of lead used in the test shown in Figure 11 is not clear, it is educationally meaningful to try to calculate the deflection by assuming mechanical property data.

6. Creep phenomena in real world problems

Creep phenomena occur in boilers, turbines, engines, vessels, piping, etc., operating at high temperatures. Design of these equipment and machines needs to consider creep essentially. For example, Frost and Ashby [9] showed some case studies on pipes, reactors, metal-forming, metal-shaping, etc., with their maps. On the other hand, Arnold et al. [13] considered creep in materials selection for aerospace systems.

The author’s course explains creep phenomena in nitriding processes where nitrogen atoms diffuse into processed metals [14]. For example, Sienz et al. [15] performed low-energy implantation with nitrogen from one side of a stainless steel (X5CrNi18.10) plate at 400 °C. Referring to the deformation mechanism map of 304 steel in Figure 6, it is considered that this specimen may creep.

Figure 10 Lead piping under creep by its own weight

(a) Test apparatus

(b) Specimen after 6 days

Figure 11 Four-point bending creep test for lead beam

Figure 12 Curvature changes in plate during nitriding


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The plate specimen provided by Sienz et al., with a size of 10 x 30 mm2 and a thickness of 1 mm, was nitrided for 63 min at 400 °C and held until 90 min at the same temperature after stopping the nitrogen injection. During the above experiment, changes of bending curvature in the plate were obtained using an in-situ measurement as shown in Figure 12. The author applied heat treatment simulation with a formation model of expanded austenite by nitriding [16] to the experiment. Figure 12 includes simulated curvature changes corresponding to presence or absence of creep phenomenon. In addition, the author clarified the generation mechanism of bending and stress generation by analysing distributions of the simulated longitudinal component of each strain: elastic, thermal expansion, diffusion expansion, and creep in the plate.

The author’s simulation did not consider the occurrence of plasticity. Also, his creep model and parameters were not derived from the condition of steel during nitriding. The author’s course describes additional simulation results considering creep for nitriding processes of steels [14].

7. Conclusion

Creep may occur in structures of machine equipment and products in processes at high temperature and in a prolonged time. Understanding the nature of this phenomenon, which is affected by temperature and time, should be use knowledge from both mechanics of materials and materials science. Although the author’s teaching material uses primitive models, it is created based on the viewpoints from the two sides. This material will contribute to enhance learning of design and manufacturing methods based on deeper understanding of the phenomena by engineers.

The author wish to express his thanks to the people involved in one-day courses for engineers for giving him the opportunity to do this work.

References 1. Arimoto, K., 2018, “Learning about Stress and Strain for Real World Applications”, JSEE International Session in

66th Annual Conference of Japanese Society for Engineering Education, pp. 59-64. 2. Bowden, F. P. and Tabor, D., 1973, “Friction: an Introduction to Tribology”, Science Study Series No. 41, Anchor

Press. (The Japanese version, translated by Hara, Y. was published in 1974) 3. Arimoto, K., 2016, “Thermally-Processed Steels: Residual Stresses and Distortion”, Encyclopedia of Iron, Steel,

and Their Alloys, Taylor and Francis, Vol. 13, pp. 3605-3633. 4. Gere, J. M. 2004, “Mechanics of Materials”, 6th Ed., Thompson Publishing. 5. Ashby, M. F. and Jones, D. R. H., 1980, “Engineering Materials: An Introduction to their Properties and

Applications”, Pergamon Press. (The Japanese version, translated by Horiuchi, R., Kaneko, J. and Otsuka, M., was published in 1985)

6. Ashby, M. F., Shercliff, H., and Cebon, D., 2014, “Materials-Engineering, Science, Processing and Design”, 3rd Ed., Butterworth-Heinemann, Elsevier.

7. Finnie, I. and Heller, W. R., 1959, “Creep of engineering materials”, McGraw-Hill (The Japanese version, translated by Suzuki, F., was published in 1965)

8. Odqvist, F. K. G. and Hult. J., 1962, “Kriechfestigkeit metallischer Werkstoffe”, Springer-Verlag (The Japanese version, translated by Murakami, S., was published in 1976)

9. Frost, H. J. and Ashby, M. F., 1982, “Deformation Mechanism Maps (The Plasticity and Creep of Metals and Ceramics)”, Pergamon Press (Webversion: http://engineering.dartmouth.edu/defmech/)

10. CES EduPack software, Granta Design Limited, Cambridge, UK, 2019 (www.grantadesign.com). 11. Tapsell, H.J. and Johnson, A. E., 1935, “An Investigation of the Nature of Creep under Stresses Produced by Pure

Flexure”, J. Inst. Metals. (British), Vol.58 (2), pp. 121-137. 12. Lubahn, J. D. and Felgar, R. P., 1961, “Plasticity and Creep of Metals”, John Wiley and Sons, Inc. 13. Arnold, S.M., Cebon, D., Ashby, M., 2012, “Materials Selection for Aerospace Systems”, NASA/TM-2012-

217411. 14. Arimoto, K., Ikuta, F., and Yamanaka, S., 2012, “Studies on Distortions in Nitrided Steel Cylinders Using

Computer Simulation”, Int. J. Microstructure and Materials Properties, Vol. 7(4), pp. 353-366. 15. Sienz, S., Mandl, S. and Rauschenbach, B., 2002, “In Situ Stress Measurements during Low-Energy Nitriding of

Stainless Steel”, Surface & Coatings Technology, Vol. 156 (1-3), pp. 185-189. 16. Arimoto, K., Ikuta, F., Yamanaka, S., and Funatani, K., 2010, “Development of Simulation Tool for Predicting

Distortion and Residual Stress in Nitrided Parts”, Int. J. Microstructure and Materials Properties, Vol. 5 (4/5), pp. 386-398.

Biography Kyozo Arimoto has been a consultant at Arimotech Ltd in Japan since 2002. He was a mechanical engineer for

Hitachi Shipbuilding Co. Ltd. after graduating from Osaka Prefectural Technical College in 1972. He moved to Century Research Center Corp. in 1986 and created some specialized computer codes until 1997. After that he was involved in developing a heat treatment module of a code at SFTC, USA, for four years. He has been developing a new system for determination of cooling characteristics of quenchants, and also introducing the Ashby’s materials selection method in Japan. He has published over 50 articles for reporting the above mentioned works. FIFHTSE.


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Learning about Stress and Strain Due to Creep for Real