# Increase in Elastic Stress Limits by Plastic Conditioning: Influence of Strain Hardening on Interference Fits

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

**p**(identical to the negative radial stress

_{F}**p**= −

_{F}**σ**) are produced during the joining process, as shown in Figure 1, along the load path $\overline{0E}\hspace{0.17em}-\hspace{0.17em}\overline{EF}$ (blue arrow/solid line).

_{r}**α**of the straight line $\overline{0E}\hspace{0.17em}$ is determined exclusively by the diameter ratio

**Q**of the component, as shown.

_{A}**F**then marks the stress state at the inner diameter of the outer part of the interference fit after the completion of the joining process. Since point

**F**lies on the graph of yield strength, increases in stress lead to changes in stress along with the yield strength, and thus, inevitably, to further plastic deformations.

**K**by increasing the radial stress. After this increased stress is relieved, elastic relief takes place along the relief path $\overline{KK\prime}\hspace{0.17em}$(brown arrow/dashed), whereby the inclination angle of the straight line of relief, in accordance with Kollmann’s work, is also defined by the diameter ratio

**Q**of the component according to Equation (1), and is, thus, always parallel to the load line. As a result of the plastic deformations, the stress state that develops thereafter must lie in a region to the right of −

_{A}**p**along the relief path (for example, at point

_{F}**K′**). Thus, the previous joint pressure

**p**can no longer be reached due to the loss of plastic interference, which may result in impairments of the operational safety or the service life of the components [6,7]. A description of the points in Figure 1 is as follows:

_{F}**E**: Purely elastically joined state at the yield strength;

**F**: Elastically–plastically joined state (conventional according to DIN 7190-1 [8]);

**K**: Stress state on the yield strength with plastic stress increase after joining;

**K′**: State of stress after relief of the plastic stress at point

**K.**

## 3. Materials and Methods

#### 3.1. Problem-Solving through the Plastic Conditioning of Interference Fits—Basic Concept of the Procedure

**p**to a hub (see Figure 2).

_{K}#### 3.2. Analytical Investigations (Two-Dimensional, Ideal Plastic Calculation Example)

**Q**and a yield strength of

_{A}= 0.45**R**, followed by an explanation of the numerical calculation methods in Section 3.3; reference is also made to the respective stress states in the principal stress plane (Figure 3). For the sake of clarity, the shaft is not shown here; the following considerations are based on the radial stress at the inner diameter of the hub. This is identical in magnitude to the joint pressure of the joined connection with the shaft and is, thus, a decisive parameter for the transmissible forces and torques. Based on the ductile ideal plastic behavior of materials according to the GEH, the ESZ, and a safety requirement against plastic deformations of

_{eL,A}= 370 MPa**S**, the achievable joint pressures of conventionally purely elastically joined interference fits were compared with those of plastically conditioned ones. The GEH was used here as the equivalent stress hypothesis because it agrees very well with the experimental results obtained in practice for a ductile isotropic material; for this reason, it is of greater importance for engineering practices than SH, which is used primarily to describe the brittle behavior of materials such as gray cast iron. The von Mises yield function (GEH) of the principal stresses is as follows Equation (2):

_{PA}= 1.5**σ**

**for a yield strength of**

_{v,lim}**R**was calculated with a safety against plastic deformation of

_{eL,A}= 370 MPa**S**as the quotient of these two, to

_{PA}= 1.5,**σ**

**. According to Equation (4), in compliance with the GEH, a yield pressure of**

_{v,lim}= 247 MPa**p**was calculated (see also point

_{N,lim}= 113 MPa**F**in Figure 3). From Equation (1) in connection with Equation (3) for

_{elast}**σ**this was as follows:

_{z}= σ_{z}= 0,**S**). For the current example assuming the ideal plastic behavior of the material, a conditioning pressure

_{PA}**p**

_{Kond,max}=**−σ**

_{r}**= 311 MPa**was used (point

**K**in Figure 3). After conditioning, this also represented the new load limit

**p**for purely elastic joint pressures with a safety against plastic deformation of

_{N,lim}**S**. Only when this pressure was exceeded at the inner diameter of the hub did further plastic deformation occur.

_{PA}= 1.0**S**, i.e., the searched stress point was located together with point

_{PA}= 1.5**F**on the common equivalent stress curve

_{elast}**σ**

**(shown by the dashed black ellipse in Figure 3), which represents the joint state (point**

_{v,lim}= 370 MPa/1.5**G**) of the hub after the previous conditioning. The equation for the relief straight line $\overline{KD}\hspace{0.17em}$ in Figure 3 is in Equation (5):

**D**in Figure 3). According to Equation (6), the joint pressure of the conditioned interference fit here is

**p**

_{E,Kond}=**−σ**

_{r}**= 246 MPa**(see point

**G**in Figure 3), which corresponds to an increase in the transmission capacity of the interference fit to

**218%**.

**D**after complete relief at the end of the conditioning process for the GEH can be calculated as follows:

#### 3.3. Numerical Verification of the Analytical Investigations (Ideal Plastic)

**p**, as shown in Figure 2, was applied as a load. This was gradually increased up to the conditioning pressure

_{i}**p**(point

_{Kond,max}= 311 MPa**K**in Figure 3), and then gradually reduced again to

**0**(point

**D**). This was followed by reloading up to

**p**

_{E,Kond}**= 246 MPa**(point

**G**) according to the load history analogous to Figure 3. The load steps examined for comparison are listed in Table 2.

**p**for load steps 1 and 4. These allow comparison of the stresses in the hub for a conventionally purely elastically joined interference fit according to DIN 7190-1 (pure elastic DIN 7190 (

_{i}**F**)) and a plastically conditioned one (conditioned ideal plastic (

_{elast}**G**)). For reasons of better clarity in direct comparison, the calculation results for the conditioned hardening material (conditioned hardening (

**G**)), which is described in more detail in Section 3.4, are presented as well. The values of the graphs at the radius coordinate of

^{+}**30 mm**show the stress state of the respective load steps at the inner diameter of the hub for the different joining states and are thus directly comparable with the analytical results shown in Figure 3.

**F**

_{elast},**G,**and

**G**), with a safety against plastic deformation of the outer part

^{+}**S**(

_{PA}= 1.5 = S^{+}_{PA}**S**→ after hardening of the material) based on a yield strength

^{+}_{PA}**R**. The values for the points

_{eL,A}= 370 MPa**F**(

_{elast}**σ**

**) and**

_{v}= 245 MPa**G**(

**σ**

**) only differed by the numerical deviations and were in good approximation to the analytical specifications.**

_{v}= 253 MPa**F**,

_{elast}**G**, and

**G**are shown, which are identical to the magnitude of the joint pressure (

^{+}**p**=

_{F}**−σ**=

_{r}**p**) at the radius coordinate at

_{i}**30 mm**.

**F**(purely elastically joined state, according to DIN 7190-1), this is

_{elast}**σ**=

_{r}**−110 MPa**and at point

**G σ**=

_{r}**−245 MPa**. When comparing the two, it is clear that the conditioned hub has a significantly higher value of radial stress for transmitting forces and torques, while the hub maintains the same level of safety against plastic deformations as in a conventionally joined purely elastic state, according to DIN 7190-1.

**G**in Figure 6) correspond to those in Figure 3 and illustrate the residual stresses caused by plastic conditioning. In contrast, the tangential stress at point

**F**of the inner diameter of the hub is

_{elast}**σ**

_{t}**= 171 MPa**.

**K, G,**and

**D**are also presented. Because of the relatively large differences in the stress gradients, the partial images in Figure 7 were each given their own scale for better graphical resolution of the stress values.

**very good agreement can be seen in the results of the numerical calculations with the analytical results shown in Figure 3**.

#### 3.4. The Influence of Strain-Hardening

**σ**to

_{0}**σ**(Figure 8). After a subsequent purely elastic relief to

_{n2}**σ**in Figure 8, renewed reloads between

_{n3}**σ**and

_{n3}**σ**were also purely elastic. New plastic stresses only occurred when the previous stress level

_{n2}**σ**was exceeded. Thus, at point

_{n2}**σ**, the same safety against plastic stress

_{n4}**S**was achieved as with the stress state

_{P}**σ**(without conditioning), but at a significantly higher stress level.

_{X}**p**. The changed values of the individual steps are shown in Table 4 and are explained in more detail below.

_{i}**p**according to Figure 2 was applied as the load. This was gradually increased up to the conditioning pressure, which in this experiment was

_{i}**p**(point

_{Kond,max}= 342 MPa**K**in Figure 3). In contrast to the ideal plastic material, the equivalent stress here increased to

^{+}**σ**due to the strain hardening of the material.

_{v}= 407 MPa,## 4. Discussion of the Results of the FE Calculations with Hardening Material and Comparison with Ideal Plastic Investigations

**-x-x-x-x-**line. The gradient of the stress change in the respective load step is mainly defined by the stress-strain curve (represented by the blue dashed line in Figure 9), which is determined by the values found in Table 3. The stress range applied during the FE calculation is represented on the stress–strain curve with a red dashed line.

**p**(point

_{i}= 0 MPa**D**), this equivalent stress then marks the new amount of the yield strength (represented by the black dash-dot line in Figure 4 and Figure 9 and also by the red ellipse in Figure 3). Each reload is then made purely elastic up to this value. Only if this new yield strength is exceeded will further plastic deformations occur.

^{+}**σ**

^{+}**for the predetermined safety against plastic deformation of**

_{v,lim}**S**with

_{PA}= 1.5**σ**

^{+}**(represented by the red line-point ellipse in Figure 3). Their point of intersection with the load straight line $\overline{{D}^{+}{K}^{+}}\hspace{0.17em}$ indicates point**

_{v,lim}= 274 MPa**G**for the desired stress state, with

^{+}**p**. The identity of the aforementioned equivalent limit stress shows that the tangential stresses disappear here.

_{E,Kond}= 274 MPa**σ**, the equivalent stress at point

_{v}= 274 MPa**G**(Figure 4) is greater than that for the ideal plastic material (

^{+}**σ**at point

_{v}= 253 MPa**G**), where the yield strength

**R**(black line-point line in Figure 4) has also increased as a result of the material strain hardening. The identity of the safety against the plastic deformation of

^{+}_{eL,A}**S**(

_{PA}= 1.5 = S^{+}_{PA}**S**→ after hardening of the material) was a prerequisite for the comparability of the investigated modes and a central point of the task (see Section 3.2). However, for the sake of clarity, the safety shown in Figure 3 and Figure 4 was symbolized as differences between

^{+}_{PA}**R**and

_{eL,A}**σ**or

_{v,lim}**R**and

^{+}_{eL,A}**σ**

^{+}**. Therefore,**

_{v,lim}**S**at this point appears to be lower than

_{PA}**S**due to the smaller amounts of

^{+}_{PA}**R**and

_{eL,A}**σ**. The values for radial stress are shown in Figure 5 (represented by the red line), which illustrates their additional reinforcement compared with the ideal plastic material (represented by the blue dashed line).

_{v,lim}## 5. Conclusions Regarding Engineering Practice and Industrial Applications

**243%**with

**the plastic conditioning method,**compared to conventional elastically joined interference fits, taking into account the material’s strain hardening. With the ideal plastic behavior of the material, this increase in potential was

**218%**. In addition to the residual stresses mentioned in Section 3.3, the elastic potential of the component is additionally increased by hardening during the conditioning, which means that higher joint pressures for the transmission of forces and torques can be realized in engineering practice.

## 6. Summary and Outlook

## 7. Patents

**plastic conditioning of**

**interference fits**in drive technology, which forms the basis for the results presented here, was published in the patent specification

**DE 10 2016 004 223 B3**. This also contains further details for practical applications and engineering implementation. In addition, the basic physical relationships and decisive influencing parameters are shown.

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Abbreviations

Abbreviation | Unit | Meaning |

d | mm | Diameter coordinate (control variable) |

D_{aA} | mm | Outer diameter of the outer part |

D_{F} | mm | Joint diameter (nominal) |

D_{iA} | mm | Inner diameter of the outer part |

D_{PA} | mm | Plasticity diameter of the outer part |

D_{σr(GEH),} D_{σt(GEH)} | MPa | Stress values at point D in the principal stress plane for von Mises yield criterion (GEH) |

E_{A} | MPa | Young’s modulus of the outer part |

F_{GEH} | MPa | Von Mises yield function |

k | MPa | Critical value (yield strength in shear) |

m | - | Factor for determining the angle of inclination |

p_{i} | MPa | Internal pressure of the disc |

p_{E} | MPa | Elastic joint pressure at the yield strength |

p_{E, Kond} | MPa | Elastic joint pressure after previous conditioning |

p_{F} | MPa | Joint pressure |

p_{K} | MPa | Conditioning pressure |

p_{Kond,max} | MPa | Maximum joint pressure when undergoing conditioning |

p_{N,lim} | MPa | Yield pressure (elastic limit) of an outer part |

Q_{A} | - | Diameter ratio of the outer part |

R_{eL,A} | MPa | Lower yield strength of the outer part |

R^{+}_{eL,A} | MPa | Lower yield strength of the outer part after hardening |

r | mm | Radius |

S_{P} | - | Safety against plastic deformation |

S_{PA} | - | Safety against plastic deformation of the outer part |

S^{+}_{PA} | - | Safety against plastic deformation of the outer part after hardening |

y | MPa | Intersection of the relief straight line with the ordinate (tangential residual stress after complete relief) |

α | ° | Inclination angle for load line and relief straight line |

ε_{v} | - | Equivalent strain |

ν_{A} | - | Poisson’s ratio of the outer part |

σ_{r} | MPa | Radial stress |

σ_{t} | MPa | Tangential stress |

σ_{v} | MPa | Equivalent stress |

σ_{v,G} | MPa | Equivalent stress at point G |

σ_{v,}_{G+} | MPa | Equivalent stress at point G^{+} |

σ_{v,lim} | MPa | Equivalent limit stress |

σ^{+}_{v,lim} | MPa | Equivalent limit stress after hardening of the material |

σ_{1}, σ_{2}, σ_{3} | MPa | Principal stresses of the stress tensor |

AT | Outer part of the PV | |

ESZ | Plane stress state | |

FE | Finite elements | |

FEM | Finite element method | |

GEH | Von Mises yield criterion | |

PV | Interference fit | |

MPV | Multiple interference fit | |

SH | Shear stress hypothesis according to TRESCA | |

IKAT | Institute of Construction and Drive Technology (TU Chemnitz) | |

IKTD | Institute for Engineering Design and Industrial Design (University of Stuttgart) |

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**Figure 1.**Principal stresses at the inner diameter of the outer part of an interference fit with an ideal plastic yield strength (GEH).

**Figure 3.**Stress states (analytically and numerically determined) at any point of the inner diameter (marked in red) of an outer part with

**Q**in the principal stress plane, based on the von Mises yield criterion during conditioning.

_{A}= 0.45**Figure 4.**Von Mises stresses in the stress states of points

**F**,

_{elast}**G**, and

**G**with a safety against the plastic deformation of the outer part

^{+}**S**(

_{PA}= 1.5 = S^{+}_{PA}**S**→ after hardening of the material).

^{+}_{PA}**Figure 5.**Radial stresses in the stress states of the points

**F**,

_{elast}**G**, and

**G**with a safety against plastic deformation of the outer part

^{+}**S**(

_{PA}= 1.5 = S^{+}_{PA}**S**→ after hardening of the material).

^{+}_{PA}**Figure 6.**Tangential stresses in the stress states of the points

**F**,

_{elast}**G**, and

**G**with a safety against plastic deformation of the outer part

^{+}**S**(

_{PA}= 1.5 = S^{+}_{PA}**S**→ after hardening of the material).

^{+}_{PA}**Figure 7.**FE stress graphics (hub quarter section) of the equivalent stresses (GEH) for the load points

**K**,

**G,**and

**D**[12]. (

**a**)

**σ**

**at point**

_{v}**K**; (

**b**)

**σ**

**at point**

_{v}**G**; (

**c**)

**σ**

**at point**

_{v}**D**.

**Figure 8.**Example of strain hardening (multilinear) [12].

**Figure 10.**Shaft–hub connection with a cone clamping element (hub clamped on the inside) from RINGSPANN GmbH (Image source: RINGSPANN GmbH).

Geometry Data | |||

Hub outer diameter | D_{aA} | 133.33 | mm |

Hub inner diameter | D_{iA} | 60.00 | mm |

Hub diameter ratio | Q_{A} | 0.45 | - |

Hub material | |||

Designation | C 45 | ||

Yield strength | R_{eL,A} | 370 | MPa (ideal plastic) |

Young’s modulus | E_{A} | 205,000 | MPa |

Poisson’s ratio | ν_{A} | 0.3 | - |

Technological Data | |||

Joint pressure for conditioning | p_{Kond,max} | 311 | MPa |

Target safety against plastic deformation | S_{PA} | 1.5 | - |

**Table 2.**Load steps of the load history of the ideal plastic behavior of a material, according to Figure 3.

Load Step | p_{i}/MPa |
---|---|

1 (F)_{elast} | 113 |

2 (K) | 311 |

3 (D) | 0 |

4 (G) | 246 |

σ_{v} | ε_{v} |
---|---|

(MPa) | (-) |

370.00 | 0.00171 |

390.00 | 0.00182 |

400.00 | 0.00189 |

410.00 | 0.01303 |

420.00 | 0.01417 |

430.00 | 0.01549 |

440.00 | 0.01665 |

470.00 | 0.02075 |

500.00 | 0.02543 |

530.00 | 0.03097 |

560.00 | 0.03775 |

590.00 | 0.04612 |

620.00 | 0.05967 |

**Table 4.**Load steps of the load history with hardening of the material, according to Figure 3.

Load Step | p_{i}/MPa |
---|---|

1 (F)_{elast} | 113 |

2 (K)^{+} | 342 |

3 (D)^{+} | 0 |

4 (G)^{+} | 274 |

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**MDPI and ACS Style**

Schierz, M.
Increase in Elastic Stress Limits by Plastic Conditioning: Influence of Strain Hardening on Interference Fits. *Appl. Mech.* **2022**, *3*, 375-389.
https://doi.org/10.3390/applmech3020023

**AMA Style**

Schierz M.
Increase in Elastic Stress Limits by Plastic Conditioning: Influence of Strain Hardening on Interference Fits. *Applied Mechanics*. 2022; 3(2):375-389.
https://doi.org/10.3390/applmech3020023

**Chicago/Turabian Style**

Schierz, Mario.
2022. "Increase in Elastic Stress Limits by Plastic Conditioning: Influence of Strain Hardening on Interference Fits" *Applied Mechanics* 3, no. 2: 375-389.
https://doi.org/10.3390/applmech3020023