# Assessment of the Durability of Threaded Joints
## 4 Shakedown of the Threaded Joints
### Crack-Free Joints  
The progressive shape change conditions in the case of repetitive variable loading are determined according to the elastic-plastic body shakedown theories and the experimental data. [PERSON] [44] formulated the kinematic theorem (two statements) in terms of kinematic definitions that determine the work of external forces and the dissipation of plastic energy over the cycle time [45] (p. 25):  
\[ \begin{array}{l}\int\limits_{0}^{T}\left\{\left\{y\right.\right.x_{i}^{0}u_{ i0}dV+\int\limits_{A_{p}}p_{i}^{0}u_{i0}dA\right\}dt\geq\int\limits_{0}^{T}dt \int\limits_{V}F(e_{ij0}^{\prime\prime})dV\\ \int\limits_{0}^{T}\left\{\left\{y\right.\right.x_{i}^{0}u_{i0}dV+\int\limits_ {A_{p}}p_{i}^{0}u_{i0}dA\right\}dt<\int\limits_{0}^{T}dt\int\limits_{V}F(e_{ ij0}^{\prime\prime})dV\end{array}  \tag{4}\]  
[PERSON] and [PERSON] [45] (p. 33) proposed representing plastic energy dissipative function \(F(e_{ij0}^{\prime\prime})\) as follows:  
\[\int\limits_{0}^{T}dt\int\limits_{V}\min\left[(\sigma_{ij}-\sigma_{ij}^{(c)})e _{ij0}^{\prime\prime}\right]dV \tag{5}\]  
Following the previous one, it can be written as:  
* a given structure shakedown: \[\int\limits_{0}^{T}dt\int\limits_{V}(\sigma_{ij}-\sigma_{ij}^{(c)})e_{ij0}^{ \prime\prime}dV>0\] (6)
* a given structure that does not undergo shakedown, i.e., the structure must collapse eventually due to cyclic plastic deformations: \[\int\limits_{0}^{T}dt\int\limits_{V}(\sigma_{ij}-\sigma_{ij}^{(c)})e_{ij0}^{ \prime\prime}dV<0\] (7)  
Therefore, the elastic-plastic body shakedown condition will be as follows:  
\[\int\limits_{0}^{T}\left\{\left\{\int\limits_{V}x_{i}^{0}u_{i0}dV+\int\limits_ {A_{p}}p_{i}^{0}u_{i0}dA\right\}dt<\int\limits_{0}^{T}dt\int\limits(\sigma_{ ij}-\sigma_{ijt}^{(e)})e_{ij0}^{\prime\prime}\right. \tag{8}\]  
The kinematic shakedown theorem is based on the fundamental assumption of the admissible cycle of plastic strain rates. According to the definition of the plastic strain increment in such a cycle over a specified time interval:  
\[\Delta e_{ij0}^{\prime\prime}=\int\limits_{0}^{T}e_{ij0}^{\prime\prime}dt \tag{9}\]  
The increments of residual translations:  
\[\Delta u_{i0}=\int\limits_{0}^{T}u_{i0}dt \tag{10}\]Considering that the integration over time and over the volume of the body (or over its surface) is mutually independent:  
\[\int\limits_{0}^{T}dt\int\limits_{V}x_{i}^{0}u_{i0}dV=\int\limits_{V}x_{i}^{0} \Delta u_{i0}dV \tag{11}\]  
\[\int\limits_{0}^{T}dt\int\limits_{A_{p}}p_{i}^{0}u_{i0}dA=\int\limits_{A_{p}}p_ {i}^{0}\Delta u_{i0}dA \tag{12}\]  
Minimum values of continuous stresses:  
\[\sigma_{ij}^{0}=\sigma_{ij}-\sigma_{ij}^{(c)} \tag{13}\]  
Equation (8) can be written as follows:  
\[\int\limits_{V}x_{i}^{0}\Delta u_{i0}dV+\int\limits_{A_{p}}p_{i}^{0}\Delta u_{ i0}dA<\int\limits_{V}\sigma_{ij*}^{0}\Delta e_{ij0}^{\mu}dV \tag{14}\]  
The shakedown theory widely uses approximate methods. The approximate kinematic method enables an approximation of the true failure mechanism and a reduction in the error of the threshold cycle parameters. The increments in plastic strain are determined according to the potential nature of the failure mechanism and the selected redistribution of displacements \(\Delta u_{i0}\). The respective fictive yield stresses \(\sigma_{ij*}^{0}\) are determined by using the associated flow rule. According to the experimental data, crack initiation and propagation take place in the weakest link of the threaded joint, i.e., in the stud, after a certain number of cycles. Plastic strains occur in the outer layers of the stud before crack initiation, under the action of repetitive variable load. As a rod with circular cross-section, the stud is subjected to axial force \(F\) and symmetrically variable bending moment \(M\) (\(-M^{*}<M_{f}<M^{*}\)). Figure 12 shows a cross-section of a stud and the stress distribution under \(M\) and \(F\) forces leading to yield stresses (\(\sigma_{y}\)--yield stress). The tension zone \(A\) is equal to the compression zone \(C\).