FEM-based shakedown analysis of hardening structures
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This paper develops a new finite element method (FEM)-based upper bound algorithm for limit and shakedown analysis of hardening structures by a direct plasticity method. The hardening model is a simple two-surface model of plasticity with a fixed bounding surface. The initial yield surface can translate inside the bounding surface, and it is bounded by one of the two equivalent conditions: (1) it always stays inside the bounding surface or (2) its centre cannot move outside the back-stress surface. The algorithm gives an effective tool to analyze the problems with a very high number of degree of freedom. Our numerical results are very close to the analytical solutions and numerical solutions in literature.
KeywordsRatchetting Kinematic hardening Two-surface plasticity Shakedown FEM
Shakedown analysis for hardening structures has been investigated by many researchers. Among hardening models, the isotropic hardening law is generally not reasonable in situations where structures are subjected to cyclic loading because it does not account for the Bauschinger effect and rejects the possibility of incremental plasticity. The unbounded kinematic hardening model has already been introduced theoretically by Melan  and later by Prager . Applications of this model have been investigated by Maier  and Ponter . The unbounded kinematic hardening model cannot estimate the plastic collapse and also incremental plasticity but only low-cycle fatigue, while low-cycle fatigue limit with the kinematical hardening model seems not to be essentially different from the perfectly plastic model, cf. Gokhfeld and Cherniavsky  and Stein and Huang .
Introducing a bounding surface in Melan-Prager's model, a two-surface model of plasticity with a fixed bounding surface is achieved which appears to be most basic, suitable and simple for shakedown analysis. Application of bounded kinematic hardening model was introduced theoretically and numerically by Weichert and Groß-Weege  who used the generalized standard material model (GSM). They used Airy's stress function to satisfy the equilibrium conditions in the interior of the structures fulfilled. Shakedown theorems for bounded linear and nonlinear kinematic hardening have been proposed by Bodovillé and de Saxcé , Pham [9, 10] and Nguyen .
Numerical investigations for bounded kinematic hardening using basic reduction technique have been introduced by Staat and Heitzer [12, 13] and Stein and Zhang . By the lower bound approach, it permits to avoid the nondifferentiability of the objective function, which must be regularized via internal dissipation energy and there is no incompressibility constraint in nonlinear programming problem, but this approach suffers from nonlinear inequality constraints.
A company of lower bound algorithm is the upper bound algorithm, which is based on Koiter theorem. For perfectly plastic structures, the upper bound algorithm has been established by Yan and Nguyen Dang [15, 16] and Yan et al. . The major numerical obstacle in this approach is the singular property of plastic dissipation function. Dealing with this difficulty, the researchers replaced the original dissipation function by , where ϵ0 is a very small number. This technique is also used in our algorithm.
By using the static approach and the criterion of the mean, Nguyen Dang and König  showed that the shakedown solution can be obtained by a maximization or a minimization problem. The yield criterion of the mean was further applied in practical computations by displacement method and equilibrium finite element by Nguyen Dang and Palgen .
A very efficient primal-dual algorithm, which can derive lower and upper bound simultaneously of shakedown limit load factor for complicated structures, has been introduced by Vu, Yan and Nguyen Dang [20, 21, 22] and Vu . In these works, dual relationship between upper bound and lower bound for shakedown analysis of perfectly plastic structures has been proven. Theoretically speaking, primal-dual algorithm helps to find a very accurate solution of shakedown analysis problem.
While using the finite element method (FEM) for limit and shakedown analysis, the stress method can be used, but this method is restricted since for certain structures, it is very difficult to find appropriate stress function, so the displacement method is preferred to make the numerical approach as general as possible.
For the structures with hardening material, it is difficult to prove the relationship between upper bound and lower bound because of the complication of the objective function. Furthermore, in the static approach, it is difficult to present alternating limit and ratcheting limit separately. In this paper, we have presented a FEM-based upper bound algorithm for shakedown analysis of bounded kinematic hardening structures with von Mises yield criterion. By the direct plasticity methods, shakedown analysis is a nonlinear programming problem. The present algorithm can deal with complicated realistic structures which are modelled by 3D, 20-node elements with huge number of degree of freedom. Two numerical examples are included to validate the algorithm and to study the influence of hardening effect.
Bounded kinematic hardening model
where π is the back stress. If hardening is unbounded, π is infinite.
- 1.Centre of subsequent yield surface cannot move outside the back-stress surface. This is expressed by(3)
Subsequent yield surface always stays inside bounding surface. This is expressed by
In the preceding conditions, Equations 3 and 4, equalities occur when the subsequent surface touches bounding surface. We have proven that bounding conditions (3) and (4) are exactly equivalent. See detail in the study of Pham and Staat .
Shakedown formulation based on Koiter's theorem
The first term in the right hand side of Equation 6 is plastic energy dissipation of perfect plasticity material, and the second term is hardening effect. Evidently, if σ u = σ y , we have ideal plastic material.
Constraint (5b) is the definition of plastic strain accumulation. The plastic strain rate may not necessarily be compatible, but Δ ϵ p must be compatible. This is expressed by constraints (5d) and (5e). Constraint (5c) is the incompressibility condition, and (5g) is the normalized condition.
Step 4: Check convergence criteria: if they are all satisfied, then stop; otherwise go to step 2.
Results and discussions
Two examples are reported. To compare the results on shakedown limit for perfectly plastic materials with other researches, we choose σ u = σ y . To investigate the effect of bounded hardening, we choose σ y < σ u < 2σ y . When σ u ≥ 2σ y , we have unbounded kinematic hardening model.
Comparison of plastic limit collapse and shakedown results
[p1, p2] = [2.0, 0.0]
[p1, p2] = [0.0, 1.0]
[p1, p2] = [1.2, 1.0]
[p1, p2] = [2.0, 1.0]
Garcea et al. 
Tran et al. 
Present (perfectly plastic)
Present (kin. hardening)
Cylindrical pipe under complex loading
The analytical solutions of plastic collapse limit for cylindrical pipe under complex loading can be cited from Vu .
Limit and shakedown load multipliers of cylindrical pipe subjected to internal pressure and bending
Limit factor (perfectly plastic)
Shakedown factor (perfectly plastic)
Shakedown factor (bounded hardening)
Shakedown factor (unbounded hardening)
Figure 8 shows that the present results of limit analysis for σ u /σ y = 1 are close to analytical solutions. Figure 9 shows that the hardening effect is clear if the applied moment is less than 0.5Mb lim. If σ u ≥ 2σ y , bounded hardening model becomes unbounded, and shakedown limit of structure cannot exceed two times of elastic limit.
The paper developed a new upper bound algorithm for shakedown analysis of elastic plastic-bounded linearly kinematic hardening structures. This is an efficient tool for practical computation, especially for complicated structures subject to mechanical loads.
The proposed algorithm gives results that are close to the results in literatures. If σ u = σ y , it leads to perfectly plastic material; if σ u ≥ 2σ y , it leads to unbounded kinematic hardening material; otherwise, σ y < σ u < 2σ y , we have bounded kinematic hardening material.
In the preceding expression, the left equality occurs if the subsequent yield surface translates inside the bounding surface, the middle equality occurs if the subsequent yield surface fixed on the bounding surface and the last equality occurs when yield surface translates unboundedly. If the structure shakes down in alternating plasticity mode, then there is no difference between perfectly plastic and kinematic hardening models.
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