On finite element implementation of cyclic elastoplasticity: theory, coding, and exemplary problems

In this work the finite element (FE) implementation of the small strain cyclic plasticity is discussed. The family of elastoplastic constitutive models is considered which uses the mixed, kinematic-isotropic hardening rule. It is assumed that the kinematic hardening is governed by the Armstrong–Frederick law. The radial return mapping algorithm is utilized to discretize the general form of the constitutive equation. A relation for the consistent elastoplastic tangent operator is derived. To the best of the author’s knowledge, this formula has not been presented in the literature yet. The obtained set of equations can be used to implement the cyclic plasticity models into numerous commercial or non-commercial FE packages. A user subroutine UMAT (User’s MATerial) has been developed in order to implement the cyclic plasticity model by Yoshida into the open-source FE program CalculiX. The coding is included in the Appendix. It can be easily modified to implement any isotropic hardening rule for which the yield stress is a function of the effective plastic strain. The number of the utilized backstress variables can be easily increased as well. Several validation tests which have been performed in order to verify the code’s performance are discussed.

A Design Optimization Study for the Die Dimensioning Using the Locking Nut Folding Simulation

An inverse analysis based on optimization process is performed to determine die curvatures for a locking nut’s flange folding process which has highly nonlinear material behaviour. The nut material is AISI C1040 steel. The ring material is polyamide 6. The Chaboche’s nonlinear kinematic hardening rule is combined with bilinear isotropic hardening model as a hardening rule for the plasticity model combined with associated flow rule and von Mises yield criterion. The inverse analysis is applied to determine the curvatures by using genetic algorithm optimization method based on dimensional accuracy. The optimum mould curvatures are determined. So a comprehensive methodology is presented for determination of curvatures.

Elaborated subloading surface model for accurate description of cyclic mobility in granular materials

The description of the cyclic mobility observed prior to the liquefaction in geomaterials requires the sophisticated constitutive formulation to describe the plastic deformation induced during the cyclic loading with the small stress amplitude inside the yield surface. This requirement is realized in the subloading surface model, in which the surface enclosing a purely elastic domain is not assumed, while a purely elastic domain is assumed in other elastoplasticity models. The subloading surface model has been applied widely to the monotonic/cyclic loading behaviors of metals, soils, rocks, concrete, etc., and the sufficient predictions have been attained to some extent. The subloading surface model will be elaborated so as to predict also the cyclic mobility accurately in this article. First, the rigorous translation rule of the similarity center of the normal yield and the subloading surfaces, i.e., elastic core, is formulated. Further, the mixed hardening rule in terms of volumetric and deviatoric plastic strain rates and the rotational hardening rule are formulated to describe the induced anisotropy of granular materials. In addition, the material functions for the elastic modulus, the yield function and the isotropic hardening/softening will be modified for the accurate description of the cyclic mobility. Then, the validity of the present formulation will be verified through comparisons with various test data of cyclic mobility.

Modifications on A-F Hardening Rule to Assess Ratcheting Response of Materials and Its Interaction with Fatigue Damage under Uniaxial Stress Cycles

Ratcheting deformation is accumulated progressively over three distinct stages in materials undergoing asymmetrical cyclic stresses. The present thesis evaluates the triphasic ratcheting response of materials from two stand points: (i) Mechanistic approach at which stages of ratcheting progress over stress cycles was related to mechanistic parameters such as stress level, lifespan, mechanical properties and the softening/hardening response of materials. Mechanistic approach formulated in this thesis was employed to assess ratcheting strain over triphasic stages in various steel and copper alloys under uniaxial stress cycles. Good agreements were achieved between the predicted ratcheting strain values based on the proposed formulation and those of experimentally reported. (ii) Kinematic hardening rule approach at which the hardening rule was characterized by the yield surface translation mechanism and the corresponding plastic modulus calculated based on the consistency condition. Various cyclic plasticity models were employed to assess ratcheting response of materials under different loading conditions. The Armstrong-Frederick (A-F) hardening rule was taken as the backbone of ratcheting analysis developed in this thesis mainly due to less complexity and number of coefficients in the hardening rule as compared with other earlier developed hardening rules in the literature. To predict triphasic ratcheting strain over stress cycles, the A-F hardening rule has been further developed by means of new strain rate coefficients γ2 and δ. These coefficients improved the hardening rule capability to calibrate and control the rate of ratcheting over its progressive stages. The modified hardening formulation holds the coefficients of the hardening rule to control stress-strain hysteresis loops generated over stress cycles during ratcheting process plus the ratcheting rates over stages I, II, and III. These coefficients were calibrated and defined based on the applied stress levels. The constructed calibration curves were employed to determine strain rate coefficients required to assess ratcheting response of materials under uniaxial loading conditions at various cyclic stress levels. The predicted ratcheting strain values based on the modified hardening rule were found in good agreements with the experimentally obtained ratcheting data over stages I and II under uniaxial loading conditions. The capability of the modified hardening rule to assess ratcheting deformation of materials under multi-step uniaxial loading spectra was also assessed. Subsequent load steps were considerably affected by previous load steps in multi-step loading conditions. Ratcheting strains for low-high stress steps were successfully predicted by the modified hardening rule. High-low loading sequences however resulted in an overestimated reversed ratcheting strain in the later load steps. The modified hardening rule proposed in this thesis was then employed to predict the ratcheting strain and its concurrent interaction with fatigue damage over stress cycles in steel alloys. The interaction of ratcheting and fatigue damage was defined based on mechanistic parameters involving the effects of mean stress, stress amplitude, and cyclic softening/hardening response of materials. The extent of ratcheting effect on the overall damage of steel samples was defined by means of the product of the average ratcheting strain rate over the stress cycles and the applied maximum cyclic stress, while fatigue damage was analysed based on earlier developed energy-based models of Xia-Ellyin and Smith-Watson-Topper. Overall damage induced by both ratcheting and fatigue was calibrated through a weighting factor at various ratios of mean stress/cyclic amplitude stress. The estimated lives based on the proposed algorithm at different mean stresses and stress amplitudes showed good agreements as compared with experiments.

Modifications on A-F Hardening Rule to Assess Ratcheting Response of Materials and Its Interaction with Fatigue Damage under Uniaxial Stress Cycles

Ratcheting deformation is accumulated progressively over three distinct stages in materials undergoing asymmetrical cyclic stresses. The present thesis evaluates the triphasic ratcheting response of materials from two stand points: (i) Mechanistic approach at which stages of ratcheting progress over stress cycles was related to mechanistic parameters such as stress level, lifespan, mechanical properties and the softening/hardening response of materials. Mechanistic approach formulated in this thesis was employed to assess ratcheting strain over triphasic stages in various steel and copper alloys under uniaxial stress cycles. Good agreements were achieved between the predicted ratcheting strain values based on the proposed formulation and those of experimentally reported. (ii) Kinematic hardening rule approach at which the hardening rule was characterized by the yield surface translation mechanism and the corresponding plastic modulus calculated based on the consistency condition. Various cyclic plasticity models were employed to assess ratcheting response of materials under different loading conditions. The Armstrong-Frederick (A-F) hardening rule was taken as the backbone of ratcheting analysis developed in this thesis mainly due to less complexity and number of coefficients in the hardening rule as compared with other earlier developed hardening rules in the literature. To predict triphasic ratcheting strain over stress cycles, the A-F hardening rule has been further developed by means of new strain rate coefficients γ2 and δ. These coefficients improved the hardening rule capability to calibrate and control the rate of ratcheting over its progressive stages. The modified hardening formulation holds the coefficients of the hardening rule to control stress-strain hysteresis loops generated over stress cycles during ratcheting process plus the ratcheting rates over stages I, II, and III. These coefficients were calibrated and defined based on the applied stress levels. The constructed calibration curves were employed to determine strain rate coefficients required to assess ratcheting response of materials under uniaxial loading conditions at various cyclic stress levels. The predicted ratcheting strain values based on the modified hardening rule were found in good agreements with the experimentally obtained ratcheting data over stages I and II under uniaxial loading conditions. The capability of the modified hardening rule to assess ratcheting deformation of materials under multi-step uniaxial loading spectra was also assessed. Subsequent load steps were considerably affected by previous load steps in multi-step loading conditions. Ratcheting strains for low-high stress steps were successfully predicted by the modified hardening rule. High-low loading sequences however resulted in an overestimated reversed ratcheting strain in the later load steps. The modified hardening rule proposed in this thesis was then employed to predict the ratcheting strain and its concurrent interaction with fatigue damage over stress cycles in steel alloys. The interaction of ratcheting and fatigue damage was defined based on mechanistic parameters involving the effects of mean stress, stress amplitude, and cyclic softening/hardening response of materials. The extent of ratcheting effect on the overall damage of steel samples was defined by means of the product of the average ratcheting strain rate over the stress cycles and the applied maximum cyclic stress, while fatigue damage was analysed based on earlier developed energy-based models of Xia-Ellyin and Smith-Watson-Topper. Overall damage induced by both ratcheting and fatigue was calibrated through a weighting factor at various ratios of mean stress/cyclic amplitude stress. The estimated lives based on the proposed algorithm at different mean stresses and stress amplitudes showed good agreements as compared with experiments.

Development of a kinematic hardening rule to assess ratcheting response of materials under various multiaxial loading spectra

The present thesis develops an Armstrong-Frederick (A-F) type coupled kinematic hardening rule to assess ratcheting response of steel alloys under various multiaxial loading paths. The hardening rule is constructed on the basis of the recently proposed Ahmadzadeh-Varvani (AV) hardening rule to further evaluate the ratcheting response of materials under multiaxial loading spectra. The modified model offers a simple framework with limited number of terms and coefficients in the dynamic recovery portion of the model. The dynamic recovery further holds inner product of plastic strain increment p dand backstress unit vector a a with different directions under multiaxial stress cycles enables the model to track different directions. Term 1/ 2 n. a a taking positive values less than unity for multiaxial loading conditions is to control the accumulation rate of ratcheting strain and to prevent the modified model to experience plastic shakedown over stress cycles in stage II. Term(2 n. a a ) taking the values between 1 and 3 under multiaxial loading, magnifies the effect of coefficient γ2 to take into account the nonproportionality effect of various loading paths and further to shift down the predicted ratcheting strain over the stress cycles. The predicted ratcheting curves by the modified rule were compared with those predicted based on earlier developed hardening rules of Ohno-Wang (O-W), Jiang-Sehitoglu (J-S), McDowell, and Chen-Jiao-Kim (C-J-K) holding relatively complex framework and more number of coefficients. The O-W, the J-S, McDowell and C-J-K models mainly deviated from the experimental ratcheting strain of steel alloys for various multiaxial loading histories, while the predicted curves of the modified model closely agreed with experimental data of steel samples over ratcheting stages. The predicted ratcheting curves based on the modified model closely agreed with experimental data of steel samples under various multiaxial step-loading histories. The modified model was also found capable of predicting ratcheting in the opposite direction as the tensile axial mean stress dropped in magnitude. The O-W, J-S, McDowell and C-J-K models holding more backstress components and coefficients require longer Central Processing Unit (CPU) time. While time required for ratcheting assessment using the modified hardening rule was found to be twice shorter due to its simpler framework and limited number of coefficients.

Development of a kinematic hardening rule to assess ratcheting response of materials under various multiaxial loading spectra

The present thesis develops an Armstrong-Frederick (A-F) type coupled kinematic hardening rule to assess ratcheting response of steel alloys under various multiaxial loading paths. The hardening rule is constructed on the basis of the recently proposed Ahmadzadeh-Varvani (AV) hardening rule to further evaluate the ratcheting response of materials under multiaxial loading spectra. The modified model offers a simple framework with limited number of terms and coefficients in the dynamic recovery portion of the model. The dynamic recovery further holds inner product of plastic strain increment p dand backstress unit vector a a with different directions under multiaxial stress cycles enables the model to track different directions. Term 1/ 2 n. a a taking positive values less than unity for multiaxial loading conditions is to control the accumulation rate of ratcheting strain and to prevent the modified model to experience plastic shakedown over stress cycles in stage II. Term(2 n. a a ) taking the values between 1 and 3 under multiaxial loading, magnifies the effect of coefficient γ2 to take into account the nonproportionality effect of various loading paths and further to shift down the predicted ratcheting strain over the stress cycles. The predicted ratcheting curves by the modified rule were compared with those predicted based on earlier developed hardening rules of Ohno-Wang (O-W), Jiang-Sehitoglu (J-S), McDowell, and Chen-Jiao-Kim (C-J-K) holding relatively complex framework and more number of coefficients. The O-W, the J-S, McDowell and C-J-K models mainly deviated from the experimental ratcheting strain of steel alloys for various multiaxial loading histories, while the predicted curves of the modified model closely agreed with experimental data of steel samples over ratcheting stages. The predicted ratcheting curves based on the modified model closely agreed with experimental data of steel samples under various multiaxial step-loading histories. The modified model was also found capable of predicting ratcheting in the opposite direction as the tensile axial mean stress dropped in magnitude. The O-W, J-S, McDowell and C-J-K models holding more backstress components and coefficients require longer Central Processing Unit (CPU) time. While time required for ratcheting assessment using the modified hardening rule was found to be twice shorter due to its simpler framework and limited number of coefficients.

A comparison of different hardening rules on a multi-step global manufacturing process modeling

The main difficulty presented by the simulation of a global process that includes different forming stages is the correct characterization of the material state at the end of each of these stages, which in turn, are the initial point of the following process. Hardening variables are capable of characterizing the state of the material, which, after a plastic transformation, varies according to the direction of the solicitation and its intensity. The present work carries out an analysis of the influence in the election of the hardening rule used in the behavior law, comparing the most used approach. For a work piece solicited by combined efforts in multiple stages, results are obtained by numerical simulation. A correct choice will allow obtaining reliable predictions, not the solicitations but also to the final geometry and the dissipated energy in the global process, allowing an eventual optimization of such process.

A non-associative macro-element model for vertical plate anchors in clay

This work proposes a new plastic hardening, non-associative macro-element model to predict the behaviour of anchors in clay for floating offshore structures during keying and up to the peak load. Building on available models for anchors, a non-associated plastic potential is introduced to improve prediction of anchor trajectory and loss of embedment at peak conditions for a large range of padeye offsets and different pull-out directions. The proposed model also includes a displacement-hardening rule to simulate the force and displacement mobilisation at the early stages of the keying process. The model is challenged and validated against different sets of numerical and centrifuge data. This extensive validation process revealed that two of the four newly introduced model parameters assume a constant value for the range of simulated cases. This suggests that only two of the newly introduced parameters may need to be calibrated for the use of the proposed macro-element model in practice.