The Linear Matching Method Framework (LMMF)
In the last decades an alternative approach, which uses a numerical direct method, has been developed and tested to solve different engineering problems. A series of numerical procedures based on the Linear Matching Method (LMM) have been developed to calculate, in the most efficient way, the cyclic plastic response of a structure subjected to a cyclic thermal and mechanical load. They are now embedded in a solid and robust framework known as the Linear Matching Method Framework (LMMF). The LMMF provides numerical procedures to accurately and efficiently calculate the elastic and plastic shakedown limits, without adopting any conservative assumption and stress linearization.
Flow chart of the Linear Matching Method Framework developed by Structural Integrity and Life Assessment research group.

Great flexibility for wide range of cyclic loading histories.

More accurate than rule based methods such as the R5

Requires simple material data to run.

Better computational efficiency than classical FEA.

Possibility to produce both upper and lower bounds.

Excellent and proven convergence algorithm.

Capability of calculating the steady state cycle.

Capability of considering full creepfatigue interaction.
Elastic Shakedown Limit
A component is said to shakedown when, on the basis of perfect plasticity, behaviour during the steady state cyclic operation is elastic at every point in the structure even though there may be some yielding during early cycles of load. The LMM was originally developed for shakedown of an elastic perfectly plastic solid and gives particularly stable solutions. The method consist of an iterative process where a sequence of upper bounds to the load parameter λUB to the exact shakedown limit λs are derived from a sequence of linear problems for the residual stress field according to an incompressible linear viscous matching model. The sequence monotonically reduces, typically in 30 iterations, to the least upper bound associated with the finite element mesh. This means that the converged shakedown limit is evaluated to the same level of accuracy as the linear elastic solution.
Typical convergence trend of the iterative process for shakedown analysis
Elastic shakedown limit of a structure subjected to a cyclic thermal and constant mechanical load. Comparison between R5 and LMM results.
Creep Rupture Limit
The LMM approach to creep rupture analysis is performed through an extended shakedown analysis (Chen et al., 2003, Ponter et al., 2000 and Ponter and Engelhardt, 2000), where the original yield stress of material in the analysis is replaced by socalled revised yield stresses at each integration points for all load instances in the finite element model. Using the strategy of extended shakedown analysis, the creep rupture limit can be assessed for both the cyclic and monotonic load conditions depending upon the number of load instances in a cycle. In the method, the revised yield stress σyR is determined by the minimum of original yield stress of material σy and a creep rupture stress σC for a predefined time to creep rupture tf, With this scheme, the creep rupture limit of a structure can be evaluated efficiently and conveniently by using the creep rupture data only, without the usage of detailed creep constitutive equations.
Finite element model of a quarter of a plate with a central hole, subjected to a cyclic thermal field and constant mechanical load.
Creep rupture limit of the right side model, obtained using the LMM and the R5 procedure.
Ratchet limit
A novel approach based upon the Linear Matching Method framework has been developed in order to directly calculate the ratchet limit of structures subjected to arbitrary thermomechanical load histories. Traditionally, ratchet analysis methods have been based upon the fundamental premise of decomposing the cyclic load history into cyclic and constant components, respectively, in order to assess the magnitude of additional constant loading a structure may accommodate before ratcheting occurs. The method developed accurately and efficiently calculates the ratchet limit with respect to a proportional variation between the cyclic primary and secondary loads, as opposed to an additional primary load only. The method is a strainbased approach and utilises a novel convergence scheme in order to calculate an approximate ratchet boundary based upon a predefined target magnitude of ratchet strain per cycle. The ratcheting failure mechanism evaluated by the method leads to less conservative ratchet boundaries compared with the traditional Bree solution. The method yields the total and plastic strain ranges as well as the ratchet strains for various levels of loading between the ratchet and limit load boundaries. The proposed methodology has been rigorously tested and verified via worked examples of increasing complexity.
a) Plane stress Bree model; b) applied cyclic thermomechanical load history; and c) load domain.
Bree problem ratchet boundaries for proportional loading compared with analytical solution, alongside original Bree limit for constant loading and the relevant ASME III Code 3Sm limits for each respective loading regime.
The LMM user Plugin
From the earliest stage of the LMM development, the efforts have been focused on the development of the ABAQUS user subroutines. The user has to interact with the FORTRAN source code to modify it accordingly to perform the desired analysis. A few actions such as the definition of the loading history need to be carefully taken by the user. This procedure is not straightforward for engineers without any programming experience and can introduce errors that invalidate the results. In order to solve this issue, and to enhance the usage of the LMMF from industries, a Graphical User Interface (GUI) and an autonomous plugin have been developed recently. The aim of the plugin is to adopt the information provided by the user through a GUI, to automatically modify the finite element model within the working environment, and the special commands required to output all the required parameters. This solution allows a convenient use of the LMM, but at the same time, it reduces the possibility to introduce errors. The user can select the model and the analysis type required through the GUI . When this is done, the user can either extract or provide the material constants required. The plugin checks each value provided by the user in order to identify any possible error. When an error occurs a dialog box appears, with an error code and a brief explanation that is useful for debugging. The definition of the loading cycle is performed by compiling a load table, where multiple load points can be created by using individual load or temperature field defined in the CAE. This approach allows the user to consider a whole class of different loading conditions. Once the load history is defined the desired convergence rule can be selected. The maximum number of increments and the name of the analysis file can be decided to generate a successful LMM job. It is worth noting that the LMM software tool has the capability to perform multiple Computer Processing Units (CPUs) analysis. This feature is very important for the large model when 3D complex geometries are considered.
Video example showing the use of plugin for a elastic shakedown analysis by LMM.