Modelling and Simulation of Mechanical Behaviour of Materials

Examples of activities

• Physically based material modelling
• Determination of material parameters, model verification
• High speed loading/cutting
• Loading in a turbine blade
• Notch in a single crystal
• Microsystem technology, flip chip
• Determination of the elastic constants of anisotropic materials

Physically based material modelling

Here in the case of Nickel base superalloys with a high volume fraction of the γ' precipitate phase. Nickel base superalloys, also as single crystals, are widely used for hot section of turbine blades in power plants or aero engines. A specific constitutive law has been developed and implemented in an FE code, which explicitly takes into account the intricate interactions between dislocations and precipitates.

Representation of the γ' precipitates (in grey) in an octahedral {111} slip plane and of the main dislocation mechanisms of a single crystalline fcc superalloy:

1. Filling of matrix channels with dislocation half-loops (blue)
2. Shearing of matrix and precipitates by large dislocation segments (red)
3. Multiple cross slip between phase boundaries, resulting in macroscopic cubic slip (purple)
4. Climbing of dislocation loops around the precipitates, resulting in a recovery of the internal stresses (green)

The complex interactions between dislocations and precipitates and the stress fields generated by the interface dislocations are responsible for essential features of the macroscopic deformation behaviour of these alloys. Some of them are listed below:

• importance of cubic slip and its relation to the crystal orientation
• recovery mechanisms
• strength degradation due to directional coarsening of the precipitates

Determination of Material Parameters, Model Verification

Simulation of the stress-response with a viscoplastic constitutive model at a non-isothermal, axial-torsional, cyclic straning for the verification of the model:

Comparison of the experiment on a hollow specimen (left) with the numerical simulation (right), 15 min hold-times at 850 °C, material: IN738LC

The model was exclusively adapted to isothermal, uniaxial tensile, LCF and creep tests. The physical control of the optimization was realized by relating of material parameter groups to different hardening and softening phenomena in the material behaviour. One set of material parameters was determined for each testing temperature.

High Speed Loading/Cutting

The behaviour of metallic materials at high loading rates is characterised primarily by a thermal softening due to the fact that the loading time is too short for a sufficient heat flow. The softening can lead to the formation of shear bands. For verification of the constitutive models (e.g. Johnson-Cook model) experiments and finite element (FE) simulations are carried out on notched flat specimens.

FE simulation (ABAQUS/explicit) of the shear band formation in a notched flat specimen, the stiffness of the test machine is considered by truss elements

An industrial application of the material behaviour under high loading rates is, beside impact problems, for example the high speed cutting process with the formation of shear bands at chip segmentation. For the optimisation of the cutting process parameter, FE simulations with appropriate constitutive models for deformation and damage are required.

Stress in a Turbine Blade

Viscoplastic FE-analysis (ABAQUS/standard) of a cross-section of an internally cooled turbine blade for the determination of the stress distribution in the blade:

Notch in a Single Crystal

Simulation of the damage behaviour at notches in single-crystalline superalloys under cyclic loading at high temperature:

Micro-System Technology, Flip-Chip

Simulation of the strain situation at a crack in the interface of a solder bump, which is placed between the silicon chip and a ceramics substrate:

Field of the accumulated inelastic strain in a cracked structure

The electrical failure of such an electronic structure occurs not before a large crack has been formed.

Determination of the elastic constants of anisotropic materials

The elastic constants of materials can be accurately determined from the measured resonance frequencies of freely vibrating specimens. The procedure is described in the ASTM standard E1875 for isotropic materials.

The case of anisotropic materials, like, e.g. single crystals, is much more demanding. First, the number of required independent constants is higher. In addition, the eigenmodes often consist of a mixture of flexural and torsional components. Hence, the classical formulae, which have been derived under the assumption of isotropy, do not apply. As an alternative, the eigenmodes can be evaluated by the Finite Element Method for given elastic constants.

By iteratively varying the independent elastic constants until an agreement is obtained with the measured resonance frequencies, the actual elastic constants of an anisotropic material can be estimated. Eventually, the experimentally obtained resonance spectrum can be completely interpreted (see Figure below).

With the apparatus available at the division, resonance measurements can be performed up to 1900 °C, allowing for the characterisation of the dependency of the elastic constants upon the temperature up to 1900 °C.

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