A Miniaturized Biaxial Deformation Rig for in Situ Mechanical Testing

The conceptual design of the new biaxial in-plane tensile testing apparatus is illustrated in Fig. 1, the corresponding relevant parameters are provided in Table 1. A cruciform shaped specimen is mounted on 4 grips with the aid of cylindrical shaped pins that are slightly tilted outwards from the centre of the apparatus. This ensures that the specimen is pushed down onto the grips during pulling. The 4 grips are sliding inside a guide system, ensuring a good alignment between loading axes and the grips. Each grip is connected to a load cell (MLP-10, Transducer Techniques) with a capacity of 44 N and a safe overload of 150%. The use of 4 load cells ensures a fully symmetric setup and, as a consequence, identical compliance of all 4 axes and proper sample alignment while applying a biaxial load. Via a carrier plate the load cells, grips and guide systems are connected to 4 piezo actuators (N-111 Nexline, PI). Each individual actuators allow for step sizes between 10 nm and 10,000 nm with a total travel range of 10 mm and maximum velocity of 0.4 mm/s. Furthermore the actuators can pull up to a 50 N force. In other words, these actuators combine large travel range with nanometre step size and can work under relatively high forces, which is crucial for the compact design of the apparatus. 8 optical limit switches (two per axis) ensure that the grips do not crash into each other. It should be noted that the system as shown in Fig. 1 was designed solely for biaxial tensile experiments. However, it is in principle also possible to perform biaxial compression experiments by replacing the grips with appropriate compression heads.

The traction system is mounted on a compact aluminium alloy frame, which also serves as housing for the cables. The outer dimensions of the complete setup are 200 mm × 190 mm × 73.2 mm. The frame is open on the backside, which allows access of an X-ray beam to the centre of the specimen. The grips are designed such that the diffracted beam does not interfere with the apparatus. Each component is vacuum tight, which allows for safe operation inside an SEM chamber. Also, the grips and pins are designed such that the sample can be approached close enough to allow for high magnification imaging.

The compact design of the biaxial apparatus and its components result in a system with relatively low stiffness (see also section on machine characterisation). Furthermore the 4 axes are not perfectly identical due to mechanical tolerances. As a consequence, under load identical displacements of the actuators may result in slightly different motion of the grips. However, in order to avoid any bending of the cruciform shaped samples it is crucial that opposite grips exhibit exactly the same amount. Therefore, an additional system was designed, which allows monitoring the position of the grips with nanometre resolution. A plate (blue in Fig. 1(c)) with 4 linear guides with integrated encoders (GDW-1720-S, SmarAct) can be mounted on top of the machine. These guides are connected to pins mounted on the grips. When a grip moves, the connected guide moves parallel to it. In this way the encoder signal of the linear guides can be used as a measure for the motion of the grips. This additional position readout system cannot be used inside the SEM but it is compatible with in situ X-ray diffraction.

An E-712 controller (PI) drives the actuators and reads out the limit switches. The load cells are read out by TEDS readout panels (DPM-3, Transducer Techniques), that are connected by a serial device server (Moxa, Nport 5410). The encoders of the linear positioners are read out by nano position sensors (MCS-3S-ES-SDS15-TAB, SmarAct) connected to a sensor module (MCS-6CC-USB, SmarAct). A custom-written LabVIEW program controls the complete setup.

The rig has been designed for mounting in a conventional, regular sized SEM. An electrical feed-through flange was designed in order to have full control over the rig from outside the SEM. Figure 2(a) and (b) display, respectively, a schematic and a real-life image of the rig mounted in a Zeiss Ultra 55 SEM. The grips that hold the sample have been designed such that the minimum working distance is 7 mm, which is sufficiently close for recording images with a resolution down to 5 nm. The full travel range of the whole machine in the horizontal plane is 24 mm. This allows obtaining images from areas away from the centre of the specimen. This is particularly useful for checking the quality of the sample mount or to control whether the sample has fractured at stress concentrations that may occur near the pins that hold the sample.

Figure 3 shows a technical drawing (Fig. 3(a) and (b) and a real image (Fig. 3(c)) of the biaxial machine attached to the goniometer of the powder diffraction station at the Materials Science (MS) beam line [37] of the Swiss Light Source (SLS). The machine is mounted on a rotational stage allowing positioning the rig for transmission (Fig. 3(a)) and reflection mode (Fig. 3(b)). Furthermore a translational stage with three degrees of freedom is used to place the centre of the sample at the position of the X-ray beam. The diffracted X-ray beam is detected by a one-dimensional (see Fig. 3(a)) or two-dimensional detector located behind the device. The in situ X-ray experiments can be performed with the additional carrier plate (see Fig. 1(c)). It has been designed such that the diffracted beam is not shadowed.

Defining the geometry of a cruciform shaped sample for biaxial testing is not straightforward [20, 22, 38‐40]. It is the result of a trade-off between various considerations:

The sample geometry has to be slightly tuned for each material, depending on its mechanical properties. The main parameters that are to be optimized include thickness of the arms, reduced thickness in the center, cross-arm shape and depth. In what follows experiments on an AlMg alloy are presented (see section on Mechanical Tests). Figure 4(a) displays a schematic of the corresponding sample geometry. In this particular case the shape was optimized for reaching high levels of plasticity during subsequent strain path changes with the aid of ABAQUS [42] finite element modelling (FEM). It has a width of 19 mm and an overall thickness of 0.2 mm. The centre part is surrounded by 4 sharp corners and thinned down in several steps down to 50 μm in a central circular area with a diameter of 400 μm. Figure 4(b), (c) and (d) show the results of an FEM simulation. Here the cruciform sample was strained uniaxially with constant strain rate up to 2.6% plastic strain in the gage section. Figure 4(b) displays the equivalent plastic strain, which shows that the plastic strain is concentrated in the central region. Figure 4(c) and (d) displays the two in-plane stress components, the out-of-plane stress component is negligible. Stress concentrations at the notches and at the steps are clearly visible. The stress state at the centre of the sample is however homogeneous. It is interesting to note that the stress component perpendicular to the loading direction is relatively high. This is related to the so-called “ring effect”. It can be rationalized as follows: when applying a force along one particular axis, the thicker area surrounding the thin circular area in the centre induces an in-plane compressive stress perpendicular to the applied stress. The nature and magnitude of this stress strongly depends on the cruciform geometry. A detailed description of this complex phenomenon is beyond the scope of this paper. An in-depth discussion on this topic can be found in references [24, 41, 43, 44].

The cruciform specimens are machined using ultra-short pulsed laser ablation (PLA). Ultra-short PLA techniques use highly localized energy bursts to evaporate material, allowing the shaping of sub-micron structures with minimal residual damage [45, 46]. For these samples, a pulsed laser with a 10 ps pulse width is used. At this pulse width, ablation is known to be accompanied by the formation of a heat affected zone (HAZ), characterized by plasma formation and ejection of plasma from the surface [47]. However, by lowering the laser fluence to just above the ablation threshold fluence, the depth of the HAZ is minimized to insignificant levels (300 nm in a 50 μm thick sample) [48]. Consequently, complex specimen shapes with reduced test sections can be made without causing significant damage in the test section. Figure 5 displays the sequential steps to fabricate the specimens.

Figure 5(g) Displays an SEM image of the gage section of an AlMg alloy. The periodic structure on the surface of the sample is due to the PLA technique. The observed features are of the order of a few hundred nanometres high and 5-10 μm wide. For most materials such small surface features are not expected to play an important role in the mechanical properties

An additional difficulty occurs when transferring the sample from one holder to another or to the biaxial apparatus. As the samples are rather thin a clamping method (e.g. with tweezers) cannot be used, in particular for low strength materials such as, for instance, pure Al. Therefore a low-vacuum tool has been developed. A schematic view of this tool is shown in Fig. 6(a). Here the complete specimen is sucked against a flat surface. This allows a safe transfer and mounting of the samples. The samples are stored in a dedicated sample holder, as shown in Fig. 6(b). With the aid of the sample transfer device the sample can be placed accurately inside the grooves of sample holder.

Several deformation tests were performed on Al-5wt%Mg samples. The chemical composition as determined by optical emission spectroscopy is listed in Table 2. This material is known to exhibit the Portevin-Le Chatelier (PLC) effect [49, 50]. This is characterized by a serrated strain-stress response as a result of dynamic strain aging

A cruciform shaped specimen is subjected to uniaxial straining along the horizontal axis. Figure 8 displays a snapshot of the centre of the sample prior to the deformation and after ~12% plastic strain. The centre of the sample exhibits three concentric circles due to the thickness reduction (see also Fig. 8(a)). In order to follow the deformation of the sample these features are simultaneously fitted with elliptical functions with a common centre, as indicated in yellow in Fig. 8(b) and (c). As expected, after plastic straining the circles have become ellipses (Fig. 8(c)). Two in-plane strain components are calculated as ε_hkl=ΔL_h,v/L₀ with L₀, ΔL_h and ΔL_v the initial length and length change of the major (horizontal) and minor (vertical) axis of the ellipses, respectively. It should be noted that these strain values are only a rough estimation. In particular, it is assumed that the strain distribution within the inner ellipse is homogeneous, which is presumably not correct, as suggested by the FEM simulations shown in Fig. 4(b). More reliable measures for the strain can be obtained by performing the experiment in situ inside an SEM. With the aid of digital image correlation the actual strain distribution within the gauge area can be determined with high resolution (see, for instance, Refs. [51‐54]). This is a matter of future research.

Figure 9 displays the evolution of the strain along the major and minor axes of the 3 ellipses as a function of time. Several interesting observations can be made:

Due to the complex geometry and the related ring effect the relationship between applied force and the local stress state at the centre of the sample cannot be expressed by analytical equations. Instead, FEM simulations are needed. ABAQUS/Standard software [42] is used to perform the FE simulations for the cruciform and dog-bone samples. In order to improve the computational efficiency, only 1/8th of the entire geometry is simulated with symmetric boundary conditions on appropriate surfaces. A structured hexahedron mesh is employed with linear 8-node mesh elements (C3D8 in ABAQUS) for both the cruciform and dog-bone geometries. Isotropic material properties of AlMg5 were used. The plastic response is modelled using the ABAQUS material model that is based on the von Mises yield criterion and the associated flow rule. The built-in rate-independent non-linear isotropic and kinematic hardening law with 5 back-stresses is used. It should be noted that such a simulation represents a first approximation only, as it does not account for, for instance, anisotropic material properties and relies on a phenomenological hardening law. In order to appropriately capture the multiaxial material behaviour crystal plasticity models are needed (see, for instance, Refs. [41, 55]). This is however beyond the scope of this paper.

The stress-strain response from a monotonic tensile loading test on a dog-bone sample is used as input to ABAQUS/Standard. In order to account for micro-plasticity, the initial yield point is taken at 350 MPa. The ABAQUS/Standard algorithm uses this experimental curve to fit the back-stress parameters. Then the experiment with the cruciform is simulated. This results in a relationship between applied force and local stress at the centre of the cruciform specimen. With the relationship the experimentally measured applied force is converted into a von Mises stress. Figure 10 displays the resulting von Mises stress as a function of the equivalent plastic strain. It clearly shows the presence of stress drops and strain bursts, related to the PLC effect.

An in situ proof-of-principle experiment was performed on the Al-5wt%Mg alloy at the MS beam line. The X-ray beam had a size of ~80 × 80 μm² and an energy of 17.5 keV. First, the sample was raster scanned in the beam with a step size of 50 μm and the intensity of the transmitted beam is recorded by a diode placed behind the sample. The resulting transmission map recorded at the centre of the specimen is shown in Fig. 11(b). The transmission is normalized to that of air. The shape of the specimen shown in Fig. 11(a) can easily be recognized in the transmission map, which allows localizing the centre of the specimen with high precision.

We report here on the result of three continuous deformation tests: 1) uniaxial deformation of a standard dog-bone shape specimen, 2) uniaxial deformation of a cruciform shaped specimen and 3) equibiaxial straining of a cruciform shaped specimen. The scattering geometry for the tests is shown in Fig. 12. For test 1) the scattering vector Q (which is parallel to the normal of the diffracting planes) is perpendicular to the loading axis σ₂₂ (Fig. 12(a)). For tests 2) the cruciform shaped specimen is strained along axis σ₁₁, which is parallel to the scattering vector (Fig. 12(b)). For test 3) the cruciform shaped specimen is strained equally along both axes σ₁₁ and σ₂₂. Here the scattering vector is parallel to the direction σ₁₁ (Fig. 12(c)).

During straining high-resolution diffraction patterns were recorded continuously at a frequency of 0.05 Hz. The diffraction peaks were fitted by asymmetric PVII functions. More details on the beam line and fitting procedures can be found in references [56, 57]. Elastic lattice strains ε_hkl were calculated from the evolution of the diffraction peak positions θ_hkl:

Figure 13 displays the evolution of the lattice strain derived from the {311} diffraction peak as a function of applied stress for the three experiments. The stress is derived from the FEM simulations. Note that we restrict the discussion here to the elastic regime only. As expected, all curves exhibit a linear behaviour. The differences in slope can be rationalized by considering the analytical expressions that relate the applied stress σ_ij with elastic strain ε_ij. For isotropic cubic materials this is given by the following Hooke’s law [58]:

with ν_hkl and E_hkl the hkl dependent Poisson’s ratio and elastic modulus, respectively. It is well accepted [58] that for the {311} diffraction peak the bulk elastic constants can be used: ν₃₁₁ = 0.335 and E₃₁₁ = 69 GPa. Considering only the strain component along the σ₁₁ axis (vertical axis in Fig. 12) and an in-plane biaxial stress state (σ₁₂ = σ₂₁ = σ₁₃ = σ₃₁ = σ₂₃ = σ₃₂ = σ₃₃ = 0) this equation considerably simplifies to:

with r the ratio between the two principle in-plane stress components σ₂₂ and σ₁₁. For tests 1) and 3) r = 0 and r = 1, respectively. The ratio r for test 2 was obtained directly from the FEM simulations and is given by r = −0.25 (in the elastic regime). The resulting curves are shown in Fig. 13 by the solid lines. A very good match between the experimental and modeled data is obtained. This is a good indication that the finite element simulations capture well the geometrically induced complex stress states that arise at the centre of the cruciform shaped specimens.