Fatigue fracture mechanics in gold-based MEMS notched specimens: experimental and numerical study

The characterization of fatigue fracture mechanics in gold-MEMS notched specimens is presented in this work. A test microstructure with a central notched specimen is specifically designed and built to perform on-chip fatigue test. The central specimen undergoes cyclic loading due to the application of alternating voltage. The variation in the microstructure deflection is measured using an optical profilometer and is attributed to the crack growth in the gold material, causing the variation in the specimen stiffness. The occurrence of pull-in condition is used as a fracture detector, then the fracture of the specimen can be recognized without performing scanning electron microscope inspections during the fatigue test. Crack propagation in the test specimen is simulated through a coupled-field electromechanical fracture finite element model and the resulting crack path is compared to the experimental measurments performed with scanning electron microscope analyses. Finally, Paris’ law is applied and the number of cycles to failure is computed by exploiting the results of the fracture model and experimental measurements. Both experimental and numerical results demonstrate that the notch acts as a stress and strain raiser, fostering crack nucleation, and that the linear elastic fracture mechanics theory is still valid to describe crack propagation in micro-size samples.


Introduction
In recent years, micro-electro-mechanical systems (MEMS) devices have become a well-established technology, given the * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. growing need for integrated and miniaturized systems with advanced functionality [1,2].
The advantages of MEMS devices lie in their low cost, small size, lightweight, low power consumption, high reliability, easy integration, and suitability for harsh working environments [3][4][5][6]. Then, MEMS devices are employed in a wide range of applications and fields, from industrial to consumer areas [7][8][9][10], being also recently attractive in the field of the Internet Of Things technology [11,12].
The problem of reliability and durability is a central theme in the field of MEMS research, especially in those applications where high-reliability performance under harsh field Electrostatic actuation Fatigue characterization of bending and torsion in microstructures [36] Tungsten and metallic glass In-situ indenter Measurement of the cyclic crack growth rate in notched specimen and Paris' law application [20] LIGA Nickel Piezoelectric actuation Fatigue characterization and lifetime prediction of notched specimen using Paris' law [38] Ni-P amorphous alloy Mechanical testing machine for micro-sized specimens Fatigue fracture test on notched specimen [37] conditions is required, such as medical, automotive, and aerospace applications. The reliability assessment in MEMS devices is a challenging task since MEMS devices are miniaturized systems integrating both electronic and mechanical components, then the device failure may be caused both by mechanical and electric reasons [13].
Fatigue is one of the major cause of failure in MEMS devices [14]. Generally speaking, fatigue is caused by repeated cyclic loads, leading to crack initiation and propagation from initial micro-defects in the material, resulting in the complete fracture of the component ultimately.
Fatigue in MEMS devices could arise due to the fluctuation in applied stress or strain [15], as well as due to temperature variation [16,17], and chemically aggressive or embrittling environment [18] which can create or accelerate crack growth.
It is well known that the fatigue behaviour of a structure strictly depends on the material characteristics. Furthermore, the mechanical properties of microstructural materials can be affected by the fabrication process and may differ from those of bulk materials [19,20]. Therefore, there is the need to properly characterize the fatigue behaviour of micro-components in order to design reliable MEMS devices.
The characterization of the fatigue fracture behaviour of samples at the micro-scale is challenging, both from an experimental and modelling point of view.
Fracture toughness and fatigue properties in macro-sized specimens are measured according to ASTM standards, however, there are no standards for micro-sized specimens.
Furthermore, the approaches commonly used in traditional macro-scale mechanics cannot be adopted to perform fatigue testing at the micro-scale because of two main reasons. Firstly, it is not easy to decouple the characterization of the specimen material from the machine soliciting it. Secondly, the development of test microstructures is challenging because proper alignment, as well as gripping, loading of the specimen, and minimization of residual stresses due to the fabrication process have to be addressed during the test. For this reasons, the so-called on-chip fatigue testing is often used instead of off-chip testing. Then, specific test microstructures are designed to load and measure the resulting deformation of the test specimen on the same chip, without employing external instrumentation.
Finally, some authors [21][22][23][24] have evidenced the breakdown of the traditional fracture mechanics theory at a very small scale.
The fatigue behaviour of the material is generally described using two different approaches: the S-N (or Wöhler) curve, which correlates the applied mechanical stress (or strain) to the fatigue life, and Paris' law, which describes the growth of cracks in the material based on the linear elastic fracture mechanics (LEFM) theory [25][26][27].
Different attempts to obtain Wöhler curves of micromechanical samples were made [28][29][30][31][32][33][34][35][36], on the other hand, few works applying Paris' law to characterize fatigue fracture process in MEMS devices exist in the literature [20,[37][38][39]. Table 1 summarizes the main works existing in the literature dealing with fatigue in MEMS devices. Different materials were analyzed, as well as several test microstructures were proposed, and different strategies to load the test specimen were adopted, according to on-chip and off-chip testing.
Modelling the fatigue fracture in MEMS devices requires solving a multiphysics problem, which is often highdemanding from the computational point of view, especially in a three-dimensional (3D) model. Different numerical methods have been proposed in the literature to solve multiphysics fracture problems, such as phase-field model (PFM) [41], cohesive elements model (CZM) [42], and extended finite element method (XFEM) [43]. Alternatively, crack propagation is simulated by computing stress intensity factors (SIFs) through the J-integral or energy release rate using finite element method (FEM), on the basis of LEFM theory [44].
This paper aims to characterize the fatigue fracture behaviour of gold-based MEMS notched specimens, both from experimental and modelling point of view. A test microstructure with a notched central specimen is specifically designed to perform on-chip fatigue test. The kinematic model, as well as the experimental setup and the multiphysics FEM fracture model, are presented. The test notched specimen is cyclically loaded at different stress levels using electrostatic actuation. The variation in the microstructure deflection is monitored during the test and correlated to the stiffness variation caused by crack propagation in the material. The occurrence of the pull-in phenomenon is used as a failure detector, then the specimen is not inspected during the test. The crack propagation path is observed using scanning electron microscope (SEM) after the specimen failure. A comparison between the crack path obtained from the fracture model and measured experimentally is performed. Finally, the fatigue life cycle is computed by applying Paris' law and compared to the experimental results.

Test microstructure design
A test microstructure is specifically designed and built to characterize the fatigue fracture in gold-based MEMS notched specimen with on-chip testing. The design of the test microstructure, as well as its fabrication process and the advantages over other designs proposed in the literature, are extensively reported in authors' previous works [32,33,[45][46][47], thus just a brief description is given below for the sake of clarity.
The schematic and the SEM image of the test microstructure are shown in figure 1.
The effective dimensions of the test microstructure are measured with the high-resolution optical interferometric profilometer ZoomSurf3D by Fogale Nanotech and are reported in table 2.
The test microstructure is composed of two suspended perforated plates, anchored at the external edges to the substrate by eight trapezoidal flexural beams.
A test specimen with a rectangular cross-section is located at the centre of the microstructure and connected with the two suspended plates with rounded corners to avoid local stress concentration effects. The specimen has a circumferential notch in the centre, which acts as local stress and strain raisers.
The suspended plates can move and act as electrostatic actuators, and the flexural beams are structural hinges subjected to a uniform stress distribution when they are loaded by bending moment thanks to their trapezoidal shape.
Two bottom actuation electrodes are fixed on the substrate without covering the full area under the top suspended plates, which allows for maximizing displacement and stress in the central specimen before the occurrence of the pull-in phenomenon. Furthermore, the bottom electrodes are located at a distance of 25 µm from the flexural beams to avoid local plasticity in the beams.
The test specimen is a clamped-clamped microbeam subjected to tensile load when the actuation voltage is applied between the top suspended plates and bottom electrodes. In this case, the suspended plates move downwards and undergo a symmetrical rigid rotation thanks to the trapezoidal flexural beams, loading the central specimen ultimately.
The microstructure is fabricated using thin film gold material according to RF Switch Surface Micromachining (RFS) process developed by Bruno Kessler Foundation [48,49]. The moving parts of the microstructure are built through a two-step gold electroplating process over a photoresist of 3 µm thickness, resulting in thicker and stiffer suspended plates compared to the central specimen. A summary of RFS microfabrication process steps is widely described in [28,50,51]. It is important to point out that the fabrication process is designed to limit the onset of residual stress, as explained in detail in previous authors' works [45,46,52]. The design of the test microstructure is addressed to correlate the degradation of the specimen stiffness due to cycling to the variation in the device deflection, as previously reported in [32,33,47]. The kinematic scheme of the test microstructure is shown in figure 2.
The electrostatic force (F e ) acting on each suspended plate and caused by the application of the actuation voltage (V) between the top suspended plates and the bottom electrodes is expressed according to equation (1).
where ε 0 is the permittivity, A = l e w e is the overlap area, l e and w e are the length and width of the bottom electrodes respectively, g 0 is the initial air gap thickness and h 1 is the downward deflection at the centre of the top suspended plates. The electrostatic force causes the plates to rigidly rotate by α angle, resulting in the horizontal (∆l/2) and vertical (h 2 ) displacement of the specimen anchors. Then, the horizontal displacement cause the axial force (P), on the other hand, the vertical displacement causes the bending moment (M). Finally, the resulting axial stress in the central specimen is computed according to equation (2).
where σ a,P is the axial stress component due to the axial force P and is computed according to equation (3a), and σ a,M is the axial stress component due to bending moment M and is computed according to equation (3b).
where E is the Young modulus, A s is the specimen crosssection, l s is the specimen length, P is the axial force, M = 3EJ is the bending moment, J = wst 3 s 12 is the bending moment of inertia, w s and t s are the specimen width and thickness respectively, and z s is the coordinate along the specimen thickness.
The maximum stress at the notch root (σ n,max ) is computed from the nominal axial stress (σ a ) as long as stress and strain at the notch root remain elastic, according to equation (4).
where K t is the elastic stress concentration factor. The vertical deflection of the centre of the specimen with respect to the edge due to bending moment can be neglected as it is much lower than the overall structure vertical displacement (h 2 ). As a consequence, the proposed test microstructure allows correlating the microstructure vertical displacement, the nominal axial stress in the central specimen (equations (2) and (3)) and the maximum axial stress at the notch root (equation (4)).
Furthermore, the variation of the vertical displacement measured during the fatigue test is attributed to the variation in the device stiffness due to the crack growth at the notch. This is quite reasonable for three reasons: (1) the central specimen is the most stressed component of the microstructure, as also demonstrated by FEM analysis; (2) the notch acts as a stress raiser, fostering crack nucleation [42,47]; (3) the plates and flexural beams are less affected by damage, as also confirmed by SEM analyses and by results from previous works [32,33].
Finally, the specimen failure is identified by the instantaneous large variation of the microstructure displacement (h 2 ) without inspecting the specimen when loaded, as the device undergoes pull-in condition when the specimen fractures [28,52,53].

Fundamentals of linear elastic fracture mechanics
(LEFM) theory. LEFM theory is used to describe the stress state near the crack tip region when the material is linear and  elastic or when the assumption of small-scale yielding holds, i.e. when the extension of the plastic zone at the crack tip is negligible as compared to the crack dimensions. According to LEFM, the stress at the crack tip goes to infinite and the SIF is generally adopted to compute the stress field (σ ij ) at the crack tip, according to equation (5) [54].
where K is the SIF, r and θ are the polar coordinates in the frame with the origin at the crack tip (figure 3(a)) and f ij (θ) is a dimensionless shape function.
In the case of arbitrary loading, the stress field is expressed as the superposition of three fracture modes (figure 3(b)), namely mode I (Opening), mode II (Sliding) and mode III (Tearing) [55], according to equation (6).
)] (6c) where K i is the SIF of mode i and ν is the Poisson ratio. For a given mode i, the K i is a function of the corresponding energy release rate (G i ), which is the decrease in the total potential energy of the system per increase in fracture length (or surface area for 3D problems), according to equation (7) [56].
where G i is the energy release rate of mode i and µ is the shear modulus. In the case of mixed-mode, the energy release rate (G) is computed by applying the superposition principle, according to equation (8).
SIF is often computed numerically from the energy release through the J-integral when LEFM theory holds, as the Jintegral is equal to the energy release rate. The J-integral for a 2D fracture problem is expressed in equation (9), considering an arbitrary counterclockwise path Γ around the crack tip (figure 4) [57].
where ψ el =´ε ij 0 σ ij dε ij is the elastic strain energy density, σ ij are the components of the stress tensor, ε ij are the components of the strain tensor, t i = σ ij n j are the components of the traction vector acting on the path Γ, n j is the versor normal to Γ, u i are the components of the displacement vector, x and y are the coordinates parallel and perpendicular to the crack direction respectively, and ds is the element length along Γ.
The contributions of different modes cannot be directly distinguished from J-integral in the case of mixed-mode fracture. In this case, the interaction energy integral method initially proposed by Yau et al [58] is largely used to compute mixedmode SIFs separately, being implemented in most commercial FEM software, such as Ansys and Abaqus. Further details on the interaction integral method and its numerical implementation are reported in [59].

Fatigue crack growth
Fatigue refers to the progressive failure of materials due to crack initiation and propagation during repeated cyclic loading. Fatigue crack growth is generally divided into three stages, as shown in figure 5 [56]. First, macro-cracks nucleate from material imperfections, such as voids and micro-defects (Stage I), then stable crack propagation occurs (Stage II), followed by unstable crack propagation (Stage III), which rapidly leads to final failure.
Paris' law is the most popular power law used to describe stable crack growth (Stage II). The crack growth rate is a function of the SIF range (∆K), defined in equation (10), according to equation (11) [25][26][27].
where K max and K min are the maximum and minimum SIF during the loading cycle, R = K min Kmax is the stress ratio, N is the number of loading cycles, and C and m are empirical material parameters.
Paris' law was initially applied just for mode-I loading. Tanaka [60] proposed a modified form of Paris' law for mixedmode I/II fracture, as defined in equation (12).
where C and m are obtained from mode-I fatigue crack growth tests, ∆K eqv is the equivalent SIF range and is a function of ∆K I (equation (13a)) and ∆K II (equation (13b)), according to equation (14).
Different expressions of ∆K eqv have been proposed in the literature and are summarized in table 3.
The fatigue cycles number (∆N) for each finite crack length increment (∆a) is estimated by integrating equation (12), according to equation (22).
Then, the number of cycles ∆N i required for the crack to grow by a very small crack increment ∆a i = a i+1 − a i is simply estimated as a function of ∆a i , according to equation (23).
where i is the crack growth step and ∆K i eqv is computed at the crack length a i . Finally, the fatigue life cycle (N f ) is calculated by summing ∆N i for each crack growth step, according to equation (24).
where n is the number of crack growth steps. The direction of crack propagation (θ c ) is equal to the initial crack orientation in the case of single fracture mode. On the other hand, three criteria are commonly used to determine θ c under mixed-mode loading: the criterion of the maximum circumferential stress [65], the criterion of the maximum energy release rate [66], and the criterion of the minimum strain energy density [67]. In the present work, the criterion of maximum circumferential stress is adopted. The reader can refer to [56] for further details about the other two criteria.
According to the maximum circumferential stress criterion, the crack propagates in the radial direction where the circumferential stress (σ θ ) component is maximum. Then, the circumferential stress expressed in equation (6b) is maximized ( ∂σ θ ∂θ = 0), according to equation (25).
where θ c is positive in the anticlockwise direction with respect to the initial crack orientation direction.

Experimental setup
The change in the microstructure deflection during the fatigue loading due to crack growth in the test specimen is monitored using the optical profilometer ZoomSurf3D by Fogale Nanotech.
The central specimen fails when the pull-in phenomenon occurs, which acts as a detector of material failure and is recognized by a large variation in the measured microstructure deflection (h 2 ) as compared to previous cycles. Then, the occurrence of pull-in is detected at fixed time intervals during the fatigue loading in order to identify the failure of the central specimen.
Generally speaking, a fatigue loading cycle is fully characterized by the following parameters: σ max and σ min , which are the maximum and minimum stress reached in the cycle, σ m = (σmax+σ min ) 2 which is the mean stress, and σ a = (σmax−σ min ) 2 which is the stress amplitude.
The fatigue load is obtained by imposing the actuation voltage between the top suspended plates and the bottom electrodes using a voltage generator. Then, the mean voltage (V DC ) and alternating (V AC ) voltage components, the stress frequency (f ) and the time of actuation (t a ) need to be properly chosen during the test.
The variation of microstructure deflection caused by the stiffness degradation due to fatigue loading is monitored by applying a constant static voltage between the electrodes.
Referring to figure 6, the testing procedure is performed according to the following steps: The procedure is repeated until the specimen failure, i.e. when the pull-in condition occurs due to the static voltage application.
During the fatigue loading phase, the alternating (V AC ) and mean voltage (V DC ) components are set based on the desired levels of stress amplitude and mean stress, exploiting the deflection-voltage and deflection-stress relations of the kinematic model, as explained in section 2.2.1. The stress frequency is equal to 30 kHz and 5 kHz and the time of each actuation step is equal to 1 s and 6 s, respectively.
The number of loading cycles for each step of actuation (n i ) is computed from the imposed stress frequency (f i ) and the time of actuation (t a,i ), as expressed in equation (27).
The stress frequency f i depends on the frequency of the applied alternating voltage (f V ). The analytical expression of the stress frequency is derived in appendix for the sake of clarity.
Finally, the number of cycles to failure (N f ) is computed according to equation (28).
where N a is the total number of performed actuation steps before the specimen failure.

FEM Model
Fatigue crack growth in the central notched specimen is simulated through a coupled-field electromechanical fracture FEM model built in Ansys.
The study of the fatigue fracture process in the proposed test specimen is a multi-physics problem, as it involves the simultaneous resolution of two coupled domains: mechanical and electrostatic. Indeed, the stress driving the crack propagation in the central specimen is caused by the electrostatic force due to the application of the voltage between the top movable plates and bottom electrodes, as explained in section 2.2.1. However, the electromechanical coupling is strongly nonlinear (as shown in equation (1)). Furthermore, fracture simulation is often high-demanding from the computational point of view, especially for three-dimensional (3D) problems.
Referring to figure 7, the procedure adopted to increase the numerical efficiency of the coupled-field electromechanical fracture simulation is described below.

• Multiphysics FEM model of the test microstructure.
Displacement field in the central specimen as a result of the applied actuation voltage is computed from coupled electromechanical simulation of the test microstructure.

• Fracture FEM model of the central notched specimen.
Simulation of fatigue crack growth in the central notched specimen due to the applied actuation voltage is performed according to the previously computed displacement field. • Paris' law. The number of cycles to failure (N f ) is computed using the equivalent SIF range (∆K eqv ) resulting from fatigue fracture simulations according to Paris' law.
Then, the modelling approach adopted in this work avoids repeating the high-demanding multiphysics simulations at each incremental crack step, as the stress and displacement fields leading to the crack growth in the central specimen are determined only once using the multiphysics FEM model of the test microstructure.   The test microstructure is modelled as a 3D solid structure using tetrahedral 10-node SOLID186 elements with three degrees-of-freedoms (DOFs) per node, namely displacement in x (u x ), y (u y ) and z (u z ) directions. A sufficiently refined mesh is used near the notch region to correctly describe stress and strain distributions. All DOFs at the end of trapezoidal beams are fixed ( figure 8(a)).

Multiphysics FEM model of the test microstructure
The electromechanical coupling is modelled using reducedorder one-dimensional (1D) TRANS126 elements. The TRANS126 element is a plane capacitor modelling the capacitive response of an electromechanical device as a function of displacement in one direction. The element has two nodes, connecting the nodes of the FE model between which the electrostatic force acts. Each node has two DOFs: the structural displacement (u x , u y , or u z ) and the electrostatic voltage (V).
TRANS126 elements connect the nodes of the top movable plates and those nodes representing the bottom fixed electrodes, which are not directly included in the FEM model as they are described by boundary conditions at TRANS126 element nodes (figures 8(b) and (c)).
The electrostatic field fringing effect is neglected, and then the capacitance C of electrodes modelled by TRANS126 elements depends on the gap g between electrodes (equation (29)), which is a function of the displacement of the top plates (h 2 ).
The set of TRANS126 elements is generated automatically in Ansys using the command EMTGEN (figures 8(b) and (c)), which also automatically creates the plane of nodes representing the bottom fixed electrodes.
The electrostatic voltage is applied on the TRANS126 nodes connected to the top plates mesh, on the other hand, the electrostatic voltage and the translational DOFs of the bottom electrodes nodes are set to zero.
Just SIFs corresponding to the maximum load, i.e. K I,max and K II,max , are required for the fatigue crack growth computation, according to equations (13) and (14) as the compression phase can be neglected. Then, multiphysics simulations are carried out at the maximum voltage V max causing the maximum stress level (σ max ) in the specimen during the cycle.
Finally, displacements of nodes at the edges of the central specimen are saved for the following fatigue crack growth analysis.

Fracture FEM model of central notched specimen model
A 3D model of the central notched specimen with a preexisting sharp crack at the notch root is built in Ansys for the fatigue crack growth analysis, assuming the initial crack length (a 0 ) and orientation (θ 0 ), as shown in figure 9.
The specimen material is considered linear elastic and isotropic, i.e. plastic effects do not cause damage in the gold material and LEFM hypotheses hold. This is quite reasonable because the maximum actuation voltage during cycling does not exceed 140 V, then the stress at the notch region is below the yield limit, as it will be demonstrated in section 4.1.
The fracture analysis is performed just modelling the notched specimen. Then, the mesh of the model is composed of SOLID186 structural elements, and no multiphysics elements are included in the model. Displacement field of the nodes at the edges of the specimen is computed with the multiphysics model described in section 3.1 and is applied as boundary conditions. Referring to figure 10, the mesh of the model is built according to the procedure summarized below.
• The 3D volume of the central notched specimen with a preexistent crack is generated and a cylindrical volume around the crack tip is created to control the mesh generation near the crack region ( figure 10(a)). • Collapsed quarter-point singular elements are used to accurately estimate the singularity in the stress field near the crack tip [70]. Singular elements are obtained by collapsing the 20-node brick version of SOLID186 elements (figure 10(f)). A spider web mesh made of 4 concentric rings is created on both faces of the cylindrical volume using 2D MESH200 elements to simplify the generation of singular elements. The mesh of these areas will provide the pattern for the solid elements mesh within the cylindrical volume. The innermost ring is composed of 24 2D singular elements automatically generated by the KSCON command in Ansys, on the other hand, the remaining rings are meshed using 8-node quadrilateral elements ( figure 10(b)). • One of the two meshed areas of the cylindrical volume is defined as the source, the other one as the target. Then, the cylindrical volume is meshed by sweeping the mesh pattern of the source to the target area along the specimen thickness ( figure 10(c)). • The rest of the model is meshed using 10-node tetrahedral version of SOLID186 elements. The mesh size is refined near the crack front, on the other hand, the size gradually increases when moving away from the crack (figures 10(d)-(f)).
The update of geometry and mesh is one of the main issues when dealing with crack propagation with FEM, especially in 3D models. However, the strategy previously described allows for automatic remeshing during the simulation based on the location of the crack tip. The crack path is determined by computing the direction of crack propagation (θ c ) at each crack growth step, considering a fixed crack length increment (∆a). The mixedmode SIFs K I,max and K II,max are computed in Ansys from the interaction energy integral using the built-in CINT command and the crack propagation direction (θ c ) is determined from equation (25). Then, the equivalent SIF range (∆K eqv ) is computed according to equations (16) and (17), and the number of cycles (∆N) required for the crack to grow by ∆a is obtained from equation (23).
Then, the mesh of the model is removed, the geometry of the model is updated according to the new crack tip position, quarter-point singular elements are created around the new crack tip and a new mesh is generated in the remaining unmeshed volume, following the strategy described above. Finally, the same procedure is repeated for the next crack growth steps, until the crack reaches one of the specimen edges.

Results and discussion
The results of the kinematic and multiphysics FEM-based models of the test microstructure, as well as the results of the fatigue crack growth analysis are presented in this section. Then, a comparison between the fatigue crack path in the central notched specimen resulting from the fracture model and measured experimentally with SEM after the specimen failure is performed. Finally, the number of cycles to failure computed  using Paris' law is validated by the fatigue life data from experimental tests. Figure 11 shows the distribution of the out-of-plane displacement and Von Mises stress in the test microstructure obtained from the multiphysics FEM model built in Ansys considering a static voltage of 140 V.

Multiphysics FEM simulation of the test microstructure
The stress in the movable plates and trapezoidal beams is much lower than in the central specimen. Then, the notch root on the bottom surface of the central specimen is the most stressed region, whereas further from the notch the highest stress occurs in the material region at an angle of 45 • to the line connecting the notch root to the centre of the specimen, as confirmed by figures 11(b) and (c). In addition, although the stress increases near the connection between the specimen and movable plates, then the increase in stress is quite small and does not alter the stress distribution in the specimen thanks to the inclusion of rounded corners in the microstructure design.
The curves of the microstructure vertical deflection (h 2 ) and the maximum Von Mises stress at the notch root obtained by applying a static voltage (V DC ) ranging 0-140 V are shown in figures 12(a) and (b), respectively. The static voltage is such that the pull-in phenomenon does not occur and the maximum Von Mises stress at the notch does not overcome the yield stress of the gold material (table 3), as also demonstrated in previous works [47]. Then, the material behaviour in the central specimen is elastic in the voltage range 0 − 140 V, and LEFM assumptions hold.
The kinematic model of the test microstructure proposed in section 2.2.1 is validated using the FEM-based model in Ansys.
The deflection of the test microstructure (h 2 ) due to the applied static voltage (V DC ) computed through the multiphysics FEM model is used in the kinematic model. Then, the nominal axial stress (σ a,nom ) is computed using equations (2) and (3) and the maximum stress (σ n,max ) at the notch root is computed using equation (4). Figure 13 shows the comparison between axial stress values computed using FEM and kinematic models.
The good correspondence between the stress values demonstrates that the deflection of the microstructure can be correlated to the stress in the central notched specimen. Then, the variation in the microstructure deflection when actuated by static voltage can be properly used as a detector of the variation in the device stiffness caused by the crack growth in the central specimen during the fatigue loading test.

Fatigue fracture simulation of central notched specimen
The fatigue fracture analysis aims to determine the crack propagation path in the central notched specimen when alternating voltage actuates the test microstructure, causing a fatigue load in the central specimen.
Cracks preferentially nucleate at defects acting as local stress and strain raisers, then several crack nucleation sites may exist across the specimen surface [33]. Fabrication processes, etching techniques, and substrate materials greatly affect the occurrence of initial defects, then a variation may exist from sample to sample. However, cracks are much more likely to start at the notch root, which is the most stressed  region in the specimen, as also demonstrated by FEM simulation results reported in section 4.1. Furthermore, different samples are inspected by SEM after some fatigue cycles (figure 14), confirming that the notch fosters crack propagation, despite the differences between samples.
The fatigue crack propagation due to the applied electrostatic voltage is simulated through the fracture FEM model built in Ansys. The crack path is determined following the procedure described in section 3.2 and the crack propagation angle (θ c ) is computed according to equation (26), using the mixed-modes SIFs (K I and K II ) evaluated at the specimen bottom surface, which is the most stressed area. Figure 15 shows the sequence of crack evolution during fatigue cycling at the alternating voltage of 140 V, and null mean voltage. The results show that the developed model is able to simulate the propagation of the crack, which grows following the direction of maximum tangential stress (as predicted by the LEFM theory), until it reaches the external edge of the specimen, causing the final failure of the microstructure.
Crack propagation in the central specimen may be affected by the initial crack length and orientation, which may depend on the microstructure of the gold material ahead of the notch root. Since the crack nucleation phase is not analyzed in this work, then a sensitivity analysis is performed to evaluate the influence of the assumed initial crack length and orientation on the resulting crack propagation path. The influence of the chosen crack incremental length (∆a) is analyzed as well.
A comparison between the crack paths obtained varying the initial crack length and orientation, considering a fixed crack increment (∆a) of 0.5 µm during cycling, is shown in figures 16(a) and (b) respectively. In the former, the initial crack orientation (θ 0 ) is 10 • , in the latter the initial crack length (a 0 ) is 1 µm. The results demonstrate that the crack path is little affected by the initial crack length and orientation. Figure 16(c) shows the variation of crack paths with different crack incremental lengths, considering initial crack length (a 0 ) of 0.5 µm and initial crack orientation (θ 0 ) of 10 • . The crack paths are similar as long as the crack incremental length is quite small, i.e ∆a is smaller than 1 µm, on the other hand, the crack path is slightly different for larger incremental length, i.e. ∆a = 2 µm.

Experimental results and discussion
The crack propagation path is experimentally observed by SEM, inspecting the central specimen just after failure, i.e. when the pull-in phenomenon occurs, as explained in section 2.4. Figure 17 shows the SEM images of two specimens after failure at different magnifications. The specimens were cycled at the alternating voltage of 140 V and null mean voltage, with  the stress frequency and total time of actuation equal to 30 kHz and 10 s, respectively (N f = 300 000).
The SEM images show that failure occurs due to crack propagation from the notch root in the central specimen, as pointed out by the green arrows in figures 17(b) and (g). Furthermore, the crack propagation path in the two specimens is very similar, as evidenced by comparing figures 17(c) and (h), although the crack initiation site in the two specimens is slightly different, as resulting by comparing figures 17(d) and (j). This is because crack nucleation is influenced by the local variation in grain morphology as well as by the local defects distribution in the region ahead of the notch, which may vary among the two specimens (figures 17(e) and (k)).
Then, different morphologies of the material ahead of the notch may be responsible for different initial crack paths, but once the crack initiates, it propagates following the direction of maximum principal stress according to LEFM theory. Figure 18 shows a comparison between the crack propagation path resulting from the fracture model (a 0 , θ 0 and ∆a are equal to 0.5 µm, 10 • and 0.4 µm, respectively) and measured experimentally. The slight deviation between the two crack paths, highlighted in green in figure 18(a), can be attributed to local defects present in the material, locally influencing the crack path. Nevertheless, the consistency of the experimental and simulated crack paths is quite good, confirming the validity of LEFM theory in characterizing the fatigue fracture behaviour of microstructures, as well as the effectiveness of the built FEM model in predicting crack growth in the notched specimen, even at micro-scale.
The number of cycles to failure (N f ) in the central specimen can be computed using Paris' law reported in equation (11), exploiting the results of fatigue fracture FEM simulations.
Paris' law is a function of empirical material coefficients C and m. However, values of C and m coefficients for the thin film gold material are not available in the literature as far as   the authors know. Moreover, C and m coefficients are usually determined by monitoring the crack increment during the fatigue cycling, which was not carried out in this work as the specimen was inspected by SEM just at failure. Indeed, only the number of cycles to failure (N f ) corresponding to different alternating voltages (V AC ) is recorded during experiments.
Nevertheless, C and m are estimated by minimizing the error between the number of cycles to failure obtained from experimental measurements (N f,exp ) and computed using equations (23) and (24), according to equation (30).
where N m is the number of experimental measurements. The equivalent SIF range (∆K eqv,i ) in equation (30) is computed at each crack length increment using the FEM fracture model, considering a crack incremental length (∆a) of 0.4 µm, an initial crack length (a 0 ) and orientation (θ 0 ) equal to 0.5 µm and 10 • , respectively. The experimental number of cycles to failure N f,exp is measured at different alternating voltage amplitude (V AC ) with the stress frequency of 5 kHz. The results of Paris' law coefficients estimation, based on Tanaka's (equation (16)) and Irwin's (equation (17) The values of Paris' coefficients computed using equation (30) are compared to those obtained from micro-scale experimental fatigue tests performed on some common materials for MEMS devices, such as Polysilicon, LIGA Nickel, and Tungsten, as reported in table 6.
The comparison between the values of m exponent, which do not depend on the load ratio R, shows that the estimated exponent m for thin film gold material is quite reasonable, as it is in the same range as LIGA Nickel and Tungsten materials. Table 6 also shows the Paris' coefficients values of metallic materials obtained from experimental tests on macro-size specimens. The comparison shows slight variability among the results, demonstrating that the specimen size, as well as the material composition and microstructure, influence the values of Paris' law coefficients. Furthermore, there is a slight difference between experimental results obtained from macroscale and micro-scale tests, which was also evidenced by Gabel et al [20]. Despite this, the values of exponents m are in the same order of magnitude, which demonstrates the comparability between macroscopic and microscopic tests, as well as the validity of Paris' law and LEFM theory in describing the fatigue fracture behaviour of materials at micro-scale.

Conclusions
The fatigue fracture characterization of MEMS gold-based notched specimens subjected to cyclic loading is presented in this work. On-chip fatigue tests are performed using a test microstructure specifically designed to load the specimen at different stress levels and with multiple loading cycles using electrostatic actuation. Numerical simulations of fatigue crack growth are performed using a coupled-field electromechanical fracture finite element model, on the basis of LEFM theory and Paris' law.
A kinematic model of the test microstructure is derived and the axial stress in the central specimen is computed through the measured microstructure deflection caused by the actuation voltage. Then, the variation in the microstructure deflection after the fatigue loading is correlated to the stiffness degradation of the central specimen due to crack propagation. The central specimen is not inspected when loaded thanks to the developed test microstructure design, and the specimen failure is identified by the instantaneous large variation in the microstructure deflection, as the pull-in condition occurs when the specimen fractures.
SEM analyses of specimens after failure confirm that cracks start at the notch root, which acts as local stress and strain raiser fostering crack propagation. Furthermore, the comparison between the crack propagation path resulting from SEM images and obtained from the fracture model shows a good agreement, with the slight deviation being attributed to local defects present in the material which can locally influence the crack path. This confirms the validity of the LEFM theory in characterizing the fatigue fracture behaviour of microstructures, as well as the effectiveness of the developed fracture model in predicting crack growth in microsize samples.
Finally, Paris' law is applied to compute the fatigue life by exploiting the results of the fracture FEM model. Paris' coefficients are estimated by minimizing the error between the number of cycles to failure obtained from experimental measurements and numerically computed, as paris' coefficients are not available in the literature for the thin film gold material. The results are quite reasonable if compared to Paris' law coefficients obtained from fatigue tests on micro-size samples and from macroscopic tests on the most common metallic materials. Then, this further demonstrates the reliability of the developed fracture model and experimental measurements.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).