The contradiction of increasing demand for accurate engineering data and insufficient existing knowledge on mechanical properties has motivated a great deal of research into the MEMS characterisation fields. Substantial research and commercial development have been made in the past years towards establishing reliable approaches for mechanical characterisation. However, common problems associated with small size of specimen of MEMS material such as sample fabrication, sample measurement and sample loading, have been major drawbacks and have led to a considerable variation among mechanical values.
Prior to designing a mechanical characterisation test-rig for MEMS materials, it is necessary to determine suitable experimental methodologies. In the past decades, most work had been devoted to developing the two primary types of characterisation tests: tensile test and bending test, which made them much more mature candidates for general mechanical characterisation than other approaches. A theoretical analysis of both tensile test and bending test is therefore presented here in order to justify the preferred practical experimental designs for this work. Two main material properties – Young’s modulus and fracture strength are focused on as they provide the most valuable information for
most occasions. A rectangular beam (with length L, width band depth ℎ) was chosen as the specimen for each case. In the tensile test, a point force ܨ௧is applied along the centreline of the beam while the concentrated bending forceܨ is applied at the mid-span of a built-in beam (Figure 3.3).
Figure 3.3: Basics of a tensile test (left) and a built-in bending test (right)
From classic strength of materials, the tensile stress (ߪ௧) and the end deflection (ߜ௧) of the test beam in the uniaxial tensile test are easily deduced as
ߪ௧=ܾ݄ܨ௧ (3.3)
ߜ௧=ߪܧ ܮ ൌ௧ ܧܾ݄ܮሺ͵ǤͶሻܨ௧
whereE denotes the Young’s modulus of the material. In the bending test, it is assumed that the plane cross-sections remain planar (also known as St Venant’s principle) and the effect of shear stress is negligible. From simple beam bending theory, it is obtained that
L
ܨ௧
ܨ
L
Tensile test on a cantilever beam Bending test on a built-in beam (concentrated loading at centre)
ܯ ܫ= ߪ ݕ = ܧ ܴ (3.5)
where Mis the applied bending moment at a transverse section, Iis the second moment of area of the beam cross-section about the neutral axis (N.A.) of the beam,ߪ is the bending stress at a distance of yfrom the N.A and Ris the radius curvature at the cross- section. Thus
ߪ =ܯ ݕܫ (3.6)
The measurement of built-in bending test is usually taken place at the mid-span where the maximum deflection occurred. The bending moment diagram of a built-in beam carrying a concentrate load at the mid-span could be as given in Figure 3.4. The total bending moment diagram of a built-in beam is a superposition of a ‘free’ moment diagram and a fixing moment diagram.
By symmetry, the bending moment at the mid-span can be concluded to be
ܯ ൌ ܨ8ܮ (3.7)
The N.A of the rectangle cross section is located at the central line and the second moment of areaI =య
ଵଶ. Hence the maximum stress occurs at the edge of mid-span where x=L/2,y=ℎ 2⁄ is given by:
Figure 3.4: The Bending Moment (B.M) diagram of a built-in beam
Then Mohr’s second theorem can be applied for the deflection at mid-span, thus the defection at the mid-span relative to the ends is given as
ߜ =ாூଵ ൈ ሾ݂݅ݎݏݐ݉ ݉ ݁݊ݐ݂ܽݎ݂ܾ݁ܽ݁݊݀݅݊݃݉ ݉ ݁݊ݐ݀݅ܽ݃ݎܽ݉ ܾ݁ݐݓ݁݁݊݉ ݅݀ െ ݏܾܽ݊ܽ݊݀݊݁݁݊݀ܽݑݐݐ݄݁ܿ݁݊ݐݎ݁ሿ
Thus, from the B.M. diagram in Figure 3.4, ߜ =ܧܫቆ1 ܨܮ ଷ 96 + െܯ ܮଶ 8 ቇ ൌ െ ܨܮଷ ͳܧܾ݄ଷ (3.9) M -M -M
+
--
+ - - ܨ LThus, a summary of the result of uniaxial tensile test and built-in bending test is concluded in Table 3.1.
Table 3.1:Maximum Stress and Maximum deflection of the tensile test and the bending test
Maximum Stress Maximum Deflection Young’s Modulus
Tensile test
ߪ௧=ܾܨℎ௧ ߜ௧=ܧܾܨ௧ℎܮ E =ߜF୲L ௧ܾℎ Bending test
(Built-in beam central load)
ߪ=38ܾܨℎܮଶ ߜ=− ܨܮ ଷ 16ܧܾℎଷ E = ܨܮଷ 16ߜܾℎଷ From Table 3.1 ߪ௧ ߪ = ͺ ܨ௧ℎ ͵ܨܮ (3.10)
For same materials, it is presumed that: σୡ୰୧୲୧ୡୟ୪= σ୲ൌ ߪThus ܨ௧
ܨ = ͺ ܮ
3ℎ (3.11)
In the case of linear simple bending beam theory, the length L is usually much larger value than h in typical designs of test beams (usually ten times more). Thus, equation 3.11 also emphasizes that a smaller force (usually an order of magnitude smaller) is required to generate the critical stress in the bending test. Another great advantage of bending test is that the maximum deflection at failure is much larger than that of a tensile test on a similar beam and can be conveniently detected with various techniques.
On the other hand, according to Table 3.1, accurate measurements of applied force, specimen dimensions and displacement are required to determine Young’s modulus. Hence
E =ߜF୲L ௧ܾ݄=
ܨܮଷ
ͳߜܾ݄ଷ (3.12)
According to equation 3.12, the main uncertainty in the tensile metrology is likely to originate from the measurement of elongation ߜ௧ because of its small magnitude compared with the tension forceܨ௧, while the dominant factor in the cantilever bending metrology is likely to be the depth of height h due to the high exponent ( ℎଷ)in the denominator and the relatively small forceܨ. Therefore, an accurate measurement of the specimen’s dimensions is critical in bending tests while a precise measurement of elongation is the top priority in tensile tests.
The primary object of this research is to conduct a solid mechanical characterisation for polymeric MEMS materials, mainly MSL materials. The major difficulties in mechanical characterisation on MEMS materials were insufficient precise models for interpreting data and the metrology errors in establishing the geometry of test devices (Senturia 1998). Hence data conversion and metrology errors were the main concerns in selecting proper characterisation methods.
In principle, the tensile test is a more straightforward method to obtain Young’s modulus or stress level than the bending test. The tensile tests are the standard procedures in ASTM and BS standards and are similar in concept to the standard definition of mechanical properties where the testing theory of bending is derived from classic bending theories based on general assumptions which may be questionable under small scales and
the interpretation of test results is much more complicated and susceptible to the uncertainties of measured quantities. A few tensile tests on MEMS materials had been carried out and exhibits high agreement of values of mechanical properties with other material tests (e.g. membrane test) (Sharpe 2003). The tensile test is therefore the preferred method for more direct data interpretation and more reliable results would be expected in general mechanical characterisation of materials.
On the other hand, the bending test has the advantage of lower requirements for force implementation and displacement measurement. However, the spatial and lateral accuracy of most MEMS specimens fabricated using MSL systems is usually limited by the nature of this technology, especially the thickness. Furthermore, it is also practically difficult to precisely measure the dimensions of typical high-aspect-ratio MEMS specimens (fibres and thin-films) at small scale. Since the accuracy of specimen dimension is crucial in bending tests, uncertainty in the geometry can result in significant errors in the whole bending metrology loop. Moreover, the surface residual stress of MSL specimen, which commonly exists but is hard to detect, also has significant effect on the strength characterisation in bending tests. Thus, the bending test is not a desirable candidate for testing MSL specimens as the dimensional error and surface residual stress in MSL specimens usually introduce severe metrology errors.
In all, because the data interpretation and metrology errors were the major concerns in designing a mechanical characterisation test-rig for MSL material, the tensile test was finally selected as the characterisation approach in the present research.