Yielding Mode in Tension and Compression

4. Uniaxial Deformation on Np-Au

4.3. Tension-compression Asymmetry

4.3.2. Yielding Mode in Tension and Compression

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Fig. 4-12. SEM images (a) after tensile tests with broken and dangling ligaments marked by yellow and dashed-blue arrows, respectively, and (b) after compressive tests with slip lines in ligaments

indicated by orange arrows [30].

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Fig. 4-13(a) presents a general open-cell model consisting of length of cell edge l and thickness of cell edge t reproduced from the Gibson-Ashby model [1]. Fig. 4-13(b) shows a schematic of the four stress components induced by the uniaxial compressive force F: (1) Axial stress 𝜎_{𝑎𝑥𝑖𝑎𝑙} for ligaments parallel to the loading axis, (2) bending stress 𝜎_{𝑏𝑒𝑛𝑑𝑖𝑛𝑔} for transverse ligaments perpendicular to the loading axis, (3) transverse shear stress 𝜏_{𝑡𝑟𝑎𝑛𝑠} for transverse ligaments perpendicular to the loading axis, (4) pure shear stress 𝜏_{𝑠ℎ𝑒𝑎𝑟} for transverse ligaments with short and thick shape. A fixed-fixed beam model [109] was adopted for calculating the stress components 𝜎_{𝑏𝑒𝑛𝑑𝑖𝑛𝑔} and 𝜏_{𝑡𝑟𝑎𝑛𝑠} induced by the bending deformation in the transverse cell edge considering the constraint of the cell edges at end of it. Fig. 4-13(c) indicates a schematic for the fixed-fixed beam model and stress distribution along the neutral axis (dashed-yellow line) that presents the maximum bending stress occurs at the outer surface while the maximum transverse shear stress by bending deformation is applied at the neutral axis.

It is noteworthy that the calculations were only considered for the maximum stress state where slip was most likely to occur. From above description, the maximum resolved shear stress of axial stress can be given by

𝜏_{𝑅𝑒𝑠_𝜎𝑎𝑥𝑖𝑎𝑙,𝑚𝑎𝑥} = 0.5 ∙^𝐹

𝐴 when 𝜏_{𝑅𝑒𝑠_𝜎𝑎𝑥𝑖𝑎𝑙} = cos 𝜙 ∙ cos 𝜆 ∙^𝐹

𝐴, (4-4)

where the highest Schmid factor of 0.5 is chosen considering applied stress on a 45°-inclined slip plane at a 45°-inclined slip direction. The maximum resolved shear stress of bending stress is expressed by

𝜏_{𝑅𝑒𝑠_𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔,𝑚𝑎𝑥} = 0.5 ∙^{𝑀𝑚𝑎𝑥}_𝐼 ^𝑐 when 𝜏_{𝑅𝑒𝑠_𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔} = ^𝑀𝑦_𝐼 ∙ cos 𝜙 ∙ cos 𝜆, (4-5)

where the highest Schmid factor of 0.5 is also used for the maximum resolved shear stress of bending stress; M is the calculated bending moment, c is the centroidal distance when y is the vertical distance away from the neutral axis, and I is the inertial moment of area. The maximum transverse shear stress by bending deformation is written as

𝜏_{𝑅𝑒𝑠_𝜏𝑡𝑟𝑎𝑛𝑠,𝑚𝑎𝑥} =^{4𝑉𝑚𝑎𝑥}

3𝐴 when 𝜏_{𝑅𝑒𝑠_𝜏𝑡𝑟𝑎𝑛𝑠} =^𝑉𝑄

𝐼𝑏, (4-6)

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where V is the calculated shear force, A is the cross-sectional area, Q is the first moment of area, and b is the width of the cross-section. The maximum shear stress by shear deformation can be simply calculated as

𝜏_{𝑅𝑒𝑠_𝜏𝑠ℎ𝑒𝑎𝑟,𝑚𝑎𝑥} =^𝐹

𝐴. (4-7)

The cross-sectional area A and the inertia moment of area I were calculated assuming a filled circular shape of cross-section for the ligament shape.

Fig. 4-14 presents the maximum resolved shear stresses calculated by axial stress 𝜏_{𝑅𝑒𝑠_𝜎𝑎𝑥𝑖𝑎𝑙,𝑚𝑎𝑥} (dark blue line), by bending stress 𝜏_{𝑅𝑒𝑠_𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔,𝑚𝑎𝑥} (red line), by bending deformation 𝜏𝑅𝑒𝑠_𝜏_{𝑡𝑟𝑎𝑛𝑠},𝑚𝑎𝑥 (pink line), and by shear deformation 𝜏𝑅𝑒𝑠_𝜏_{𝑠ℎ𝑒𝑎𝑟},𝑚𝑎𝑥 (light blue line) as a function of distance from cell edge to loading point as shown in Fig. 4-13(c), which regards as a random point by an applied force. For the comparison, resolved shear stresses were normalized by F/A and these were calculated with aspect ratio t to l of 2, 1, and 0.5, which correspond to relative densities of 48%, 28%, and 13%, respectively, using the relative density of open-cell structure given by ^𝜌

𝜌_𝑠=

(𝑡 𝑙⁄ )²+0.766(𝑡 𝑙⁄ )³

0.766(1+𝑡 𝑙⁄ )³ [109]. This range of relative density is to consider the irregular structure in np-Au.

Figs. 4-14(a), 4-14(b), and 4-14(c) present the maximum resolved shear stresses for the aspect ratios 2, 1, and 0.5, respectively. 𝜏_{𝑅𝑒𝑠_𝜎𝑎𝑥𝑖𝑎𝑙,𝑚𝑎𝑥} (dark blue line), 𝜏_{𝑅𝑒𝑠_𝜏𝑡𝑟𝑎𝑛𝑠,𝑚𝑎𝑥} (pink line), and 𝜏_{𝑅𝑒𝑠_𝜏𝑠ℎ𝑒𝑎𝑟,𝑚𝑎𝑥} (light blue line) appear to be independent of the aspect ratio, and 𝜏_{𝑅𝑒𝑠_𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔,𝑚𝑎𝑥} (red line) is increased with increasing the aspect ratio, which indicates that the greater resolved shear stress is applied for longer and thinner ligaments. 𝜏_{𝑅𝑒𝑠_𝜏𝑡𝑟𝑎𝑛𝑠,𝑚𝑎𝑥} shows the greatest applied stress at relatively both end regions for all aspect ratio cases. In the middle region, 𝜏_{𝑅𝑒𝑠_𝜏𝑠ℎ𝑒𝑎𝑟,𝑚𝑎𝑥} is the greatest for aspect ratio 2, and 𝜏_{𝑅𝑒𝑠_𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔,𝑚𝑎𝑥} is the greatest for aspect ratios 1 and 0.5. For aspect ratio 0.5 in Fig. 4-14(c), 𝜏𝑅𝑒𝑠_𝜎_{𝑏𝑒𝑛𝑑𝑖𝑛𝑔},𝑚𝑎𝑥 is much higher than the others and has the maximum resolved shear stress for a wide middle region. It is noteworthy that the resolved shear stress of axial stress is always lower that other stress components; the resolved shear stress of bending stress, transverse shear stress by bending deformation, and shear stress by shear deformation. This result implies that when uniaxial loading is applied, the yielding seems to initiate in transverse ligaments by bending and/or shear rather than in vertical ligaments parallel to the loading direction.

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In compressive loading, as described above, initial yielding will occur by bending, transverse shear by bending, and pure shear depending on local aspect ratio and distance between vertical ligaments supporting normal forces. Plastic collapse by bending and shear during compression generally leads to thickening and strengthening of the deformed ligaments; a higher stress is required to induce additional deformation in this ligament and the applied force is also dispersed to other weaker ligaments rather than concentrated on this deformed ligament. The former case results in the strain- hardening in the compressive stress-strain curves, and the latter induces a uniform distribution of strain throughout the entire gauge volume as shown in Fig. 4-15(b). In tensile loading, yielding could also firstly initiate by bending or shear of transverse ligaments since the amount of applied shear stress is the same and the sign is opposite under tensile loading. However, axial deformation of ligaments along the tensile loading direction is the dominant deformation mode. This is because, even after initial yielding, once the weak ligaments being aligned to the tensile loading axis are deformed, they progress to necking and failure quickly by stress concentration, resulting in catastrophic failure by stress concentrations at neighboring ligaments of the fractured ligaments (Fig. 4-15(a)). Although yielding could be initiated by bending and shear of the ligaments, this behavior could not be attributed to strain- hardening plasticity in the tensile stress-strain curves. Instead, necking and failure of vertical ligaments aligned to the tensile loading axis will govern the macroscopic yielding and failure. In other words, when the same force is applied in the tensile and compressive directions, the compressive loading can take the higher resolved stress and attain yielding earlier. It is believed that the compressive state is mainly accompanied by bending or shear deformation of ligaments corresponding to 𝜏_{𝑅𝑒𝑠_𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔}, 𝜏𝑅𝑒𝑠_𝜏_{𝑡𝑟𝑎𝑛𝑠},𝑚𝑎𝑥, and 𝜏𝑅𝑒𝑠_𝜏_{𝑠ℎ𝑒𝑎𝑟},𝑚𝑎𝑥, which are subject to relatively high resolved shear stress. In tensile loading, initial bending and shear of ligaments can also occur first, which not leads to tensile failure but to subsequent axial deformation of ligaments, considered as the dominant deformation behavior, and to tensile yielding and failure. This can be represented as the 𝜏_{𝑅𝑒𝑠_𝜎𝑎𝑥𝑖𝑎𝑙,𝑚𝑎𝑥} taking the least resolved shear stress, resulting in the delayed attainment of the macroscopic yield. Even though the np-Au structure is more complicated and irregular [22, 23, 27, 64, 122], the simple periodic open-cell structure model explains why compressive yield strength is lower than tensile yield strength except for dL of 868 nm.

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Fig. 4-13. (a) unit cell consisting of cell edge length l and thickness t represented by Gibson-Ashby model [1]. (b) Schematics for 4-types of stress components under the uniaxial compressive force F to

the unit cell; (1) axial stress 𝝈_{𝒂𝒙𝒊𝒂𝒍} (dark blue arrow) in ligaments parallel to the loading direction, (2) bending stress 𝝈_{𝒃𝒆𝒏𝒅𝒊𝒏𝒈} (red arrow) and (3) transverse shear stress 𝝉_{𝒕𝒓𝒂𝒏𝒔} (pink arrow) in ligaments perpendicular to the loading direction, and (4) pure shear stress 𝝉_{𝒔𝒉𝒆𝒂𝒓} (light blue arrow)

especially in short and thick transverse ligaments. (c) Schematic for fixed-fixed beam model for bending deformation with loading point F away from the cell edge as x at random and cross-section of

the beam which represents stress distribution for bending stress (indicated by red) and transverse shear stress (indicated by pink) as the neutral axis (indicated by dashed yellow line) [30].

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Fig. 4-14. Maximum resolved shear stress for 4-types of stress components normalized by F/A as a function of distance from cell edge to loading point with aspect ratio t to l of (a) 2, (b) 1, (c) 0.5; the

maximum resolved axial shear stress 𝝉_{𝑹𝒆𝒔_𝝈𝒂𝒙𝒊𝒂𝒍,𝒎𝒂𝒙} (dark blue line), the maximum resolved shear stress of bending stress 𝝉_{𝑹𝒆𝒔_𝝈𝒃𝒆𝒏𝒅𝒊𝒏𝒈,𝒎𝒂𝒙} (red line), the maximum transverse shear stress by bending

deformation 𝝉_{𝑹𝒆𝒔_𝝉𝒕𝒓𝒂𝒏𝒔,𝒎𝒂𝒙} (pink line), and the maximum shear stress by shear deformation 𝝉_{𝑹𝒆𝒔_𝝉𝒔𝒉𝒆𝒂𝒓,𝒎𝒂𝒙} (light blue line) [30].

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