Recent Advancements in Fractal Geometric-Based Nonlinear Time Series Solutions to the Micro-Quasistatic Thermoviscoelastic Creep for Rough Surfaces in Contact

<table>The fractal middle-third Cantor structure <svg height="10.9125" id="M226" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 34.200001 10.9125" width="34.200001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑠</tspan>
<tspan style="font-size: 12.50px; " x="8.9646511" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.005039" y="0">3</tspan>
</text>
</g>
</g>
</svg>, where <svg height="14.3875" id="M227" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 17.375 14.3875" width="17.375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.3875)">
<g transform="translate(72,-60.49)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan>
</text>
<text transform="matrix(1,0,0,-1,-63.02,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
</text>
</g>
</g>
</svg> is the initiator step, <svg height="14.6625" id="M228" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 56.150002 14.6625" width="56.150002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6625)">
<g transform="translate(72,-60.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.45)">
<tspan style="font-size: 12.50px; " x="0" y="0">{</tspan>
<tspan style="font-size: 12.50px; " x="6.00144" y="0">𝐸</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.02,60.32)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-52.15,63.45)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝐸</tspan>
</text>
<text transform="matrix(1,0,0,-1,-38.01,60.32)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-33.14,63.45)">
<tspan style="font-size: 12.50px; " x="0" y="0">}</tspan>
</text>
</g>
</g>
</svg> are the other generated step of cantor structure, <svg height="10.325" id="M229" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 11.1875 10.325" width="11.1875" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
<g transform="translate(72,-63.74)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan>
</text>
</g>
</g>
</svg>'s are the lengths of the <svg height="10.325" id="M230" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 12.225 10.325" width="12.225" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
<g transform="translate(72,-63.74)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan>
</text>
</g>
</g>
</svg>'s steps, <svg height="10.95" id="M231" style="vertical-align:-0.1254pt" version="1.1" viewbox="0 0 9.1374998 10.95" width="9.1374998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.95)">
<g transform="translate(72,-63.24)">
<text transform="matrix(1,0,0,-1,-71.95,63.41)">
<tspan style="font-size: 12.50px; " x="0" y="0">ℎ</tspan>
</text>
</g>
</g>
</svg>'s are the heights of <svg height="10.325" id="M232" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 12.225 10.325" width="12.225" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
<g transform="translate(72,-63.74)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐸</tspan>
</text>
</g>
</g>
</svg>'s steps, and <svg height="10.325" id="M233" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 12.05 10.325" width="12.05" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
<g transform="translate(72,-63.74)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐹</tspan>
</text>
</g>
</g>
</svg> is the applied load.</table>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: Recent Advancements in Fractal Geometric-Based Nonlinear Time Series Solutions to the Micro-Quasistatic Thermoviscoelastic Creep for Rough Surfaces in Contact 

Figure 3 | Recent Advancements in Fractal Geometric-Based Nonlinear Time Series Solutions to the Micro-Quasistatic Thermoviscoelastic Creep for Rough Surfaces in Contact 