Analytical Solutions for Corrosion-Induced Cohesive Concrete Cracking

<table>Propagation of cracked front <svg height="16.6" id="M249" style="vertical-align:-4.74141pt" version="1.1" viewbox="0 0 22.9625 16.6" width="22.9625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,16.6)">
<g transform="translate(72,-58.72)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑟</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.69,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑦</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.85,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
</text>
</g>
</g>
</svg> and critical crack front <svg height="14.7125" id="M250" style="vertical-align:-3.22281pt" version="1.1" viewbox="0 0 26.0375 14.7125" width="26.0375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7125)">
<g transform="translate(72,-60.23)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑟</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.69,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">c</tspan>
<tspan style="font-size: 8.75px; " x="3.8858881" y="0">r</tspan>
</text>
<text transform="matrix(1,0,0,-1,-55.39,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
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</g>
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</svg> with time <svg height="9.125" id="M251" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 5.0124998 9.125" width="5.0124998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.125)">
<g transform="translate(72,-64.7)">
<text transform="matrix(1,0,0,-1,-71.95,64.86)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑡</tspan>
</text>
</g>
</g>
</svg>.</table>

Journal of Applied Mathematics

fig6

Figure 6

Figure 6: Analytical Solutions for Corrosion-Induced Cohesive Concrete Cracking 