Exact Solution for an Anti-Plane Interface Crack between Two Dissimilar Magneto-Electro-Elastic Half-Spaces

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td align="left">properties</td><td align="center">BaTiO<sub>3</sub> piezoelectric</td><td align="center">CoFe<sub>2</sub>O<sub>4</sub> piezomagnetic</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td align="left"><span style="width: 99.324997px;"><svg height="19.924999" id="M205" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 99.324997 19.924999" width="99.324997" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.017,-0,0,-.017,53.488,15.775)"><path d="M719 650v-28q-43 -2 -62 -15t-22 -44q-6 -47 -6 -169v-403h-31l-426 524h-2v-251q0 -111 6 -169q4 -37 24 -50.5t72 -16.5v-28h-237v28q45 2 64.5 16t23.5 49q6 62 6 171v220q0 54 -3 68.5t-17 32.5q-16 19 -34.5 26.5t-54.5 10.5v28h147l418 -502h3v246q0 117 -7 166
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<g transform="matrix(.017,-0,0,-.017,93.387,15.775)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg></span></td><td align="center">43,00</td><td align="center">45,30</td></tr><tr><td align="left"><span style="width: 73.5px;"><svg height="20.0625" id="M206" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 73.5 20.0625" width="73.5" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350
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<g transform="matrix(.012,-0,0,-.012,7.1,19.863)"><use xlink:href="#x31"></use></g><g transform="matrix(.012,-0,0,-.012,12.811,19.863)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
l-26 -65h-233z" id="x35"></path></g>
<g transform="matrix(.017,-0,0,-.017,23.113,15.775)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,28.994,15.775)"><path d="M614 175l29 -10q-33 -109 -57 -154q-121 -26 -184 -26q-90 0 -160.5 29t-112.5 77t-63.5 105.5t-21.5 119.5q0 157 108 253t277 96q36 0 71.5 -5t69 -13.5t36.5 -8.5q15 -102 20 -150l-29 -8q-20 79 -66.5 114t-128.5 35q-119 0 -187.5 -86t-68.5 -207
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<g transform="matrix(.012,-0,0,-.012,61.225,7.613)"><use xlink:href="#x32"></use></g>
<g transform="matrix(.017,-0,0,-.017,67.562,15.775)"><use xlink:href="#x29"></use></g>
</svg></span></td><td align="center">11,60</td><td align="center">0,00</td></tr><tr><td align="left"><span style="width: 109.2125px;"><svg height="19.924999" id="M207" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 109.2125 19.924999" width="109.2125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M387 375q0 -16 -17 -31t-29 -15q-9 0 -12 13q-13 74 -80 74q-36 0 -61 -24.5t-25 -56.5t24 -51.5t68 -19.5q33 0 47 2l2 -7l-32 -42q-20 2 -54 2q-45 0 -74 -21.5t-29 -60.5q0 -41 28 -65.5t73 -24.5q78 0 145 67l17 -23q-33 -47 -84.5 -75t-111.5 -28q-70 0 -114.5 35.5
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<g transform="matrix(.017,-0,0,-.017,23.05,15.775)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,28.932,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,37.091,15.775)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.012,-0,0,-.012,45.25,7.613)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.012,-0,0,-.012,52.235,7.613)"><use xlink:href="#x39"></use></g>
<g transform="matrix(.017,-0,0,-.017,60.55,15.775)"><use xlink:href="#x43"></use></g><g transform="matrix(.017,-0,0,-.017,71.854,15.775)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,78.858,15.775)"><path d="M687 650v-28q-52 -6 -71.5 -23t-47.5 -83q-142 -346 -208 -527h-31q-119 325 -207 545q-21 54 -39.5 68.5t-65.5 19.5v28h245v-28q-52 -5 -61.5 -15.5t3.5 -43.5q55 -157 166 -437h2q120 303 158 421q14 43 3.5 55.5t-71.5 19.5v28h225z" id="x56"></path></g><g transform="matrix(.017,-0,0,-.017,89.346,15.775)"><use xlink:href="#x6D"></use></g><g transform="matrix(.017,-0,0,-.017,103.268,15.775)"><use xlink:href="#x29"></use></g>
</svg></span></td><td align="center">11,20</td><td align="center">0,08</td></tr><tr><td align="left"><span style="width: 81.5px;"><svg height="16.5625" id="M208" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 81.5 16.5625" width="81.5" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,8.413,16.212)"><use xlink:href="#x31"></use></g><g transform="matrix(.012,-0,0,-.012,14.124,16.212)"><use xlink:href="#x35"></use></g>
<g transform="matrix(.017,-0,0,-.017,24.425,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,30.307,12.138)"><use xlink:href="#x4E"></use></g><g transform="matrix(.017,-0,0,-.017,42.937,12.138)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,49.94,12.138)"><path d="M673 0h-245v28q47 6 58 16t1 39l-55 154h-213q-31 -87 -36 -129q-16 -47 -5 -61t68 -19v-28h-226v28q50 7 68.5 23.5t43.5 80.5l214 524l24 9l193 -536q23 -63 41.5 -79.5t68.5 -21.5v-28zM418 280l-81 255h-2q-74 -197 -101 -255h184z" id="x41"></path></g><g transform="matrix(.017,-0,0,-.017,61.635,12.138)"><use xlink:href="#x6D"></use></g><g transform="matrix(.017,-0,0,-.017,75.557,12.138)"><use xlink:href="#x29"></use></g>
</svg></span></td><td align="center">0,00</td><td align="center">550,00</td></tr><tr><td align="left"><span style="width: 106.675px;"><svg height="20.35" id="M209" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 106.675 20.35" width="106.675" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,9.5,19.925)"><use xlink:href="#x31"></use></g><g transform="matrix(.012,-0,0,-.012,15.211,19.925)"><use xlink:href="#x31"></use></g>
<g transform="matrix(.017,-0,0,-.017,25.512,15.85)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,31.394,15.85)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,39.553,15.85)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.012,-0,0,-.012,47.713,7.688)"><use xlink:href="#x2212"></use></g><g transform="matrix(.012,-0,0,-.012,54.697,7.688)"><path d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5
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<g transform="matrix(.017,-0,0,-.017,63.012,15.85)"><use xlink:href="#x4E"></use></g><g transform="matrix(.017,-0,0,-.017,75.643,15.85)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,82.646,15.85)"><use xlink:href="#x41"></use></g>
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<g transform="matrix(.017,-0,0,-.017,100.725,15.85)"><use xlink:href="#x29"></use></g>
</svg></span></td><td align="center">5,00</td><td align="center">590,00</td></tr><tr><td align="left"><span style="width: 112.875px;"><svg height="19.924999" id="M210" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 112.875 19.924999" width="112.875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg></span></td><td align="center">0,00</td><td align="center">0,00</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

The material constants for BaTiO<sub>3</sub> and CoFe<sub>2</sub>O<sub>4</sub>.

Smart Materials Research

tab1

Table 1

Table 1: Exact Solution for an Anti-Plane Interface Crack between Two Dissimilar Magneto-Electro-Elastic Half-Spaces 