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<p>In nonlinear elasticity, the square root of a tensor arises in the polar decomposition of the deformation gradient, and in many other applications in other areas as well. In this work, given a positive integer <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10659_2025_10149_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> <EquationSource Format="TEX">$p$</EquationSource> </InlineEquation>, we derive an explicit expression for the principal <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10659_2025_10149_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> <EquationSource Format="TEX">$p$</EquationSource> </InlineEquation>-th root of a real-valued second-order tensor, which is not necessarily diagonalizable, whose eigenvalues do not lie on the closed negative real axis, but which is otherwise arbitrary, for any underlying space dimension <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10659_2025_10149_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> <EquationSource Format="TEX">$n$</EquationSource> </InlineEquation>. We also present a method for the explicit evaluation of the derivative of the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10659_2025_10149_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> <EquationSource Format="TEX">$p$</EquationSource> </InlineEquation>-th root of a tensor.</p>

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The Explicit Determination of the \(p\)-th Root of an Arbitrary Second-Order Tensor and Its Derivative

  • 期刊论文

  • 发表日期：

    2025年6月25日

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  • C. S. Jog

摘要

In nonlinear elasticity, the square root of a tensor arises in the polar decomposition of the deformation gradient, and in many other applications in other areas as well. In this work, given a positive integer p $p$ , we derive an explicit expression for the principal p $p$ -th root of a real-valued second-order tensor, which is not necessarily diagonalizable, whose eigenvalues do not lie on the closed negative real axis, but which is otherwise arbitrary, for any underlying space dimension n $n$ . We also present a method for the explicit evaluation of the derivative of the p $p$ -th root of a tensor.