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- W4293194885 abstract "Abstract We consider Morrey’s open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies $$W{:}{text {GL}}^+(n)rightarrow mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mtext>GL</mml:mtext> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> with an additive volumetric-isochoric split, i.e. $$begin{aligned} W(F)=W_{mathrm{iso}}(F)+W_{mathrm{vol}}(det F)={widetilde{W}}_{mathrm{iso}}bigg (frac{F}{sqrt{det F}}bigg )+W_{mathrm{vol}}(det F),, end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>iso</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>vol</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>det</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mi>iso</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mfrac> <mml:mi>F</mml:mi> <mml:msqrt> <mml:mrow> <mml:mo>det</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:msqrt> </mml:mfrac> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>vol</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>det</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of “least” rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function $$begin{aligned} W_{mathrm{magic}}^{+}(F)=frac{lambda _{mathrm{max}}}{lambda _{mathrm{min}}}-log frac{lambda _{mathrm{max}}}{lambda _{mathrm{min}}}+log det F=frac{lambda _{mathrm{max}}}{lambda _{mathrm{min}}}+2log lambda _{mathrm{min}} end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>magic</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfrac> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>max</mml:mi> </mml:msub> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>min</mml:mi> </mml:msub> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mo>log</mml:mo> <mml:mfrac> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>max</mml:mi> </mml:msub> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>min</mml:mi> </mml:msub> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mo>log</mml:mo> <mml:mo>det</mml:mo> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>max</mml:mi> </mml:msub> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>min</mml:mi> </mml:msub> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>log</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>min</mml:mi> </mml:msub> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> is quasiconvex. In addition, we demonstrate that under affine boundary conditions, $$W_{mathrm{magic}}^+(F)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>magic</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of Burkholder and Iwaniec in the field of complex analysis." @default.
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- W4293194885 date "2022-08-24" @default.
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- W4293194885 title "Morrey’s Conjecture for the Planar Volumetric-Isochoric Split: Least Rank-One Convex Energy Functions" @default.
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