Geometry of Logarithmic Strain Measures in Solid Mechanics
Geometry of Logarithmic Strain Measures in Solid Mechanics
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DOI：
10.1007/s00205-016-1007-x
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发表时间：
2016-11-01
影响因子：
2.5
通讯作者：
Martin, Robert J.
中科院分区：
文献类型：
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作者：
Neff, Patrizio;Eidel, Bernhard;Martin, Robert J.
我们考虑Hencky应变张量的各向同性不变量，即两个对数应变度量(Iso)=平行于dev(N)log U平行于=平行于dev(N)对数根F-T F平行于andomega(Vol)=平行于tr(Log U)平行于=平行于trlog根F-T F)，它们是Hencky应变张量的各向同性不变量，并证明了它们可以用基于一般线性群上测地距离的纯几何方法唯一地刻画.这里，是变形梯度，U=根F-T F是右Biot伸展张量，表示主矩阵对数，平行于是Frobenius矩阵范数，是迹算子，dev(N)X=X-1/ntr(X)。1是X的n维偏量，是R-nxn的一个元素。该特征将Hencky(或真)应变张量标识为线性(无穷小)应变张量epsilon=sym del u的自然非线性扩展，其是位移梯度del u的对称部分，揭示了线弹性力学中平行于dev(N)sym del u平行于(2)+kappa/2[tr(Sym Del U)](2)=u平行于dev(N)epsilon平行于(2)+kappa/2[tr(Epsilon)](2)的经典二次各向同性能量势u与平行于dev(N)log U平行于(2)+kappa/2[tr(LogU)]的几何非线性二次各向同性Hencky能量体操，(2)=u omega(2)(Iso)+kappa/2其中是剪切模数，表示体积模数。我们的推导涉及到正交极因子的一个新的基本对数最小化性质，其中是的极分解。我们还将我们的方法与先前试图将对数Hencky应变张量直接确定为非线性各向同性弹性力学中的首选应变张量的方法进行了对比。
We consider the two logarithmic strain measuresomega(iso) = parallel to dev(n) log U parallel to = parallel to dev(n) log root F-T F parallel to andomega(vol) = parallel to tr(log U)parallel to = parallel to trlog root F-T F)parallel to = parallel to log(det U)parallel to,which are isotropic invariants of the Hencky strain tensor , and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group . Here, is the deformation gradient, U = root F-T F is the right Biot-stretch tensor, denotes the principal matrix logarithm, parallel to.parallel to is the Frobenius matrix norm, is the trace operator and dev(n) X = X - 1/n tr(X) . 1 is the n-dimensional deviator of X is an element of R-nxn. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor epsilon = sym del u, which is the symmetric part of the displacement gradient del u, and reveals a close geometric relation between the classical quadratic isotropic energy potentialmu parallel to dev(n) sym del u parallel to(2) + kappa/2 [tr(sym del u)](2) = mu parallel to dev(n) epsilon parallel to(2) + kappa/2 [tr(epsilon)](2)in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energymu parallel to dev(n) log U parallel to(2) + kappa/2 [tr(log U)](2) = mu omega(2)(iso) + kappa/2 omega(2)(vol),where is the shear modulus and denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor , where is the polar decomposition of . We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity.