Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations) and work-conjugate stress measures are analyzed and compared for each formulation. While the Lagrangian formulation has been used in prior work for constitutive modeling of anisotropic and hyperelastic materials, the Eulerian formulation, which may be advantageous for modeling isotropic solids and fluids with no physically identifiable reference configuration, does not seem to have been used elsewhere in a continuum mechanical setting for the purpose of constitutive development, though it has been introduced before in a purely kinematic setting.

研究了变形梯度张量的两个三角分解。第一个称为拉格朗日公式，由上三角拉伸量乘以旋转张量组成。第二个称为欧拉公式，由后三角拉伸后乘以不同的旋转张量组成。相应的拉伸张量分别表示为拉格朗日拉伸和欧拉拉普拉斯拉伸。针对每种配方分析并比较了运动学（具有物理解释）和共轭应力测量方法。尽管拉格朗日公式已用于各向异性和超弹性材料的本构模型的先前工作中，但欧拉公式对于在没有物理可识别参考配置的情况下对各向同性固体和流体进行建模可能是有利的，