Twinned Finite Plane Deformations Generated by Prescribed Rotation Fields

In previous work, it has been shown that any suitably smooth plane proper-orthogonal tensor field can serve as a rotation tensor for generating a plane finite deformation. In this paper, this previous analysis is used to study plane finite twin deformations. We show that given a defined smooth curve which separates two arbitrarily prescribed rotation fields, a twin deformation field can be generated in a neighborhood surrounding such curve. Examples are presented for cases where the Jacobian of the finite deformation field is discontinuous or continuous across the defined curve. Twinning in an elastic region is also analyzed in some detail.     

Temperature field formation in a planet due to tidal deformations

A model of a planet considered as a homogeneous viscoelastic sphere in the gravitational field of a point mass is discussed. Tidal deformations occur in the process of their mutual motion. The deformation rate tensor and the corresponding dissipative function are found. The time variation of the deformation tensor components accompanied by the heat release at each point of the planet causes the formation of a temperature field described by the inhomogeneous heat conduction equation. The temperature field is determined by averaging with respect to the proper rotation angle and is estimated for the Moon.     

Dynamic rotation and stretch tensors from a dynamic polar decomposition

The local rigid-body component of continuum deformation is typically characterized by the rotation tensor, obtained from the polar decomposition of the deformation gradient. Beyond its well-known merits, the polar rotation tensor also has a lesser known dynamical inconsistency: it does not satisfy the fundamental superposition principle of rigid-body rotations over adjacent time intervals. As a consequence, the polar rotation diverts from the observed mean material rotation of fibers in fluids, and introduces a purely kinematic memory effect into computed material rotation. Here we derive a generalized polar decomposition for linear processes that yields a unique, dynamically consistent rotation component, the dynamic rotation tensor, for the deformation gradient. The left dynamic stretch tensor is objective, and shares the principal strain values and axes with its classic polar counterpart. Unlike its classic polar counterpart, however, the dynamic stretch tensor evolves in time without spin. The dynamic rotation tensor further decomposes into a spatially constant mean rotation tensor and a dynamically consistent relative rotation tensor that is objective for planar deformations. We also obtain simple expressions for dynamic analogues of Cauchy's mean rotation angle that characterize a deforming body objectively.     

On the compatibility condition of displacement field for finite deformation of continuum

The vanishing of Riemann-Christoffel tensor is usually adopt-ed as the compatibility condition of finite deformation.However,we prove in this paper by the method of Cesaro that this condition is necessary but not sufficient for guarantee of a single-valued,continuous displacement field.A new general compatibility condi-tion,based on the theorem of strain-rotation decomposition(Chen[4])is derived.The displacement compatible condition reduces to Saint-Venant's condition when strain and rotation are infinitesimal.     

有限转动张量的保角参量及在多柔体系统中的应用

采用保角转动参数描述了多体系统中的大转动张量．该方法消除了传统的欧拉参数描述所必需的约束方程，并且适于大变形部件的建模需要．利用以上结果建立了含大变形梁状部件的多体系统的力学模型．     

Determination of the Rotation Tensor in the Polar Decomposition

An explicit representation for the rotation tensor which contains the lower powers of deformation gradient is proposed and used to evaluate the angle and axis of the rotation tensor. Some related equations about the rotation tensor are established. Through the approximate analysis, the relation between the S-R decomposition and the polar decomposition is examined. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some results for approximate strain and rotation tensor formulations in geometrically non-linear Reissner-Mindlin plate theory

Finite element deflection and stress results are presented for four flat plate configurations and are computed using kinematically approximate (rotation tensor, strain tensor or both) non-linear Reissner-Mindlin plate models. The finite element model is based on a mixed variational principle and has both displacement and force field variables. High order interpolation of the field variables is possible through p-type discretization. Results for some of the higher order approximate models are given for what appears to be the first time. It is found that for the class of example problems examined, exact strain tensor but approximate rotation tensor theories can significantly improve the solution over approximate strain tensor models such as the von Kármán and moderate rotation models when moderate deflections/rotations are present. However, for each of the problems examined (with the exception of a postbuckling problem) the von Kármán and moderate rotation model results compared favorably with the higher order models for deflection magnitudes which could be reasonably expected in typical aeroelastic configurations.     

The explicit representation to the principal rotation angle and the principal rotation axis

By using Cayley-Hamilton theorem, two kinds of explicit representation for the rotation tensor are proposed. One contains the lower powers of deformation gradient, by which the formula of the principal rotation angle and the explicit representation of principal axis are obtained; the other, a high efficient method to obtain the rotation tensor, does not contain the complicated coefficients and uses few variables. Some properties about the principal rotation angle and the principal rotation axis are obtained.     

近场动力学框架下的键基对应模型

传统键基近场动力学模型存在泊松比限制的问题,为了解决这一问题发展了态基近场动力学模型。其中非常规态的近场动力学模型通过定义非局部的变形梯度将近场力和传统应力关联起来,方便使用传统本构,但是态基近场动力学计算效率低于键基近场动力学。结合态基模型和键基模型的优势,提出键基对应模型,定义了基于键的变形梯度,参考连续介质力学中变形梯度的极分解过程,将键的变形分为转动部分和伸长部分。从而进一步定义了应变,通过物理方程求应力,进而计算键传递的近场力。键基对应模型解决了键基近场动力学的泊松比限制问题,也不需要进行近场动力学微观材料常数的计算。数值算例和理论推导证明了键变形梯度定义以及近场力计算方式的正确性。     

Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition

We show how to determine the midsurface of a deformed thin shell from known geometry of the undeformed midsurface as well as the surface strains and bendings. The latter two fields are assumed to have been found independently and beforehand by solving the so-called intrinsic field equations of the non-linear theory of thin shells. By the polar decomposition theorem the midsurface deformation gradient is represented as composition of the surface stretch and 3D finite rotation fields. Right and left polar decomposition theorems are discussed. For each decomposition the problem is solved in three steps: (a) the stretch field is found by pure algebra, (b) the rotation field is obtained by solving a system of first-order PDEs, and (c) position of the deformed midsurface follows then by quadratures. The integrability conditions for the rotation field are proved to be equivalent to the compatibility conditions of the non-linear theory of thin shells. Along any path on the undeformed shell midsurface the system of PDEs for the rotation field reduces to the system of linear tensor ODEs identical to the one that describes spherical motion of a rigid body about a fixed point. This allows one to use analytical and numerical methods developed in analytical mechanics that in special cases may lead to closed-form solutions.     

Experimental lower bounds on geometrically necessary dislocation density

A single nickel crystal is indented with a wedge indenter such that a two-dimensional deformation state with three effective plane strain slip systems is induced. The in-plane lattice rotation of the crystal lattice is measured with a three micrometer spatial resolution using Orientation Imaging Microscopy (OIM). All non-zero components of the Nye dislocation density tensor are calculated from the lattice rotation field. A rigorous analytical expression is derived for the lower bound of the total Geometrically Necessary Dislocation (GND) density. Existence and uniqueness of the lower bound are demonstrated, and the apportionment of the total GND density onto the effective individual slip systems is determined. The lower bound solution reduces to the exact solution under circumstances in which only one or two of the effective slip systems are known to have been activated. The results give insight into the active slip systems as well as the dislocation structures formed in the nickel crystal as a result of the wedge indentation.     

A direct representation of the rotation tensor in polar decomposition of deformation gradient and its applications

Through a variational approach, an explicit connection between the additive and the polar decompositions of deformation gradient has been established. An exact formula for determining the rotation tensor in polar decomposition is obtained. The formula is fundamental in continuum mechanics and can be used to separate the rotation and the pure strain in deformation, by which various approximate expressions can be easily obtained.     

On several new methods to polar decomposition computation

In this paper, the polar decomposition of a deformation gradient tensor is analyzed in detail. The four new methods for polar decomposition computation are given: (1) the iterated method, (2) the principal invariant's method, (3) the principal rotation axis's method, (4) the coordinate transformation's method. The iterated method makes it possible to establish the nonlinear finite element method based on polar decomposition. Furthermore, the material time derivatives of the stretch tensor and the rotation tensor are obtained by explicit and simple expressions. The authors gratefully acknowledge the support rendered by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi of China in 1998.     

On 2D Viscoelasticity with Small Strain

An exact two-dimensional rotation–strain model describing the motion of Hookean incompressible viscoelastic materials is constructed by the polar decomposition of the deformation tensor. The global existence of classical solutions is proved under smallness assumptions only on the size of the initial strain tensor. The proof of global existence utilizes the weak dissipative mechanism of motion, which is revealed by passing the partial dissipation to the whole system.     

Digital volume correlation including rotational degrees of freedom during minimization

Digital volume correlation is a new experimental technique that allows the measurement of the full-field strain tensor in three dimensions. We describe the addition of rotational degrees of freedom into the minimization problem for digital volume correlation in order to improve the overall performance of the strain measurement. A parameterization of rotations that is particularly suited to the minimization problem is presented, based on the angle-axis representation of finite rotations. The partial derivative of both a normalized cross-correlation coefficient and the sum-of-squares correlation coefficient are derived for use with gradient-based minimization algorithms. The addition of rotation is shown to greatly reduce the measurement error when even small amounts of rigid body rotation are present in an artificially rotated test volume. In an aluminum foam sample loaded in compression, including rotational degrees of freedom produced smoother contours of minimum principal strain. Renderings of the aluminum foam architecture in areas of low, medium and high rotation showed material deformation pattern in detail.     

Anholonomic configuration spaces and metric tensors in finite elastoplasticity

Deformation mappings are considered that correspond to the motions of lattice defects, elastic stretch and rotation of the lattice, and initial defect distributions. Intermediate (i.e., relaxed) configuration spaces associated with these deformation maps are identified and then classified from the differential-geometric point of view. A fundamental issue is the proper selection of coordinate systems and metric tensors in these configurations when such configurations are classified as anholonomic. The particular choice of a global, external Cartesian coordinate system and corresponding covariant identity tensor as a metric on an intermediate configuration space is shown to be a constitutive assumption often made regardless of the existence of geometrically necessary crystal defects associated with the anholonomicity (i.e., the non-Euclidean nature) of the space. Since the metric tensor on the anholonomic configuration emerges necessarily in the definitions of scalar products, certain transpose maps, tensorial symmetry operations, and Jacobian invariants, its selection should not be trivialized. Several alternative (i.e., non-Euclidean) representations proposed in the literature for the metric tensor on anholonomic spaces are critically examined.     

Finite deformation analysis of mechanism-based strain gradient plasticity: torsion and crack tip field

A finite deformation theory of mechanism-based strain gradient (MSG) plasticity is developed in this paper based on the Taylor dislocation model. The theory ensures the proper decomposition of deformation in order to exclude the volumetric deformation from the strain gradient tensor since the latter represents the density of geometrically necessary dislocations. The solution for a thin cylinder under large torsion is obtained. The numerical method is used to investigate the finite deformation crack tip field in MSG plasticity. It is established that the stress level around a crack tip in MSG plasticity is significantly higher than its counterpart (i.e. HRR field) in classical plasticity.     

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.