X x x is a primary measure of deformation characterizes the variation of relative placements in the neighbourhood of a spatial point. Introduction to continuum mechanics computational solid. The concepts of displacement gradient and deformation gradient are introduced to quantify the change in shape of infinitesimal line elements in a solid body. A configuration is a set containing the positions of all particles of the body. However, only local tensor information such as voxel tensor similarity is utilized in these algorithms, which can be regarded as a counterpart of the traditional intensity similaritybased image registration in the dti case. To see this, imagine drawing a straight line on the undeformed configuration of a solid, as shown in the figure. Polar decomposition theorem states that the deformation gradient tensor f may be decom.
Note that the velocity gradient l, the rate of deformation tensor d, and the vorticity tensor w depend only on the present configuration, and have no connection to any reference or previous configuration. Inverse spatial deformation gradient tensor the spatial or inverse deformation gradient tensor. The principal components of the right or left cauchygreen tensors are. The rateofdeformation tensor d and spin tensor w are the symmetric and antisymmetric parts of the spatial velocity gradient l, respectively. Continuum mechanics lecture notes zden ek martinec department of geophysics. The structure of the evolution equation for growthinduced stretch ratio is discussed. Be familiar with linear vector spaces relevant to continuum mechanics and able to perform vector and tensor manipulations in cartesian and curvilinear coordinate systems. Simultaneous consideration of spatial deformation and. A tensor is called rotation if it is orthogonal with determinant equal to one. Ch apter 4 covers basis and coordinate transformations, and it provides a gentle introduction to the fact that base vectors can vary with position. The displacement gradient and the lagrangian strain tensor revision b by tom irvine email. For uniaxial state the strain is simply the displacement gradient. Continuum mechanicsstrains and deformations wikiversity. In this study, transparent soil surrogates are used in model tests instead of natural soils.
A tensor is called stretch if it is symmetric and positive definite. The deformation gradient tensor graphical introduction. For the deformation gradient this means f ru v r, where r is the rotation tensor. The first section introduces the material and spatial coordinates and.
A deformation may be caused by external loads, body forces such as gravity or electromagnetic forces, or changes in temperature, moisture content, or chemical reactions, etc. Strain rate tensor and rotation rate tensor or spin tensor. Our focus is on relating deformation to quantities that can be measured in the. The spatial rate of deformation tensor or stretching tensor. On surfaces embedded in 3d space, infinitesimal squares deform into infinitesimal parallelograms. C f t f and b ff t, where f is the deformation gradient. We develop an iterative, spatialgradient based algorithm, which uses only. Simpleexamplesillustratingtheuseofthedeformationgradient tensor. This is in contrast to the deformation measures f, f. Linearisation of the virtual work wiley online library. After deformation the square will be a parallelogram. In general, bodiesstructures movedeform when forces are acting on them, but we are not concerned for now about the causes of this motion deformation. A deformation gradient tensor and strain tensors for atomistic simulations article in modelling and simulation in materials science and engineering 161.
For the position vectors, the gradients result in the unity. Considering two successive deformations equation 6 is applied twice. Properties of the deformation gradients the spatial deformation gradient tensor is the inverse of the material deformation gradient tensor. Deformation gradient tensor material and spatial gradient of displacements tensors greenlagrange and almansi strain tensors volumetric strain variation of the area polar decomposition of the deformation gradient. Cven 7511 computational mechanics of solids and structures instructor. Continuum mechanics lecture notes zden ek martinec. Topic 3 lagrangian continuum mechanics variables for general nonlinear analysis the principle ofvirtualworkinterms ofthe 2nd piolakirchhoffstress and greenlagrangestraintensors deformation gradient tensor physicalinterpretationofthe deformation gradient change ofmass density polardecomposition ofdeformation gradient. Simultaneous consideration of spatial deformation and tensor orientation in diffusion tensor image registration using local fast marching patterns. Deformation, stress, and conservation laws in this chapter, we will develop a mathematical description of deformation. The gradient of uand the deformation of a rectangle dotted line, the sides of which were originally parallel to the coordinate axes. Then, by the implicit function theorem, the jacobian determinant, must be nonsingular, i.
The right cauchygreen deformation tensor is defined by. This partitioning of the deformation gradient into the product of a rotation matrix and stretch tensor is known as a polar. Numerical analysis of large deformation processes at elevated temperatures a. The rateofdeformation tensor d and spin tensor w are the symmetric and antisymmetric parts of. A deformation gradient tensor and strain tensors for. Similarly, a is a second order tensor in the material form and a is the equivalent spatial form. Particularly, from the previous time step the deformation gradient f n, the elastic part of the cauchygreen tensor be n, the plastic multiplier n and the value of the damage history parameter nare given. Infinitesimal strain tensor and infinitesimal rotation tensor the general lagrange strain tensor e written out is 2 2 1 1 11 10 22 01 1 1 2 1 2 x y xx t x y yy xx x xxyyyy xxxy u. Every part of the solid body deforms as the whole does.
This paper addresses the need for nonintrusively measuring spatial deformation pattern inside soils. Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. Numerical analysis of large deformation processes at. Spatial description an overview sciencedirect topics. The deformation gradient is the derivative of the configuration of the body. The coordinates of the two diagonally opposite corners are used to fill the deformation matrix, also known as the deformation gradient tensor. Using the gradient operator with respect to the undeformed state, the deformation tensor can be written as a gradient, which explains its much used name. Accordingly, the ijth component of x represents the mean rotation about the kth coordinate axis, where ijk. From wikiversity pdf available in ieee transactions on robotics 292.
The deformation of soft tissues is often described by means of the right and left cauchygreen tensors defined as. Deformation gradient, second order tensors, rotation, rigid. The spatial cristoffel symbol is used above and we substitute x with x x,t to get a x spatial acceleration. B if the lagrangian deformation gradient and displacement gradient tensors are symmetric about the main diagonal, the deformed body will be symmetric about axes parallel to. C right cauchygreen deformation tensor d rateofdeformation tensor e relative greenlagrange strain tensor ea cartesian basis vectors in reference con. In general, bodiesstructures movedeform when forces are acting on them, but we are not concerned for now about the causes of this motiondeformation. There will be exceptions to this, but it should be clear from the context what is implied. I general comments on the deformation gradient and displacement gradient tensors a the tensors that describe deformation of a body depend on derivatives of displacements. We pose the deformation transfer problem as a regular shape deformation problem, where a sparse set of. Cven 7511 computational mechanics of solids and structures. Divergence or spatial divergence of a vector or tensor. The line would be mapped to a smooth curve on the deformed configuration. It decomposes the transformation represented by t in a twostage process. In this chapter the main aspects of nonlinear kinematics of deformation of a continuum medium are introduced.
The discussion below begins with a definition of the deformation gradient, then proceeds in the following order. Cauchygreen tensor an overview sciencedirect topics. Kinematics of deformation in this chapter, we will study how bodiesstructures movedeform and how can this motiondeformation be described mathematically. Because of the onetoone nature of the deformation map, f is nonsingular and it can thus be expressed. Div material divergence of a vector or tensor grad gradient or spatial gradient of a scalar or vector grad material gradient of a scalar or vector tr trace of a tensor inner product of two vectors or tensors. Subpixel displacement and deformation gradient measurement. This paper proposes a novel dti image registration algorithm, called fast marchingbased simultaneous registration. It is not the spatial description of the material deformation gradient tensor 1 1, i, 1. The tensor fx x is called the deformation gradient. Simpleexamplesillustratingtheuseofthedeformationgradient. At each step, a gradient of the displacement field is applied to analyze the situation. Chapter 3 shows how cartesian formulas for basic vector and tensor operations must be alte red for noncartesian systems. Kinematics of deformation in this chapter, we will study how bodiesstructures movedeform and how can this motion deformation be described mathematically.
F x xx xy z yy y xy z zz z xy z and g uu u xy z vv v xy z ww w xy z the deformation gradient f describes stretches and rigid. Identifying regional cardiac abnormalities from myocardial strains using spatiotemporal tensor analysis zhen qian 1, qingshan liu, dimitris n. C ftf the right cauchygreen deformation tensor is thus a measure of the change in ds2. It is a twopoint tensor gives the mapping between points in the original geometry x to current geometry x. Simulated images are then used to verify this algorithm, as well as to study the impact of speckle size on the accuracy. Identifying regional cardiac abnormalities from myocardial. Pdf spatial deformation measurement using transparent soil. I is the unit tensor and g is called the displacement gradient tensor the components of the deformation gradient f, and the displacement gradient tensor g, thus becomes. Hunter department of mathematics university of california, davis, ca 95616, usa 1. Generally an antisymmetric tensor has three independent components.
Deformation gradient tensor an overview sciencedirect topics. Due to the assumption of continuity of, has the inverse, where is the spatial deformation gradient tensor. Notes on strain and deformation tensors marcel frehner september 2006. Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the spatial displacement gradient tensor. Simultaneous consideration of spatial deformation and tensor. You can calculate the deformation gradient at other points.
The deformation gradient, like any invertible secondorder tensor, can be decomposed, using the polar decomposition theorem, into a product of two secondorder tensors truesdell and noll, 1965. The previous two examples of simple shear and constant extension were both homogeneous deformations, and many experimental testing procedures attempt to maintain this type of deformation within the specimen under study. Then, by the implicit function theorem lubliner, the jacobian determinant. If f is not dependent on the space coordinates, the deformation is said to be homogeneous. I view nonlinear continuum mechanics as a vital tool for mathematical modeling of many physical events.
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