![]() ![]() Tensor, S 11, S 12, S 13, S 21 … S 33 (Einstein convention): If an index is repeated in a product of notation, we would express x andĬonvention: Lower case Latin subscripts ( i, j, k…) have the range ( 1, 2, 3 ) The symbol x i three components of a vector x 1, x 2 x 3 The symbol S i j nine components of a second order X in this basis by ( x 1, x 2, x 3 ) denote the components of S by On Cartesian components of vectors and tensors may be expressed very We’ll see the physical meaning of and when we discuss “polar decomposition.2.2 Index Notation for Vector and Tensor Operations This is analogous to the previously derived Where the Eulerian Strain Tensor (sometimes called the Almansi Strain Tensor) is: Similar to the way that we derived, let’s consider the difference in lengths of any particular element, or fiber, within our strain potato, before and after deformation. Presentation on theme: Review for Final Exam Basic knowledge of vector & index notation, matrix-tensor theory, coordinate transformation, principal value. The coordinate system in which is calculated (the “spatial” coordinate system) remains constant. In other words, the coordinate system in which is calculated (the “material” coordinate system) rotates with rigid body rotation. Under rigid body rotation, the Eulerian strain values will change, whereas the Lagrangian strain tensor is invariant to rigid body rotation. Here, “Eulerian strain” is simply referring to a measure of strain that is defined in spatial coordinates. Recall the definition of from the figure at the beginning of this chapter, and recall that is the transpose of 5) that, in general, for linear infinitesimal deformation (higher order terms are neglected). We can also see from the following equation (eq. Note: is “zero” on the LHS of the above equation because and are originally orthogonalĭoes our shear strain reduce to this value for infinitesimal deformation? As in the theory of 3D nonlinear wave equations in three space dimensions, global existence hinges on two basic assumptions. if is in the plane of interest (orthogonal to ), then is the shear strain The unbounded elastic medium is assumed to be homogeneous, isotropic, and hyperelastic. ![]() Since is the strain vector on a plane whose unit normal is , Suppose and are orthogonal in the undeformed configuration. Strain vector on a plane whose normal vector is (the actual location in space is specified within ) These are categorized into equilibrium, strain. ![]() Infinite solution trajectories are expressed analytically and classified in terms of only three parameters related to physical quantities. There are 15 partial differential equations that must be simultaneously solved to get the state of stress at any point in an arbitrary structure. We investigate the equilibrium configurations of the ideal 3D elastica, i.e., inextensible, unshearable, isotropic, uniform, and naturally straight and prismatic rods, with linear elastic constitutive relations. This method is used for analyzing structures that behave in a linearly elastic fashion. Where the Lagrangian Strain Tensor (or Green-Lagrange Strain Tensor) is: 3-D elasticity is one of three methods of structural analysis. The reason for this name will become clear once we begin discussion our on “polar decomposition” theory. Note: “Right C – G” (Cauchy – Green) deformation tensor. This is a convenient choice because all points on such a hemisphere satisfy the condition ·x0. ![]() Away from the origin, we can choose the hemisphere having the direction x as the zenith. Note: you can’t do this transpose manipulation as easily if multiplying two tensors, but it works for two vectors or a vector and a tensor (use indices to easily prove) This noteworthy result shows that the Green tensor is non-singular at the origin, in contrast to classical elasticity. To find such length “magnitudes” we can take dot products as follows: Now, consider how the length of any element or fiber within the continuum may change under deformation. The tensor,, is typically referred to as the “deformation gradient.” To see how is found in 2D and 3D in an actual FEA application, see Finite Element Coordinate Mapping. ![]()
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