By Anthony E. Armenàkas
CARTESIAN TENSORS Vectors Dyads Definition and principles of Operation of Tensors of the second one Rank Transformation of the Cartesian elements of a Tensor of the second one Rank upon Rotation of the method of Axes to Which they're Referred Definition of a Tensor of the second one Rank at the foundation of the legislations of Transformation of Its parts Symmetric Tensors of the second one Rank Invariants of the Cartesian parts of a Symmetric Tensor of the second one Rank desk bound Values of a functionality topic to a Constraining Relation desk bound Values of the Diagonal elements of a Symmetric Tensor of the Second. Read more...
summary: CARTESIAN TENSORS Vectors Dyads Definition and principles of Operation of Tensors of the second one Rank Transformation of the Cartesian parts of a Tensor of the second one Rank upon Rotation of the approach of Axes to Which they're Referred Definition of a Tensor of the second one Rank at the foundation of the legislations of Transformation of Its elements Symmetric Tensors of the second one Rank Invariants of the Cartesian elements of a Symmetric Tensor of the second one Rank desk bound Values of a functionality topic to a Constraining Relation desk bound Values of the Diagonal elements of a Symmetric Tensor of the second one
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Extra info for Advanced Mechanics of Materials and Applied Elasticity
1 Example of a Tensor of the Second Rank In this section we give an example of a quantity which is a tensor of the second rank. Consider a body subjected to external forces. In general the particles of this body will be stressed. 8 we show that the state of stress of a particle of a body is completely specified by a tensor of the second rank, which referring to Fig. 67) J is called the stress tensor and Jij(i, j = 1, 2, 3) are its components with respect to the system of orthogonal axes x1, x2, x3.
81) are not independent. 87) Since dx1, dx2 are independent, their coefficients in the above relation must vanish. 89). Thus, the stationary values of f (x1, x2, x3) subjected to the constraining relation g(x1, x2, x3) = 0 and the stationary values of F(x1, x2, x3) without a constraining relation occur at the same points. 92). The multiplying constant 8 is called the Lagrange multiplier. Example 3 Using the method of Lagrange multipliers, find the point on the plane specified by the following relation which is the nearest to the origin of the system of axes x1, x2, x3: (a) Solution The distance d(x1, x2, x3) of any point P (x1, x2, x3) from the origin is given as 26 Cartesian Tensors (b) Thus, we must establish the point at which the function d 2 assumes a stationary value under the constraining relation (a).
1, 2, 1). The units are in meters. (a) Determine the unit vector acting from point P1 to point P2. (b) Determine the angles pP1OP2 and pOP1P2, where O is the origin of the axes of reference. (c) Determine the unit vector normal to the plane specified by the points O, P1, P2. (d) Compute the volume of the parallelepiped whose edges are OP1, OP2 and OP3. Ans. 27o (c) (d) 8 m3 2. Consider the rectangular system of axes xNi (i = 1, 2, 3) specified with respect to the rectangular system of axes xj by the transformation matrix 45 Problems The cartesian components of a vector referred to the rectangular system of axes xj are a = 4i1 + 3i2.
Advanced Mechanics of Materials and Applied Elasticity by Anthony E. Armenàkas