## Computational Continuum MechanicsThis book presents the nonlinear theory of continuum mechanics and demonstrates its use in developing nonlinear computer formulations for large displacement dynamic analysis. Basic concepts used in continuum mechanics are presented and used to develop nonlinear general finite element formulations that can be effectively used in large displacement analysis. The book considers two nonlinear finite element dynamic formulations: a general large deformation finite element formulation and a formulation that can efficiently solve small deformation problems that characterize very stiff structures. The book presents material clearly and systematically, assuming the reader has only basic knowledge in matrix and vector algebra and dynamics. The book is designed for use by advanced undergraduates and first-year graduate students. It is also a reference for researchers, practising engineers, and scientists working in computational mechanics, bio-mechanics, computational biology, multibody system dynamics, and other fields of science and engineering using the general continuum mechanics theory. |

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¼ ð absolute nodal coordinate algorithm assumed beam element body coordinate system Cauchy stress tensor chapter coefﬁcients constant constitutive equations continuum mechanics current conﬁguration decomposition deﬁned deﬁnition deformation tensor deviatoric differential equations discussed dyadic product dynamic ðÞ elastic forces equal to zero equation shows equations of motion Euler angles Eulerian example expressed in terms ﬁnite element ﬁnite element formulations ﬁrst ﬂoating frame ﬂow ﬂuid frame of reference gradient vectors Green–Lagrange strain tensor inertia forces inertia shape integrals inﬁnitesimal kinematic linear mass matrix material points matrix of position modulus multibody system nodal coordinate formulation nonlinear obtained orthogonal matrix parameters Piola–Kirchhoff stress tensor polar decomposition polynomial position vector gradients preceding equation problems rate of deformation reference conﬁguration reference formulation respect rigid-body rotation second Piola–Kirchhoff stress second-order tensor Shabana shear strain energy strain tensor symmetric symmetric matrix transformation matrix variables viscoelastic written in terms

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Page 3 - The determinant of the product of two matrices is equal to the product of the determinants of the two matrices.

Page 3 - One can show that the determinant of a matrix is equal to the determinant of its transpose, that is.

Page 6 - Therefore, the number of columns in A must be equal to the number of rows in B. If A is an mxn matrix and B is an nxp matrix, then C is an mxp matrix.

Page 2 - Use the last exercise to show that every square matrix can be written as the sum of a symmetric matrix and a skew—symmetric matrix.

Page 2 - ... is said to be square; otherwise the matrix is said to be rectangular or nonsquare.