02 - introduction to vectors and tensors. me338 - syllabus. introduction tensor calculus. continuum mechanics. introduction. continuum mechanics

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1 02 - introduction to vectors and tensors me338 - syllabus holzapfel nonlinear solid mechanics [2000], chapter , pages introduction 2 the potato equations is the branch of mechanics concerned with the stress in solids, liquids and gases and the deformation or flow of these materials. the adjective continuous refers to the simplifying concept underlying the analysis: we disregard the molecular structure of matter and picture it as being without gaps or empty spaces. we suppose that all the mathematical functions entering the theory are continuous functions. this hypothetical continuous material we call a continuum.! malvern introduction to the mechanics of a continuous medium [1969] kinematic equations - what s strain? general equations that characterize the deformation of a physical body without studying its physical cause balance equations - what s stress? general equations that characterize the cause of motion of any body constitutive equations - how are they related? material specific equations that complement the set of governing equations introduction 3 4

2 the potato equations kinematic equations - why not? inhomogeneous deformation» non-constant finite deformation» non-linear inelastic deformation» growth tensor balance equations - why not? equilibrium in deformed configuration» multiple stresses constitutive equations - why not? finite deformation» non-linear inelastic deformation» internal variables tensor the word tensor was introduced in 1846 by william rowan hamilton. it was used in its current meaning by woldemar voigt in was developed around 1890 by gregorio ricci-curbastro under the title absolute differential calculus. in the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of einsteins's theory of general relativity around tensors are used also in other fields such as continuum mechanics. 5 6 vector algebra - notation vector algebra notation, euklidian vector space, scalar product, vector product, scalar triple product tensor algebra notation, scalar products, dyadic product, invariants, trace, determinant, inverse, spectral decomposition, sym-skew decomposition, vol-dev decomposition, orthogonal tensor tensor analysis derivatives, gradient, divergence, laplace operator, integral transformations einstein s summation convention summation over any indices that appear twice in a term 7 8

3 vector algebra - notation kronecker symbol vector algebra - euklidian vector space euklidian vector space is defined through the following axioms permutation symbol zero element and identity linear independence of is the only (trivial) solution to if 9 10 vector algebra - euklidian vector space vector algebra - euklidian vector space euklidian vector space equipped with norm euklidian vector space euklidian norm equipped with norm defined through the following axioms representation of 3d vector with coordinates (components) of relative to the basis 11 12

4 vector algebra - scalar product euklidian norm enables definition of scalar (inner) product vector algebra - vector product vector product properties of scalar product properties of vector product positive definiteness orthogonality vector algebra - scalar triple product scalar triple product area volume properties of scalar triple product tensor algebra - second order tensors second order tensor with coordinates (components) of relative to the basis transpose of second order tensor linear independency 15 16

5 tensor algebra - second order tensors second order unit tensor in terms of kronecker symbol with coordinates (components) of relative to the basis matrix representation of coordinates identity tensor algebra - third order tensors third order tensor with coordinates (components) of relative to the basis third order permutation tensor in terms of permutation symbol tensor algebra - fourth order tensors fourth order tensor tensor algebra - fourth order tensors symmetric fourth order unit tensor with coordinates (components) of relative to the basis fourth order unit tensor transpose of fourth order unit tensor screw-symmetric fourth order unit tensor volumetric fourth order unit tensor deviatoric fourth order unit tensor 19 20

6 tensor algebra - scalar product tensor algebra - scalar product of second order tensor zero and identity positive definiteness and vector of two second order tensors and zero and identity properties of scalar product properties of scalar product tensor algebra - scalar product tensor algebra - dyadic product dyadic (outer) product of two second order tensors of two vectors introduces second order tensor properties of dyadic product (tensor notation) of fourth order tensors and second order tensor zero and identity 23 24

7 tensor algebra - dyadic product dyadic (outer) product of two vectors introduces second order tensor properties of dyadic product (index notation) 25