02 - introduction me338 - syllabus to vectors and tensors
holzapfel nonlinear solid mechanics [2000], chapter 1.1-1.5, pages 1-32
02 - tensor calculus 1 introduction 2
continuum mechanics the potato equations continuum mechanics is the branch of mechanics concerned with the • kinematic equations - what s strain? stress in solids, liquids and gases and the general equations that characterize the deformation deformation or flow of these materials. the of a physical body without studying its physical cause adjective continuous refers to the simpli- fying concept underlying the analysis: we • balance equations - what s stress? disregard the molecular structure of matter general equations that characterize the cause of and picture it as being without gaps or motion of any body empty spaces. we suppose that all the mathe- matical functions entering the theory are • constitutive equations - how are they related? continuous functions. this hypothetical material specific equations that complement the set continuous material we call a continuum.! of governing equations malvern introduction to the mechanics of a continuous medium [1969]
introduction 3 continuum mechanics 4 the potato equations tensor calculus
tensor the word tensor was introduced • kinematic equations - why not ? in 1846 by william rowan hamilton. it was inhomogeneous deformation » non-constant used in its current meaning by woldemar finite deformation » non-linear voigt in 1899. tensor calculus was deve- inelastic deformation » growth tensor loped around 1890 by gregorio ricci-curba- stro under the title absolute differential • balance equations - why not ? calculus. in the 20th century, the subject equilibrium in deformed configuration » multiple stresses came to be known as tensor analysis, and achieved broader acceptance with the intro- • constitutive equations - why not ? duction of einsteins's theory of general finite deformation » non-linear relativity around 1915. tensors are used inelastic deformation » internal variables also in other fields such as continuum mechanics.
continuum mechanics 5 tensor calculus 6
tensor calculus vector algebra - notation
• vector algebra notation, euklidian vector space, scalar product, vector • einstein’s summation convention product, scalar triple product • tensor algebra notation, scalar products, dyadic product, invariants, trace, determinant, inverse, spectral decomposition, sym-skew • summation over any indices that appear twice in a term decomposition, vol-dev decomposition, orthogonal tensor • tensor analysis derivatives, gradient, divergence, laplace operator, integral transformations
tensor calculus 7 tensor calculus 8 vector algebra - notation vector algebra - euklidian vector space
• euklidian vector space • kronecker symbol
• is defined through the following axioms
• permutation symbol • zero element and identity
• linear independence of if is the only (trivial) solution to
tensor calculus 9 tensor calculus 10
vector algebra - euklidian vector space vector algebra - euklidian vector space
• euklidian vector space equipped with norm • euklidian vector space equipped with euklidian norm
• norm defined through the following axioms • representation of 3d vector
with coordinates (components) of relative to the basis
tensor calculus 11 tensor calculus 12 vector algebra - scalar product vector algebra - vector product
• euklidian norm enables definition of scalar (inner) product • vector product
• properties of scalar product
• properties of vector product
• positive definiteness • orthogonality
tensor calculus 13 tensor calculus 14
vector algebra - scalar triple product tensor algebra - second order tensors
• scalar triple product • second order tensor
area volume with coordinates (components) of relative to • properties of scalar triple product the basis • transpose of second order tensor
• linear independency
tensor calculus 15 tensor calculus 16 tensor algebra - second order tensors tensor algebra - third order tensors
• second order unit tensor in terms of kronecker symbol • third order tensor
with coordinates (components) of relative to the basis with coordinates (components) of relative • matrix representation of coordinates to the basis
• third order permutation tensor in terms of permutation symbol • identity
tensor calculus 17 tensor calculus 18
tensor algebra - fourth order tensors tensor algebra - fourth order tensors
• fourth order tensor • symmetric fourth order unit tensor
• screw-symmetric fourth order unit tensor with coordinates (components) of relative to the basis • volumetric fourth order unit tensor • fourth order unit tensor
• transpose of fourth order unit tensor • deviatoric fourth order unit tensor
tensor calculus 19 tensor calculus 20 tensor algebra - scalar product tensor algebra - scalar product
• scalar (inner) product • scalar (inner) product ·
of second order tensor and vector of two second order tensors and • zero and identity • positive definiteness • zero and identity • properties of scalar product • properties of scalar product
tensor calculus 21 tensor calculus 22
tensor algebra - scalar product tensor algebra - dyadic product
• scalar (inner) product • dyadic (outer) product
of two vectors introduces second order tensor of two second order tensors • properties of dyadic product (tensor notation) • scalar (inner) product
of fourth order tensors and second order tensor • zero and identity
tensor calculus 23 tensor calculus 24 tensor algebra - dyadic product
• dyadic (outer) product
of two vectors introduces second order tensor • properties of dyadic product (index notation)
tensor calculus 25