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