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