106 - kinematic equations

06 - kinematic equations -

2introduction

continuum mechanics is abranch of physics (specifically mechanics)that deals with continuous matter. the factthat matter is made of atoms and that itcommonly has some sort of heterogeneousmicrostructure is ignored in the simplify-ing approximation that physical quantities,such as energy and momentum, can be handledin the infinitesimal limit. differentialequations can thus be employed in solvingproblems in continuum mechanics.

continuum mechancis

3introduction

continuum mechanicscontinuum mechanics isthe branch of mechanics concerned with thestress in solids, liquids and gases and thedeformation or flow of these materials. theadjective continuous refers to the simpli-fying concept underlying the analysis: wedisregard the molecular structure of matterand picture it as being without gaps orempty spaces. we suppose that all themathematical functions entering the theoryare continuous functions. this hypotheticalcontinuous material we call a continuum.

Malvern „Introduction to the mechanics of a continuous medium“ [1969]

4introduction

continuum mechanics

continuum hypothesiswe assume that the characteristic lengthscale of the microstructure is much smallerthan the characteristic length scale of theoverall problem, such that the propertiesat each point can be understood as averagesover a characteristic length scale

example: biomechanics

the continuum hypothesis can be applied when analyzing tissues

5introduction

the potato equations

• kinematic equations - what‘s strain?

• balance equations - what‘s stress?

• constitutive equations - how are they related?

general equations that characterize the deformationof a physical body without studying its physical cause

general equations that characterize the cause ofmotion of any body

material specific equations that complement the setof governing equations

6introduction

the potato equations

• kinematic equations - why not ?

• balance equations - why not ?

• constitutive equations - why not ?

inhomogeneous deformation » non-constantfinite deformation » non-linearinelastic deformation » growth tensor

equilibrium in deformed configuration » multiple stress measures

finite deformation » non-linearinelastic deformation » internal variables

7kinematic equations

kinematic equations de-scribe the motion of objects without theconsideration of the masses or forces thatbring about the motion. the basis of kine-matics is the choice of coordinates. the1st and 2nd time derivatives of the posi-tion coordinates give the velocities andaccelerations. the difference in placementbetween the beginning and the final stateof two points in a body expresses the nu-merical value of strain. strain expressesitself as a change in size and/or shape.

kinematic equations

8kinematic equations

kinematics is the study of motionper se, regardless of the forces causingit. the primitive concepts concerned areposition, time and body, the latterabstracting into mathematical termsintuitive ideas about aggregations ofmatter capable of motion and deformation.

kinematic equations

Chadwick „Continuum mechanics“ [1976]

9kinematic equations

potato - kinematics

• nonlinear deformation mapwith

• spatial derivative of - deformation gradientwith

10kinematic equations

potato - kinematics

• transformation of line elements - deformation gradient

• uniaxial tension (incompressible), simple shear, rotationwith

11kinematic equations

potato - kinematics of finite growth

• transformation of volume elements - determinant of

• changes in volume - determinant of deformation tensor

12kinematic equations

potato - kinematics

• temporal derivative of - velocity (material time derivative)

with

• temporal derivative of - acceleration

with