# Nonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess

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1 Nonlinear Theory of Elasticity Dr.-Ing. Martin Ruess

2 geometry description Cartesian global coordinate system with base vectors of the Euclidian space orthonormal basis origin O point P domain of a deformable body closed domain surface

3 geometry description solid body rigid body deformable body neighboring points remain neighboring points independent of time distant between points remains constant during displacement distance between neighboring points may change with time

4 configurations reference configuration often: state at time material points instant configuration often: state at time material points

5 kinematics state of displacements

6 kinematics state of displacements displacement is a combination of rigid body movement/rotation (AB = const for any two points) deformation (AB const) A, B neighboring points of the deformable body

7 kinematics deformation state infinitesimal volume considered base vectors of reference and instant configuration

8 kinematics deformation state method of Lagrange material particle identification in the reference configuration particle location at time is a function of and analog for state variables

9 kinematics deformation state method of Euler material particle identification in the instant configuration particle location at time is a function of and analog for state variables

10 kinematics deformation gradient F material deformation gradient F representation of diagonal as function of diagonal columns of F are the instant base vectors b k (k=1,2,3)

11 kinematics deformation gradient F use the deformation gradient F to show the change in volume for the infinitesimal volume in instant and reference configuration

12 kinematics displacement gradient H split of the deformation gradient F into a unit matrix I and a matrix H H contains the partial derivatives of u w.r.t. coordinates of ref. config.

13 kinematics state of strain consider the change the length of deformation measure referred to the reference configuration results in the strain tensor of Green

14 kinematics state of strain consider the change the length of deformation measure referred to the reference configuration results in the strain tensor of Green-Lagrange LINEAR THEORY

15 kinematics Green strain tensor diagonal coefficients e ii off-diagonal coefficients e im stretch: measure of fibre elongation shear: measure of the angle between fibre angles

16 kinematics strain-displacement relation... applying Voigt notation

17 statics state of stress stress vector stress tensor stress vector in direction of the surface normal n action of subfield C 1 on C 2 is replaced by a fictitious force dp stress vector at point

18 statics stress tensor of Cauchy axis parallel cube of the deformed configuration stress tensor of the deformed configuration columns diagonal of the coefficients Cauchy stress e ii tensor are the stress vectors on the positive faces of the element

19 statics stress vector stress tensor D stress vector on a section C B

20 statics equilibrium

21 statics equilibrium sum of the moments acting on the element is null in the state of equilibrium from this follows the symmetry of the Cauchy stress tensor

22 statics 1st Piola-Kirchhoff stress tensor consider a surface element of the reference configuration replaced to the instant configuration which is 1 st Piola-Kirchhoff tensor causes the same force on & (definition!) 1 st Piola Kirchhoff tensor stress coordinates are referred to the global base vectors 1 st PK stress tensor is unsymmetric, in general, not in use!

23 statics 2nd Piola-Kirchhoff stress tensor PK1 force vector referred to the basis of the instant configuration bases vectors in are the columns of the deformation gradient relation between Cauchy stress tensor and 2 nd PK tensor 2 nd Piola Kirchhoff tensor is symmetric! energetically conjugate stress tensor to the Green strain tensor

24 statics stress-strain relation... linear elasticity Hooke s law

25 governing equations partial differential equation / Green-Lagrange/Cauchy strain/stress tensor coordinates

26 solution approach FEM weighted residual approach, cf linear theory of elasticity choice of a suited approximation rule for the displacement state definition of residuals which are not a priori satisfied choose of admissible/suited weight functions here: Bubnov-Galerkin approach: variation of displacements multiply residuals with weight functions integration over volume of the instant configuration

27 spatial integral form 1 st integral form 2 nd integral form (Principle of virtual work) spatial integral form derived for volume elements of the instant config. volume of the body in is unknown!

28 material integral form integral equation is referred to the known reference configuration replace... unknow volume Cauchy coordinates instant coordinate with known volume with 2 nd Piola-Kirchhoff coordinates with on the left hand follows on the right hand follows in analogy

29 material integral form Principle of virtual work strains are nonlinear functions of the derivatives (Green-Lagrange) stresses (2 nd PK) are referred to base vectors of reference & instant config. conservative loads are assumed independent of the displacements

30 incremental equations strategy stepwise solution for the nonlinear equations initial configuration is assumed to be known solution at the end of each step governing equations are incremental equations consistent linearization leads to incremental equations

31 incremental equations reference configuration instant configuration I, known from previous step instant configuration II, unknown unknown displacement state at the end of step i known displacement state at beginning of step i displacement increment from to

32 incremental equations state variables Total Lagrangian (TL) formulation Updated Lagrangian (UL) formulation referred to referred to

33 state variables incremental equations Incremental strain-displacement relationship

34 state variables incremental equations Incremental strain-displacement relationship

35 state variables incremental equations Variation of the state of displacements Variation of the state of strain

36 incremental equations Governing equations in vector notation

37 incremental equations Governing equations in vector notation

38 Governing equations in vector notation incremental equations

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