1. CH-4 Plane problems in linear isotropic elasticity
HUMBERT Laurent
Thursday, march 18th 2010
Thursday, march 25th 2010

2. The basic equations of elasticity (appendix I) :
4.1 Introduction
Framework : linear isotropic elasticity, small strains assumptions, 2D problems (plane strain , plane stress)
The basic equations of elasticity (appendix I) :
f : body forces (given)
- Equilibrium equations (3 scalar equations) :
s : Cauchy (second order) stress tensor
z, x3
Explicitly,
x, x1
symmetric
Cartesian basis
y, x2

3. - Linearized Strain-displacement relations (6 scalar equations) :
Equations of compatibility: f
displacement imposed on Gu
tractions applied on Gt
because the deformations are defined as partial derivative of the displacements

4. - Hooke’s law (6 scalar equations) :
Isotropic homogeneous stress-strain relation
Lamé’s constants
with
Inversely,
15 unknowns
→ well-posed problem
→ find u 4

5. Young’s modulus for various materials :

6. 1D interpretation : 1 2 3
Before elongation
traction
but

7. - Navier’s equations:
3 displacement components taken as unknowns
- Stress compatibility equations of Beltrami - Michell :
6 stress components considered as unknowns

8. Boundary conditions :
Displacements imposed on Su
Surface tractions applied on St f
n outwards unit normal to St
→ displacement and/or traction boundary conditions to solve the previous field equations

9. 4.2 Conditions of plane strain
Assume that
Strain components :
“thick plate”
Thus,
functions of x1 and x2 only

10. Associated stress components
and also (!),
Inverse relations,
rewritten as

11. n t
- 2D static equilibrium equations :
: body forces
Surface forces t are also functions of x1 and x2 only
: components of the unit outwards vector n

12. - Non-zero equation of compatibility (under plane strain assumption)
implies for the relationship for the stresses:
That reduces to by neglecting the body forces,
Proof ?

13. - Differentiate the equilibrium equation and add
Proof:
- Introduce the previous strain expressions in the compatibility equation
one obtains
(1)
- Differentiate the equilibrium equation and add
(2)
- Introduce (2) in (1) and simplify

14. 4.3 Conditions of plane stress
thin plate
From Hooke’s law,
Similar equations obtained in plane strain :
and also,
functions of x1 and x2 only

15. Inverse relations,
and the normal out of plane strain ,

16. Airy’s stress function :
introduce the function as
equations of equilibrium automatically satisfied !
then substitute in
leads to
Biharmonic equation
Same differential equation for plane stress and plane strain problems
Find Airy’s function that satisfies the boundary conditions of the elastic problem

17. 4.4 Local stress field in a cracked plate :
- Solution 2D derived by Williams (1957)
- Based on the Airy’s stress function
Notch / crack tip
: polar coordinate system
Crack when
, notch otherwise

18. Local boundary conditions :
Remote boundary conditions
Find
stress field
displacement field
Concept of self-similarity of the stress field (appendix II) :
Stress field remains similar to itself when a change in the intensity (and scale) is imposed
Stress function in the form

19. Biharmonic equation in cylindrical coordinates:
Consider the form of solution
with
Solutions of the quadratic equation :
complex conjugate roots

20. Consequently, :
ci (complex) constants
Using Euler’s formula
A, B, C and D constants
to be determined …
according to the symmetry properties of the problem !

21. F F Modes of fracture : A crack may be subjected to three modes
More dangerous !
Notch, crack F
Example : Compact Tension (CT) specimen : F
Mode I loading
natural crack

22. Mode I – loading
with
symmetric part of
Stress components in cylindrical coordinates
Use of boundary conditions,

23. Non trivial solution exits for A, C if
… that determines the unknowns l
For a crack, it only remains
n integer
infinite number of solutions

24. Relationship between A and C
For each value of n → relationship between the coefficients A and C
→ infinite number of coefficients that are written:
From,
(crack)
with

25. Airy’s function for the (mode I) problem expressed by:
Reporting

26. Expressions of the stress components in series form (eqs 4.32):
Starting with,
and recalling that,

27. From,
and using,

28. Range of n for the physical problem ?
The elastic energy at the crack tip has to be bounded
but,
is integrable if or

29. Singular term when or
Mode - I stress intensity factor (SIF) :

30. In Cartesian components,
→ does not contain the elastic constants of the material
→ applicable for both plane stress and plane strain problems :
n is Poisson’s ratio

31. Asymptotic Stress field:x y O θ r x y O θ r
Similarly,
singularity at the crack tip
+ higher–order terms (depending on geometry)
fij : dimensionless function of q in the leading term
An amplitude , gij dimensionless function of q for the nth term

32. Stresses near the crack tip increase in proportion to KI
Evolution of the stress normal to the crack plane in mode I :
Stresses near the crack tip increase in proportion to KI
If KI is known all components of stress, strain and displacement are determined as functions of r and q (one-parameter field)

33. Singularity dominated zone :
→ Admit the existence of a plastic zone small compared to the length of the crack

34. Expressions for the SIF :
Closed form solutions for the SIF obtained by expressing the biharmonic function in terms of analytical functions of the complex variable z=x+iy
Westergaard (1939)
Muskhelishvili (1953), ...
Ex : Through-thickness crack in an infinite plate loaded in mode -I:
Units of

35. → Stress intensity solutions gathered in handbooks :
For more complex situations the SIF can be estimated by experiments or numerical analysis
Y: dimensionless function taking into account of geometry (effect of finite size) , crack shape
→ Stress intensity solutions gathered in handbooks :
Tada H., Paris P.C. and Irwin G.R., « The Stress Analysis of Cracks Handbook », 2nd Ed., Paris Productions, St. Louis, 1985
→ Obtained usually from finite-element analysis or other numerical methods P

36. Examples for common Test Specimens
B : specimen thickness

37. Mode-I SIFs for elliptical / semi-elliptical cracks
Solutions valid if
Crack small compared to the plate dimension
a ≤ c
When a = c
Circular:
(closed-form solution)
Semi-circular:

38. Associated asymptotic mode I displacement field :
Polar components :
Cartesian components :
with
shear modulus
E: Young modulus n: Poisson’s ratio
Displacement near the crack tip varies with
Material parameters are present in the solution

39. Ex: Isovalues of the mode-I asymptotic displacement:
x= r cosq
y=r sinq
crack
plane strain, n=0.38
crack
x= r cosq
y=r sinq

40. Mode II – loading
Same procedure as mode I with the antisymmetric part of
Asymptotic stress field :

41. Cartesian components:
Associated displacement field :

42. Mode III – loading
Stress components :
Displacement component :

43. Closed form solutions for the SIF
Mode II-loading :
Mode III-loading :

44. Principe of superposition for the SIFs:
With n applied loads in Mode I,
Similar relations for the other modes of fracture
But SIFs of different modes cannot be added !
Principe of great importance in obtaining SIF of complicated specimen loading configuration
Example:
(a)
(b)
(c)
values of G are not additive for the same

45. 4.5 Relationship between KI and GI:
Plane strain
Plane stress
Mode I only :
When all three modes apply :
Self-similar crack growth
Values of G are not additive for the same mode but can be added for the different modes

46. Proof in load control (ch 3) Work done by the closing stresses : with
but,
slide 38, with
and also
slide 32 for
Calculating
and injecting in GI

47. 4.6 Mixed mode fracture
in global frame ( )
biaxial loading
→ expressed in local frame ( )
Q = Rotation tensor
Thus,
Stress tensor components :

48. Mode I loading :
→ Principe of superposition :
Mode II loading :

49. Propagation criteria
Mode I
Crack initiation when the SIF equals to the fracture toughness or
Mixed mode loading
Self-similar crack growth is not followed for several material
Useful if the specimen is subjected to all three Modes, but 'dominated' by Mode I
General criteria:
explicit form obtained experimentally

50. Examples in Modes I and II
m , n and C0 parameters determined experimentally
Erdogan / Shih criterion (1963):
Crack growth occurs on directions normal to the maximum principal stress
Condition to obtain the crack direction

51. 4.7 Fracture toughness testing
Assuming a small plastic zone compared to the specimen dimensions,
a critical value of the mode-I SIF may be an appropriate fracture parameter :
→ plane strain fracture toughness KIC
Specimen Thickness KC
plane stress
plane strain
KC : critical SIF, depends on thickness
KI > KC : crack propagation
KIC : Lower limiting value of fracture toughness KC
Material constant for a specific temperature and loading speed
Apparent fracture surface energy

52. How to perform KIC measurements ?
→ Use of standards:
- American Society of Testing and Materials (ASTM)
- International Organization of Standardization (ISO) CT
ASTM E 399 first standardized test method for KIC :
- was originally published in 1970
- is intended for metallic materials
- has undergone a number of revisions over the years
- gives specimen size requirements to ensure measurements in the plateau region
ASTM D is used for plastic materials:
- Many similarities to E 399, with additional specifications important for plastics.
KI based test method ensures that the specimen fractures under linear elastic conditions
(i.e. confined plastic zone at the crack tip)

53. Chart of fracture toughness KIC and modulus E (from Ashby)
Large range of KIC >100 MPa.m1/2
At lower end, brittle materials that remain elastic until they fracture

54. Chart of fracture toughness KIC and yield strength sY (from Ashby)
Materials towards the bottom right : high strength and low toughness
→fracture before they yield
Materials towards the top left : opposite
→ yield before they fracture
Metals are both strong and tough !

55. Typical KIC values:

56. Ex Aircraft components
Fuselage made of alloy (Al + 4% Cu + 1% Mg)
Thickness of the sheet ~ 3mm
(elastic limit)
AIRBUS A330
Plane stress criterion with Kc is typically used here in place of KIC