CH-4 Plane problems in linear isotropic elasticity HUMBERT Laurent laurent.humbert@epfl.ch laurent.humbert@ecp.fr Thursday, march 18th 2010 Thursday, march 25th 2010

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

- 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

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

Young’s modulus for various materials :

1D interpretation : 1 2 3 Before elongation traction but

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

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

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

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

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

- 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 ?

- 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

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

Inverse relations, and the normal out of plane strain ,

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

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

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

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

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 !

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

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

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

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

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

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

From, and using,

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

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

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

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

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)

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

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

→ 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

Examples for common Test Specimens B : specimen thickness

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:

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

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

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

Cartesian components: Associated displacement field :

Mode III – loading Stress components : Displacement component :

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

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

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

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

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

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

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

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

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

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 5045 -99 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)

Chart of fracture toughness KIC and modulus E (from Ashby) Large range of KIC 0.01->100 MPa.m1/2 At lower end, brittle materials that remain elastic until they fracture

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 !

Typical KIC values:

Ex Aircraft components Fuselage made of 2024 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