Damage and Residual Life Prediction of Vehicle Structures Chi L. Chow Department of Mechanical Engineering University of Michigan-Dearborn
Table of Contents • Introduction • Theory of Damage Mechanics • Example Projects i) Bumper Damage under multiple Impact Loading ii) Crashworthiness of High Strain Rate Plastics iii) Fatigue Damage of Strain-Rate and Temperature Dependent Solder Alloys iv) Forming Limit Diagram (FLD) of Strain Rate Dependent Metals
1. Introduction Damages in a vehicle structure are caused by material degradation due to initiation, growth and coalescence of mirco-cracks/voids in a ‘real-life’ material element from monotonic, cyclic/fatigue, thermo-mechanical loading or dynamic/explosive impact loading.
Fracture/Rupture Process • Before Initial Subsequent Final • Loading Loading Loading Loading
Damage Mechanics The theory of damage mechanics takes into account the process of material degradation due to the initiation, growth and coalescence of micro-cracks/voids in a ‘real-life’ material element under monotonic or cyclic or impact or thermo-mechanical loading
Fracture/Rupture Criteria A valid material failure criterion must therefore take into account the process of progressive material degradation/damage under either static or dynamic/fatigue loading. Unfortunately, all conventional failure criteria including fracture mechanics ignore the process and thus unrealistic and unreliable.
P P Rupture Criterion - Conventional For Smooth Specimen (with/without notches) • Static loading • Stress, strain or energy-based criteria • Fatigue loading • S-N curve (Due to Wohler in 1858) for constant amplitude loading • Miner’s rule for variable amplitude loading • Rainfall counting method for ‘real life’ fatigue loading
P crack P Rupture Criterion – Conventional For Cracked Specimen • Static loading • Fracture Mechanics • Fatigue loading • Paris Law • Others based on G, J
Rupture Criterion – Damage Mechanics A material element is failed when cumulative damage reaches its critical value. Unified criterion for different conditions: • Smooth and cracked specimen • Static and fatigue loading • Crack initiation and propagation
Major Advantages • Ability to quantify material damage and predict residual life after impact loading • Capable of providing a unified rupture criteria for macro-crack initiation or propagation for either brittle and ductile fracture. This includes fatigue damage, localized necking, multi-phase composite failure, etc
Past Project: Crash Mechanics • Bumper under Multiple Impact Loading • Crashworthiness of High Strain Rate Plastics • Head Impact Mechanics • Design of Seat Impact • Knee Bolster Design Optimization under Impact
Past Projects: Electronic Packaging • Fatigue Damage of Strain-Rate and Temperature dependent Solder Alloys • Scale Effect of Solder Joints • Micro-structural Evolution of Solders
Past Projects: Sheet Metal Formability • Forming Limit Diagram (FLD) for Strain Rate Dependent Metals • Formability of Tailor-Welded Blanks of Aluminum, Steels and Titanium • FLD of Warm and Hot Forming • FLD of Multiple Stamping Processes • Warm and Hot Magnesium Tube Hydroforming
Past Projects: Fatigue and Fracture • Failure Analysis of Rubber-like Materials • Thermo-mechanical Fatigue of Engine Block Cracking • Fractures in Composite Structures • Fracture in Aluminum Weld Components • Mechanics of Fracture in Tires
n A Definition of scalar damage variable where A0 = original surface area (with defects); A = surface area excluding defects
Damage-coupled Elasticity • True stress was replaced by effective stress • Based on strain equivalent principle • Based on energy equivalent principle • Damage evolution equation
Relationship Between Scalar Damage Variable and Young’s Modulus Undamaged material Damaged Material or
Tensor Damage Variable Relationship between Effective stress and Cauchy Stress where M(D) is damage effect tensor. For isotropic damage, D becomes a scale damage variable. M(D) then becomes I is a unit tensor .
Damage Effect Tensor For multi-axial stress state, damage effect tensor is where D and are scalar damage variables
Free Energy Equivalence The free energy for a damaged material may be expressed in a form similar to that for a material without damage except that all stresses are replaced by their corresponding effective stresses. • Without damage • With damage
Damage-Coupled Elastic Equation where C is effective/damage stiffness matrix and
Damage Energy Release Rate The conjugates of the damage variables, known as damage energy release rate, are defined as
Damage-Coupled Yield Surface where is defined as and and R(p) are yield stress and strain hardening threshold. p is overall effective plastic strain.
Damage-Coupled Plastic Equation where S is the true stress deviator tensor, λ is a Lagrange multiplier.
2 plastic damage surface no damage 1 fatigue damage surface Fatigue and Plastic Damage Surfaces
Plastic Damage surface The expanding plastic damage surface is expressed in terms of the damage energy release rates and as
Plastic Damage Evolution Equations where dwp is overall plastic damage increment, λpd is the Lagrange multiplier
Fatigue Damage Fatigue damage surface Fatigue damage evolution equations where dwf is overall fatigue damage increment, λfd is a Lagrange multiplier
Total Damage Total damage is the summation of fatigue damage and plastic damage
Damage Failure Criterion A material element is said to have rupture when the total cumulative overall damage (w) has reached the measured critical value (wc) of the material under investigation.
Finite Element Analysis The damage coupled material model has been implemented in ABAQUS and LS-DYNA through UMAT and user specified material subroutine respectively. It has also been programmed in FCRASH of Ford.
UMAT: Variables to be defined • Jacobian Matrix of the material model It must be defined accurately if rapid convergence is to be achieved. However, an incorrect definition only affects the convergence rate. The results (if obtained) are unaffected. • Stress tensor • Elastic and plastic strain tensor • Damage variables defined as the solution-dependent state variables
Damage-Coupled Material Model FEA Damage Index Damage Analysis Approach
Bumper Damage under Multiple Impact Loading Objectives • To evaluate crashworthiness of bumpers under multiple low speed impact. • To quantify accumulative damages in two bumpers, one made of SAE 950 and another, martensitic sheet steel. • To predict overall damages sustained in the bumpers and their potential sites of failure using FCRASH programmed with the damage model and then compare the simulation results with those of drop-weight testing.
Damage Variables D and µ From the equations of E and in an earlier slide, we can evaluate D and μ with measured data (E0, E, 0, ) as and
Critical Damage Critical overall damage for SAE 950 steel was measured to be wc=0.112 and martinsite sheet steel, wc=0.04.