Introduction to Continuum Damage g Mechanics By: Dr. Curtis F F. Berthelot P.Eng. g Department of Civil and Geological Engineering Centre of Excellence for Transportation and Infrastructure

Engineering Modeling Frameworks  

Engineers require a methodology that accurately predicts ((models)) whole life p p performance of the system. y Three basic-modeling techniques: • Purely-Empirical (Experience based) • Phenomenological-Empirical (fudge tests that make us feel better, but don’t really tell us anything conclusive) • Mechanistic-based M h i i b d (future) (f )

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Purely-Empirical Modeling Framework 



Empiricism is defined as: • “Relying on experience or observation without due regard for system and/or theory.” Employs observed field performance to derive performance based design relationships. • Classical statistical regression analysis to derive road performance relations based solely on judgment inferred from road performance observations. observations • Engineer’s intuition-dangerous in a litigious society.

Introduction to Continuum Mechanics 3

Purely-Empirical Modeling Framework Repeated Performance Observations (provide the necessary large quantities of data to calibrate the regression model)

Assume Functional Form of Performance Model (usually linear)

Perform Regression Analysis (quantify predictor variable coefficients and eliminate statistically insignificant predictors) Test Predictive Capability of Model (quantify goodness of fit, confidence intervals, and coefficient of determinism) Good Model (High R2)

Predicted Performance

Predicted Performance

Iterativve Model Reformulation and Calibratioon

Assume Predictor Variables to Include in Performance Model (independent and dependent variables based on theoretical principles)

Poor Model (Low R2)

Observed Performance

Observed Performance

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PhenomenologicalEmpirical Modeling 



Phenomenological materials tests are used to simulate field conditions and performance. • Make us feel better, but do not really tell us anything of engineering significance. • Often used to “rank” alternatives in terms of performance. • May give us the wrong answer or prioritization scheme. Phenomenological tests do not provide mechanistic measures of behavior that can be used in a mechanistic modeling framework. Introduction to Continuum Mechanics 5

Limitations of Phenomenological Empirical Models   

Modern engineered systems are comprised of diverse structures that are subjected j to variable loadings. g Significant models and supporting databases are required to empirically predict performance. Traditional phenomenological material tests provide test dependent material behavior indicators that do not characterize the fundamental thermomechanical constitutive behavior of the materials used to construct the system.

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Limitations of Phenomenological Empirical Models 





Performance predictions based exclusively on judgment and/or experience may be biased and/or inaccurate (usually both). Changing field state conditions; ongoing developments in engineering, materials, system functionality requirements, construction methods etc, are beyond the inference of traditional performance models. As an engineer looking at the next 30 years of your career begin to develop skills and knowledge that career, establish a more reliable check and balance system for your engineer’s intuition.

Introduction to Continuum Mechanics 7

Scientific-Engineering Process 1. 2. 3. 4. 5.

Observe Measure Model Validate Economic Evaluation.

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Mechanistic Modeling 

To overcome the limitations of purely-empirical and phenomenological-empirical p g p techniques, q , mechanisticmodeling methods that draw upon materials science, engineering thermomechanics, and computational capabilities are being used: • Mechanistic-Empirical • Purely Mechanistic (closed form solutions-ideal). • Damage mechanics: • Flow • Fracture Introduction to Continuum Mechanics 9

Mechanistic Modeling 



Continuum Mechanics: • Modeling materials/systems as homogeneous matter (ignores internal cracks, voids, asperities and anomalies at link scales much smaller than the boundary geometry). • Encodes mechanical material behavior with constitutive relationships (i.e. stress & strain). Micro Mechanics: • Study S d off materials i l and d systems at an atomic i or iinterparticulate level.

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Mechanistic Modeling 

Mechanics is the application of the principles of thermodynamics y to characterize the mechanical behavior of materials and structures: • Based on Natural Laws: • First and Second Laws of Thermodynamics. • Conservation of linear and angular momentum. • Conservation of charge • Powerful and reliable framework for engineers

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Galileo Galilee      

1564-1642 Originally a professor of medicine at Pisa. Pisa Focused on mathematics (work of Euclid and Archimedes). Later became professor of mathematics and focused on falling bodies (dynamics). Galileo went against many theories of Aristotle, however hi new science his i drew d lectures l off over 2000 students. d Worked in strength of materials for ship building industry. Mechanistic Road Materials Characterization and Road Structural Analysis 12

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Galileo Galilee

Mechanistic Road Materials Characterization and Road Structural Analysis 13

Cosimo Medici-Florence

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Santa Maria del Fiore1296 to 1436

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Robert Hooke    

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1635-1703 Curator of experiments at Royal Society. Society Performed many experiments evaluating the strength and performance of common materials. 1660 – Ceiiinosssttuv-“ut tensio sic vis” • “As the load so does the displacement” Hooke’s Model (called Law) established the foundation of observed linear elastic behaviour.

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Sir Isaac Newton   

1642-1727 (Born on Xmas day of the year that Galileo died)) 1665 – Apple fell on Newton’s head in Hampton Court, Woolsthorpe, England (natural gravity). Studied work of Galileo, Copernicus and Keppler and derived 3 axioms of nature to describe the natural laws. • Law I: Every body remains in a state of rest or steady motion in a straight line unless acted on by a force. • Law II: The quantity of altered motion is proportional to the force acting upon it (F=ma). Mechanistic Road Materials Characterization and Road Structural Analysis 17

Sir Isaac Newton



• Law III: For equilibrium to exist, every action must have an equal and opposite action for equilibrium to exist, i ii.e. C Conservation i off Li Linear M Momentum. Newton had to develop temporal calculus to provide time wise derivatives of velocity to give acceleration which was related to the force required to move mass ‘a’.

F  ma dV d 2 s a  2 dt dt Mechanistic Road Materials Characterization and Road Structural Analysis 18

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Sir Isaac Newton 

These laws were published in 1686, “Philosophiae Naturalis Principia p Mathematica.”

“If it is I that has seen farthest, it is because I stood on the shoulders of giants”.

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Leonard Euler       

1707-1783 Father of Strength of Materials. Materials Most regarded 18th Century scientist. Studied under John Bernoulli. Worked at Russian Academy of Science in St. Petersburg when it opened in 1725. Jacob and Daniel Bernoulli worked with Euler. Wrote over 400 papers and textbooks; 200 of which were during last 20 years of his life when he was blind. Mechanistic Road Materials Characterization and Road Structural Analysis 20

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Kinematics: Strain Displacement Relations  

Leonard Euler (1707-1783). Engineering Uniaxial Strain (xx) is the square of the d f deformed d llength th minus i th the square off th the iinitial iti l llength th divided by the square of the initial length:  2 1  l f  lo xx  lim  2  lo 2 



2

   

For infinitesimal deformations, uniaxial strain may be approximated by: 

l f lo l  lo lo

Kinematics: Strain-Displacement 21

Infinitesimal Strain Displacement Relation 

Normal and shear strains:

u x v  yy  y w  zz  z

 xx 

u v  y x u w   z x v w   z y

 xy   xz  yz

6 unknowns

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Napoleon

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Conservation Laws 

Conservation of mass (trivial): d d dV  0   u i ,i  0 dt V dt



Conservation of linear momentum:

 F 0   ji , j  f i   

d 2 ui 0 dt 2

Conservation of angular momentum:

 M 0 Conservation

  ij  ji

of charge for C.E. (Trivial)

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Conservation Laws 

Conservation of Energy (1st Law of Thermodynamics): dU dU dh dw d       ij  ij qi ,i  r dt dt dt dt



Entropy Inequality (2nd Law of Thermodynamics): 

dS  qi  r  , i  0 dt  T  T

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Conservation Laws 

Given St. Venants rules, the field equations are significantly reduced: • Conservation C ti off E Energy  ij  ij qi ,i  r   •

  ij  ij  

dU dt

Entropy Inequality: 

•

dU dt

dS  qi  r  ,i  0 dt  T  T

 

dS 0 dt

Conservation of Linear Momentum:  ji , j  f i  

d 2ui  0   ji , j  0 dt 2

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Engineering Properties of Materials 

Thermomechanical Approach • Applicable to all engineered systems (multidisciplinary approach) • Natural laws do not vary in time, therefore, thermomechanical models provide a stable platform for future model improvements • Thermomechanical material characterization provides direct mapping p pp g to p primaryy response p and performance of the material in the field • Thermomechanics can be applied as a uniform theory to all materials and systems Introduction to Continuum Mechanics 27

Cauchy’s Uniqueness Theorem of Stress  

Augustin Louis Cauchy, 1789-1857. Cauchy applied his knowledge as a hydrodynamics engineer to formulate force over unit area for strength of solid materials purposes.

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Cauchy’s Uniqueness Theorem of Stress X2

 2 2 (x 2 +  x 2 )

 2 1 (x 2 +  x 2 )  2 3 (x 2 +  x 2 )

X2

 1 3 (x 1 )

 1 1 (x 1 )

 1 2 (x 1 )  3 2 (x 3 +  x 3 )

 3 3 (x 3 +  x 3 )

x3

 3 3 (x 3 )  3 1 (x 3 )

X1

X3

 1 2 (x 1 +  x 1 )

 3 2 (x 3 )

 1 1 (x 1 +  x 1 )

 1 3 (x 1 +  x 1 )

 3 1 (x 3 +  x 3 )

x1

 2 3 (x 2 )

 2 1 (x 2 )

 2 2 (x 2 )

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Differential Equations of Equilibrium 

Conservation of linear momentum must hold for every point in the material body. p y

 σ 11  σ 21  σ 31    X1 0 x 1 x 2 x 3  σ 12  σ 22  σ 32    X 2  0  σ ji, j  X i  0 in V x 1 x 2 x 3  σ 13  σ 23  σ 33    X 3 0 x1 x 2 x 3

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Barre St. Venant    

1797-1886 Father of elasticity. elasticity Derived compatibility equations between deformations and strain. Derived 7 rules for characterizing deformation of materials.

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Barre St. Venant 1. Uniform stress-strain field in sample: St. Venant’s principle p p of sample p ggeometry. y 2. Principle of Link scale of asperities: Specimen size must be an order of magnitude larger than maximum asperity size contained in the material: • Very important for particulate composite road materials 3. Eliminate thermal gradients in specimen (test at constant temperature: DT=0

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Barre St. Venant 4. Applied load rate should be much slower than natural frequency of material (quasi-static): ffsteel  10,000 Hz; ffhumans  2 Hz 5. Eliminate body forces (creep) in specimen: fbody