# dim - Overview of Topics Dimensional Stability EARLY AGE...

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Dr. Kimberly Kurtis School of Civil Engineering Georgia Institute of Technology Atlanta, Georgia Dimensional Stability Overview of Topics EARLY AGE CONCRETE Plastic shrinkage – shrinkage strain associated with early moisture loss Thermal shrinkage – shrinkage strain associated with cooling LATER AGE CONCRETE Drying shrinkage -shrinkage strain associated with moisture loss in the hardened material Deformations occur under loading - Elastic - Viscoelastic When does concrete crack? To understand this, we must consider the way concrete deforms under loading: Concrete exhibits both elastic and viscous (time-dependent deformation) behavior Concrete is a viscoelastic material Viscoleastic behavior can be described using rheological models with 2 components: linear elastic spring linear viscous dashpot Pure Linear Elastic Behavior Visualized by a spring Deformation quantified by Hooke’s Law Stress(t) = E x strain(t) where E is the spring constant Deformation is instantaneous; no continued deformation with time E σ σ ε = σ /E Pure Linear Elastic Behavior for for constant σ constant ε Pure Viscous Behavior Visualized by a dashpot Piston displaces a viscous fluid in a cylinder with a perforated bottom Deformation quantified by Newton’s Law of viscosity ε (t) = σ (t) where ε is the strain rate d ε /dt η is the viscosity coefficient . . σ σ η

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Linear elastic vs. Viscous Behavior for for constant σ constant ε Constant strain rate ε (t) = σ (t) . ε (t)= σ (t)/E Stress →∞ Combinations of these two elements can be used to describe more complex behavior. For example, the Maxwell model: Equilibrium equation: σ E (t) = σ η (t) = σ (t) Compatibility equation: ε (t) = ε E (t) + ε η (t) Constitutive relationships: σ E (t) = E ε E (t) spring σ η (t) = ηε η (t) dashpot σ . Maxwell Model Differentiating ε (t) = ε E (t) + ε η (t), we get ε (t) = ε E (t) + ε η (t) Differentiating σ E (t) = E ε E (t), we get σ E (t) = E ε E (t) Assuming σ E (t) = σ η (t) = σ (t) and remembering that σ η (t) = ηε η (t) substituting into we get ε (t) = σ (t)/E + σ (t)/ η Integrating, we get, for an initial applied stress, σ ο ε (t) = σ ο /E + ( σ ο / η )t Which predicts that for a constant applied stress σ ο (i.e., creep), there will be an instantaneous strain and then strain will increase without bounds .. . . . . σ Maxwell Model ε (t) = σ ο /E + ( σ ο / η )t If after loading, the system is unloaded, there will be an instantaneous recovery ( σ ο /E) in the spring, while a permanent strain [( σ ο / η )t] remains in the dashpot. For relaxation, where the strain ε o is constant, integration of ε (t) = σ (t)/E + σ (t)/ η gives σ (t) = E ε o e -Et/ η The ratio η /E is defined as the relaxation time; a small relaxation time indicates that the relaxation process will be fast σ Maxwell Model for for constant σ constant ε Kelvin Model Equilibrium equation: σ (t) E (t) + σ η (t) Compatibility equation: ε (t) = ε E (t) = ε η (t) Constitutive relationships: σ E (t) = E ε E (t) spring σ η (t) = ηε η (t) dashpot Resulting in: σ (t) = E ε E (t) + ηε η (t) .
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## This note was uploaded on 08/23/2011 for the course CEE 8813b taught by Professor Kurts during the Spring '07 term at Georgia Tech.

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dim - Overview of Topics Dimensional Stability EARLY AGE...

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