Why is viscoelastic behavior a topic for civil engineers?

Transcription

1 Why is viscoelastic behavior a topic for civil engineers? Under small load, the deformation is linearly reversible. Under constant higher load, the material behaves however non linear and irreversible with damage accumulation and failure. In epoxy, one observed molecular rearrangements resulting in continuous creep. The creep rate and strain depend on the molecular structure, crosslinking density and temperature. One can make an experimental time temperature superposition to map short term experimental results to long term creep. After the Big Dig accident such experiments were made and results are given in the plot for normal and fast hardening epoxy resin. Fast hardening ones form a network with significantly lower cross linking density and hence tend to creep at much higher rates than normal hardening ones. The big dig accident was a combination of fast hardening epoxy, insufficient processing and a too small depth of anchoring. 1 2

2 Polymers can exhibit excessive viscoelastic behavior and in principle the thermo dynamic theory of polymer physics can be used to describe it. Some phenomena are rate dependence, effective stiffness. For cyclic loading one observed hysteresis, what means dissipation of elastic energy. As well as a type of conditioning. Opposite to elastic materials, viscoelastic ones have an elastic and a viscos component. From this follows time dependence, energy dissipation in form of heat (hysteresis with the dissipated energy as area in the loop) In general all materials exhibit viscoelastic behavior, in metals or quartz these are however at room temperature rather small. Polymers, young concrete, wood and other biologic tissues show however an excessive viscoelastic effect. A complete description of the materials hence always contains viscoelastic behavior, that is mainly determined in experiments. Linear viscoelasticity means always a linear relation between force and displacement properties. 3 4

3 If suddenly a load is applied and the stress is kept constant, one observes an increase in strain called creep. Take putty or rubber as an example. The time interval of the load application is considered to be small compared with the observation time. The corresponding strain eps(t) with respect to the stress is called creep or retardation function. For t=0 one talks of a momentary compliance, for t=inf of an equilibrium compliance. If the system is constantly strained, the stresses decrease with time. Bolted connections with pre stress are such an example, where the pre stress decreases. The retardation function describes, how the strain is retarded and develops in time. Both functions characterize the behavior of linear viscoelastic bodies. For creep one distinguished three creep regimes, that are defined by the creep rate: Primary, secondary and tertiary creep. Creep is always related to damage. The most important creep mechanisms are diffusion and dislocation creep. They dominate the regime of secondary creep and when we talk about creep rates, we refer to this regime. Of course the activated creep mechanisms always depend on temperature and stress level. One defines the creep modulus like a material parameter mainly for design purposes of polymer components to consider the influence of relation in constructions. To mathematically describe the creep strain, often the Norton s creep law is used. From examinations of secondary creep in 1929 by Norton, the first purely stress dependent creep description was developed. It describes the minimal creep rate as power law of the stress. The law has two material parameters: the stress exponent n and a temperature dependent material constant K. The stress exponent n is an indicator for the failure or creep mechanism. In literature vales of n=4 7 are related to dislocation creep and n=1 2 to grain boundary creep. For low stress or low creep rate, exponents close to 1 can describe mainly diffusion creep. The Norton s creep law is used nowadays mainly due to its simplicity for estimating creep deformation or stress redistribution and relaxation in components. It is however only valid for low and moderate stress in the second creep regime and the parameters K and n have to be identified for each 5 6

4 temperature level separately. For more accurate descriptions or high temperature deformations, much better descriptions are needed. Those are time and temperature dependent and can be phenomenological equations, that mathematically describe measured creep curves and constitutive equations based on continuum mechanical or micro structural approaches that couple deformations to damage. Polymers: Stress induced molecular rearrangements. Chain failure: As long as intact, they take load. Degradation like oxidation or hydrolysis leads to polymer chain failure. Bond Interchange: portions of polymers may change partners causing release of stress. In presence of stress, rearrangements tend to reform so as to reduce stress Viscous flow: caused by linear chains that eventually slip past one another Thirion Relaxation: reversible relaxation of cross links or entanglements Molecular Relaxation: below Tg, molecules move very slowly, thus the material appears as an amorphous solid. As temperature approaches Tg, chains move and relax at an observable rate on the lab time scale. For chains under stress, motions will tend to relieve the stress. Metals: Mainly dislocation movements and other interactions. trans cristalline processes: Dislocation movement; void diffusion inter cristalline processes: Grain boundary gliding; grain boundary diffusion Creep in concrete: Creep and shrinkage of concrete are two physical properties of concrete. The creep of concrete, which originates from the calcium silicate hydrates (H M D) in the hardened Portland cement paste (which is the binder of mineral aggregates), is 6 7

5 fundamentally different from the creep of metals and polymers. Unlike the creep of metals, it occurs at all stress levels and, within the service stress range, is linearly dependent on the stress if the pore water content is constant. Unlike the creep of polymers and metals, it exhibits multi months aging, caused by chemical hardening due to hydration which stiffens the microstructure, and multi year aging, caused by long term relaxation of self equilibrated micro stresses in the nano porous microstructure of the H M D. If concrete is fully dried, it does not creep, but it is next to impossible to dry concrete fully without severe cracking. Changes of pore water content due to drying or wetting processes cause significant volume changes of concrete in load free specimens. They are called the shrinkage (typically causing strains between and , and in low strength concretes even ) or swelling (< in normal concretes, < in high strength concretes). To separate shrinkage from creep, the compliance function, defined as the stressproduced strain (i.e., the total strain minus shrinkage) caused at time t by a unit sustained uniaxial stress applied at age, is measured as the strain difference between the loaded and load free specimens. The multi year creep evolves logarithmically in time (with no final asymptotic value), and over the typical structural lifetimes it may attain values 3 to 6 times larger than the initial elastic strain. When a deformation is suddenly imposed and held constant, creep causes relaxation of critically produced elastic stress. After unloading, creep recovery takes place, but it is partial, because of aging. In practice, creep during drying is inseparable from shrinkage. The rate of creep increases with the rate of change of pore humidity (i.e., relative vapor pressure in the pores). For small specimen thickness, the creep during drying greatly exceeds the sum of the drying shrinkage at no load and the creep of a loaded sealed specimen (Fig. 1 bottom). The difference, called the drying creep or Pickett effect (or stress induced shrinkage), represents a hygro mechanical coupling between strain and pore humidity changes. Drying shrinkage at high humidities (Fig. 1 top and middle) is caused mainly by compressive stresses in the solid microstructure which balance the increase in capillary tension and surface tension on the pore walls. At low pore humidities (<75%), shrinkage is caused by a decrease of the disjoining pressure across nano pores less than about 3 nm thick, filled by adsorbed water. The chemical processes of Portland cement hydration lead to another type of shrinkage, called the autogenous shrinkage, which is observed in sealed specimens, i.e., at no moisture loss. It is caused partly by chemical volume changes, but mainly by selfdesiccation due to loss of water consumed by the hydration reaction. It amounts to only about 5% of the drying shrinkage in normal concretes, which self desiccate to about 97% pore humidity. But it can equal the drying shrinkage in modern high strength concretes with very low water cement ratios, which may self desiccate to as low as 75% humidity. The creep originates in the calcium silicate hydrates (C S H) of hardened Portland cement paste. It is caused by slips due to bond ruptures, with bond restorations at adjacent sites. The C S H is strongly hydrophillic, and has a colloidal microstructure disordered from a few nanometers up. The paste has a porosity of about 0.4 to 0.55 and an enormous internal surface area, roughly 500 m 2 /cm 3. Its main component is the tri calcium silicate hydrate gel (3 CaO 2 SiO 3 3 H 2 0, in short C 3 S 2 H 3 ). The gel forms particles of colloidal dimensions, weakly bound by van der Waals forces. Creep is signifcant when designing pre stressed concrete parts like slender columns or floors. 7 7

6 Rheology is the science of deformation and flow behavior of bodies Rheology lecture Werkstoffe IV. Fundamental models are springs, damper, frictional elements that can be operating in parallel or series, similar to connection diagrams in electrical engineering. The demonstrative models and their resulting model rheology are in particular usable for a qualitative description. Linear viscoelasticity means always a linear relation between force and displacement properties. 8 9

7 Let s first look at a parallel organization of spring and damper a so called Kelvin Voigtmodel. From the diagram it is obvious, that strain of spring and damper are identical, hence the sum of stress from the Hook s spring and the Newton s damper. The constant tau has the dimension of a time and is called retardation time. To determine the creep function, we define the stress evolution via the Heavysidefunction. The 1st order differential equation now hast to be solved, what leads with the initial condition epsilon(0)=0 to the creep function. Since the Kelvin Voigt body has a liquid type initial behavior but a solid type final one, it is in principal a solid body. If we put damper and spring in series, we obtain a Maxwell element. It is clear that both elements experience the same stress and the strains add up. However, since in the material law not strains but their rates appear, we add strain rates. For creep we obtain by integration and the BC eps(0)=sig_0/e. And consequently the creep function J(t)=eps(t)/sig_0:.. After an instantaneous jump of the strain due to the elastic deformation of the Hooks element, the body continues to creep infinitely in principle like a fluid. The relaxation function ( G(t)=sig(t)/eps_0 ) for given strain jump stems from the differential equation with the initial condition time 0 (what corresponds to the Hooks reply)

8 The tangent in point t=0 intersects the asymptote J(inf)=1/E at the position called retardation time. This characteristic time hence characterized the point, where 63.2% of the equilibrium compliance is reached. For the Maxwell model the relaxation time is reached, when the stress decreases to 36.8% of the initial value

9 2P models are in general not capable of predicting the viscoelastic material sufficiently good. In other words, one needs more parameters und finally the combination of several dampers and spring. As a first version, we can put our KV in a series with an additional spring or make a parallel arrangement with a Maxwell element and a spring. One calls these models linear Standard bodies or Zener Models. If we set up the constitutive equation for both, we realize, that both approaches are equivalent and can be written in a general form. Only the interpretation of parameters is different. For completeness we want to make the same combination, but now with two dampers. Such models are for example used for modeling the earth crust but also blood. One can again derive a general form with diverse parameters. It is interesting to note that the retardation time is smaller than the relaxation time, hence just opposite to the linear standard body. Looking at the creep function one can see, that the linear standard body has a momentary elasticity but also a limit elasticity. It is hence a solid body. The dashed line shows the unloading behavior. It is also interesting to note, that the retardation time tau and relaxation time tau_bar are not identical. They relate tau_bar<tau

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11 By respective consideration of elements in brackets, we obtain a solid body or a fluid for an arbitrary number of elements. One calls such groups also Prony series, and in FEprograms, viscoelastic behavior is expressed in this very general way. For an infinite number of elements and infinitesimal neighbored retardation times, the discrete retardation spectrum merges into a continuous one f1(tau). This holds for the creep function as well as for the relaxation function. These functions can be determined inversely from experimental data. The total strain can summed up following the superposition principle from all stress jumps, starting at the loading (t=0) up to the present state. This is possible only for linear creep, hence Newton elements or Stokes elements. Since the behavior is linear, the total answer of a number of actions equals the sum of the individual answers to each action (Boltzmanns superposition principle). Starting from 2 loads, we can generalize it to an infinite number of infinitesimal load steps dsigma and hence to arbitrary load profiles. With dsigma=dsigma/dtau*dtau we obtain the integral way of writing. The integral is also called convolution integral (Faltungsintegral ) and is a different type of constitutive relation for linear viscoelastic materials

12 The behavior of specimen at a force can be classified into 3 different types: 1. Purely elastic behavior (Hooks body) with instantaneous reaction of the sample to the action φ = 0 2. Purely viscos behavior (Newton body) Largest strain for zero crossing of action φ = π/2 = Viscoelastic behavior marks a case between 1st and 2nd one, where the deformation follows the force with a certain phase shift angle 0 < φ < π/2 = 90. The smaller φ the more elastic the material is. With the Euler equation, we can express oscillations in a complex plane and obtain. One defines the storage modulus as the property of stored elastic energy in a viscoelastic material, as well as the loss modulus as the part of the dissipated energy, e.g. by heat. For the tensile storage and loss modulus one can write: - The storage modulus E characterizes the stiffness of a viscoelastic material and is proportional to the maximum elastic stored work during the loading period. The loss modulus E is proportional to the work, dissipated during a loading period and his thus irreversible. The ratio of loss to storage modulus is called loss factor. Its value of 0 corresponds to entirely elastic behavior, higher values for dissipative processes

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14 The sample is loaded with a sinusoidal oscillating force. The temporal evolution of the deformation is measured. This way stiffness can be evaluated via the storage modulus E and the damping behavior by the loss modulus E'' or by the loss factor tan Phi. Different deformation types like bending, tension or penetration are used, depending on the material, specimen shape and practical load case. One can obtain information on: mechanical properties, temperature limitations, crosslinking degree / hardening state. The DMA is used for obtaining temperature and frequency dependence of storage modulus, loss modulus and loss factors. Furthermore on can study glass transition of polymers and its morphology via DMA. An other important field is the characterization of aging e.g. in polymers

15 Transformation into the frequency domain results to a simplified way of writing the constitutive relation and a superposition of various loads. Additionally it is simpler than the direct solution of ODEs. The Laplace transform is very useful in constructing and analyzing linear viscoelastic models. The Laplace transformation is a one sided integral transformation that shifts a given function f from the real time domain into a function F in frequency domain. The Function F is called Laplace transformed of spectral function. There are similarities with the Fourier transformation. There is as well an inverse transformation, also called Bromwich integral

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