International Symposium (NDT-CE 2003)Non-Destructive Testing in Civil Engineering 2003
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Elastic Properties of Reactive Powder ConcreteGlenn Washer, Federal Highway Administration, Turner Fairbank Highway Research Center, McLean, VA 22101
Paul Fuchs, Fuchs Consulting, Inc., Turner Fairbank Highway Research Center, McLean, VA 22101
Benjamin Graybeal, PSI, Inc, Turner Fairbank Highway Research Center, McLean, VA 22101
Concrete is a critical material for the construction of infrastructure facilities throughout the world. A new material known as reactive powder concrete (RPC) is becoming available that differs significantly from traditional concretes. RPC has no large aggregates, and contains small steel fibers that provide additional strength and in some cases can replace traditional mild steel reinforcement. Due to its high density and lack of aggregates, ultrasonic inspections at frequencies ten to twenty times that of traditional concrete inspections are possible. These properties make it possible to evaluate anisotropy in the material using ultrasonic waves, and thereby measure quantitatively the elastic properties of the material. The research reported in this paper examines elastic properties of this new material as modeled as an orthotropic elastic solid and discusses ultrasonic methods for evaluating Young's modulus nondestructively. Calculation of shear moduli and Poisson's ratio based on ultrasonic velocity measurements are also reported. Ultrasonic results are compared with traditional destructive methods.
A new material has recently become available in the United States that demonstrates greatly improved strength and durability characteristics compared with traditional or even high-performance concrete. Classified as Ultra-High Performance Concrete (UHPC), or Reactive Powder Concrete (RPC), the material consist of a concrete using sand as its largest aggregate and fine steel fibers distributed within the concrete. Compressive strengths of 200 to 800 MPa have been achieved with RPC, compared with maximum compressive strength of 50 to 100 MPa for high-performance concretes. Young's modulus of 50 to 60 GPa are common for RPC, as compared with values of 14 to 42 GPa of normal weight concrete (Mindness and Young, 1981). Additionally, the material has a tensile strength of between 6 - 13 MPa that is maintained after first cracking, whereas traditional concrete has tensile strengths on the order of 2 to 4 MPa that is lost when cracking occurs.
The improved properties of RPC are obtained by improving the homogeneity of the concrete by eliminating large aggregates, increasing compactness of the mixtures by optimizing packing density of fine particles, and using fine steel fibers to provide ductility (Richard, 1995, Graybeal, 2002). The largest aggregates typically used for RPC are fine sands with diameters from 100 to 600 m m with other solids within the mixture orders of magnitude smaller (Graybeal, 2002, Bonneau, 1997). Steel fiber reinforcement is also fine, with diameter on the order of 0.2 mm and typical lengths of 12 mm (Richard, 1995, Roux, 1996, Graybeal, 2002). Figure 1 presents X-ray computed tomography images of the RPC material. Figure 1A is a three dimensional reconstruction of a section of a 40 mm diameter RPC core, showing the random distribution of steel fibers. Figure 1B displays a single slice of RPC data that indicates air voids in the core and the distribution of the fibers along a single horizontal plane.
Traditional concrete has time-dependant, nonlinear elastic behavior due to the complex interaction of large, hard aggregates with a distributed, relatively soft cement matrix. Standard methods for determining the modulus of traditional concrete (ASTM C496) require incremental loading at high strains to determine an estimate of the modulus. Moduli determined by small-strain, dynamic testing such as ultrasonic pulse velocity measurements can be up to 30% higher than those determined by high strain, static testing (Mindness and Young, 1981). As such, ultrasonic pulse velocity measurements have not provided a satisfactory measure of the modulus of hardened concrete and are not recommended (ASTM C597). The homogeneous nature of RPC results in a more linear elastic behavior, less scattering and more easily propagated ultrasonic waves and as such these recommendations may not be supported.
This paper presents an initial analysis of RPC assuming that the material is an anisotropic elastic solid. The possibility of orthotropic elastic constants is examined by measuring ultrasonic wave velocity along different directions in an orthogonal coordinate system. Anisotropic and isotropic elastic constants are reported.
Theoretical BackgroundThe method of mixture, placement and finishing of the RPC specimens suggests that if there is anisotropy in the material, one principle axis could be coincident with the force of gravity. While the RPC is in the plastic state (prior to hardening), it is placed into the forms and subsequently vibrated externally. It is possible during this process that the force of gravity acts to align steel fibers, or to segregate constituent materials. Alternatively, the magnetic field of the earth will exert some force on the ferromagnetic fibers. This force will be tangential to the earth's surface, and generally orthogonal to the gravitational force. Although these forces are assumed to have negligible effect of the physical properties of RPC, they provide a starting point for a rational to define a coordinate system and symmetry for evaluating elastic anisotropy.
Perhaps more importantly, the dead load of a civil structure is typically the largest load applied, and it is convenient to utilize a coordinate system with one axis coincident with gravity. The second direction in which elastic properties would be of importance would lie along a beam axis, defining a second axis of symmetry. It is therefore significant to design engineers generally to discover if elastic properties vary between principle axes defined by the beam axis and gravity, i.e. to discover if the material has orthotropic elastic properties.
Cube-shaped specimens make analyses of the material properties based on a coordinate system with one axis defined by the gravitational force, and the other two axes orthogonal, convenient and practical. As such we define a coordinate system as shown in figure 2. We assume here that if the material were anisotropic, and the natural phenomena of gravitation and magnetic field are the only assumed external forces acting during solidification, then the material would exhibit orthotropic symmetry of elastic properties. We also assume that for practical engineering applications orthotropic properties would be of significant interest.
A material with orthotropic elastic properties has 9 independent elastic constants compared with 2 independent constants for an isotropic material. To determine if RPC has orthotropic elastic properties, and to quantify the significance of that anisotropy, ultrasonic velocity measurement can be used in conjunction with the mass density of the subject materials to quantify the elastic compliance, cij.
For the case of a homogeneous, linear-elastic anisotropic solid the equation of motion is (Green, 1973) :
where cijkl are the second-order elastic constants, is the strain tensor expressed in terms of displacements, uk and direction, xl, is the time differential of displacement (acceleration) and r is density. A plane wave solution is assumed resulting in a determinant equation:
where wave velocity v = w/k and dik is the Kronecker delta. This determinant equation, along with the direction cosines of the plane wave and its normal, li lj, can be solved to relate ultrasonic wave velocities to elastic constants. For orthotropic materials, there would be nine independent, non-zero elastic constants, with the relations (Mason, 1964, Rose, 1999)
For an isotropic material, there are only two non-zero independent elastic constants according to the relations (Green, 1973):
where l and m are the Lamé constants. Lamé constants have direct relation to engineering constants Young's modulus, E, shear modulus G, and Poisson's ratio, n (Green, 1973).
There are several fundamental differences between moduli determined by ultrasonic wave velocities and those determined by traditional engineering methods (ASTM C496). Most important among these is the effect of strain rate and the non-linear behavior of concrete. The importance of these differences is not yet known and is one of the topics being explored in this research.
Relation to Ultrasonic Velocities
Solutions to the determinate equation (2) can be used to define relationship between the velocity of ultrasonic waves and the second-order elastic constants cij. This leads to three solutions for waves propagating in a particular principle direction. The wave velocities vij (where i is the direction of propagation and j is the direction of particle displacement or polarization) are:
By this means the diagonal components of the compliance matrix can be determined by propagating longitudinal and polarized shear waves along principle directions of the solid. To determine the off-diagonal components, off-axis measurements are required. For the present study, we are most interested in determining if the material exhibits anisotropy relative to the principal directions of the assumed coordinate system, and hence it is sufficient as a preliminary measure to determine if C11=C22=C33, C44= C55=C66. If so, the material may exhibit cubic symmetry or may have truly isotropic elastic properties. For the RPC material, there is no known reason so suspect cubic symmetry and therefore it is not considered.
If the material is in fact isotropic then the direction of propagation or polarization is irrelevant. In this case, a single shear wave and a single longitudinal wave velocity would be all that was required to define moduli in terms useful for engineering applications. The relations between the velocity of ultrasonic waves and the Lamé constants are (Green 1973):
where VL is the longitudinal wave velocity and VS is the shear wave velocity. However, it is important to note that these equations are derived assuming a homogeneous solid and an absence of dispersion. RPC is more homogeneous than traditional concrete, but is still a composite material consisting of a cement paste, sand and steel fibers.
Ultrasonic pulses were generated using a RITEC RAM-10000 high-power ultrasonic instrument. Single-cycle pulses of appropriate frequency were generated and transmitted to the ultrasonic transducer through a diplexer. Ultrasonic waveforms were collected using an HP digital oscilloscope. Piezoelectric shear and longitudinal wave transducers 25 mm in diameter with a center frequency of 500 KHz were used to launch and receive waves in a pulse-echo configuration.
RPC cubes were used to determine basic wave propagation characteristics of RPC and to measure anisotropic elastic behavior. Cubes with and without steel fibers were cast as shown in Table 1. The cubes were fabricated by placing RPC within steel forms and vibrating externally using a vibrating table. The exposed surface of the cubes was leveled at the time of placement. Cubes L3-17,18 and 19 containing steel fibers were air cured in a laboratory environment (22° C, ~ 50 % humidity), while specimen L12-4 (without steel fibers) was steam cured at 90° C and 95% humidity for 48 hours. After removal from the form, the exposed surface was mechanically ground using a grinding machine to provide a level and uniform surface that was parallel to the opposing surface. This surface was finished by grinding with a 120-grit wheel.The steel fibers utilized were 0.2 mm in diameter and approximately 12 mm in length. The volume percent of these fibers is approximately 2% as shown in Table 1.
Cylinders of RPC were cast as shown in Table 2. These specimen were consolidated using a vibrating table, and the exposed surface leveled. The specimens were then steam cured at 90°C and 95 % humidity for 48 hours. The exposed surface was again ground to provide a level surface parallel to the opposing surface.
Elastic Constants From RPC Cube Data
Velocities of polarized shear waves are also shown in table 3 along with the standard deviations for measurements along principle axes. Again, differences between velocities are small and likely result from minor variations in the properties of different cubes and measurement error.
Also in table 3 are the results from cube L12-4 that show this specimen without fibers has consistently higher velocities than L3-17, L3-18 and L3-19. This increased wave velocity is primarily related to the curing conditions of the cube. The steam curing used results in a higher modulus material than air-curing. Measurements on cylindrical specimens reported in the following section indicated that specimens containing fibers have a higher velocity than those without fibers if the curing conditions are identical.
Based on the results shown in table 3, it was concluded that RPC demonstrated essentially isotropic elastic properties when measured by an ultrasonic pulse. Table 4 indicates the calculated elastic constants. As shown in the table, C11 " C22 " C33, and C44 " C55 " C66, and this data supports the conclusion that the material exhibits isotropic elastic properties.
Elastic Constants from RPC Cylinder Data
An examination of ultrasonic wave velocities along assumed principle directions in RPC has been conducted. Testing revealed consistent elastic constants along principle directions. RPC appears to have linear-elastic, isotropic properties within some limits. It is possible to use this characteristic to estimate elastic properties of the material, and there was close correlation with statically measured properties in the very small number of specimens tested.
It is important to note that the results reported here are from a very small sample set. The conclusions drawn from this data are therefore of limited applicability. For example, the difference between averaged E values measured ultrasonically and statically is less than 2%. It is possible that this falls beneath the natural dispersion of static measurement results, and therefore could be difficult to repeat with a larger sample set. This is the subject of ongoing research.