ABSTRACT
The objective is the ability to produce useful information on the strain/stress states from the theory of acoustoelasticity (Murnaghan F. D., 1951) applied to sound waves acquired through transducers placed on the concrete surfaces. Cross correlation is used to compare waves at different loading stages to evaluate changes in sound velocity. The block was loaded up to 70 kN through the tendon and unloaded thrugh the nuts (Figure 1). Geometry of the test setup. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/fig80_1.tif"/>
Ultrasound signal were acquired on the opposite side each 1 μ s. Changes in the sound velocity with respect to changes in the strains are computed from the equations: (Egle & Bray, 1976) () 1 v 11 0 d V 11 d ε = 1 2 ( λ + 2 μ ) ⋅ [ 2 l + 4 m + 5 λ + 10 μ + ( 2 l + λ ) ( β + γ ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq145.tif"/>
V 11: Wave speed with initial value V 11 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq146.tif"/>
λ, μ: Lamé constants, Poisson’s ratio
l, m, n: Murnaghan’s constants
ε, βε, λε: triaxial principal strains
For uniaxial stress loading (i.e., β = λ = — ν): () δ ε = − 1 2 + μ + 2 m + v μ ( 1 + 2 l λ ) λ + 2 μ τ t k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq147.tif"/> τ results from maximization of the cross correlation factor C C t k k ( τ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq148.tif"/> (Equation 3)
δε is the strain variation between the unperturbed and perturbed waves () C C t k k ( τ ) = ∫ t k + T 2 t k + T 2 h ′ ( t − τ ) h ( t ) d t ∫ t k + T 2 t k + T 2 h ′ 2 ( t − τ ) d t ∫ t k + T 2 t k + T 2 h 2 ( t ) d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq149.tif"/>
h(t): sound wave signals (unloaded specimen)
h’(t): sound wave signals (loaded specimen).
tk : center of a window in the time axis
T: half width of the time window
Average τ (μs) as function of the load (kN). https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/fig80_2.tif"/>Preliminary conclusions:
Results indicate that the stress level at the wave path A is higher than at the path B, reproducing the expected tendency;
Nonlinearity of the load versus τ (or Δt) curve, attributed to local damage around the duct.
