Comments on the Validity of Test Conditions for Kolsky Bar Testing of Elastic-Brittle Materials

Abstract

Analytic solutions for the time it takes an elastic Kolsky bar sample to achieve stress equilibrium and constant-strain-rate are presented. Using these solutions simple analytic criterion are derived that can be used either a posteriori to check the validity of a conducted experiment or a proiri to provide envelopes of validity under which tests could be conducted correctly on elastic-brittle materials.

Introduction

Kolsky bar¹ experimental methods [1–3] are well known techniques for performing high-rate material testing, usually to produce a family of stress-strain relationships at strain-rates from 10²/s to 10⁴/s. A minimal Kolsky apparatus consists of an incident, transmission, and striker bar, all cylindrical in geometry, and typically with length-to-diameter ratios of greater than 20 to ensure one-dimensional wave propagation. Figure 1 shows a schematic of a Kolsky bar for reference.

Fig. 1

Schematic of Kolsky bar

The purpose of this technical note is to derive, in closed-form, expressions for the time it takes a Kolsky bar elastic sample to reach both stress equilibrium and a state of constant-strain-rate; then, using these times provide simple analytic criterion that can be used to check the validity of a Kolsky bar test of an elastic-brittle material a posteriori or even develop envelopes of validity that can be used a priori for a given set of test conditions without having to make assumptions about how many sample stress wave transits it will take for a given sample/bar configuration to achieve equilibrium. This work provides an alternative definition of time-to-equilibrium that has been presented elsewhere [4, 5]. It’s notably different from Ravichandran and Subash [5] who provided a critical appraisal of the upper-limit of valid strain-rates for ceramic materials by observing and estimating from an x − t diagram the number of wave reflections in a sample needed for the sample to achieve a dynamic stress equilibrium condition. This work also extends the work of a more recent paper by Pan et al. [6], who considered the upper-limit of constant-strain-rates by analyzing equations derived in Frew et al. [7], by providing an analytic expression for the time-to-constant-strain-rate.

Using these analytic expressions for time-to- equilibrium and time-to-constant-strain-rate, interesting observations are made about the timing of these events and simple analytic criterion for valid test conditions are presented. Only by using a uniform approach to the derivation of these two critical time events can these observations be made.

Time-to-Equalibrium

In Frew et al. [7], the authors present two sets of equations that describe the stress (or strain) response in a Kolsky bar elastic sample as a result of a ramp incident pulse loading. It is shown that for elastic materials a ramp incident pulse loading will produce an asymptotic constant-strain-rate. One set of equations are derived assuming the sample undergoes uniform deformation and results in the following solution for sample stress

$$ \sigma_s(t) = \gamma M \left [ t - \frac{r t_0}{2} \left (1 - \exp \left({-\frac{2 t}{r t_0}} \right) \right ) \right ], $$

(1)

where M corresponds to the slope of the incident pulse (i.e. stress rate). Equation (1) uses the following definitions for the impedance ratio, r, the area ratio, γ, and the sample wave-transit time, t ₀,

$$ r = \frac{ \rho c A }{ \rho_0 c_0 A_0 }, $$

(2a)

$$ \gamma = \frac{ A }{ A_0 }, $$

(2b)

$$ t_0 = \frac{ l_0 }{ c_0}, $$

(2c)

where ρ and c are the material density and characteristic rod wave speed, respectively, A represents cross-sectional area with the subscripted constants referring to sample parameters and the unsubscripted constants referring to bar parameters. l ₀ is the sample length.

The second set of equations presented in [7] which describe the sample stress are derived from the wave mechanics and are periodic solutions which describe the stress histories at stations 1 and 2 in Fig. 1.

If we average the wave mechanics solutions over the sample the result is equation (3).

$$ \begin{array}{rll} \sigma_{avg}(t) &=& \frac{\sigma_1(t) + \sigma_2(t)}{2}\\ &=& \frac{\gamma M}{r + 1} \left( t + \sum_{i=1}^{n} \left[ \left ( \frac{r-1}{r+1} \right )^{i-1} \right.\right.\\ &&\quad\; \left.\left.+\left( \frac{r-1}{r+1} \right)^{i} \right] \left( t - i t_0 \right) \right), \end{array} $$

(3)

for n t ₀ ≤ t < (n + 1) t ₀, ∀ n = 0, 1, 2, ... where we use the convention that the summation term $\sum_{i=1}^0 \left [ \cdot \right ] = 0$. We will now seek to solve for a time-to-equilibrium, τ _eq , defined as the time in which equations (1) and (3) are sufficiently near² each other for all time. The periodic nature of equation (3) makes this task difficult without further simplification because there in no way to know a priori which sample wave transit n a given sample/bar combination will cause our definition of “near” to be valid for all future time. However, we can use the convergence of a finite geometric series to simplify summation term in equation (1). Then, assuming that for all materials sample equilibrium will take several wave transits, n, to occur, we can allow the ((r − 1)/(r + 1))ⁿ term to → 0 because this term decays rapidly for any r > 0 and increasing n. In a Kolsky bar test r > 1 is always true otherwise the incident and transmission bars will not remain in contact with the sample. The simplification described above is illustrated below resulting in equation (4)

(4)

Now we substitute the expression arrived at in equation (4) back into equation (3) for the summation term and simplify to get the asymptotic stress, σ _asym, shown in equation (5)

$$ \sigma_{\rm asym}(t) = M \gamma \left( t - \frac{r t_0}{2} \right). $$

(5)

If we now assume that there is a small threshold value, ξ, that defines when equation (1) is sufficiently near the asymptotic stress defined by equation (5) we can setup the following equation to solve for the τ _eq

$$ \sigma_{\rm asym}(t) = \left( 1 - \xi \right) \sigma_s(t), $$

(6)

which has the solution

$$ \tau_{eq} \equiv t = \frac{r t_0}{2} \left[ 1 - W\left(\frac{1-\xi}{e \xi} \right) \right], $$

(7)

W represents the Lambert-W function, or the function W(z) satisfying the equation

$$ z = W(z) e^{W(z)}. $$

(8)

Notice that τ _eq only has terms that involve the sample and bar geometries and characteristic wave speed; therefore, it is independent of the magnitude of the incident stress rate. It is also not dependent on an a priori estimate of which wave transit n the sample will equilibrate. This n will be different for a given sample/bar configuration, although as pointed out by other authors it typically occurs within 4 to 5 wave transits [5, 7, 8]. The analysis provided to develop this time-to-equilibrium is quite different from what has been previously published [5, 7] and when combined with a similar criterion for time-to-constant-strain-rate leads to a few interesting observations to be developed in the subsequent sections.

Time-to-Constant-Strain-Rate

We can proceed with a similar analysis to the one presented in the last section to determine a time at which the sample reaches near constant-strain-rate, τ _csr . We’ll start by presenting the equilibrium solution originally presented in [7] for strain-rate shown in equation (9)

$$ \dot{\varepsilon}_{s}(t) = \frac{M \gamma}{E_0} \left[ 1 - \exp \left( -\frac{2 t}{r t_0} \right) \right]. $$

(9)

Since the asymptotic stress solution has already been developed we can arrive at the asymptotic strain-rate by simply dividing σ _asym by the sample elastic modulus, E ₀ and taking the time derivative of the result. This is shown in equation (10)

$$ \dot{\varepsilon}_{\rm asym}= \frac{d}{dt} \left( \frac{\sigma_{\rm asym}}{E_0} \right) = \frac{M \gamma}{E_0}. $$

(10)

It is interesting to observe that the asymptotic strain-rate does not involve any terms that involve the sample length. This shows when designing experiments we can increase the maximum achievable strain-rate, not by making the sample shorter, but by making the area ratio, γ, larger or by increasing the magnitude of incident stress-rate. However, as we shall soon observe that the time to reach this asymptotic strain-rate is a function of the sample length; therefore, by making the sample shorter we can reach the desired strain-rate sooner.

We now set up a similar equality to equation (6) this time to determine when $\dot{\varepsilon}_{s}$ is sufficiently near $\dot{\varepsilon}_{\rm asym}$ for all time. The solution to this equation will be the time in which the sample will be assumed to arrived at constant-strain-rate. This is shown in equation (11)

$$ \left( 1 - \xi \right) \dot{\varepsilon}_{\rm asym} = \dot{\varepsilon}_{s}(t). $$

(11)

Equation (11) has the term (1 − ξ) multiplying the asymptotic solution instead of the equilibrium solution because of the way the equilibrium solution approaches the asymptotic solution and a desire to use the same definition for the small threshold parameter ξ. For the case of constant-strain-rate the equilibrium solution approaches the asymptotic solution from a lower position on the ordinate axis, whereas for the case of sample equilibrium, the equilibrium solution approaches the asymptotic solution from a higher position on the ordinate axis. We can solve equation (11) for τ _csr

$$ \tau_{csr} \equiv t = \frac{r t_0}{2} \ln \left( \xi \right) $$

(12)

The approach presented in this section is similar to the one used by Pan et al. [6]; however, they did not solve for τ _csr in closed-form, instead choosing to take it from experimental observations while developing their criterion for an upper-limit of valid strain-rates for the testing of brittle materials.

Criterion for Valid Test Conditions

By using a uniform approach to the derivation and solution of τ _eq and τ _csr we can make an interesting observation about which event will occur first, sample equilibrium or the sample achieving a state of constant-strain-rate. Let us start by determining the condition on ξ in which the following inequality holds.

$$ \begin{array}{rllc} \tau_{eq} &<& \tau_{csr} \\ W \left( \frac{1 - \xi}{e \xi} \right) &<& -1 - \ln \left( \xi \right) \end{array} $$

(13)

If we solve equation (13) for ξ we have the following inequality which we can evaluate numerically.

$$ \xi < -W \left( - \frac{1}{e^2} \right ) \approx 15.86\% $$

(14)

The result of equation (14) implies that as long as ξ is chosen to be less than ≈ 15.86%, then τ _eq will always occur before τ _csr . If we assume that any reasonable definition of “sufficiently near” will have ξ in the range of 0–10% as other authors have indicated [5, 6], we can come to the conclusion that the sample will always come into equilibrium before a constant-strain-rate is reached (when subject to a ramp incident stress pulse). This conclusion is independent of the sample geometry and/or material properties for elastic materials. This conclusion suggests that any criterion checking for valid test conditions or limiting criterion should be derived with consideration of the time it takes for the sample to reach a constant-strain-rate, if this criterion is satisfied, the sample will already be in equilibrium. This conclusion has been observed in the laboratory, but has never been presented in a closed-form expression³ because it requires analytic values of τ _eq and τ _csr .

If we now assume there is some compressive stress at which an elastic brittle material will fail, σ _f , we can substitute it into equation (1) allowing σ _s  = σ _f , we can combine the result with equation (9) by eliminating M. If we evaluate this single equation at t = τ _eq and solve for σ _f we have equation (15a) and if we evaluate at t = τ _csr we have equation (15b)

$$ \sigma_f |_{t=\tau_{eq}} = \frac{r t_0 E_0}{2} \left[ \xi \exp \left(1\!+\!W \left( \frac{1-\xi}{e \xi} \right) \right) \!-\! \xi \right]^{-1} \dot{\varepsilon}_s(\tau_{eq}), $$

(15a)

$$ \sigma_f |_{t=\tau_{csr}} = \frac{r t_0 E_0}{2} \left[ \frac{1 + \ln \left( \xi \right) - \xi }{\xi - 1} \right] \dot{\varepsilon}_s(\tau_{csr}). $$

(15b)

Choosing ξ = 0.05 as before we can evaluate equations. (15a) and (15b) to give the approximations shown in equation (16)

$$ \sigma_f |_{t=\tau_{eq}} \approx 0.8717\, r t_0 E_0 \dot{\varepsilon}_s(\tau_{eq}), $$

(16a)

$$ \sigma_f |_{t=\tau_{csr}} \approx 1.0767 \, r t_0 E_0 \dot{\varepsilon}_s(\tau_{csr}). $$

(16b)

Equations (15) and (16) provide a simple condition to check the validity of a Kolsky bar test on elastic-brittle materials. The sample failure stress must be larger than the term on the right-hand side of equation (16a) for the sample to have reached equilibrium at the time of failure. If constant-strain-rate is desired for the reporting of the data the sample failure stress must be larger than the term on the right-hand side of equation (16b). If we choose ξ = 0.05 and plug in the data from the Indiana limestone experiments presented by Frew et al. [7], namely, M =3.3 MPa, γ = 1, r = 3.56, t ₀ = 4.3105 μs, and E ₀ = 25 GPa, with the failure stress being plotted as a function of strain-rate, we can a priori define envelopes of validity for our experiments. These envelopes of validity are compared to the final results of Frew et al. in Fig. 2. Figure 2 illustrates that the data presented by Frew et al., at the highest strain-rates, was very near the highest strain-rate achievable for Indiana limestone with these sample dimensions. Even an order of magnitude increase in failure strength would only extend the range of valid strain-rates achievable by a few hundred per second.

Fig. 2

Comparison limiting curves for constant-strain-rate and sample equilibrium to the data presented in Frew et al. [7] on Indiana limestone.

Summary and Final Observations

Analytic criterion for valid test conditions based on the time for a sample to achieve equilibrium and the time for a sample to achieve a constant-strain-rate where developed for Kolsky bar tests of elastic-brittle materials. These criterion were compared against experiments reported on Indian limestone. The analysis showed that for all elastic materials the sample will achieve a equilibrium state before reaching constant-strain-rate. Using the criterion for constant-strain-rate as a limiting factor for a valid test, the experimentalist is offered a simple check for validity based on equation (16b).

If we assume a failure strain, ε _f , and substitute the relationship σ _f  = E ₀ ε _f into equation (16b) while solving for $\dot{\varepsilon}_s(\tau_{csr}) \equiv \dot{\varepsilon}_{csr}$ we can also use this equation to calculate a limiting strain-rate. If we assume ε _f  = 0.01 we can compare this limiting strain-rate $\dot{\varepsilon}_{csr}$ to the criterion proposed by Ravishandran and Subash [5], $\dot{\varepsilon}_l$, shown in Table 1. For almost all cases the two criterion are agreeable. The criterion proposed herein, $\dot{\varepsilon}_{csr}$, is slightly smaller because this definition is based on the sample reaching a constant-strain-rate whereas Ravichandran and Subash [5] where considering sample equilibrium only, which was shown to occur before the sample reaches a constant-strain-rate.

Table 1 Comparison of limiting strain-rate criterion $\dot{\varepsilon}_{csr}$ with $\dot{\varepsilon}_{l}$ reported from Ravishandran and Subash [5] assuming a sample length of l ₀ = 9.52 mm and a failure strain of 1%