## Abstract

Steels are usually stronger at low temperatures than at high temperatures. But low temperatures are, particularly in combination with high strain rates and high stress triaxiality ratios, known to cause embrittlement. The common understanding is that the ductility of steels decreases dramatically below a threshold temperature known as the ductile-to-brittle transition temperature. This study explores the ballistic performance of Strenx 960 Plus steel plates at both low temperatures and room temperature. We describe a ballistic setup where target plates were cooled down to as low as −60 °C before we present results from ballistic impact tests with three different projectile types. The ballistic limit velocities from tests at low temperatures were higher than the ballistic limit velocities from tests at room temperature, indicating that brittle fracture does not take place. An analytical approach based on the Johnson–Cook constitutive relation, the Cockcroft–Latham ductile failure criterion, and a simple brittle fracture criterion is presented. The model suggests that ductile fracture prevails for most realistic material state histories, both in the ballistic impact tests as well as for quasi-static and dynamic tensile tests. This supports previous observations that brittle fracture is unlikely to occur in modern steels even when subjected to rapid loading and low temperatures.

## 1 Introduction

Steel gets stronger when it gets colder, and weaker when it gets hotter. However, low temperatures can be detrimental to the ductility of steel because of the ductile-to-brittle transition phenomenon, which means that some steels quickly lose their ductility below a certain temperature [1]. Recent studies on modern martensitic steels show that the transition from ductile fracture to brittle fracture occurs at very low temperatures—even lower than what is relevant for human activity in the Arctic region [2,3]. This is important from a design point of view since the transition from ductile to brittle behavior can severely affect predictions of fracture.

Cleavage is a typical brittle fracture mode. According to Anderson [4], cleavage occurs when the plastic flow is restricted. At least three factors restrict plastic flow [1]: low temperature, high strain rate, and high stress triaxiality ratio. The stress triaxiality ratio is defined as
$σ*=σI+σII+σIII3σeq$
(1)
where σI > σII > σIII are the ordered principal stresses and σeq is the equivalent (von Mises) stress. Along with the lode parameter [57] defined as
$L=2σII−σI−σIIIσI−σIII$
(2)
the stress triaxiality ratio has been shown to be extremely important for the strain to fracture.

Perez-Martin et al. [3] investigated the combined effect of temperature, strain rate, and stress triaxiality ratio by stretching smooth and pre-notched tensile specimens of Strenx 960 Plus at quasi-static and dynamic loading rates. Charpy V-notch impact tests were also conducted. These tests were carried out at temperatures between +20 °C and −90 °C. The most important results from that study are revisited later in this paper, but the main conclusion was that ductile fracture was by far the dominating fracture mode regardless of test configuration and temperature.

Similar results were also found by Tu et al. [2] who looked at the quasi-static response of a structural steel down to −60 °C. The material was stronger at low temperatures than at room temperature, and the ductility did not deteriorate even at a test temperature of −60 °C. Xie et al. [8] observed the same for a high-strength steel strand used to prestress concrete. The ultimate tensile strength and failure strain increased as the temperature decreased until about −100 °C when brittle fracture started occurring for this steel alloy.

Jia et al. [9] conducted a series of perforation tests of 1.5 mm 304 stainless-steel plates at temperatures between +200 °C and −163 °C. The projectiles had a conical nose, a diameter of 12.8 mm, and a mass of 29 g. In terms of ballistic limit velocity, the capacity was 96 m/s at room temperature and 103 m/s at −20 °C and below. This constitutes a 7.3% increase of the ballistic capacity which means that more energy is absorbed at low temperatures. The authors attributed this increase to both the temperature sensitivity of the 304 stainless-steel and to the martensitic transformation effect that increases at lower temperatures. Investigations showed that the martensite content in the steel that was initially austenitic was highest at low test temperatures. Rodríguez-Martínez et al. [10] subjected TRIP 1000 steel plates to low velocity impacts and found that the target absorbed more energy at low temperature than at room temperature. But in that study, the increased energy absorption was not attributed to the transformation to martensite.

Other studies showing ballistic tests for sub-ordnance and ordnance velocities at low temperatures are as far as we know not available in the open literature. However, marine structures are frequently designed to operate under Arctic conditions. Some material and component studies are listed below.

Paik et al. [11] conducted a study of ASTM A500-type carbon steel. Tensile tests on dog bone specimens and crush tests of square tubes were done at room temperature (+23 °C) and at temperatures as low as −180 °C. The low-temperature tests were conducted both in a liquid nitrogen chamber and in a dry-ice chamber. Interestingly, they found that the fracture strain of the dog bone specimens remained constant all the way down to −80 °C for specimens tested in a liquid nitrogen cooled chamber, but when dry ice was used as cooling agent the fracture strain dropped even from 0 °C. The peak and mean forces increased by 8.4% from +23 °C down to −60 °C. They observed brittle fractures at the corners of the square tube crush tests at temperatures between −40 °C and −60 °C. The crush tests were only conducted with liquid nitrogen as the cooling agent. Paik et al. [12,13] further reported full-scale collapse tests of plated AH32 steel structures and they concluded that the steel got stronger with decreasing temperature, but that the ductile-to-brittle-transition temperature exists somewhere between −80 °C and −100 °C.

Ehlers and Østby [14] looked at the collision resistance of ships at subzero temperatures (0 °C, −30 °C, −60 °C, and −90 °C) with particular interest in the temperature sensitivity of shipbuilding steel grade Det Norske Veritas Grade A (NVA) with a yield stress of approximately 350 MPa. They concluded that subzero temperatures would increase the energy absorption in the collision and thereby increase the structural safety.

Ductile fracture is often described as a function of the stress state in terms of the stress triaxiality ratio and the lode parameter and is driven by plastic straining. Brittle fracture of steel can on the other hand be described in terms of a critical stress criterion. Ritchie et al. [15] described a criterion in which cleavage due to slip will propagate in an unstable manner if the major principal stress (σI) exceeds a critical stress (σC) over a critical distance, that is
$σI≥σC⇒brittlefracture$
(3)

The critical stress was assumed to be constant regardless of temperature and strain rate, while the critical distance was twice the grain size. The value of σC will typically be between three and five times the yield stress at room temperature. The criterion by Ritchie et al. [15] (denoted the RKR (Ritchie, Knott, and Rice) criterion) is often utilized to predict the transition from ductile to brittle behavior of steels at low temperatures. Needleman and Tvergaard [16] combined a porous plasticity model with the RKR criterion and successfully described brittle and ductile fracture as well as the transition. A similar approach was applied by Nam et al. [17,18] where embrittlement of steel in Charpy V-notch tests and in large-scale drop tests was simulated.

Most of the studies above show that the strength and the ductility of steel either increase or remain constant when the temperature decreases. In the following, we present a test setup with a cooling chamber for ballistic testing at low temperatures. Then, test series on a high-strength steel with three different projectile types at different temperatures are conducted with the described setup. An analytical approach based on well-known and relatively simple constitutive and fracture models is finally used to shed some light on the results from the experimental tests.

## 2 Material Behavior

### 2.1 Strenx 960 Plus.

Strenx 960 Plus is a hot-rolled, quenched, and tempered strip steel with a martensitic crystalline structure that is produced by SSAB [19]. Figure 1 shows the grain structure on three orthogonal planes. The grains are irregularly arranged with a size of 10–20 µm. The minimum nominal yield stress of Strenx 960 Plus is 960 MPa which makes the steel suitable for challenging load-bearing structural applications. It meets the requirements for the S960QL steel grade. Table 1 summarizes the chemical composition and nominal mechanical properties.

Fig. 1
Fig. 1
Close modal
Table 1

Material data for Strenx 960 Plus

Chemical composition (wt%)
CSiMnPSCrNiMoVTiCuAlNbBN
Nominal0.180.501.700.0200.0100.018
Certificate0.1590.281.280.0090.0010.150.050.400.040.010.010.0410.0020.00150.003
Chemical composition (wt%)
CSiMnPSCrNiMoVTiCuAlNbBN
Nominal0.180.501.700.0200.0100.018
Certificate0.1590.281.280.0090.0010.150.050.400.040.010.010.0410.0020.00150.003
Nominal mechanical properties
Yield strengthTensile strengthElongationImpact properties (Charpy V-notch)
ReH (min MPa)Rm (MPa)A (min %)T (°C)Absorbed energy (J)
960980–11507−4027
Nominal mechanical properties
Yield strengthTensile strengthElongationImpact properties (Charpy V-notch)
ReH (min MPa)Rm (MPa)A (min %)T (°C)Absorbed energy (J)
960980–11507−4027

### 2.2 Behavior of Strenx 960 Plus.

Perez-Martin et al. [3] characterized the mechanical behavior of Strenx 960 Plus in detail both at room temperature and low temperatures. All the specimens were taken from a plate with a nominal thickness of 8 mm. The most important results are summarized as follows.

Quasi-static tension tests were conducted at +20 °C and −40 °C. Smooth cylindrical specimens (Fig. 2(a)) as well as pre-notched specimens with notch radii of 2.0 mm (Fig. 2(b)) and 0.8 mm (Fig. 2(c)) were tested to investigate the influence of the stress triaxiality ratio. Edge tracing with the digital image correlation (DIC) code eCorr [20] enabled measurement of true stress (σI) and logarithmic strain (ɛl) all the way to specimen fracture through
$σI=FA,εl=ln(A0A),A=πD⊥Dt4$
(4)
where F is the measured force, A is the cross-sectional area, A0 is the initial cross-sectional area, while D and Dt are the measured diameters in the transverse direction of the specimen and the thickness direction of the plate, respectively. Equation (4) assumes plastic incompressibility and negligible elastic strains and the calculated values must be considered average values over the minimum cross section for the notched specimens and after diffuse necking for the smooth specimens. Figure 3(a) shows the representative true stress–strain curves from these tests. The true stress is higher at −40 °C than at +20 °C, but the failure strain is almost the same.
Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

Dynamic tension tests were conducted in a split Hopkinson tension bar (SHTB) at +20 °C and −40 °C using the same specimen geometries as for the quasi-static tests. The initial strain rate was between 100 1/s and 1000 1/s for the smooth specimens. The strain rate varies in the smooth specimens after necking and in the pre-notched specimens from the start of the test. Figure 3(b) shows representative true stress–strain curves from the high-rate tests. Just like in the quasi-static tests, the stress is higher at −40 °C than at +20 °C, and the failure strain is almost the same. Strenx 960 Plus exhibits positive strain rate sensitivity for all the investigated temperatures, which implies that the stress level increases with strain rate when all the other state variables are unchanged.

A common test method to determine the ductile-to-brittle-transition temperature is the Charpy V-notch impact test. Results from such tests are only meaningful if a large test series is conducted over a wide temperature range. The transition temperature can be found by either considering the energy absorbed in the tests or by examining the fracture surface in a microscope and evaluating the relationship between ductile and brittle fracture modes. Sub-size Charpy V-notch tests (see Fig. 2(e)) were conducted from +20 °C to −90 °C. Figure 3(c) shows the absorbed energy as a function of temperature. The absorbed energy decreases gradually with decreasing temperature. It was not possible to identify a distinct ductile-to-brittle transition temperature from these tests.

The material test results indicate that Strenx 960 Plus behaves in a ductile manner and that brittle fracture does not occur even when the material is subjected to low temperatures, high strain rates, and high stress triaxiality ratios—three factors that should restrict plastic flow and promote brittle fracture. The quantitative results in Perez-Martin et al. [3] were corroborated by a fractographic study which showed that the fracture surfaces were predominantly covered by dimples, a clear sign of ductile fracture.

### 2.3 Testing of Sharp-Notched Tension Specimens.

As an expansion of the test program from Ref. [3], we performed a new set of tension tests where the stress triaxiality ratio was believed to be even higher than in the tests summarized in Sec. 2.2. Figure 2(d) shows the sharp V-notched tension specimens used in these tests. The notch radius at the notch root in the specimens were measured in a microscope and found to be roughly 0.2 mm which results in additional hampering of the plastic flow compared to the blunter notches.

The V-notched specimens were tested under quasi-static and dynamic loading conditions at +20 °C and −40 °C, while additional tests at –60 °C were conducted under dynamic loading in the SHTB. For brevity, no stress–strain curves from these tests are shown here. The tests were mainly conducted to see if the material would behave brittle at more extreme conditions than already investigated.

The fracture surfaces were investigated using a scanning electron microscope and are shown in Figs. 4(a) and 4(b) for one of the tests subjected to dynamic loading at −40 °C. Ductile fracture was identified in the images, where the shallow dimple structure is similar to what was observed for the pre-notched specimens in Ref. [3]. Despite the combination of high stress triaxiality, high strain rate, and low temperature, cleavage was not observed on the fracture surfaces in any of the V-notched tests.

Fig. 4
Fig. 4
Close modal

In a last attempt to further evoke brittle fracture/cleavage, a fatigue crack was created in the V-notched specimens. Using spark erosion, a small groove of 0.3 mm thickness was cut into the center of the notch root. Then, a fatigue crack was created in the specimen using an Instron 100 kN Resonance Fatigue testing machine operating at a frequency of ∼100 Hz, before it was tested under dynamic loading conditions at –40 °C. The fatigue crack significantly increases the stress triaxiality at the crack tip. The given conditions are considered to be close to a worst-case scenario that should be capable of generating brittle fracture behavior. Figures 4(c) and 4(d) present the fracture surface of one specimen with a fatigue crack, and it is clearly seen that the failure mode is completely different from that in Figs. 4(a) and 4(b). Here, cleavage dominates the fracture surface, which is evident from the observed river patterns. Apparently, brittle fracture can also occur for Strenx 960 Plus under extreme and enforced conditions.

## 3 Ballistic Impact Test Setup

Figure 5 shows the three different projectile types that are used in this paper: 7.62 mm armor piercing (AP) bullets, 20 mm-diameter ogive-nosed projectiles, and 20 mm-diameter blunt-nosed projectiles. The relatively high impact velocity, low mass, and small diameter of 7.62 mm AP bullets compared to the target thickness ensure that the target response is predominantly local. Global deformation phenomena like bulging and dishing might, in general, occur when the ratio between the diameter of the bullet (Dp) and the thickness of the target plate (tt) is relatively low, which in this study is the case for the 20 mm-diameter projectiles. The response is also dependent on the strength and ductility of the target material. The nominal thickness of the Strenx 960 Plus target plates used in the ballistic tests is 8 mm (the same as that used in the material tests) and the in-plane dimensions are 300 mm × 300 mm.

Fig. 5
Fig. 5
Close modal

### 3.1 7.62 mm Armor Piercing Bullets.

The total mass of 7.62 mm AP bullet is approximately 10.5 g. Its hardened steel core has a mass of 5.0 g, a diameter of 6.1 mm, and an ogive shaped nose with a caliber radius head (CRH) of 3 [21]. In the ballistic tests, the distance from the muzzle to the target plate was about 1.0 m and a magnetic trigger made it possible to conduct the tests from a safe distance. The ratio DP/tt (Dp being the diameter of the core) is 6.1 mm/8 mm = 0.7625.

The bullets were fired from a smooth-bored Mauser gun where we can control the impact velocity by varying the amount of powder in the cartridge. Velocities between 300 m/s and 950 m/s were obtained with this setup. The high velocity and low mass of the projectiles suggest that the global deformation in the plates will be limited, and thus as many as five tests were conducted on each plate.

### 3.2 20 mm-Diameter Projectiles.

The ogive and blunt-nosed projectiles made of Arne tool steel have a diameter of 20 mm and a mass of 197 g. The blunt-nosed projectile is 80 mm long, while the total length of the ogive-nosed projectile is 95 mm out of which 33 mm has a CRH of 3 (the same as the 7.62 mm AP bullet). For the 20 mm-diameter projectiles, the ratio between the diameter of the striker (Dp) and the thickness of the target plate (tt) is 20 mm/8 mm = 2.50.

The projectiles were mounted in a sabot and fired using a gas gun where the inner diameter of the barrel is 50 mm. Air was used as propellant. The air was compressed in a pressure chamber before it was rapidly released by the rupture of membranes. These were designed to withstand a certain pressure, thus making it possible to accurately predetermine the striking velocity of the projectile [22]. Velocities between approximately 100 m/s and 350 m/s were obtained with this setup. Because of the relatively low velocity and high mass of the 20 mm-diameter projectiles, the target plates exhibited noticeable global deformation and only a single test was conducted on each plate.

### 3.3 Measurements and Data Analysis.

Phantom v1610 and v2511 high-speed cameras with a recording rate of up to 120,000 fps captured the entire perforation process from well before impact until the test was finished by either complete perforation or embedment. The resulting images were used to investigate the perforation process and to measure the impact and residual velocities.

The objective of the ballistic testing was to identify the ballistic limit velocity (vbl) of the target plates at different temperatures when they were struck by the various projectiles. The ballistic limit velocity is here defined as the minimum velocity required for complete perforation of the target.

After obtaining the impact (vi) and residual (vr) velocities of the projectile from the high-speed camera measurements, the ballistic limit velocity was estimated by a generalized version of the Recht–Ipson equation [23] (also known as the Lambert–Jonas equation [24]), given by
$vr=a(vip−vblp)1/p⇒vbl=(vip−(vra)p)1/p$
(5)
where a, p, and vbl were found from best fits to the experimental data using the method of least squares.

### 3.4 Test Fixture and Cooling Chamber.

Figure 6 shows how the 300 mm × 300 mm × 8 mm thick target plates were placed in a rigid test fixture. Two steel beams that were attached to the test fixture by four M12 bolts were used to clamp the top and bottom edge of the plate; hence, the vertical sides of the plate were free with a span of 180 mm. This setup has been successfully used in previous experimental campaigns for ballistic impacts with both 7.62 mm AP bullets and 20 mm-diameter projectiles [21,25,26].

Fig. 6
Fig. 6
Close modal

The tests at low temperature were conducted in a transparent thermal chamber made of polymethyl methacrylate (also known as PMMA, acrylic glass, or plexiglass). To cool down the target plates, we flushed liquid nitrogen into the chamber. Two holes were made in the thermal chamber to allow the projectiles to strike the target plate without destroying the chamber walls. The entry hole for the projectile was small (D ≈ 30 mm ) and covered with aluminum foil to prevent leakage of liquid nitrogen. To make sure that fragments from the impact would not destroy the polycarbonate, the exit hole was larger and covered with a wooden block. There were three thermocouples in the chamber. Two of them were spotwelded to the target plate to measure the surface temperature and the third measured the ambient temperate in the chamber. The amount of liquid nitrogen that was flushed into the chamber was controlled by the thermocouple measurements allowing the temperature to remain constant for a long time. To ensure that the entire target plate reached the desired test temperature before we conducted the impact tests, the plates had to be kept inside the thermal chamber for some time before testing. In every case, the surface temperature of the plates was measured to be at the desired level for at least 16 min (2 min/mm plate thickness), meaning that the plates usually were inside the chamber for more than 20 min prior to each impact test.

## 4 Ballistic Impact Test Results

### 4.1 7.62 mm Armor Piercing Bullets.

Figure 7 shows the residual velocity plotted as a function of the impact velocity for the tests with 7.62 mm AP bullets. At least four tests were conducted at each of the four temperatures: +20 °C, −20 °C, −40 °C, and −60 °C. The ballistic limit velocity (vbl) as defined in Eq. (5) is found to be 590 m/s at +20 °C and it increases to 604 m/s at −60 °C. This constitutes a slight increase of vbl of approximately 2.4%. The results are summarized in Table 2.

Fig. 7
Fig. 7
Close modal
Table 2

Overview of the ballistic results

TemperatureStrenx 960 PlusNVE36Armox 500T
7.62 mm APOgive noseBlunt nose7.62 mm AP7.62 mm AP
+20 °Cvbl = 590 m/s
a = 1.00
p = 2.10
vbl = 223 m/s
a = 1.00
p = 1.90
vbl = 138 m/s
a = 0.79
p = 3.30
vbl = 402 m/s
a = 1.00
p = 2.22
vbl = 525 m/s
a = 1.00
p = 2.09
−20 °Cvbl = 596 m/s
a = 1.00
p = 2.08
−40 °Cvbl = 588 m/s
a = 1.00
p = 2.03
vbl = 235 m/a
a = 1.00
p = 1.88
vbl = 142 m/s
a = 0.66
p = 5.20
vbl = 419 m/s
a = 1.00
p = 2.17
vbl = 549 m/s
a = 1.00
p = 2.26
−60 °Cvbl = 604 m/s
a = 1.00
p = 2.05
TemperatureStrenx 960 PlusNVE36Armox 500T
7.62 mm APOgive noseBlunt nose7.62 mm AP7.62 mm AP
+20 °Cvbl = 590 m/s
a = 1.00
p = 2.10
vbl = 223 m/s
a = 1.00
p = 1.90
vbl = 138 m/s
a = 0.79
p = 3.30
vbl = 402 m/s
a = 1.00
p = 2.22
vbl = 525 m/s
a = 1.00
p = 2.09
−20 °Cvbl = 596 m/s
a = 1.00
p = 2.08
−40 °Cvbl = 588 m/s
a = 1.00
p = 2.03
vbl = 235 m/a
a = 1.00
p = 1.88
vbl = 142 m/s
a = 0.66
p = 5.20
vbl = 419 m/s
a = 1.00
p = 2.17
vbl = 549 m/s
a = 1.00
p = 2.26
−60 °Cvbl = 604 m/s
a = 1.00
p = 2.05

The images presented in Fig. 8 reveal that the perforation process is similar regardless of temperature, and there is no evidence to claim that the behavior of the target plate is less ductile at low temperatures compared to room temperature. Note that some of the projectiles broke into two large pieces during impact, but this did not appear to be temperature dependent.

Fig. 8
Fig. 8
Close modal

The total yaw angle is defined as the angle between the longitudinal axis of the projectile and its velocity vector, and the oblique angle is defined as the angle between the target normal and the velocity vector of the projectile. For the ballistic setup used to launch the 7.62 mm AP bullets, the total yaw angle has in previous studies found to be below 3.5 deg while the oblique angle in general is below 0.5 deg.

Because we used one camera in this study (and no mirror), we could only monitor the vertical component of the total yaw (also called the pitch) and the obliquity. The highest pitch angle in the tests was 5.0 deg which occurred in a test where the impact velocity was far below the ballistic limit velocity, otherwise the trajectories were within the typical ranges given above.

According to Goldsmith [27], total yaw angles that are lower than 5.0 deg do not affect the ballistic limit velocity of the target configuration. Furthermore, oblique angles of almost 30 deg were needed to reduce the ballistic limit velocity in a study by Børvik et al. [28]. Hence, the effects of the trajectory imperfections in this study are assumed to be negligible.

### 4.2 20 mm Ogive Projectiles.

Tests with 20-mm-diameter ogive-nosed projectiles were conducted because a high projectile diameter to target thickness ratio has previously been shown to promote fragmentation of the target plate [26]. Thus, the possibility of observing brittle fracture should be higher for this configuration. Tests were done at +20 °C and −40 °C. Figure 9 and Table 2 show that the ballistic limit velocity increases from 223 m/s to 235 m/s when the temperature is reduced to −40 °C. Thus, vbl increases by 5.4%.

Fig. 9
Fig. 9
Close modal

Figure 10 shows the perforation process at +20 °C and −40 °C. The tip of the nose shattered in two of the tests at low temperature but was always intact at room temperature. At room temperature, the plate opened in a petaling pattern before the petals broke off as fragments. The debris cloud seen behind the projectile in the images is just the soft foam part of the sabot package that went through the sabot trap in this particular test, and it has no influence on the perforation process. At −40 °C, a plug-like fragment was ejected from the plate. The highest pitch angle (component in the vertical plane) that was measured was 4.5 deg, but it was lower than 1.5 deg in most of the tests.

Fig. 10
Fig. 10
Close modal

### 4.3 20 mm Blunt Projectiles.

Figure 11 shows the residual velocities plotted as a function of the impact velocity for the tests with 20-mm-diameter blunt-nosed projectiles. Tests were done at +20 °C and −40 °C. The ballistic limit velocity increases from 138 m/s to 142 m/s when the temperature is reduced to −40 °C, which represent a 2.9% increase of vbl. Table 2 summarizes the results. The slope of the ballistic limit curve for +20 °C is steeper than the slope of the ballistic limit curve for −40 °C (Fig. 11). This means that a lower residual velocity is obtained at the low temperature for a given impact velocity.

Fig. 11
Fig. 11
Close modal

Figure 12 shows that the perforation mechanism is very similar at both temperatures for this nose shape. A clear distinct plug is ejected from the plate, with little global deformation and no additional fragmentation. Thus, adiabatic shear banding and plugging is the main perforation mode at both room and low temperatures. The highest pitch angle (component in the vertical plane) that was measured was 3.2 deg, but it was lower than 1.5 deg in most of the tests.

Fig. 12
Fig. 12
Close modal

The estimated ballistic limit velocities for the blunt-nosed projectile are determined by the Recht–Ipson parameters from Eq. (5), and there is some uncertainty to whether vbl at −40 °C really is higher than at +20 °C. But for impact velocities higher than 200 m/s, the residual velocity differs significantly. We attribute this difference to the material strength since the perforation mechanism and projectile mushrooming are similar at the two temperatures.

## 5 Analytical Approach for Fracture Mode Evaluation

### 5.1 Model Description.

We can qualitatively study how the strain to failure is affected by ambient temperature, plastic strain rate, and stress triaxiality ratio, and further estimate whether the failure is ductile or brittle with the following analytical derivation. The material behavior is modeled with a modified version of the Johnson–Cook (JC) constitutive relation [29] in which the equivalent stress is given by
$σeq=(A+Bpn)(1+(p˙p˙0))C(1−(T−T0Tm−T0)m)$
(6)
where p is the equivalent plastic strain, $p˙$ is the equivalent plastic strain rate, T is the temperature, A is the yield stress at T = T0, B and n are parameters governing work-hardening, C and $p˙0$ are parameters controlling rate sensitivity, the parameter m determines the thermal softening, T0 is a reference temperature, and Tm is the melting temperature of the material.
Self-heating is accounted for by assuming that the temperature increases according to
$dT=ω(p˙)βTQσeqcερdp$
(7)
with initial condition T = Ti, where Ti is the initial temperature. Furthermore, cɛ is the specific heat capacity, ρ is the density, βTQ = 0.9 is the Taylor–Quinney coefficient, and $ω(p˙)$ is a strain rate dependent weighing factor introduced by Roth and Mohr [30] to account for the transition from isothermal to adiabatic conditions:
$ω(p˙)={0forp˙p˙ad$
(8)

The factor ω assumes a value of zero when the strain rate is lower than the isothermal limit $p˙iso$ and unity when the strain rate is above the adiabatic limit $p˙ad$. For intermediate strain rates, the factor ω varies smoothly, as shown in Fig. 13. According to Johnson et al. [31], the transition from isothermal to adiabatic conditions occurs roughly between 10 1/s and 100 1/s for steels.

Fig. 13
Fig. 13
Close modal
Using the trapezoidal rule for numerical integration, assuming constant plastic strain rate, we get
$Tk+1=Tk+12ω(p˙)βTQcερ(σeq,k+1+σeq,k)(pk+1−pk)$
(9)
Subscripts k and k + 1 denote quantities at time-steps tk and tk+1, respectively. Assuming small time-steps (or, equivalently, small plastic strain increments), we may approximate the equivalent stress at tk+1 by
$σeq,k+1=(A+Bpk+1n)(1+(p˙p˙0))C(1−(Tk−T0Tm−T0)m)$
(10)

Note that the temperature lags one step behind the equivalent stress and the equivalent plastic strain, which is assumed sufficiently accurate for small plastic strain increments Δpk+1 = pk+1pk.

Ductile fracture is modeled by the Cockcroft–Latham (CL) criterion
$D=1WC∫0pmax(σI,0)dp≥1⇒ductilefracture$
(11)
where WC is the fracture parameter and ductile fracture occurs when Eq. (11) is fulfilled. The major principal stress σI is given by
$σI=(σ*+3−L33+L2)σeq$
(12)
where $σ*$ is the stress triaxiality as given by Eq. (1) and L is the lode parameter as given by Eq. (2). In particular, L is equal to −1 for generalized axisymmetric tension, 0 for generalized shear, and +1 for generalized axisymmetric compression. Thus,
$D=1WC∫0pmax(σ*+3−L33+L2,0)σeqdp$
(13)
Applying once more the trapezoidal rule for constant values of $σ*$ and L, we get
$Dn+1=Dn+12WCmax(σ*+3−L33+L2,0)(σeq,n+1+σeq,n)(pn+1−pn)$
(14)
and ductile failure takes place when Dn+1 becomes equal to or greater than unity.
Brittle failure is assumed to take place when σI reaches a critical value σC which is invariant to changes in temperature and strain rate as described by the RKR model and shown in Eq. (3). This can alternatively be written in terms of the stress invariants $(σ*,L,σeq)$ as
$(σ*+3−L33+L2)σeq≥σC⇒brittlefracture$
(15)

For a given material, the parameters A, B, n, C, $p˙0$, m, Tm, ρ, cɛ, WC, and σC have prescribed values. The reference temperature is taken as T0 = 0 K, since we are considering low-temperature behavior. The loading is defined by given values of plastic strain rate $p˙$, initial temperature Ti, stress triaxiality $σ*$, and lode parameter L.

### 5.2 Calibration of the Constitutive Model.

We used the experimental results from Perez-Martin et al. [3] and some standard parameters for steel from the literature to calibrate the modified Johnson–Cook constitutive model and Cockcroft–Latham failure criterion. Table 3 lists the relevant model parameters for Strenx 960 Plus. The procedure to obtain them is listed below.

Table 3

Model parameters for Strenx 960 Plus used in the analytical approach

Strenx 960 plus
Density, ρ (kg/m3)7850
Young’s modulus, E (MPa)210,000
Poisson ratio, ν0.33
Specific heat capacity, cɛ (J/(kg K))452
Taylor–Quinney coefficient, βTQ0.9
Initial yield stress, A (MPa)1024
Hardening coefficient, B (MPa)425
Hardening exponent, n0.43
Strain rate sensitivity, C0.0114
Reference strain rate, $p˙0$ (1/s)0.0865
Thermal softening exponent, m1.0
Reference temperature, T0 (K)293
Melting temperature, Tm (K)1800
CL failure parameter, WC (MPa)1525
Strenx 960 plus
Density, ρ (kg/m3)7850
Young’s modulus, E (MPa)210,000
Poisson ratio, ν0.33
Specific heat capacity, cɛ (J/(kg K))452
Taylor–Quinney coefficient, βTQ0.9
Initial yield stress, A (MPa)1024
Hardening coefficient, B (MPa)425
Hardening exponent, n0.43
Strain rate sensitivity, C0.0114
Reference strain rate, $p˙0$ (1/s)0.0865
Thermal softening exponent, m1.0
Reference temperature, T0 (K)293
Melting temperature, Tm (K)1800
CL failure parameter, WC (MPa)1525
First, we determine the strength and hardening parameters A, B, and n in Eq. (6) and the ductile failure parameter WC in Eq. (13) using the quasi-static uniaxial tension tests at room temperature. Figure 14(a) illustrates the process. WC was found by integrating the measured true stress over the plastic strain until failure as follows:
$WC=∫0pfσIdp$
(16)
Fig. 14
Fig. 14
Close modal

The strength and hardening parameters were found by curve fitting the first term of Eq. (6) to a true stress–strain curve corrected using an approach proposed by Bridgman [32] and Le Roy et al. [33]. The Bridgman correction is a pragmatic way of removing the notch effect on the flow stress after necking in tension tests of cylindrical specimens and provides the equivalent uniaxial stress unaffected by the increased stress triaxiality ratio.

Second, the rate sensitivity parameters C and $p˙0$ were found from the dynamic uniaxial tension tests at room temperature. Figure 14(b) shows the measured stress at 2% plastic strain $(σ2%)$ and the curve defined by the following equation derived from Eq. (6):
$σ(p˙)=σ2%(1+(p˙p˙0))C$
(17)
where $σ2%$ is the true quasi-static stress at 2% plastic strain.
Third, we evaluate the temperature exponent m. Figure 14(c) shows $σ2%$ from the quasi-static uniaxial tension tests at +20 °C and −40 °C compared to the following equation:
$σ(T)=σ2%(1−(T−T0Tm−T0)m)$
(18)

The relation is plotted for m = 1.0 which implies that the stress varies linearly with the temperature, which is often assumed in simulations of ballistic impact [21,25,26]. It has been used here because only two data points are available, but the prediction with m = 1.0 is reasonably close to the test data.

The other parameters in Table 3 are readily available in the open literature (ρ, E, cɛ, Tm) or they are varied in the exploration of the model (σC).

### 5.3 Material State History.

Accurately evaluating the local stress state, temperature evolution, and strain rate for the test configurations in Secs. 2 and 3 is difficult without conducting finite element simulations. Simulations where this type of data has been extracted are not reported in this study but can be found in the open literature, see, for example, Refs. [30,3436].

In the following, we will conduct analytical evaluations using a lode parameter of L = − 1, strain rates $p˙$ between 0.005 1/s and 500,000 1/s, and stress triaxiality ratios $σ*$ between 0.0 and 3.0. These values should be representative for the material and ballistic tests in Secs. 2 and 3.

The material state usually varies significantly during a tensile test. For instance, the stress triaxiality for uniaxial tension tests starts out at 0.33 but can increase to more than 0.8 in the center of the necked cross section at fracture [35]. The material in the center of the notch of pre-notched specimens can experience stress triaxiality ratios of more than 1.5. If the initial strain rate in a dynamic uniaxial tension test is around 600 1/s, the final strain rate has been reported to be over 5000 1/s in the center of the necked cross section and the temperature can increase by as much as 300 °C by the time the specimen fractures.

The material state histories in ballistic tests depend on the target configuration (thickness and material), the impact velocity, and the projectile nose shape. Strain rates as high as 1,000,000 1/s have been reported based on finite element simulations, while the temperature may approach the melting point of the material for high impact velocities.

Parts of the target plate will be dominated by compressive stresses in ballistic tests with an ogive-nosed projectile, but brittle fracture is unlikely to occur under severe compression. Since we are interested in the transition from ductile to brittle fracture, we focus on tension dominated stress states where the stress triaxiality ratio is higher than 0.33.

In Sec. 2.3, we presented tension tests with a fatigue-induced crack through the cross section. In this case, the stress triaxiality ratio close to the crack tip can be as high as 3.0. In these tests, we identified brittle fracture from the fracture surfaces.

### 5.4 Analysis.

By varying the mechanical state Ti, $p˙$, and $σ*$ of the material, we can study the influence of ambient temperature, plastic strain rate, and stress triaxiality on the strain to failure pf and further decide whether the failure mechanism is ductile (D ≥ 1) or brittle (σIσC).

A series of analyses with stress triaxialities between 0.33 and 3.0, initial temperatures between −80 °C and +80 °C, and strain rates of 0.005 1/s (quasi-static and isothermal), 50 1/s (dynamic and transition zone), 500 1/s, and 5000 1/s (dynamic and adiabatic) were conducted.

The procedure outlined in Sec. 5.1 is run with the inputs from Table 3. Figures 1517 illustrate the outcome of the analytical procedure by considering three response parameters: the ratio between the major principal stress and the yield stress at room temperature (σI/σ0), the temperature (T), and the ductile damage parameter (D). The ratio σI/σ0 can be used to evaluate the likelihood for brittle fracture to occur according to the RKR criterion. As discussed in the introduction to this paper, if σIσC, then brittle fracture occurs unless ductile fracture occurs prior. The critical stress σC is typically between three and five times the yield stress at room temperature. This means that brittle fracture cannot be ruled out if σI/σ0 ≥ 3. However, the exact number is not known and could be higher. The ratio σI/σ0 = 3 is plotted as a dashed line in Figs. 1517.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal

Figure 15 presents the effects of stress triaxiality at low (0.005 1/s) and high (5000 1/s) strain rates at an initial temperature of +20 °C. It is clear that σI/σ0 is significantly influenced by the stress triaxiality ratio $(σ*)$ and that brittle fracture can occur for $σ*$ above approximately 2.0 since in this case σI/σ0 is higher than 3. The stress triaxiality ratio also affects the evolution of ductile damage. Ductile failure will occur sooner for high stress triaxiality ratios than for low stress triaxiality ratios. Because the temperature evolution is a function of the equivalent plastic strain rate (see Eq. (7)), the predicted increase in temperature is the same for all the stress triaxiality ratios at high strain rate. There is no predicted self-heating for the low strain rate.

Figure 16 presents the effects of initial ambient temperature at low (0.005 1/s) and high (5000 1/s) strain rate for a stress triaxiality ratio of 2.0. Because the stress triaxiality ratio is quite high, σI/σ0 will start out at about 3.0 which means that brittle fracture might occur for all temperatures at all the evaluated strain rates. Ductile fracture is predicted slightly earlier for low temperatures. At the time ductile fracture occurs, the temperature in the material has increased by more than 100 °C due to self-heating when the strain rate is high.

Figure 17 presents the effects of strain rate at initial ambient temperatures of +20 °C and −80 °C for a stress triaxiality ratio of 2.0. The effect of strain rate is similar at low and at high temperatures.

The model shows that the temperature can rise significantly due to self-heating in the dynamic tests. This suggests that even though the temperature is low initially, it quite quickly increases to above room temperature. This can counteract some of the expected effects from low initial temperature.

By interpreting these results in the context of the experimental tests presented in this paper, we see that if the stress triaxiality ratio is lower than 2.0 we do not have to expect brittle fracture, regardless of the temperature and strain rate. This fits well with our experimental results that brittle fracture did not occur except that we saw indication of brittle fracture for the tests with extreme notches and fatigue cracks.

## 6 Discussion

### 6.1 Armox 500T and NVE36 Steel.

We conducted additional ballistic impact tests using 7.62 mm AP bullets and two other steel alloys as targets. This was done to examine if the increased ballistic capacity of Strenx 960 Plus at low temperatures is material specific or if it is a more general result.

The first material, Armox 500T (martensitic), was chosen due to its high strength and application area as an armor steel. According to the manufacturer, the nominal yield stress should be at least 1250 MPa. However, quasi-static, uniaxial tension tests using dog bone specimens revealed that the actual yield stress was 1350 MPa. The second material, NVE36 (hot-rolled structural steel), was chosen because it is a typical steel used in marine and structural applications. Its nominal yield stress is 355 MPa, but tests showed that the yield stress of our batch was around 400 MPa [37]. A plate thickness of 6 mm was tested for NVE36 while a total plate thickness of 7 mm (2 × 3.5 mm) was tested for Armox 500T.

The results presented in Fig. 18 show that the conclusions from Strenx 960 Plus are also valid for the two other steels. For NVE36, vbl increased from 402 m/s to 419 m/s when the temperature decreased from +20 °C to −40 °C. For Armox 500T, vbl increased from 525 m/s to 549 m/s when the temperature decreased from +20 °C to −40 °C. The relative jumps in vbl are 4.2% and 4.6%, respectively.

Fig. 18
Fig. 18
Close modal

### 6.2 Assessment of the Ballistic Results.

Figure 19 presents the ballistic limit velocities from all the test series as functions of the test temperature. The ballistic capacity increases for all the configurations when the temperature is low; 7.62 mm AP bullets striking Strenx 960 Plus plates at −40 °C is the only outlier.

Fig. 19
Fig. 19
Close modal

The reason why vbl for the 20 mm-diameter ogive projectile changes more with temperature than the 20 mm-diameter blunt-nosed projectile might be because the tip of the projectile breaks off and thus the premise for perforation changes. This might have occurred because the target material (due to the reduced temperature) was hard enough to crack the tip.

Børvik et al. [21] presented many impact tests on five different steel alloys. They showed that the ballistic limit velocity increases almost linearly as a function of the initial yield strength of the material. In Sec. 2.2 of this paper, we summarize the results from Ref. [3] where the initial yield stress of the steel increases with decreasing temperature. This increase in yield stress with decreasing temperature may explain why the ballistic limit velocity is higher at low temperatures for Strenx 960 Plus, NVE36, and Armox 500T.

## 7 Conclusion

The results presented in this study suggest that the ballistic capacity of steel is slightly higher at low temperatures than at room temperature. The premise of this study was that steel becomes embrittled at low temperatures. Our tests show that Strenx 960 Plus retained its ductility also at low temperatures. Brittle fracture did not take place even at very low temperatures combined with extreme strain rates and high stress triaxiality ratios. The exception was when a fatigue crack was introduced into a sharp-notched specimen and tested dynamically at low temperature. Only then, evidence of brittle fracture was found. For this high-strength steel, it appears that low temperature improves the ballistic resistance instead of degrading it.

In this study, the ductility did not decrease enough to compensate for the increased strength at low temperature. This resulted in an almost unchanged fracture strain in the material tests, a close to linear decrease in absorbed energy in the Charpy V-notch tests, and a higher ballistic limit velocity in the ballistic tests with decreasing temperature.

We presented an analytical approach to investigate how the mechanical state affects the strain to failure. Based on well-known models, we can estimate whether brittle fracture or ductile fracture is likely to occur for a given condition. The model suggests that high stress triaxiality ratios are required for brittle fracture, and it also shows that self-heating makes the temperature rise significantly for high strain rates and that this might counteract the effect of initially low temperatures.

The results are in line with what is expected for martensitic steels and with studies on other applications than ballistic impact, for instance ship collision. This type of steel can probably be used with confidence in protective structures in arctic environments. A fatigue crack might raise the stress triaxiality ratio sufficiently to introduce brittle fracture in structures, but this is generally not relevant for protective structures.

## Acknowledgment

The financial support for this work comes from the Research Council of Norway through Centre for Advanced Structural Analysis (SFI CASA, project 237885). The Master’s theses of Susanne Thomesen (2016), Piraveena Gunathasan (2017), and Daniel Gulbrandsen (2017) were connected to this work. We thank SSAB for providing the Strenx 960 Plus steel plates.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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