Experimental Characterization and Material-Model Development for Microphase-Segregated Polyurea: An Overview

Abstract

Numerous experimental investigations reported in the open literature over the past decade have clearly demonstrated that the use of polyurea external coatings and/or inner layers can substantially enhance both the blast resistance (the ability to withstand shock loading) and the ballistic performance (the ability to defeat various high-velocity projectiles such as bullets, fragments, shrapnel, etc. without penetration, excessive deflection or spalling) of buildings, vehicles, combat-helmets, etc. It is also well established that the observed high-performance of polyurea is closely related to its highly complex submicron scale phase-segregated microstructure and the associated microscale phenomena and processes (e.g., viscous energy dissipation at the internal phase boundaries). As higher and higher demands are placed on blast/ballistic survivability of the foregoing structures, a need for the use of the appropriate transient nonlinear dynamics computational analyses and the corresponding design-optimization methods has become ever apparent. A critical aspect of the tools used in these analyses and methods is the availability of an appropriate physically based, high-fidelity material model for polyurea. There are presently several public domain and highly diverse material models for polyurea. In the present work, an attempt is made to critically assess these models as well as the experimental methods and results used in the process of their formulation. Since these models are developed for use in the high-rate loading regime, they are employed in the present work, to generate the appropriate shock-Hugoniot relations. These relations are subsequently compared with their experimental counterparts in order to assess the fidelity of these models.

Appendix: Split Hopkinson Pressure Bar

The SHPB setup is used for determination of the high-strain-rate (unconfined or confined) compression stress versus strain data. As schematically represented in Fig. A1, an SHPB setup consists of a striker bar, an incident bar, the specimen being tested and the transmitter bar. A rectangular compression wave of well-defined amplitude and duration is generated in the incident bar when the incident bar is struck by the striker bar. When the wave reaches the incident bar/specimen interface, it is partially transmitted into the specimen (and subsequently) into the transmitter bar, and partially reflected back to the incident bar.

Fig. A1

A schematic of a Split Hopkinson Pressure Bar (SHPB) setup

Using a one-dimensional wave propagation analysis, it is possible, as shown below, to determine high-strain rate stress-strain curves from measurements of the temporal evolution of strain in the incident and transmitter bars.

Propagation of waves along the length of the bar (x direction) is, to a good approximation, governed by the one-dimensional wave equation:

$$ {\frac{{\partial^{2} u}}{{\partial x^{2} }}} - {\frac{1}{{c^{2} }}}\,{\frac{{\partial^{2} u}}{{\partial t^{2} }}} = 0 $$

(A.1)

which has the solution within the incident bar:

$$ u = f(x - ct) + g(x + ct) = f(z_{1} ) + g(z{}_{2}) = u_{\text{i}} + u_{\text{r}} , $$

(A.2)

where u _i is the incident wave, u _r is the reflected wave, c is the speed of propagation of the wave, t time and z ₁ and z ₂ are single variables that f and g depend on, respectively. The corresponding axial strain in the incident bar is then defined as:

$$ \upvarepsilon = {\frac{\partial u}{\partial x}} = {\frac{{\partial (u_{\text{i}} + u_{\text{r}} )}}{\partial x}} $$

(A.3)

Substituting Eq A.3 into Eq A.2 yields:

$$ \upvarepsilon = {\frac{df}{{dz_{1} }}}{\frac{{\partial z_{1} }}{\partial x}} + {\frac{dg}{{dz_{2} }}}{\frac{{\partial z_{2} }}{\partial x}} = {\frac{df}{{dz_{1} }}} + {\frac{dg}{{dz_{2} }}} = f^{\prime} + g^{\prime} = \upvarepsilon_{\text{i}} + \upvarepsilon_{\text{r}} $$

(A.4)

The velocity of any material point in the incident bar can be determined by differentiating Eq A.2 with respect to time as:

$$ \dot{u} = {\frac{{\partial (u_{\text{i}} + u_{\text{r}} )}}{\partial t}} = {\frac{df}{{dz_{1} }}}{\frac{{\partial z_{1} }}{\partial t}} + {\frac{dg}{{dz_{2} }}}{\frac{{\partial z_{2} }}{\partial t}} = - cf^{\prime} + cg^{\prime} = c( - f^{\prime} + g^{\prime}) = c( - \upvarepsilon_{\text{i}} + \upvarepsilon_{\text{r}} ) $$

(A.5)

In the case of the transmitter bar, there is only one wave, the transmitted wave, propagating through it (until the wave reflects at the free end of the transmitter bar, which takes place at a post data-collection time). Therefore, the velocity of the material points within the transmitter bar is obtained by differentiating, with respect to time, displacement field $u(t) = h(z_{1} ) = h(x - ct)$ as:

$$ \dot{u} = {\frac{dh}{{dz_{1} }}}{\frac{{\partial z_{1} }}{\partial t}} = - ch' = - c\upvarepsilon_{\text{t}} $$

(A.6)

The strain rate in the specimen is next calculated as:

$$ \dot{\upvarepsilon } = {\frac{{\left( {\dot{u}_{\text{I/S}} - \dot{u}_{\text{S/T}} } \right)}}{{l_{\text{S}} }}}, $$

(A.7)

where l _s is the instantaneous length of the specimen and subscripts I/S and S/T are used to, respectively, denote the velocities of the incident bar/specimen, Eq A.5 and the specimen/transmitter bar, Eq A.6, interfaces. Combining Eq A.5, A.6, and A.7, the instantaneous (average) strain rate in the specimen can be calculated from the measured time-dependent strains in the incident and the transmitter bars as:

$$ \dot{\upvarepsilon } = {\frac{c}{{l_{\text{s}} }}}( - \upvarepsilon_{\text{i}} + \upvarepsilon_{\text{r}} + \upvarepsilon_{\text{t}} ) $$

(A.8)

The corresponding strain in the specimen can be obtained by integrating over time the specimen strain rate.

To determine the axial stress in the specimen, the forces in the incident and transmitter bars are first determined. Since the two bars are deformed elastically, the corresponding forces are defined as:

$$ F_{\text{I}} = A_{\text{I}} E_{\text{I}} (\upvarepsilon_{\text{i}} + \upvarepsilon_{\text{r}} ) $$

(A.9)

$$ F_{\text{T}} = A_{\text{T}} E_{\text{T}} \upvarepsilon_{\text{t}} $$

(A.10)

where A and E are, respectively, used to denote the cross-sectional area and the Young’s modulus of the incident (I) and the transmitter (T) bars.

The SHPB analysis presented here is based on the following two assumptions:

• (a) after some initial ringing, the specimen is deformed uniformly. This assumption was already used in the case of Eq A.7; and

• (b) the specimen is in the state of dynamic equilibrium, i.e., the forces acting at the I/S and the S/T interfaces are equal.

Under the conditions A _I = A _T and E _I = E _T, this assumption via Eq A.9 and A.10 yields:

$$ \upvarepsilon_{\text{t}} = \upvarepsilon_{\text{r}} + \upvarepsilon_{\text{i}} $$

(A.11)

Substituting Eq A.11 into Eq A.8 yields:

$$ \dot{\upvarepsilon } = {\frac{{2c\upvarepsilon_{\text{r}} }}{{l_{\text{s}} }}} $$

(A.12)

suggesting that the specimen strain rate can be determined solely from the measured strains associated with the reflected wave within the incident bar.

The corresponding stress in the specimen is then calculated by dividing the force in the sample by the sample cross-sectional area as:

$$ \upsigma (t) = {\frac{{A_{\text{T}} E_{\text{T}} \upvarepsilon_{\text{t}} }}{{A_{\text{s}} }}} $$

(A.13)

Equation A.13 was obtained using Eq A.10. The same could be done using Eq A.11 or a mean of the Eq A.11 and A.12.