pressure recordings, the loading was still defined as impulsive and the pressure measurements in the clamping frame were assumed reasonable to evaluate potential FSI effects. In a similar way as for the steel plates, the pressure measurements in Table 3.6 should be treated with some caution since the pressure may change as the plate deforms.
3.4 Concluding remarks
The influence of stand-off distance on the dynamic response of thin steel and aluminium plates subjected to airblast loading has been investigated experimentally. The loading was generated by detonating spherical charges of C-4 at various stand-off distances relative to the centre point of the plates, while the structural response was measured using two high-speed cameras in a stereovision setup combined with 3D-DIC analyses. The observations covered the entire range of structural response from complete ductile failure at the supports to CIB in terms of RSB at larger stand-off distances.
The overall trends in the experimental results were increased mid-point dis-placement of the plates and increased impulse as the intensity of the blast loading increased. This is shown in Figure 3.15 where the measured mid-point deflection-thickness ratio and scaled impulse as a function of scaled distance are plotted for all tests. Both the mid-point deflection and the impulse seem to have a rather linear decrease with increasing stand-off distance. However, the impulse at the closest stand-off distance differs somewhat from this linear trend. This is probably due to reduced accuracy of the pressure sensors due to high temperatures, which is supported by the observation in Figure 3.10a where it is evident that the plate is slightly burned by the fireball at the closest stand-off distance. It is also noted that the plates experiencing RSB deviate from the linear trend in Figure 3.15a.
The increased mid-point displacement with increasing impulse is intuitive as long as the final deflection is in the same direction as the external load. This behaviour is also in accordance with the theory of impulsively loaded plates as discussed by Jones [61]. First, a phase with plastic hinges that starts at the boundary corners of the plate and propagate along the diagonals toward the centre is observed (see Figure 3.11a and Figure 3.14a). Then, when the plastic hinges meet in the centre of the plate (see Figure 3.11b and Figure 3.14b), a final phase develops with oscillations around a permanent deformed shape.
However, the response at the largest stand-off distance for both materials was counter-intuitive as the plates experience RSB and the final deflection was in the opposite direction of the incident blast wave.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Scaled distance Z [m/kg1/3]
Deflection − thickness ratio
Scaled distance Z [m/kg1/3] Scaled impulse [kPa ms/kg1/3]
Docol 600DL EN AW 1050H14 Rigid steel plate
(b)
Figure 3.15: Measured mid-point deflection-thickness ratiodz,p/t(a) and scaled impulseirα/W1/3from a representative sensor in the frame (b) versus scaled distance Z=R/W1/3. In (a) the open symbols are from the 3D-DIC measurements, while the closed symbols are from the manual measurements using a slide caliper.
The RSB attracted special attention as it occurred both during and subsequent to the elastic rebound. The RSB observed in Figure 3.9 during the elastic rebound is also observed in previous studies [45,46,54]. However, to the author’s best knowledge there are no previous experimental studies on metallic plates observing RSB during the free vibrations around its new equilibrium position after the elastic rebound as shown in Figure 3.12. Based on the tests carried out, it is challenging to give an explanation of the observed phenomenon and also on how the negative phase will influence the observed response. The influence of the negative phase needs further investigations to determine the loading properties where this may dominate the response. This will be investigated in a subsequent numerical study in Chapter 5. It is also interesting to note that the response of the steel and aluminium plates at the same stand-off distance of 375 mm (S31-S33 in Figure 3.9 and A11-A13 in Figure 3.12) results in a completely different final deformed shape, as the steel plates experience RSB whereas the permanent displacement of the aluminium plates is in the intuitive direction.
These experiments also illustrate the possibilities of using finite element-based 3D-DIC for a thorough examination of the displacement field of blast-loaded structures. A comparison of the final mid-point deflections measured with DIC (dz,p1) and the corresponding permanent displacement (dz,p2) measured manually after the experiment indicated that there were some deviations in the two measurements. This was mainly due to the slight movement of the mounting frame during the tests and the DIC measurements were in very good agreement when corrected for this movement (see Tables 3.5 and 3.6). It is therefore assumed that the DIC measurements are more accurate than the
3.4. Concluding remarks 69 manually measured permanent displacement with the sliding caliper. The permanent displacement (dz,p2) was included to indicate the accuracy in the DIC measurements and to give experimental results when difficulties were observed with DIC. Thus, considering the potential sources of error in the manual measurements, DIC measurements should be used when available. The DIC technique also enabled a synchronization of the loading and response histories during the entire experiment, which provided new and accurate results under such extreme loading conditions.
4
Material modelling
Before performing the numerical study on the blast-loaded plates presented in Chapter 3, it is necessary to ensure that the computational model is able to predict the material behaviour in these types of loading environments. The observations in Chapter 3 showed that the dynamic response of the plates may become significantly different under varying blast intensity and structural properties. The dynamic response may therefore be highly dependent upon a proper treatment of the material modelling. This chapter presents the computational framework, constitutive equations and the implementation of an elastic-thermoviscoplastic material model in the explicit non-linear finite element code EUROPLEXUS.
4.1 Introduction
Blast events often involve large strains, elevated strain rates, temperature softening and ductile failure. A widely used design tool for this type of problems is the non-linear finite element (FE) method. In the FE method, the formulation and numerical solution of non-linear problems in continuum mechanics rely on the weak form of the momentum balance equation (also known as the principal of virtual power [171]). The integral form of the principal of virtual power is well suited for direct application to the FE method, and by spatial discretization the solution is integrated in time. In the particular case of blast events, it is often used an explicit time integration scheme based on the central difference method. Iterations for FE analyses involving non-linear material behaviour can in general be divided into two levels, hereby denoted the global and local level. The global level involves the explicit establishment of global equilibrium between internal stresses and external loads, while the local level updates the corresponding stress state in each FE integration point (for a given strain increment) in terms of the governing constitutive equations.
Constitutive equations (also known as material models) are mathematical descriptions of the material behaviour which gives the stress as a function of the deformation history of the body [171]. This implies that the local
integration of the constitutive equations controls the accuracy and stability of the global equilibrium, and that the overall solution in the FE model may be considerably influenced by the accuracy, robustness and effectiveness of the integration algorithm.
A commonly used scheme for the local stress integration is the predictor-corrector method (often called return mapping). The basis for this return mapping is two successive steps, i.e., the predictor step and the corrector step. The predictor step is used to estimate a trial stress state, while the corrector step applies a flow rule by using a return mapping scheme to ensure the consistency condition. That is, ensuring that the stress state is on the yield surface. This thesis uses the explicit cutting plane method proposed by Ortiz and Simo [172]. During this approach an elastic predictor uses the total strain increment to obtain a trial stress state. Then, if the elastic trial stress is outside the plastic domain, the predicted stresses are corrected to return to the updated yield surface by iterations of the plastic corrector. The basis for the cutting plane method is to utilize the known stress state at the last converged time increment to determine the normal to the yield surface.
This work considers a well-known constitutive relation proposed by Johnson and Cook [173] where von Mises plasticity and associated flow are used to update the stresses. A model including elastic-thermoviscoplasticity and duc-tile failure is implemented as a new material calledVPJC in EUROPLEXUS (EPX) [130]. The implementation uses a fully vectorized version of the forward Euler integration algorithm. Ductile failure is included in the model through the Cockcroft-Latham criterion [174] which is uncoupled from the constitu-tive equations. Large deformations are accounted for by using a hypoelastic-thermoviscoplastic formulation of the constitutive equations and a corotational formulation in EPX. This implies that the constitutive equations are defined in a local coordinate system in which the basis system rotates with the material.
The subroutine interface uses the Cauchy stress components and all stress and strain quantities are defined in terms of the rate of deformation tensor. The model is applicable for one- (1D), two- (2D) and three-dimensional (3D) stress analyses (i.e., for bar, shell, solid, axisymmetric and plane strain elements).
The model is also coupled with the element deletion options available in EPX by introducing a state variable controlling the element erosion. That is, as the failure variable reaches its critical valueDC= 1.0 the element is removed.
Finally, material parameters are identified and the performance of the model is evaluated based on the tension tests in Section 3.2.2. A complete description of the formulation and implementation of the model is given in [175], while most of the theory presented herein is based on Hopperstad and Børvik [176].