Numerical simulations of high velocity impact phenomena in composite structures D. Vinckier, K. Thoma

Transcription

1 Numerical simulations of high velocity impact phenomena in composite structures D. Vinckier, K. Thoma Fernhag, Germany ABSTRACT A Lagrangian finite element method with explicit time integration is used to simulate high velocity impact and penetration processes into fiber reinforced materials. A material model, originally developed to simulate the crash behavior of high strength UD-fiber-matrix systems in automotive applications, was extended to include fabric prepregs and describes the elastic behavior, failure, and the post-failure behavior up to complete material crushing. The failure model is based on a Tsai-Wu criterion with an added failure mode recognition system. Simulations of the impact of a metallic fragment with 750 m/s on a protective wall with a front layer of high strength steel and three backing layers of fabric prepregs illustrate the capabilities of the model in the design and analysis of lightweight protection systems. INTRODUCTION Contrary to the analysis of a high velocity impact on materials such as steel or aluminum, the simulation of impact and penetration processes into nonisotropical fiber reinforced materials still is a research task The simulation of these phenomena is important in the design process of lightweight protective systems and of the composite materials used in these structures, as they may help to understand the complex dynamic interactions between structural components undergoing large deformations and nonlinear material behaviour including failure, delamination, erosion, etc. The numerical analysis tool could then be used to identify the most important material parameters and to propose the characteristics of an "ideal" material for a particular protective structure. To be able to simulate the dynamic behavior of composite materials, the constitutive equations must cover the elastic regime, the material failure with identification of the failure mode, and the behavior of the partially or completely failed material. These equations may then be integrated in a Finite Element code, suitable for impact/contact problems. Typically this is done using non-

2 184 Composite Material Technology IV linear Lagrange codes with explicit time integration and an automatic contact algorithm. The basis for such a material model has been developed by CONDAT and BASF as part of the European research project CARMAT 2000 [1,7]. The goal of this particular part of the project was to simulate the crash behavior of fiber reinforced composite structures in automotive applications. Since then, the model was also applied to simulate material testing [8,9].The next section describes the main components of the model which has now been extended to cover also fabric materials. A MATERIAL MODEL FOR FIBER REINFORCED COMPOSITES General characteristics. In order to accurately follow the inelastic material behavior with large displacements and rotations, large strains and multiple damage modes, the constitutive equations are implemented in a 3D volume element, using an updated Lagrangian Jaumann stress rate (U.L.J.) formulation. The U.L.J. formulation [2], which is intensively used in explicit Lagrangian FE codes, is very effective in large strain analysis, because it describes the material response in a natural way. The stresses are computed in an incremental fashion, using the Jaumann stress rate and velocity strain tensors, which are respectively defined as f \ I *j Xi) where CL is the spin tensor and Ty the time derivative of the Cauchy stress. The stress-strain law in the elastic regime would then be evaluated in the local fiber reference coordinate system using to compute the stress increments where (T.^ is the elastic stress-strain matrix. Failure model. The material failure is described using a quadratic interaction failure criterion in the stress space (Tsai-Wu criterion, [3]), with the failure function defined as /j i. TT~» ^ 9 i 7 i ^^ ^^ 1 (A\ r h i /T L/n + r h ij rr LSy + h I njj rr (_/;/ rr CJyy ~ i (~t) ^ ' Elastic behavior is assumed when / < 0, and failure occurs for / = 0. The strength parameters FJ, Fy, and F,^ can be expressed as functions of the uniaxial material strengths in tension and compression in the longitudinal and transverse directions, and the in-ply and interlaminar shear strengths. The coefficients of the interaction terms F^ are chosen in a way that for an isotropic material, the failure criterion is reduced to the von Mises yield formula. Although the criterion was originally developed for unidirectionally fiber reinforced plastics, it may be used also for bidirectionally reinforced

3 Composite Material Technology IV 185 materials, provided the strength parameters can be fitted to the test data for these materials. At failure the stresses are computed using the same approach of flow theory which is used in plasticity analysis [4]. When the damage strain increments, which are defined as are computed using the proportionality factor A and the flow rule then A may be obtained from the consistency equation by Prager * (8) by substituting (5), (6)and (7) in (8) : % ^E, x ijrs &&rs (9) To model the behavior of the damaged material, the failure modes must be identified. If the directions (11, 22) define the ply plane, with 11 the fiber direction for the unidirectionally (UD) reinforced composite, and 1 1 and 22 the fiber directions for the birectionally (BD) reinforced composite, and 33 is the direction orthogonal to the ply-plane, then the following failure modes, which in this model are directly related to the directions of the fiber reference system, are taken into account :fiberfailure (11 for UD, 1 1 and 22 for BD), matrix failure (22, only for UD), in-plane shear (12), and interlaminar shear (23, 31). As damage modes are not automatically provided by the Tsai-Wu criterion, a failure mode identification is added, using normalized stress components, which are defined as N '</? do) O/y where Sy are the material strenghts associated with direction ij (e.g. S^ is the in-plane shear strength, Sjj is the strength in fiber direction, tensile when a/,- > 0, compressive when a,y < 0, etc. ). The coefficients a^ may be used tofitthe numerical model to test data. The failure mode kl is then defined by the largest normalized stress component. A graphical interpretation of the failure surface in the (cj;;,o'22^12) stress space y=0) is shown in fig. 1 for a typical UD fiber reinforced material.

4 186 Composite Material Technology IV The failure modes were computed for a,y=l and are indicated by different gray shades on the failure surface. Matrix Failure (03) Shear Failure (a^) Fiber Failure (a-j) Figure 1 : Tsai-Wu failure surface in the (^11^22^12) stress space Damage Factors. To model the behavior of the damaged material, which includes the tracking of crack planes and the modification of the original failure surface, the damage must be quantified for each damage mode. The most straighforward approach is to introduce damage factors, based on the damage energy, which is dissipated when failure occurs. The damage energy increment is obtained by The damage energy is accumulated for each activated damage mode individually. The damage factors are then defined by the Weibull distribution Wi -y ^^ijj where /^. is the scale factor and % the shape factor. (12) Modified failure surface. The reduced strength of the damaged material is taken into account by modifications to the failure function, which are quantitatively controlled by the damage factors. For the damage modes ij (i^j) (shear failure), the failure surface is "shrinked" by modifying the corresponding strength Fy (/>?) parameters in the Tsai-Wu equation (3). For the damage modes // (normal failure) the Tsai-Wu failure function is not modified, but additional failure functions are introduced, i (13)

5 Composite Material Technology IV 187 which limit the normal stress components in tension. As the introduction of these failure functions generates corners in the resulting failure surface, care must be taken when more than one failure criterion is satisfied. In the latter case, Koiter's generalization of the flow theory, as described in [4] is used to return the stresses to the failure surface. No tensile stresses are allowed when the damage factors D,, approach 0. By adjusting the shape factors ay and scale factors fly in equation (12) brittle or ductile failure may be simulated. Crack Planes. Failure or crack planes, associated with the failure modes are generated when damage is initiated. In this version of the model, three possible crack planes exist, which are normal to the directions ii in the local fiber reference system. The model keeps track of the state of the crack (open or closed) and allows only compressive stresses normal to the failure plane when the crack is closed. THE FINITE ELEMENT PROGRAM CONDAT-DYNA3D The material model for fiber reinforced composites that was described in the previous section has been implemented in CONDAT-DYNA3D [5], a 3D Lagrangian Finite Element program with explicit time integration for the analysis of large deformation dynamic response of inelastic solids and structures. Typical applications include low-, high- and hyper-velocity impacts and penetrations, shock wave studies, vehicle crash, metal forming, and damage analysis in composites. The program is based on the public domain version of DYNA3D [6] from Lawrence Livermore National Laboratories (author J.Hallquist). Several modules have been added or rewritten by CONDAT. A key option is the contact algorithm [10],.with fully automatic definition of all candidate contact nodes and segments, automatic erosion options, tracking of eroded masses, and additional time step criteria to ensure accuracy and stability. 200 x^x 3.51^ High Strength Steel 200 x 200 x 5.4 mm Fabric Prepreg E922E9PJ^AlJ]2!]lS^^^f!^ BKBS^WmaA^a^MaBa 200 x 200 x 5.4 mm Fabric Prepreg Figure 2 : Materials, Dimensions and initial conditions for the example analysis.

6 188 Composite Material Technology IV EXAMPLE : IMPACT ON A LIGHTWEIGHT PROTECTIVE WALL The capabilities of the material model are illustrated by the analysis of an impact process on a lightweight protective wall (figure 2). A steel fragment, which might have been generated after an accidental explosion of a steel container, impacts with 750 m/s on a wall, which acts as a protection against fragments. The wall consists of a 3.5 mm front layer of high strength steel, and three 5.4 mm fabric prepreg backing layers. The overall dimensions of the modeled section of the wall are 200 x 200 x 19.7 mm. Material strength in MPa In-plane tensile 1-plane tensile In-plane compressive 1-plane compressive In-plane shear 1-plane shear Snt,S22t $33t Sl1c, $22c S33c Sl2s S23s, S31s Composite A Composite B Table 1 : Material strengths as used in the impact analysis Two simulations were carried out, one with composite material A, and one with composite material B, which is characterized by higher tensile and compressive strengths in fiber direction and higher compressive strength in the out-of-plane direction. Figure 3 : Impact process on composite A (dark zones are damaged composite)

7 Composite Material Technology IV 189 The fragment impacts on the high strength steel, flattens and begins to erode (eroded material is indicated by small cubes in fig. 3). Because of the angular velocity, the fragment continues to rotate, impacts the wall on its long side, and breaks into several parts. The front steel plate fails at the impact point and the high stresses in the backing layers cause damage in the composite material. linterlaminar shear failure BUmatrix compressive failure IHfiber rupture Composite B Figure 4 : Main damage modes in the composites A and B, 100 jis after impact Directly under the first impact point, the composite matrix material fails in compression (fig. 4). This failure is more pronounced for the weaker composite B, where the upper composite layer is eroded completely in the center under the impact point. The dominant failure in both cases is interlaminar shear failure, which occurs in all three composite layers. Additionally, fiber failure is seen to occur in all three layers of the material B, which, in summary is more damaged and causes larger defomations of the wall than material A. Both structures are

8 190 Composite Material Technology IV however capable of stopping the fragment without ejecting debris at the inner side of the wall. CONCLUSION The numerical analyses with CONDAT-DYNA3D of the high velocity impact on a protective structure with backing layers of fabric prepreg composites have shown that the material model presented in this paper is able to simulate the damage initiation and growth in composite materials for dynamic loading conditions with large displacements, large strains and material erosion. The failure and the post-failure models have proven to be a sound basis for the description of the constitutive behavior from the elastic up to the completely failed state with ample room tofitthe model parameters to test data. Several aspects of the model (e.g. the Tsai-Wu failure criterion for fabric materials, the Weibull distribution for the post-failure behavior, and the fixed orientations of the failure planes ) need to be verified through well defined material tests. Future developments, which will take place in parallel to experiments, will include extensions for tridirectionally reinforced materials, arbitrary failure directions, the strain-rate dependency of the failure surface, and the shock behavior of composites. REFERENCES [1] M.Maier, Entwicklung integrierter Automobilteile aus Faserverbundwerkstoffen, BMFT- Bericht 03M 1021-A9, BASF, Ludwigshafen, [2] K. J. Bathe, Finite Element Procedures in Engineering Analysis, Chapter 6, Finite Element Nonlinear Analysis in Solid and Structural Mechanics, pp , Prentice-Hall, Englewood Cliffs, NJ, [3] S.W: Tsai, H.,T.Hahn, Introduction to Composite Materials, Technomic Publishing Company, Lancaster, Pennsylvania, [4] Y.C. Fung, Foundations of Solid Mechanics, Chapter 6, Elastic and Plastic Behavior of Materials, pp , Prentice-Hall, Englewood Cliffs, NJ, [5] D. Vinckier, User's Manual of CONDAT-DYNA3D, a 3D Lagrangian Finite Element Program for Nonlinear Dynamics of Solids and Structures, CONDAT report CB08301, August [6] J.O. Hallquist, D.J. Benson, DYNA3D User's Manual (Nonlinear Dynamic Analysis of Structures in Three Dimensions), Lwarence Livermore National Laboratory, Livermore, CA, UCID Rev. 3, [7] M.Maier, V. Altstadt, D.Vinckier, K.Thoma, Hartetest fiir Faserverbundwerkstoffe im Computer, Werkstoffund Innovation, 6, [8] M.Maier, V. Altstadt, D. Vinckier, K,.Thoma, Numerical and Experimental Analysis of the Compact Tension Test for a Group of Modified Epoxy Resins, Journal of Polymer Testing, 13, pp , [9] D.W. Wilson, V.Altstadt, M.Maier, J,Prandy, K.Thoma, D.Vinckier, An Analytical and Experimental Evaluation of 0/90 Laminate Tests for Compression Characterization, Journal of Composites Technology & Research, Vol.16, No.2, pp , April [lojk.thoma, D. Vinckier, Numerical Analysis of a High Velocity Impact on Fiber Reinforced Materials, IMPACT IV, Post-Conference Seminar of 12th Int.Conf. on Strutural Mechanics in Reactor Technology (SMiRT), BAM Berlin, August 1993.