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An Overview of Multicontinuum Theory with Application to Progressive Failure ofLarge Scale Composite Structures Don Robbins Chief Engineer Firehole Technologies, Inc. Laramie, Wyoming
TM Based on MultiContinuum Theory Simple to Use Proven Accuracy for Progressive Failure Simulation Extremely Robust Convergence Progressive Failure Simulation Helius:MCT is a software module that integrates seamlessly with commercial F.E. codes, providing accurate multiscale material response for progress failure analysis of composite structures.
Requirements for Effective Finite Element Analysis of Progressive Failure of Composite Structures: a)Deformation must be represented at the appropriate scale (dictated entirely by mesh density and element type) b)Material Response must be predicted accurately c)Loading and Constraints must be realistic d) Must use an Effective Nonlinear Solution Strategy (e.g., incrementation scheme, regularization, etc.) Common Difficulties: • Mesh discretization typically can not reach the • material ply level must use ply grouping (sublaminates) • Lack of practical constitutive relations that accurately • represent material degradation (damage/failure) • Convergence is very difficult to achieve
OUTLINE • Idealization of Failure in Composite Materials • Independent Variables for Predicting Failure • FailureCriteria, andthe Consequences of Failure • MCTCharacterization of Composite Materials • Selected Demonstration Problems
Idealization of Failure in Composite Materials In a heterogeneous composite material, failure is assumed to occur at the constituent material level. Matrix damage/failure occurs due to stresses in the Matrix Matrix The homogenized composite stress state does not provide this info. Fiber Fiber damage/failure occurs due to stresses in the Fibers Matrix damage/failure degraded Matrix properties Fiber damage/failure degraded Fiber properties
Idealization of Failure in Composite Materials A micromechanical finite element model is used to establish consistency between the homogenized composite properties and the damaged or failed constituent properties. Degraded Matrix Properties + Degraded Fiber Properties Micromechanical Finite Element Model Degraded Composite Properties
Idealization of Failure in Composite Materials How should constituent material stiffness be reduced in the event of a constituent damage/failure event? Matrix Failure Fiber Failure Should matrix stiffness be degraded Isotropically or Orthotropically? Should fiber stiffness be degraded Isotropically or Orthotropically?
Consequences of Matrix Damage/Failure Isotropic Stiffness Degradation Idealization of Failure in Composite Materials 1. Isotropic degradation requires less experimental data, and there is usually a lack of available experimental data 2. Isotropic degradation does degrade the stiffness of the primary load path. causes redistribution of the primary load path. 3. Isotropic degradation also degrades the stiffness of non-primary load paths. largely inconsequential for monotonic loading.
1. Fiber breakage produces a very strong reduction in the axial stiffness of the fiber constituent. largedegradation ofE11. 2. Fiber breakage produces a significant reduction in the longitudinal shear stiffness of the fiber constituent. intermediate degradation ofG12, G13. 3. Fiber breakage does not produce a significant reduction in the transverse normal or transverse shear stiffness of the fiber constituent. insignificantdegradation of E22, E33, G23. f f f f f f Idealization of Failure in Composite Materials Consequences of Fiber Damage/Failure Orthotropic Stiffness Degradation
1 2 3 4 5 Idealization of Failure in Composite Materials In Situ Matrix Properties In Situ Fiber Properties Measured Composite Properties Micromechanical F.E. Model Hypothesize the modes and consequences of constituent damage or failure 1. Measured Composite Properties 2. In Situ Constituent Properties 3. Constituent Damage or Failure Model 4. Degraded In Situ Constituent Properties 5. Degraded Composite Properties Degraded Matrix Properties Degraded Fiber Properties Degraded Composite Properties Micromechanical F.E. Model
Independent Variables for Predicting Damage/Failure in Composites Constituent Failure = f(?) Logical Candidates: Stress, Strain, or Both … But exactly which measures of stress or strain? i.e., at what scale should strain & stress be represented? Homogenized Laminate-Level Stress (Laminate Average Stress) Homogenized Composite Stress (Composite Average Stress) . . Constituent Average Stress . Actual stress field within the constituents of the microstructure
Independent Variables for Predicting Damage/Failure in Composites Desired attributes for the independent variables used to predict composite material response The variables chosen for predicting material response must be physically relevant to the material considered. Calculation of the variables should be efficient, adding minimal computational burden to the structural level finite element analysis. Calculation of the variables should be consistent; the calculated variables should not be overly sensitive to a) idealization of the microstructural architectural, or b) micromechanical mesh-related issues. Method used to calculate the variables should be scalable as the microstructural architecture becomes more complex (e.g. unidirectional woven braided, etc.).
Independent Variables for Predicting Damage/Failure in Composites Cij Constituent Average Strain & Stress States (j j T) i = i,j = 1,2,…,6 =1,2…,# of constituents + Retains a significant level of physical relevance + Efficient Calculation via MCT decomposition + Consistent(stability w/r to idealization and meshing of microstructure) + Scalable (same basic calculation method for unidirectional, woven, braided composites, etc.) MCT decomposition requires linearized constitutive relations
Filtering Characteristics of Volume Average Stress States Composite Average Stress States Constituent Average Stress States Filters out all stress components that are self-equilibrating over the entire RVE Filters out all stress components that are self-equilibrating over each individual constituent material. RetainsPoisson interactions between constituents. Filters outPoisson interactions between constituents. + Retainsthermal interactions between constituents caused by differences in thermal expansion coefficients. Filters outthermal interactions between constituents caused by differences in thermal expansion coefficients. + Filters out self-equilibrating shear stresses that arise solely to satisfy local equilibrium. Filters out self-equilibrating shear stresses that arise solely to satisfy local equilibrium.
Computation of Constituent Average Stress States matrix average stress state Composite RVE m= 1 dv ij ij Vm Dm fiber average stress state f = 1 dv Dc = Dm Df ij ij Vf Df It is NOT necessary to integrate stresses and strains over the micromechanical F.E. model. MCT Decomposition Instead, we use transfer functions [Hill (1963), Garnich & Hansen (1990s)] to accurately & efficiently decompose the composite average strain state into the constituent average strain states. composite average stress state c = 1 dv ij ij Vc Dc
MCT Decomposition matrix average strain state composite average strain state transfer function [Hill (1963), Garnich & Hansen (1990s)] m c cTm (C, C, C, , , , ) c m f c m f m linearized about as many different discrete damaged states as desired c= mm + ff fiber average strain state matrix average stress state fiber average stress state f= Cf(f f) m= Cm(m m) This process adds less than 3% to the overall cost of an equilibrium iteration In a typical F.E. analysis of a composite structure!
Idealization of Failure in Composite Materials A Simple Case: Three Discrete Damaged States matrix failure event fiber failure event DamagedState 2 Failed matrix, Undamaged fibers Damaged State 1 Undamaged matrix, Undamaged fibers Damaged State 3 Failed matrix, Failed fibers matrix failure event c Response of the composite to imposed deformation 1 fiber failure event 2 3 c
Constituent Failure Criteria Matrix Failure Criterion Fiber Failure Criterion Ifm (m) 1, Iff (f) 1, Then Matrix properties are isotropically degraded by a user-specified amount. Then Fiber properties are orthotropically degraded by a user-specified amount. Micromechanical F.E. Model Degraded Composite Properties 18
c ccccc E11, E22, E33, G12, G13, G23 MCT Material Characterization Step 1. Optimize the in situ constituent properties so that the micromechanical finite element model matches the measured properties of the composite material in situ constituent properties homogenized composite properties Cm,m Cf ,f Micromechanical Finite Element Model of RVE Cc,c measured composite properties c ccccc 12, 13, 23, 11, 22, 33
MCT Material Characterization Step 2. Determine the coefficients of the constituent failure criteria so that the micromechanical finite element model matches the measured strengths of the composite material constituent failure criteria measured composite strengths m1 f1 Decomposition measured composite strengths c+ c c+ c c c S11, S11, S22, S22, S12, S23
Summary • Micromechanical F.E. models are only used during the material characterization process, not during the actual structural-level finite element analysis. • During the material characterization process, the micromechanical F.E. model is used to establish in situ constituent properties and homogenized composite properties for a finite set of discrete damage states (3). • These properties are stored in a database and can be quickly accessed by the structural-level finite element model as dictated by the outcome of the constituent failure criteria. • The coefficients of the constituent failure criteria are determined using only industry standard strength tests. • The entire process of computing the constituent average stress states, evaluating the constituent failure criteria, and identifying the damaged properties of the composite material adds less than 3% to the total cost of an equilibrium Iteration in a structural-level finite element analysis.
Example: Atlas V CCB Conical ISA (Used on all Atlas V 400 series launches) Diameter: 12.5’ to 10’ Height: 65 inches Graphite/epoxy and honeycomb core Loading: Combined vertical compression & horizontal shear, designed to drive failure in the top corner of the access door.
Structural Response Predicted with Helius:MCTTM % of flight load vertical displacement of load head (in)
structural response softening becomes detectable impending global failure 185% Flight Load 170% Flight Load 190% Flight Load
The modified ISA was tested to failure at AFRL Kirtland (Oct. 2008). Ultimate failure measured at 183% of Flight Load The ISA exhibited a nearlylinear response up to ultimate failure Final failure process was very rapid (almost instantaneous) Failure initiated at door corners and progressed circumferentially Failure initiated at door corners Rapidly propagated around circumference
measured global failure load structural response predicted with MCT % of flight load vertical displacement of load head (in)
190% Flight Load Excellent agreement was achieved for: 1) Location of Failure Initiation 2) Failure Evolution Behavior 3) Ultimate Load
Leak Pressure Measured: 1233 psi Helius:MCT: 1215 psi (-1.5%) LARC 02: 900 psi (-27.0%) Example: Unlined Cryogenic Composite Pressure Vessel Loading: 1. Submerge tank in liquid nitrogen (T = -216C) 2. Pressurize tank until a constant leakage rate was detected Six tanks were tested with an average leak pressure of 1233 psi. No Damage Matrix Damage Fiber Damage c c
Good correlation between predicted region of permeation and observed region of permeation crack saturation observed permeation Matrix Damage Fiber Damage No Damage c c
Sponsorship The work presented herein has been sponsored by numerous DoD agencies as well as internal R&D at Firehole Technologies. Current government sponsorship includes: AFRL & AFOSR via contract number FA9550-09-C-0074. Directors: Dr. David Stargel & Dr. Victor Giurgiutiu. AFRL(Space Vehicles Directorate) via contract number FA9453-07-C-0191. Directors: Dr. Tom Murphey &Dr. Jeff Welsh. NASA’s Exploration Systems Mission Directorate Director: Wyoming NASA Space Grant Consortium
The End The Firehole River(Yellowstone National Park, Wyoming)
Idealization of Failure in Composite Materials Issues to Consider • Failure of the Homogenized Composite Material vs. • Failure of theHeterogeneous Composite Material • Modes of Damage/Failure Addressed • Consequences of Damage/Failure (i.e. stiffness degradation) • Isotropic Degradation vs. Orthotropic Degradation • Continuous Degradation vs. Discrete Degradation • Local vs. non-local damage/failure
Use of Micromechanical Finite Element Models Why do we need In Situ Constituent Properties? Aren’t Bulk Constituent Properties good enough? The Micromechanical F.E. Model represents an idealized microstructure that is unlikely to accurately represent a) the actual fiber distribution statistics b) the actual distribution of micro-defects caused by manufacturing & curing The Micromechanical F.E. Model does not accurately represent the fiber/matrix interphase a) the model often does not explicitly include the interphase b) knowledge of interphase properties is typically absent or incomplete The properties of the Matrix constituent material are sensitive to curing conditions (e.g., temperature, pressure, deformation, chemical environment). It is unlikely that a sample of bulk matrix material has been subjected to the same curing conditions as the matrix material in a fiber reinforced composite.
11 Imposed Uniform Temperature Reduction in an Unconstrained Composite Constituent Average Stress Composite Average Stress Micromechanical Stress Field tension 2 3 zero 2 2 3 3 compression Thermal interactions between constituents are self-equilibrating over the entire RVE, but not self-equilibrating within each individual constituent. Constituent averaging processretains thermal interactions. Composite averaging process filters out thermal interactions.
22 Imposed Uniform Temperature Reduction in an Unconstrained Composite Micromechanical Stress Field Composite Average Stress Constituent Average Stress tension 2 3 zero 2 2 3 3 compression Thermal interactions between constituents are self-equilibrating over the entire RVE, but not self-equilibrating within each individual constituent. Constituent averaging processretains thermal interactions. Composite averaging process filters out thermal interactions.
33 Imposed Uniform Temperature Reduction in an Unconstrained Composite Constituent Average Stress Composite Average Stress Micromechanical Stress Field tension 2 3 2 2 zero 3 3 compression Thermal interactions between constituents are self-equilibrating over the entire RVE, but not self-equilibrating within each individual constituent. Constituent averaging processretains thermal interactions. Composite averaging process filters out thermal interactions.
23 2 3 Imposed Uniform Temperature Reduction in an Unconstrained Composite Constituent Average Stress Composite Average Stress Micromechanical Stress Field 2 2 2 2 zero zero 3 3 3 3 Both the constituent averaging process and the composite averaging processfilter out the transverse shear stress since it is self-equilibrating within each individual constituent as well as self-equilibrating over the entire RVE.
matrix average stress state m = m = 17.25 MPa 22 all c = 0 33 m = 44.5 MPa ij 11 Example: Cryogenic cooling of a composite T= 217C fiber average stress state f = f = 25.87 MPa 22 33 f = 66.75 MPa 11 The constituent average stress states are inherently triaxial due to the thermal interactions between constituents!
fiber average stress state f = f = 10.9 MPa 22 33 f = +3.8 MPa 11 matrix average stress state m = m = 8.7 MPa 22 33 m = 5.73 MPa 11 Example: Composite under biaxial compression c = 10 MPa 33 c = 10 MPa 22 all other c = 0 ij The constituent average stress states are inherently triaxial due to the Poisson interactions between constituents!
Example: Composite Adapter for Shared PAyload Rides Two identical laminated composite monocoque shells IM7/8552 unidirectional tape (up to 64 plies thick) 60 inches tall, 74 inches in diameter CASPAR
Helius:MCT Progressive Failure Simulation Observed Predicted
Helius:MCT Prediction vs. Experiment Observation First significant slope increase Helius:MCT Ultimate Failure Test Stopped Lower Radius Failure Initial Fiber Failure Door Debonding Compressive Displacement (in) Lapband Gapping Initial Matrix Failure Initial Matrix Cracking Continuous Matrix Cracking Noise Occasional Matrix Cracking Noise Flight Load Limit (%)
Lower radius failure at 792% of FLL CASPAR was successfully tested to ultimate failure on April 14, 2008 Fiber Failure predicted in lower radius at 800% of FLL
Helius:MCT Prediction Experimental Observation Hashin Damage Evolution 1st Signif. Slope Change: 980 Initiation of Fiber Failure : 740 Test Stopped: 847 Initiation of Matrix Failure : 260 Hashin Ultimate: 1950 ??? Hashin Fiber: 1300 Hashin Matrix: 1100