Scaling of Strength of Metal-Composite Joints—Part II: Interface Fracture Analysis

2009 ◽  
Vol 77 (1) ◽  
Author(s):  
Jia-Liang Le ◽  
Zdeněk P. Bažant ◽  
Qiang Yu
Keyword(s):  
Size Effect ◽  
Crack Length ◽  
Interface Crack ◽  
Fiber Composite ◽  
Preceding Paper ◽  
Cohesive Crack ◽  
Crack Model ◽  
Metal Composite ◽  
Composite Joint ◽  
Nominal Strength

The effect of the size of hybrid metal-composite joint on its nominal strength, experimentally demonstrated in the preceding paper (part I), is modeled mathematically. Fracture initiation from a reentrant corner at the interface of a metallic bar and a fiber composite laminate sheet is analyzed. The fracture process zone (or cohesive zone) at the corner is approximated as an equivalent sharp crack according to the linear elastic fracture mechanics (LEFM). The asymptotic singular stress and displacement fields surrounding the corner tip and the tip of an interface crack emanating from the corner tip are calculated by means of complex potentials. The singularity exponents of both fields are generally complex. Since the real part of the stress singularity exponent for the corner tip is not −12, as required for finiteness of the energy flux into the tip, the interface crack propagation criterion is based on the singular field of the interface crack considered to be embedded in a more remote singular near-tip field of the corner from which, in turn, the boundaries are remote. The large-size asymptotic size effect on the nominal strength of the hybrid joint is derived from the LEFM considering the interface crack length to be much smaller than the structure size. The deviation from LEFM due to finiteness of the interface crack length, along with the small-size asymptotic condition of quasiplastic strength, allows an approximate general size effect law for hybrid joints to be derived via asymptotic matching. This law fits closely the experimental results reported in the preceding paper. Numerical validation according to the cohesive crack model is relegated to a forthcoming paper.

scholarly journals Size Effect of Cohesive Delamination Fracture Triggered by Sandwich Skin Wrinkling

2007 ◽  
Vol 74 (6) ◽  
pp. 1134-1141 ◽  
Author(s):  
Zdeněk Bažant ◽  
Peter Grassl
Keyword(s):  
Size Effect ◽  
Impact Analysis ◽  
Cohesive Crack ◽  
Crack Model ◽  
Element Analysis ◽  
Simple Description ◽  
Nominal Strength ◽  
Delamination Fracture ◽  
Special Cases ◽  
Asymptotic Matching

Because the observed size effect follows neither the strength theory nor the linear elastic fracture mechanics, the delamination fracture of laminate-foam sandwiches under uniform bending moment is treated by the cohesive crack model. Both two-dimensional geometrically nonlinear finite element analysis and one-dimensional representation of skin (or facesheet) as a beam on elastic-softening foundation are used. The use of the latter is made possible by realizing that the effective elastic foundation stiffness depends on the ratio of the critical wavelength of periodic skin wrinkles to the foam core thickness, and a simple description of the transition from shortwave to longwave wrinkling is obtained by asymptotic matching. Good agreement between both approaches is achieved. Skin imperfections (considered proportional to the the first eigenmode of wrinkling), are shown to lead to strong size dependence of the nominal strength. For large imperfections, the strength reduction due to size effect can reach 50%. Dents from impact, though not the same as imperfections, might be expected to cause as a similar size effect. Using proper dimensionless variables, numerical simulations of cohesive delamination fracture covering the entire practical range are performed. Their fitting, heeding the shortwave and longwave asymptotics, leads to an approximate imperfection-dependent size effect law of asymptotic matching type. Strong size effect on postpeak energy absorption, important for impact analysis, is also demonstrated. Finally, discrepancies among various existing formulas for critical stress at periodic elastic wrinkling are explained by their applicability to different special cases in the shortwave-longwave transition.

scholarly journals Size Effect in Fracture of Concrete Specimens and Structures: New Problems and Progress

10.14311/608 ◽  
2004 ◽  
Vol 44 (5-6) ◽  
Author(s):  
Z. P. Bažant ◽  
Q. Yu
Keyword(s):  
Size Effect ◽  
Fracture Energy ◽  
Damage Model ◽  
Apparent Size ◽  
Shear Capacity ◽  
Cohesive Crack ◽  
Smooth Transition ◽  
Crack Model ◽  
Cohesive Crack Model ◽  
Nominal Strength

Presented is a concise summary of recent Northwestern University studies of six new problems. First, the decrease of fracture energy during crack propagation through a boundary layer, documented by Hu and Wittmann, is shown to be captured by a cohesive crack model in which the softening tail slope depends on the distance from the boundary (which causes an apparent size effect on fracture energy and implies that the nonlocal damage model is more fundamental than the cohesive crack model). Second, an improved universal size effect law giving a smooth transition between failures at large cracks (or notches) and at crack initiation is presented. Third, a recent renewed proposal that the nominal strength variation as a function of notch depth be used for measuring fracture energy is critically examined. Fourth, numerical results and a formula describing the size effect of finite-angle notches are presented. Fifth, a new size effect law derivation from dimensional analysis coupled with asymptotic matching is given. Finally, an improved code-type formula for shear capacity of R.C. beams is proposed. 

Scaling of Strength of Metal-Composite Joints—Part III: Numerical Simulation

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Qiang Yu ◽  
Zdeněk P. Bažant ◽  
Jia-Liang Le
Keyword(s):  
Size Effect ◽  
Fracture Energy ◽  
Peak Load ◽  
Adhesive Layer ◽  
Cohesive Crack ◽  
Crack Model ◽  
Mixed Mode Fracture ◽  
Stress Profile ◽  
Metal Composite ◽  
Cohesive Stress

The size effect in the failure of a hybrid adhesive joint of a metal with a fiber-polymer composite, which has been experimentally demonstrated and analytically formulated in preceding two papers, is here investigated numerically. Cohesive finite elements with a mixed-mode fracture criterion are adopted to model the adhesive layer in the metal-composite interface. A linear traction-separation softening law is assumed to describe the damage evolution at debonding in the adhesive layer. The results of simulations agree with the previously measured load-displacement curves of geometrically similar hybrid joints of various sizes, with the size ratio of 1:4:12. The effective size of the fracture process zone is identified from the numerically simulated cohesive stress profile at the peak load. The fracture energy previously identified analytically by fitting the experimentally observed size effect curves agrees well with the fracture energy of the cohesive crack model obtained numerically by optimal fitting of the test data.

Stability of Cohesive Crack Model: Part II—Eigenvalue Analysis of Size Effect on Strength and Ductility of Structures

1995 ◽  
Vol 62 (4) ◽  
pp. 965-969 ◽  
Author(s):  
Z. P. Bazˇant ◽  
Yuan-Neng Li
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Size Effect ◽  
Preceding Paper ◽  
Stability Limit ◽  
Maximum Load ◽  
Cohesive Crack ◽  
Crack Model ◽  
Linear Softening ◽  
Strength And Ductility

The preceding paper is extended to the analysis of size effect on strength and ductility of structures. For the case of geometrically similar structures of different sizes, the criterion of stability limit is transformed to an eigenvalue problem for a homogeneous Fredholm integral equation, with the structure size as the eigenvalue. Under the assumption of a linear softening stress-displacement relation for the cohesive crack, the eigenvalue problem is linear. The maximum load of structure under load control, as well as the maximum deflection under displacement control (which characterizes ductility of the structure), can be solved explicitly in terms of the eigenfunction of the aforementioned integral equation.

Universal size effect of concrete specimens and effect of notch depth

2017 ◽  
Vol 3 (1) ◽  
pp. 47 ◽  
Author(s):  
Sıddık Şener ◽  
Kadir Can Şener
Keyword(s):  
Size Effect ◽  
Crack Length ◽  
Asymptotic Properties ◽  
Experimental Program ◽  
Three Point Bending ◽  
Large Size ◽  
Nominal Strength ◽  
Notch Length ◽  
Point Bend

The universal size effect law of concrete is a law that describes the dependence of nominal strength of specimens or structure on both its size and the crack (or notch) length, over the entire of interest, and exhibits the correct small and large size asymptotic properties as required. The main difficulty has been the transition of crack length from 0, in which case the size effect mode is Type 1, to deep cracks (or notches), in which case the size effect mode is Type 2 and fundamentally different from Type 1. The current study is based on recently obtained comprehensive fracture test data from three-point bending beams tested under identical conditions. In this test, the experimental program consisted of 80 three-point bend beams with 4 different depths 40, 93, 215 and 500mm, corresponding to a size range of 1:12.5. Five different relative notch lengths, a/D = 0, 0.02, 0.075, 0.15, 0.30 were cut into the beams. A total of 20 different geometries (family of beams) were tested. The present paper will use these data to analyze the effects of size, crack length. This paper presents a studying to improve the existing universal size effect law, named by Bazant, using the experimentally obtained beam strengths for various different specimen sizes and all notch depths. The updated universal size effect law is shown to fit the comprehensive data quite well.

scholarly journals Critical comparison of the boundary effect model with cohesive crack model and size effect law

2019 ◽  
Vol 215 ◽  
pp. 193-210 ◽  
Author(s):  
Christian Carloni ◽  
Gianluca Cusatis ◽  
Marco Salviato ◽  
Jia-Liang Le ◽  
Christian G. Hoover ◽  
...  
Keyword(s):  
Size Effect ◽  
Boundary Effect ◽  
Cohesive Crack ◽  
Crack Model ◽  
Cohesive Crack Model ◽  
Critical Comparison ◽  
Effect Model

Stability of Cohesive Crack Model: Part I—Energy Principles

1995 ◽  
Vol 62 (4) ◽  
pp. 959-964 ◽  
Author(s):  
Z. P. Bazˇant ◽  
Yuan-Neng Li
Keyword(s):  
Potential Energy ◽  
Crack Length ◽  
Crack Opening ◽  
Cohesive Crack ◽  
Crack Model ◽  
Cohesive Crack Model ◽  
Equilibrium Equations ◽  
Complementary Energy ◽  
Maximum Displacement ◽  
Derivatives Of

The paper deals with a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement. Under the assumption that no part of the crack undergoes unloading, the complementary energy and potential energy of an elastic structure which has a cohesive crack and is loaded by a flexible elastic frame is formulated using continuous influence functions representing compliances or stiffnesses relating various points along the crack. By variational analysis, in which the derivatives of the compliance or stiffness functions with respect to the crack length are related to the crack-tip stress intensity factors due to various unit loads, it is shown that the minimizing conditions reduce to the usual compatibility or equilibrium equations for the cohesive cracks. The variational equations obtained can be used as a basis for approximate solutions. Furthermore, the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy. They have the form of a homogeneous Fredholm integral equation for the derivatives of the cohesive stresses or crack opening displacements with respect to the crack length. Loadings with displacement control, load control, or through a flexible loading frame are considered. Extension to the analysis of size effect on the maximum load or maximum displacement are left to a subsequent companion paper.

Scaling of Strength of Metal-Composite Joints—Part I: Experimental Investigation

2009 ◽  
Vol 77 (1) ◽  
Author(s):  
Qiang Yu ◽  
Zdeněk P. Bažant ◽  
John Bayldon ◽  
Jia-Liang Le ◽  
Ferhun C. Caner ◽  
...  
Keyword(s):  
Size Effect ◽  
Peak Load ◽  
Maximum Load ◽  
Lap Joints ◽  
Interface Fracture ◽  
Metal Composite ◽  
Interface Failure ◽  
Interface Shear ◽  
Nominal Strength ◽  
The University

Knowledge of the size effect on the strength of hybrid bimaterial joints of steel and fiber composites is important for new designs of large lightweight ships, large fuel-efficient aircrafts, and lightweight crashworthy automobiles. Three series of scaled geometrically similar specimens of symmetric double-lap joints with a rather broad size range (1:12) are manufactured. The specimens are tested to failure under tensile displacement-controlled loading, and at rates that ensure the peak load to be reached within approximately the same time. Two series, in which the laminate is fiberglass G-10/FR4, are tested at Northwestern University, and the third series, in which the laminate consists of NCT 301 carbon fibers, is tested at the University of Michigan. Except for the smallest specimens in test series I, all the specimens fail by propagation of interface fracture initiating at the bimaterial corner. All the specimens fail dynamically right after reaching the maximum load. This observation confirms high brittleness of the interface failure. Thus, it is not surprising that the experiments reveal a marked size effect, which leads to a 52% reduction in nominal interface shear strength. As far as the inevitable scatter permits it to see, the experimentally observed nominal strength values agree with the theoretical size effect derived in Part II of this study, where the size exponent of the theoretical large-size asymptotic power law is found to be −0.459 for series I and II, and −0.486 for series III.

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