Continuum Theory for a Laminated Medium

1968 ◽  
Vol 35 (3) ◽  
pp. 467-475 ◽  
Author(s):  
C.-T. Sun ◽  
J. D. Achenbach ◽  
George Herrmann
Keyword(s):  
Strain Energy ◽  
Equations Of Motion ◽  
Continuum Theory ◽  
Laminated Composite ◽  
Harmonic Waves ◽  
Approximate Theory ◽  
Lagrangian Multipliers ◽  
Phase Velocities ◽  
Wave Numbers ◽  
The Matrix

A system of displacement equations of motion is presented, pertaining to a continuum theory to describe the dynamic behavior of a laminated composite. In deriving the equations, the displacements of the reinforcing layers and the matrix layers are expressed as two-term expansions about the mid-planes of the layers. Dynamic interaction of the layers is included through continuity relations at the interfaces. By means of a smoothing operation, representative kinetic and strain energy densities for the laminated medium are obtained. Subsequent application of Hamilton’s principle, where the continuity relations are included through the use of Lagrangian multipliers, yields the displacement equations of motion. The distinctive trails of the system of equations are uncovered by considering the propagation of plane harmonic waves. Dispersion curves for harmonic waves propagating parallel to and normal to the layering are presented, and compared with exact curves. The limiting phase velocities at vanishing wave numbers agree with the exact, limits. The lowest antisymmetric mode for waves propagating in the direction of the layering shows the strongest dispersion, which is very well described by the approximate theory over a substantial range of wave numbers.

Time-Harmonic Waves Traveling Obliquely in a Periodically Laminated Medium

1971 ◽  
Vol 38 (2) ◽  
pp. 477-482 ◽  
Author(s):  
C. Sve
Keyword(s):  
Dispersion Relation ◽  
Fourth Order ◽  
Continuum Theory ◽  
Numerical Results ◽  
Harmonic Waves ◽  
Two Dimensional ◽  
Arbitrary Direction ◽  
Arbitrary Angle ◽  
Phase Velocities ◽  
Time Harmonic

The dispersion relation is presented for time-harmonic waves propagating in an arbitrary direction in a periodically laminated medium. The analysis is based on two-dimensional equations of elasticity. Limiting phase velocities are presented for infinite wavelength for any angle of propagation in the form of a fourth-order determinant that illustrates the influence of an arbitrary angle. For the cases when the propagation is along or across the layers, the determinant reduces to two determinants of second order that yield the limiting phase velocities directly. Numerical results are presented to indicate the dependence of dispersion upon the angle of propagation. Also, a comparison with an approximate continuum theory is included; agreement is satisfactory for those angles where the dispersion is the strongest.

Theory of Laminated Plates

1971 ◽  
Vol 38 (1) ◽  
pp. 231-238 ◽  
Author(s):  
C. T. Sun
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Continuum Theory ◽  
Laminated Plates ◽  
Harmonic Waves ◽  
Two Dimensional ◽  
Laminated Plate ◽  
Governing Equations ◽  
Dimensional Theory ◽  
Rotatory Inertia

A two-dimensional theory for laminated plates is deduced from the three-dimensional continuum theory for a laminated medium. Plate-stress equations of motion, plate-stress-strain relations, boundary conditions, and plate-displacement equations of motion are presented. The governing equations are employed to study the propagation of harmonic waves in a laminated plate. Dispersion curves are presented and compared with those obtained according to the three-dimensional continuum theory and the exact analysis. An approximate solution for flexural motions obtained by neglecting the gross and local rotatory inertia terms is also discussed.

scholarly journals Efficient Improvement in Fracture Toughness of Laminated Composite by Interleaving Functionalized Nanofibers

Polymers ◽  
2021 ◽  
Vol 13 (15) ◽  
pp. 2509
Author(s):  
Seyed Mohammad Javad Razavi ◽  
Rasoul Esmaeely Neisiany ◽  
Moe Razavi ◽  
Afsaneh Fakhar ◽  
Vigneshwaran Shanmugam ◽  
...  
Keyword(s):  
Fracture Toughness ◽  
Fracture Behavior ◽  
Laminated Composite ◽  
Carbon Phase ◽  
Reinforced Composite ◽  
Reinforcing Agent ◽  
The Matrix ◽  
As Stress ◽  
Double Cantilever Beam Test ◽  
Pan Nanofiber

Functionalized polyacrylonitrile (PAN) nanofibers were used in the present investigation to enhance the fracture behavior of carbon epoxy composite in order to prevent delamination if any crack propagates in the resin rich area. The main intent of this investigation was to analyze the efficiency of PAN nanofiber as a reinforcing agent for the carbon fiber-based epoxy structural composite. The composites were fabricated with stacked unidirectional carbon fibers and the PAN powder was functionalized with glycidyl methacrylate (GMA) and then used as reinforcement. The fabricated composites’ fracture behavior was analyzed through a double cantilever beam test and the energy release rate of the composites was investigated. The neat PAN and functionalized PAN-reinforced samples had an 18% and a 50% increase in fracture energy, respectively, compared to the control composite. In addition, the samples reinforced with functionalized PAN nanofibers had 27% higher interlaminar strength compared to neat PAN-reinforced composite, implying more efficient stress transformation as well as stress distribution from the matrix phase (resin-rich area) to the reinforcement phase (carbon/phase) of the composites. The enhancement of fracture toughness provides an opportunity to alleviate the prevalent issues in laminated composites for structural operations and facilitate their adoption in industries for critical applications.

Effect of temperature on matrix multicracking evolution of C/SiC fiber-reinforced ceramic-matrix composites

2020 ◽  
Vol 39 (1) ◽  
pp. 189-199
Author(s):  
Longbiao Li
Keyword(s):  
Strain Energy ◽  
Ceramic Matrix Composites ◽  
Critical Value ◽  
Matrix Cracking ◽  
Interface Debonding ◽  
Matrix Composites ◽  
Temperature Dependent ◽  
Damage State ◽  
The Matrix ◽  
Sic Composites

AbstractIn this paper, the temperature-dependent matrix multicracking evolution of carbon-fiber-reinforced silicon carbide ceramic-matrix composites (C/SiC CMCs) is investigated. The temperature-dependent composite microstress field is obtained by combining the shear-lag model and temperature-dependent material properties and damage models. The critical matrix strain energy criterion assumes that the strain energy in the matrix has a critical value. With increasing applied stress, when the matrix strain energy is higher than the critical value, more matrix cracks and interface debonding occur to dissipate the additional energy. Based on the composite damage state, the temperature-dependent matrix strain energy and its critical value are obtained. The relationships among applied stress, matrix cracking state, interface damage state, and environmental temperature are established. The effects of interfacial properties, material properties, and environmental temperature on temperature-dependent matrix multiple fracture evolution of C/SiC composites are analyzed. The experimental evolution of matrix multiple fracture and fraction of the interface debonding of C/SiC composites at elevated temperatures are predicted. When the interface shear stress increases, the debonding resistance at the interface increases, leading to the decrease of the debonding fraction at the interface, and the stress transfer capacity between the fiber and the matrix increases, leading to the higher first matrix cracking stress, saturation matrix cracking stress, and saturation matrix cracking density.

scholarly journals Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

2021 ◽  
Author(s):  
Javier Bonet ◽  
Antonio J. Gil
Keyword(s):  
Kinetic Energy ◽  
Crack Propagation ◽  
Mathematical Models ◽  
Linear Elasticity ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Discontinuous Solutions ◽  
Linear Elasticity Theory ◽  
Intersonic Crack Propagation

AbstractThis paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

scholarly journals The Influence of Filler Size and Crosslinking Degree of Polymers on Mullins Effect in Filled NR/BR Composites

Polymers ◽  
2021 ◽  
Vol 13 (14) ◽  
pp. 2284
Author(s):  
Miaomiao Qian ◽  
Bo Zou ◽  
Zhixiao Chen ◽  
Weimin Huang ◽  
Xiaofeng Wang ◽  
...  
Keyword(s):  
Particle Size ◽  
Strain Energy ◽  
Mathematical Expression ◽  
Rubber Matrix ◽  
Mullins Effect ◽  
Butadiene Rubber ◽  
Crosslinking Degree ◽  
Test Range ◽  
Two Factors ◽  
The Matrix

Two factors, the crosslinking degree of the matrix (ν) and the size of the filler (Sz), have significant impact on the Mullins effect of filled elastomers. Herein, the result. of the two factors on Mullins effect is systematically investigated by adjusting the crosslinking degree of the matrix via adding maleic anhydride into a rubber matrix and controlling the particle size of the filler via ball milling. The dissipation ratios (the ratio of energy dissipation to input strain energy) of different filled natural rubber/butadiene rubber (NR/BR) elastomer composites are evaluated as a function of the maximum strain in cyclic loading (εm). The dissipation ratios show a linear relationship with the increase of εm within the test range, and they depend on the composite composition (ν and Sz). With the increase of ν, the dissipation ratios decrease with similar slope, and this is compared with the dissipation ratios increase which more steeply with the increase in Sz. This is further confirmed through a simulation that composites with larger particle size show a higher strain energy density when the strain level increases from 25% to 35%. The characteristic dependence of the dissipation ratios on ν and Sz is expected to reflect the Mullins effect with mathematical expression to improve engineering performance or prevent failure of rubber products.

scholarly journals Mesh size sensitivity analysis for interlaminar fracture of the fiber-reinforced laminated composites

2019 ◽  
Vol 14 ◽  
pp. 155892501988346 ◽  
Author(s):  
Fatih Daricik
Keyword(s):  
Crack Closure ◽  
Strain Energy ◽  
Three Dimensional ◽  
Laminated Composite ◽  
Mesh Size ◽  
Strain Energy Release ◽  
Virtual Crack Closure Technique ◽  
Closure Technique ◽  
Interlaminar Fracture ◽  
Element Size

The virtual crack closure technique is a well-known finite element–based numerical method used to simulate fractures and it suits well to both of two-dimensional and three-dimensional interlaminar fracture analysis. In particular, strain energy release rate during a three-dimensional interlaminar fracture of laminated composite materials can successfully be computed using the virtual crack closure technique. However, the element size of a numerical model is an important concern for the success of the computation. The virtual crack closure technique analysis with a finer mesh converges the numerical results to experimental ones although such a model may need excessive modeling and computing times. Since, the finer element size through a crack path causes oscillation of the stresses at the free ends of the model, the plies in the delaminated zone may overlap. To eliminate this problem, the element size for the virtual crack closure technique should be adjusted to ascertain converged yet not oscillating results with an admissible processing time. In this study, mesh size sensitivity of the virtual crack closure technique is widely investigated for mode I and mode II interlaminar fracture analyses of laminated composite material models by considering experimental force and displacement responses of the specimens. Optimum sizes of the finite elements are determined in terms of the force, the displacement, and the strain energy release rate distribution along the width of the model.

scholarly journals Matrix integrals & finite holography

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Dionysios Anninos ◽  
Beatrix Mühlmann
Keyword(s):  
Critical Exponents ◽  
Continuum Theory ◽  
Point Of View ◽  
Leading Order ◽  
Minimal Models ◽  
Point Evaluation ◽  
Brst Cohomology ◽  
Matrix Integrals ◽  
The Matrix ◽  
The Continuum

Abstract We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

Concise Equations for Rotor Dynamics Analysis

1995 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Memory Storage ◽  
The Other ◽  
Dynamics Analysis ◽  
Real Domain ◽  
The Matrix ◽  
System Equations ◽  
Ordering Algorithms ◽  
Matrix Bandwidth

Abstract Concise equations for rotor dynamics analysis are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two proposed ordering algorithms lead to more compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. This numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

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