A continuum elastic–plastic model for woven-fabric/polymer-matrix composite materials under biaxial stresses

In this study a simple continuum model for the macro-mechanical prediction of the elastic–plastic behavior of woven-fabric/polymer-matrix composites has been proposed. This model uses a scalar hardening parameter (which is a function of the current applied stress state) instead of an effective stress-strain relation to determine plastic strain increments. For simplicity, the stresses are expressed as invariants based on the material symmetry. It has been shown, by the use of experimental data for two different woven-fabric/polymer-matrix composite materials, that the newly proposed model accurately describes the non-linear mechanical behavior for different in-plane biaxial stress states ranging from pure shear to pure tension.     

Local stress–strain field intensity approach to fatigue life prediction under random cyclic loading

According to the characteristic of the local behavior of fatigue damage, on the basis of stress field intensity approach, a theory of local stress–strain field intensity for fatigue damage at the notch is developed in this paper, which can take account of the effects of the local stress–strain gradient on fatigue damage at the notch. In order to calculate the local stress–strain field intensity parameters, an incremental elastic-plastic finite element analysis under random cyclic loading is used to determine the local stress–strain response. A local stress–strain field intensity approach to fatigue life prediction is proposed by means of elastic-plastic finite element method for notched specimens. This approach is used to predict fatigue crack initiation life, and good correlation was observed with U-shape notched specimens for normalized 45 steel.     

Nonlinear Viscoelastic,Viscoplastic Characterization of Unidirectional GF/EP Composite

Viscoplastic strains of unidirectional continuous fiber composite (HEXCEL GF/EPprepreg system) are studied experimentally and theoretically. Creep and strainrecovery tests are used. Schapery's nonlinear viscoelastic viscoplasticconstitutive equations are used and generalized to describe inelastic behavior ofunidirectional composite under isothermal creep and strain recovery conditions. Themethodology to quantify the viscoplastic strains with respect to applied stress isproposed. Viscoplastic strains of composite are described by plastic shear strain inmaterial symmetry axis. Assumptions has been used and validated that the functiondescribing the stress and time dependence of viscoplastic strain can be presented asa product of two, time and correspondingly stress dependent, master curves.     

Stress–strain curves for steel-fiber reinforced concrete under compression

Steel-fiber reinforced concrete is increasingly being used day by day as a structural material. The complete stress–strain curve of the material in compression is needed for the analysis and design of structures. In this experimental investigation, an attempt has been made to generate the complete stress–strain curve experimentally for steel-fiber reinforced concrete for compressive strength ranging from 30 to 50 MPa. Round crimped fibers with three volume fractions of 0.5%, 0.75% and 1.0% (39, 59, and 78 kg/m³) and for two aspect ratios of 55 and 82 are considered. The effect of fiber addition to concrete on some of the major parameters namely peak stress, strain at peak stress, the toughness of concrete and the nature of the stress–strain curve is studied. A simple analytical model is proposed to generate both the ascending and descending portions of the stress–strain curve. There exists a good correlation between the experimental results and those calculated based on the analytical model. Equations are also proposed to quantify the effect of fiber on compressive strength, strain at peak stress and the toughness of concrete in terms of fiber reinforcing parameter.     

Relationship between lifetime under creep and constant stress rate for polymer-matrix composites

The present article reviews two existing theoretical approaches for creep failure criteria of viscoelastic materials. One criterion is based on the continuum damage mechanics (CDM) and the other is based on the fracture mechanics extended to viscoelastic materials. Although both theoretical frameworks are based on different physical concepts, the deduced lifetime expressions turn out to be equivalent even though its parameters have different physical interpretation. It is proved that both theoretical frameworks, when extended to variable stress loading cases, imply the linear cumulative damage (LCD) law. Additionally the relationship obtained between the creep–rupture and constant stress rate until failure is very simple. Moreover this simple relationship is obtained independently by two different cumulative damage laws, which do not obey the LCD law, and by experimental evidence using published data for two different polymer-matrix composites (PMC). Finally a micromechanical model, used for creep–rupture of unidirectional composites, is extended for constant stress rate until failure to corroborate the simple relationship obtained between the creep–rupture and constant stress rate until failure.     

Validation of full 3D and equivalent tape laminate modeling of plasticity induced non-linearity in 2 × 2 braided composites

Fiber reinforced polymer matrix braided composites exhibit considerable non-linear response. Plasticity induced non-linear behavior of 2 × 2 braided composites was investigated using a two scale finite element modeling approach based on Hill’s yield function for orthotropic materials. The analysis was validated by comparing the macroscopic stress–strain predictions for a variety of 2 × 2 biaxial braids with experimental data. Experimental results were also compared with equivalent tape laminate analyses, which required only three elements and much less computational time. All the braids show considerable plasticity induced non-linearity. Predictions of both full 3D and equivalent tape laminate plastic analyses agree reasonably well with the experiments. Performance of braided composites was compared with tape laminates and it was seen that equivalent tape laminates have a better performance than their braid counterparts in terms of longitudinal modulus, but in terms of percentage moduli degradation due to plasticity, both braids and tapes have similar performance.     

Thermal-stress concentration near inclusions in viscoelastic random composites

The determination of thermal-stress concentrations near inclusions in viscoelastic random composites is concerned with the prediction of the overall response of random nonlinear viscoelastic multi-component media. The continuum considered here is assumed to be subjected to a finite deformation. First Piola’s stress tensor and deformation gradient are used as conjugate field variables in a fixed reference state. A nonlinear problem is investigated in a second-order approximation theory when the gradient deformation terms higher than second order are neglected. A convex potential function in a thermo-elastic problem and time functionals in a viscoelastic one are used to construct overall constitutive relations. The technique of surface operators developed by R. Hill and others is used to determine stress concentrations near inclusions for nonlinear matrix creep.     

Models of viscoplasticity some comments on equilibrium (back) stress and drag stress

Summary Phenomenological and microstructural motivations for the terms appearing in the title are found in a literature survey. Although the interpretations differ with various investigators a strong tendency is observed to consider plastic flow as rate dependent. It is stated that plastic strain takes time to develop and the existnce of an equilibrium stress is postulated at which plastic strain is fully developed. It is similar to the back stress used in materials science. The drag stress introduced from microdynamical studies performs the same function as the isotropic variable in plasticity. Most of the theories that describe the transient and steady-state behavior of metallic alloys make the inelastic strain rate a function of the over (effective) stress. It is shown that this concept has considerable advantages in the modeling of changes of viscous (time- or rate-dependent) and plastic (time- or rate-independent) contributions to hardening that are observed in cyclic loading and dynamic plasticity.With 4 Figures     

Comparison of the effects of hydrostatic pressure and the third invariant of deviatoric stress tensor on non-linear viscoelastic deformation in cellulose nitrate

The behaviour of polymers is qualitatively known to be remarkably influenced by the hydrostatic pressure on the general plastic deformation, unlike most metals. However, as polymers behave differently in simple tension and compression, they may be influenced by the effect of the third invariant of deviatoric stress tensor as well as hydrostatic pressure. In this paper, a comparison of the effects of the hydrostatic pressure and the third invariant of deviatoric stress tensor on the non-linear viscoelastic deformation of cellulose nitrate are discussed quantitatively, following experiments of torsion of tubular specimens and simple tension of uniaxial specimens. As the result, the effect of the third invariant of deviatoric stress tensor on the non-linear viscoelastic deformation in cellulose nitrate was found to be much smaller than that of the hydrostatic pressure.     

Werkstoffanstrengung bei zeitabhängiger Spannung-Verformung-Beziehung

Equivalent Stress and Strain for Time dependent Materials Behaviour This paper deals with the generalized plastomechanics and the flow behavior of materials whose volume does mot remain constant during plastic deformation. The usual assumption of plastic incompressibility is treated as a special case. The relations deduced are formally applied to creep of polymers with nonlinear stress-strain behaviour. The concept of equivalent stress and strain is examined for this group of plastics and compared with experimental results under combined stress. A modified superposition principle and a power function of time are employed to describe creep in the nonlinear range under abrupt changes in state of combined stress. The usual assumption of linear viscoelastic analogy of Hookes law is treated as a special case.     

Finite element modeling of transient ultrasonic waves in linear viscoelastic media

Linear viscoelasticity offers a minimal framework within which to construct a causal model for wave propagation in absorptive media. Viscoelastic media are often described as media with `fading memory,' that is, the present state of stress is dependent on the present strain and the complete time history of strain convolved with appropriate time-dependent shear and bulk stress relaxation moduli. An axisymmetric, displacement-based finite element method for modeling pulsed ultrasonic waves in linear, homogeneous, and isotropic (LHI) viscoelastic media is developed that does not require storage of the complete time history of displacement at every node. This is accomplished by modeling stress relaxation moduli as discrete or continuous spectra of decaying exponentials and relaxation times. Details of the construction and computation of the time-dependent stiffness matrix are presented. As an application of the finite element method, a finite number of exponentials (amplitudes and relaxation times) are employed to represent a typical model for a continuous relaxation spectrum. It is demonstrated that a small number of discrete exponentials are required to model ultrasonic wave propagation of a typical band-limited pulse in a model material accurately. Previous work has shown this model to be consistent with other analytic models for wave propagation in viscoelastic media     

A micro-to-meso sublaminate model for the viscoelastic analysis of thick-section multi-layered FRP composite structures

Classical ply-by-ply analysis of multi-layered thick-section composite structures with tens of layers through the cross-section is often impractical, especially when material nonlinearity and time-dependent effects are included. This study introduces an integrated micromechanical-sublaminate modeling approach for the nonlinear viscoelastic analysis of thick-section and multi-layered composite structures. The sublaminate model is used to generate three-dimensional (3D) effective nonlinear responses at through-thickness material integration points with given spatial variations of strains determined from the trial strain increments of the standard displacement-based finite-element (FE). The number of material integration points is determined by the resolution of the FE discretization of the composite structure. The sublaminate model at a selected material point represents the effective nonlinear continuum behavior in its neighborhood using the 3D lamination theory with uniform in-plane strain and out-of-plane stress patterns through the representative layers. Therefore, the sublaminate has first-order stress and strain paths and cannot recognize the local sequence of the layers. While this approach is very effective approximation especially in the case of a very large number of repeating layers using relatively few elements (integration points) through the thickness, it cannot be used to represent the interlaminar stresses or bending/extension/twisting coupling effects within a sublaminate. A previously developed micromechanical model by the authors for a nonlinear viscoelastic unidirectional lamina is used for each layer in the sublaminate. The proposed modeling approach is first calibrated and verified against creep tests on off-axis glass/epoxy performed by Lou and Schapery (J. Compos. Mater. 5:208–271, 1971). Analyses for different thick-section laminated structures are presented using the integrated sublaminate with both shell and 3D continuum elements. The proposed 3D nonlinear time-dependent sublaminate model is computationally efficient and robust in analyzing multi-layered composite structures having large number of plies.     

A plasticity model for calculating stress–strain sequences under multiaxial nonproportional cyclic loading

There is increasing demand for analytical methods that estimate the fatigue life of engineering components and structures with a high degree of accuracy. The fatigue life is determined by the stress–strain sequences at the critical locations. Therefore, these sequences have be calculated with sufficient accuracy for arbitrary nonproportional cyclic loading. Based on the experience with a variety of material models following macroscale continuum mechanics approaches, an improved set of constitutive equations is proposed. The stress–strain behaviour of a commercial structural steel has been investigated experimentally. Firstly, the results of this experimental study serve to identify the material parameters comprised in the model. Secondly, the predicted stress–strain paths are compared to their experimentally determined counterparts as well as to paths predicted by other models. The overall accuracy of the proposed model is quite satisfying, especially as far as calculated amplitudes are concerned.     

Strain Measures for Rigid Crushable Foam in Uniaxial Compression

Abstract:  Progressive collapse deformation mechanisms in Rohacell-51WF foam during uniaxial compression has been studied. Measures of a macroscopic engineering strain are identified. The elastic and plastic parts of a macroscopic engineering strain can be predicted by using the compression failure strain, lock-up strain, and time dependent elastic and plastic parts of lock-up strain, which are material parameters. Identification of strain measures in a uniaxial compression test is essential to get material parameters for an elastoplastic model. The viscoelastic recovery property of Rohacell-51WF foam is also described.     

Prediction of local hygroscopic stresses for composite structures – Analytical and numerical micro-mechanical approaches

The aim of this article is to propose an analytical micro-mechanical self-consistent approach dedicated to mechanical states prediction in both the fiber and the matrix of composite structures submitted to a transient hygroscopic load. The time and space dependent macroscopic stresses, at ply scale, are determined by using continuum mechanics formalism. The reliability of the new approach is checked, for carbon–epoxy composites, through a comparison between the local stress states calculated in both the resin and fiber according to the new closed-form solutions and the equivalent numerical model.     

Mechanical behaviour and formulation of stress-strain relation for non-linear viscoelastic cellulose nitrate under cyclic loadings

The cyclic deformations under various repeated stresses are quantitatively investigated using non-linear viscoelastic cellulose nitrate heated to 60° C. The non-elastic strain or creep-plastic strain is remarkably influenced by the repeated stress and the stress rate. The cyclic deformations corresponding to the repeated stress less than a certain stress level attain the saturated state called the shake down after some cycles. The stress-strain relations of the non-linear viscoelastic media in the loading and unloading processes are deduced from the invariant theory using an hypothesis of creep potential. The non-linear viscoelastic observations obtained on the cellulose nitrate at 60° C under cyclic loadings are found to fit the deduced relations for the loading and unloading processes independent of the repeated stress and the stress rate.     

Constitutive laws for ice

A theoretical discussion of models which describe the transient and secondary creep response of polycrystalline ice is presented, including a hypothesis which incorporates temperature dependence in the rate laws by the introduction of a reduced time scale. The secondary response is described by a general non-linear incompressible viscous fluid law, and it is shown that bi-axial stress experiments are insufficient but combined shear and compression experiments are sufficient to determine, in principle, the general response functions. Transient creep can be described qualitatively by a viscoelastic fluid model, and the most simple material memory influence is given by dependence on the current creep acceleration which leads to a first order differential relation between stress and strain rate. The secondary creep response is incorporated as a steady asymptotic limit with the time scale of significant transient creep governed by the response coefficients. Quantitative tests of such transient response require data from experiments at short time intervals, and in particular the determination of an initial strain rate to complement the differential law.     

The effect of annealing on the elastoplastic and viscoelastic responses of isotactic polypropylene

Observations are reported on isotactic polypropylene (i) in a series of tensile tests with a constant strain rate on specimens annealed for 24 h at various temperatures in the range from 110 to 150 °C, (ii) in two series of creep tests in the subyield region of deformations on samples not subjected to thermal treatment and on specimens annealed at 140 °C, and (iii) in a series of tensile relaxation tests on non-annealed specimens. Constitutive equations are derived for the elastoplastic and non-linear viscoelastic responses of semicrystalline polymers. A polymer is treated as an equivalent transient network of macro-molecules bridged by junctions (physical cross-links, entanglements and lamellar blocks). The network is assumed to be highly heterogeneous, and it is thought of as an ensemble of meso-regions with different activation energies for separation of strands from temporary nodes. The elastoplastic behavior is modelled as sliding of junctions in meso-domains with respect to their reference positions driven by macro-deformation. The viscoelastic response is attributed to detachment of active strands from temporary junctions and attachment of dangling chains to the network. Constitutive equations for isothermal deformations with small strains are derived by using the laws of thermodynamics. Adjustable parameters in the stress–strain relations are found by fitting the experimental data.     

Plastic instabilities analysis during T-shaped tubes hydro-forming process

During T-shaped or Y-shaped tubes hydroforming the tube wall is flatten on the external die. This “in die” forming involves an important compressive stress along the outer normal direction directly related to the inflating pressure. In this particular case plastic instabilities and damage evolution are strongly dependent on the sign of hydrostatic stress. Hydroforming simulations of T-shaped and Y-shaped thick tubes are performed using an elastoplastic model implemented in Forge-2005 (fully implicit Finite Element Method code). A three dimensional analysis is ran using tetrahedral solid elements. An augmented Modified Maximum Force Criterion coupled to an undamage behavior law is expressed in the sheet tangent plan basis. This criterion is compared to a fully coupled Lemaitre damage model. Due to low triaxiality state, damage only growth in specifical regions. This study highlights the interest of continuum damage modeling for the formability and the final part load carrying capacity prediction for “in die” hydroforming.     

Description of depth-dependent nonlinear viscoelastic behavior for articular cartilage in unconfined compression

An optimized digital image correlation (DIC) technique was applied to investigate the depth-dependent nonlinear viscoelastic properties of articular cartilage and simultaneously the biphasic nonlinear viscoelastic relaxation model of cartilage was proposed and validated. The stress relaxation tests were performed with different strain levels and it is found that the initial stress and relaxed stress at any time increase with increasing strain levels. The depth-dependent strain of cartilage was obtained by analyzing the images acquired using the optimized DIC technique and moreover the inhomogeneous relaxation modulus distributions within the tissues were determined at different relaxation time points under strain of 11.35, 19.35 and 30% respectively. The strain rate dependent nonlinear stress and strain curves were obtained for articular cartilage through uniaxial compression tests. It is noted that the Young's modulus exhibits a slight increase near the cartilage surface, and then increases fast with depth and both the magnitude and the variation of the Young's modulus are affected by increasing strain rates. A biphasic nonlinear viscoelastic relaxation model was proposed to predict the depth-dependent relaxation behavior of cartilage under unconfined compression and the results show that there are good agreements between the experimental data and predictions.