Softening Functions Research Articles

Assessment of the tensile strength through size effect curves

Abstract When assessing the tensile strength of a cohesive material for a wide range of test geometries, the change of peak loads as a function of size — the well-known size effect — is inconclusive to determine the initial behavior of the softening curve and, in particular, the true value of the tensile strength ft. Two different softening functions, such as linear and trapezoidal, can give the same accurate predictions for the maximum load, but different values of ft must be used. Brazilian splitting tests exhibit the same behavior and are less dependent on the initial shape of the softening curve, so this kind of test is recommended to assess the tensile strength. To know the actual softening curve, one needs additional experimental information, such as the load–displacement curve or the load–CMOD curve.

Modified Fracture Mechanics Approach in Structural Analysis of Solid-Rocket Motors

A deŽ ciency in current failure initiation criteria used in the structural design and service-life prediction of solid-rocket motors, where the critical failure mode is structural failure, is the uncertainties associated with the criteria used in predicting crack initiation and propagation. Also, the constitutive models that describe the propellants’ nonlinear behavior require an extensive experimental testing program to characterize the softening functions. A modiŽ ed fracture mechanics method that accounts for bulk inelastic behavior in the calculation of a critical strain energy release rate and a relatively simple method for nonlinear viscoelastic analysis, where a three-dimensional interpolation scheme is used to solve the Prony series equations that represent Young’s and shear relaxation moduli as functions of time, temperature, and strain level, have been developed and implemented into a Ž nite element code. Predictions for a thermal shock loading on an end-burning research and development motor were made using the fracture mechanics and nonlinear viscoelastic analysis capabilities developed in this study. Reasonable agreement between the measured stresses from instrumented rocket motors and the predicted von Mises stresses was obtained. The regions of high propensity for crack propagation corresponded to the critical regions of maximum tensile principal stress in the motor.

Constitutive Model for Concrete in Strain Space

Concrete exhibits significant strain softening beyond the peak or initial failure stress. To incorporate this behavior and also reflect the entire failure process, a complete constitutive model is proposed within the framework of strain‐space plasticity, based on available experimental data and the known behavioral characteristics of concrete. It includes the initial yield and failure surfaces and the specific elastoplastic stress‐strain relationships in each stage, as well as the loading criterion and the hardening and softening functions. Criteria for multiaxial states of stress and strain are used to identify the final failure mode as either cracking, crushing, or mixed type. Based on the proposed constitutive model, a computer program for the three‐dimensional analysis of concrete structures was coded. In this program, a mixed‐iterative procedure is employed that combines variable stiffness and initial stress iterations. Two numerical examples demonstrate the proposed model's ability to reflect the elastic, elastoplastic hardening and softening, and final failure behavior of concrete.

Stepped softening functions for concrete fracture in finite element analysis

Abstract This paper presents the findings of a study on the use of stepped softening functions in the analysis of concrete fracture. Four different softening functions are studied and the results of the analysis of six experiments compared. The study shows that, for a certain type of stepped function, a significant improvement in both the stability and efficiency of solutions is achieved over those obtained with a standard bilinear function. Furthermore the study demonstrates that the same type of stepped function gives good mesh convergence.

Microscopically based constitutive relations for damage mechanics and creep-plasticity interaction

In this paper we put forward a new formulation of constitutive relations based on micromechanical analysis to describe the behavior of damaged materials and the creep-plasticity interaction effects at high temperature. The constitutive theory of Continuum Damage Mechanics (CDM) previously proposed by Fan Jinghong has been further developed on the foundation of large deformation endochronic plasticity and an elaborated analysis of the mechanical behavior of a microelement of damaged material in which different damage mechanisms are involved. The softening functions, obtained by the comparative study of a microelement with and without a hole, have been introduced into the macroscopic constitutive equations to model the damage effects on material response. The comparison between the calculated results, based on the new constitutive description and corresponding finite element code, and the experimental data of carefully conducted tests on a notched specimen, has shown reasonable agreement. In this paper we also put forward microscopically based constitutive relations to describe the important creep-plasticity interaction for transient creep and steady-state creep. The theoretical analysis developed here has been compared with Ohashi's experimental results. It seems that the present theory could satisfactorily describe the creep-plasticity effects reported.

Analysis of Mixed‐Mode Fracture in Concrete

A smeared crack model that covers tensile softening in mode I and shear softening in mode II fracture is described. In addition, the model provides for unloading and reloading and for multiple crack formation. Particular forms of tension and shear softening functions and relations with more conventional models are discussed. Two examples involving mixed‐mode fracture in notched, unreinforced concrete beams have been included to demonstrate the versatility of the model. The results indicate that the addition of shear softening is essential to obtain realistic results in the post‐peak regime since the classical approach based on a constant shear retention factor then results in a too stiff behavior. The results furthermore demonstrate that snap‐back behavior may occur in strain‐softening concrete under quasistatic loading conditions. Attention is also given to the possibility of hour‐glass formation when constitutive laws involving softening are deployed in a finite element model.

Viscoplasticity of Transversely Isotropic Clays

A viscoplastic constitutive relation of the endochronic type (i.e., the inelastic strain increments are characterized by an intrinsic time) is formulated to describe the behavior of transversely isotropic clays produced by one-dimensional consolidation. The formulation contains eight material parameters in addition to those needed for isotropic clays. The hardening and softening functions and the densification-dilatancy function are assumed to be given by the same expressions previously found for isotropic clays, but the invariants involved in these expressions are replaced by the proper transversely isotropic invariants. The pore pressure is determined from the volume change and the compressibility of the water, and the constitutive relation is written in terms of the effective stresses. The elastic moduli are assumed to be functions of hydrostatic stress and inelastic dilatancy, and they are correlated with the consolidation stress. Experimental curves of axial strain for various anisotropically consolidated clays have been fit by a time-independent version of the theory, and a satisfatory agreement has been achieved.

Constitutive Law for Nonlinear Creep of Concrete

Endochronic theory, previously proposed and verified for multiaxial time-independent experimental data, is extended to nonlinear long-time creep at high stress and is compared with available uniaxial creep data. The extension is based on a Maxwell chain model, each unit of which is characterized by its own intrinsic time, an independent variable whose increments depend both on time and strain increments. The dependence on the latter involves the previously determined hardening and softening functions. Aging is included and the previously established Maxwell chain model for low-stress creep with aging is obtained as a special case. The theory also describes the decrease of strength with load duration when the compression is high, gives an increase when the compression is low, and yields the additional inelastic strain accumulation due to cyclic load. An effective and numerically stable algorithm for timestep integration of structural response, permitting the time steps to be increased with the load duration, is presented.