Arbitrary Loading Histories Research Articles

Estimation of the long-term strength of polymer materials under arbitrary loading histories

A nonlinear version of the phenomenological theory of long-term strength of polymer materials (viscoelastic bodies) is proposed. It is based on the introduction of a function accounting for the damage accumulation connected with changes in the load intensity. The form of this function may be determined from the results of testing the material with a load changing with time in a certain way, for instance, periodically. As a parameter, the function contains the rate of the changing load or the frequency for periodic loads. For a quasi-isotropic material, the basic relationships of the theory proposed are generalized to the case of combined stresses. The durability (failure time) calculations of the material based on this theory are compared with experimental data for a number of polymer and composite materials in a wide range of loading modes.

Effective Elastoplastic Algorithms for Ductile Matrix Composites

A three-dimensional computational algorithm is proposed to implement the micromechanical framework derived by the writers. The proposed formulation is capable of predicting effective elastoplastic behavior of two-phase particle reinforced ductile matrix composites (PRDMCs) containing may randomly dispersed interacting elastic spheres. A strain-driven algorithm is presented to determine the stress for a given strain by using the two-step operator splitting methodology applicable to any arbitrary loading and unloading histories. Furthermore, continuum and consistent tangent moduli are derived in closed form to facilitate the assembly of the global tangent stiffness. Extension is also made to predict effective rate-dependent elasto-viscoplastic behavior of particle-reinforced ductile matrix composites. Finally, finite-element implementation and numerical examples are presented.

Life Prediction Methodology for Composite Structures. Part I—Constant Amplitude and Two-Stress Level Fatigue

A strength-based wearout model is presented for predicting the residual strength and life of composite structures subjected to constant amplitude or two-stress level fatigue loadings. It is assumed that the structure undergoes proportional loading, that its residual strength is a monotonically decreasing function of the number of loading cycles, and that both the life distribution due to continuous constant amplitude cycling and the residual strength distribution after an arbitrary load history may be represented by two parameter Weibull functions. The model also incorporates a "cycle mix factor" to account for the degradation of strength and life that may be associated with frequent changes in the stress amplitude of the loading. Experimental results from two-stress level, uniaxially loaded laminates are used to evaluate the model's predictive capability for sequencing effects, as in a low-high or high-low test, and for cycle mix effects, as in a low-high-low-high repeating test. Good correlation between theory and experiment is obtained for all loadings and laminates studied.

A yield surface and flow rule for orientationally hardening polymers subjected to arbitrary deformations

The phenomenological anisotropic flow rule and yield surface introduced previously were shown to reproduce fairly well the orientational hardening behavior of the glassy polymer polycarbonate. Rigidplastic, rate independent behavior was assumed, and attention was restricted to loading histories in which the principal axes of stress and strain remained fixed. Here, this previous work is extended to the case of arbitrary, albeit still rate independent, elastic-plastic finite deformations. Specifically, arbitrary loading histories are considered, and formulae for modeling the resulting elastic and plastic anisotropy development are presented. Additional formulae, required for obtaining linear time-incremental numerical solutions using the theory, also are derived.

Time dependent degradation of fiber/matrix interfaces under fatigue of SCS-6/TIMETAL 21S : Blatt, D., Jara, J.R. and Larsen, J.M. Script Metall. Mater. (1995) 33 (6), 939–944

In the present study, an analytical model of fatigue crack growth based on the plastic-blunting mechanism is developed. Both the plastic and reversed plastic flow during the process of cycling are considered. In each loading cycle, the size of the plastic and reversed plastic zone, the crack-tip opening displacement, and the stress intensity factor at crack-closure are determined. With an assumption of a crack-extension law, the crack-growth behavior under arbitrary load history is traced by making a step-by-step analysis. For the crack growing under constant-amplitude loading, the analysis of the steady-state crack-growth behavior is carried out with a simplified formulation. Regarding the crack-growth behavior near the fatigue threshold, although all of the near-threshold mechanisms are not incorporated into the model, the promoted crack-closure due to corrosion debris and surface roughness is introduced in terms of a material parameter CTOD∗. Under constant-amplitude loading, the results from the step-by-step analysis demonstrate that the closure level of the stress intensity considerably increases during the growth process due to accumulation of the residual strains. This may in part provide an explanation for the arrest a crack at a distance far from the root of a sharp notch, which is frequently observed in experiments. After a sufficiently large number of cycles, the crack-growth behavior from the step-by-step analysis reaches the steady state, and the results agree with the ones from the steady-state analysis. The crack growth law of second power is derived as a consequence of the assumed linear relation between Δa and ΔCTOD. The crack growth curves from the analysis agree with the ones obtained from experiments for Al 2024-T3 and Al 7075-T6. the ratio of Δa to ΔCTOD chosen for the calculation is in agreement with the direct, experimental measurement of Δa and ΔCTOD. With the presence of CTOD∗, the results show that the crack-growth behavior at low load changes drastically, and for each value of CTOD∗ a threshold exists below which no crack growth occurs at all. It is shown that under the action of variable-amplitude loading, the crack-growth, behavior from the model exhibits a transient behavior upon altering the loading condition. The study shows that the residual plastic deformation in the wake of an advancing crack has a large influence on the crack-growth behavior. The results also lend a theoretical support to the existence of the controversial plasticity-induced crack-closure under plane-strain conditions.

On Microcracking in Composite Laminates under Thermal and Mechanical Loading

Composite laminates used in many applications are exposed to both thermal and mechanical loads. These loads can cause cracks in the matrix, changing the laminates’ properties. An analytical methodology is developed to predict microcrack density in a general laminate exposed to an arbitrary thermomechanical load history. The analysis uses a shear lag stress solution in conjunction with an energy-based cracking criterion. It is found that crack densities depend on the stress in the cracking ply group, regardless of the source of the stress. Crack densities are a nonlinear function of stress, however, so if both thermal and mechanical loads are present, the cracking response is coupled. Experimental investigation is used to verify the analysis. Correlation between analysis and experiment is generally excellent. The method successfully predicts the initiation of cracking and crack accumulation under both thermal and mechanical loads, in most ply groups in a variety of laminates, using a single set of ply material properties. The cases where the analysis and experiment do not agree shed some light on the limits of current modelling techniques. Given an understanding of these limits, the analysis appears to be a useful preliminary design tool.

On Microcracking in Composite Laminates under Thermal and Mechanical Loading

Composite laminates used in many applications are exposed to both thermal and mechanical loads. These loads can cause cracks in the matrix, changing the laminates’ properties. An analytical methodology is developed to predict microcrack density in a general laminate exposed to an arbitrary thermomechanical load history. The analysis uses a shear lag stress solution in conjunction with an energy-based cracking criterion. It is found that crack densities depend on the stress in the cracking ply group, regardless of the source of the stress. Crack densities are a nonlinear function of stress, however, so if both thermal and mechanical loads are present, the cracking response is coupled. Experimental investigation is used to verify the analysis. Correlation between analysis and experiment is generally excellent. The method successfully predicts the initiation of cracking and crack accumulation under both thermal and mechanical loads, in most ply groups in a variety of laminates, using a single set of ply material properties. The cases where the analysis and experiment do not agree shed some light on the limits of current modelling techniques. Given an understanding of these limits, the analysis appears to be a useful preliminary design tool.

Microstructural examination of fatigue crack tip in high strength steel: Fukuoka, C., Yoshizawa, H., Nakagawa, Y.G. and Lapides, M.E. Metall. Trans. A (Oct. 1993) 24A (10), 2209–2216

In the present study, an analytical model of fatigue crack growth based on the plastic-blunting mechanism is developed. Both the plastic and reversed plastic flow during the process of cycling are considered. In each loading cycle, the size of the plastic and reversed plastic zone, the crack-tip opening displacement, and the stress intensity factor at crack-closure are determined. With an assumption of a crack-extension law, the crack-growth behavior under arbitrary load history is traced by making a step-by-step analysis. For the crack growing under constant-amplitude loading, the analysis of the steady-state crack-growth behavior is carried out with a simplified formulation. Regarding the crack-growth behavior near the fatigue threshold, although all of the near-threshold mechanisms are not incorporated into the model, the promoted crack-closure due to corrosion debris and surface roughness is introduced in terms of a material parameter CTOD∗. Under constant-amplitude loading, the results from the step-by-step analysis demonstrate that the closure level of the stress intensity considerably increases during the growth process due to accumulation of the residual strains. This may in part provide an explanation for the arrest a crack at a distance far from the root of a sharp notch, which is frequently observed in experiments. After a sufficiently large number of cycles, the crack-growth behavior from the step-by-step analysis reaches the steady state, and the results agree with the ones from the steady-state analysis. The crack growth law of second power is derived as a consequence of the assumed linear relation between Δa and ΔCTOD. The crack growth curves from the analysis agree with the ones obtained from experiments for Al 2024-T3 and Al 7075-T6. the ratio of Δa to ΔCTOD chosen for the calculation is in agreement with the direct, experimental measurement of Δa and ΔCTOD. With the presence of CTOD∗, the results show that the crack-growth behavior at low load changes drastically, and for each value of CTOD∗ a threshold exists below which no crack growth occurs at all. It is shown that under the action of variable-amplitude loading, the crack-growth, behavior from the model exhibits a transient behavior upon altering the loading condition. The study shows that the residual plastic deformation in the wake of an advancing crack has a large influence on the crack-growth behavior. The results also lend a theoretical support to the existence of the controversial plasticity-induced crack-closure under plane-strain conditions.

Algorithm for Hysteresis Analysis of Prestressed‐Concrete Frames

The descending part of the hysteretic load-deformation history of a structure with softening behavior discloses important structural properties such as strength, ductility, residual deformation, hysteresis loop shape, and energy-dissipation capacity, especially under earthquake conditions. In the present paper, a finite-element algorithm for hysteresis analysis of bonded and unbonded prestressed-concrete frames subjected to arbitrary loading history including postultimate descending portion is put forward. One-dimensional constitutive models for concrete, tendon, and reinforcing steel are proposed. The crack-contact effect is included in the constitutive model for concrete. A bond-slip relation between tendon and grout is suggested. A method for implementing the bond-slip relation into the element in which tendons are imbedded is also put forward. Being of self-correcting incremental type, the algorithm incorporates both increment-force and increment-displacement procedures whose transition from one into the other is automatically performed according to the variations of the hysteresis curve. The results of laboratory tests of three prestressed-concrete frames are reported. Comparison of computed results with experimental ones shows good agreement.

Experimental study and BEM analysis of fatigue crack growth in a cracked heterogeneous body: Sun, D.-C.,and Ma, X.-R. Acta Mechanica Solida Sinica 1992 5, (1), 1–12

In the present study, an analytical model of fatigue crack growth based on the plastic-blunting mechanism is developed. Both the plastic and reversed plastic flow during the process of cycling are considered. In each loading cycle, the size of the plastic and reversed plastic zone, the crack-tip opening displacement, and the stress intensity factor at crack-closure are determined. With an assumption of a crack-extension law, the crack-growth behavior under arbitrary load history is traced by making a step-by-step analysis. For the crack growing under constant-amplitude loading, the analysis of the steady-state crack-growth behavior is carried out with a simplified formulation. Regarding the crack-growth behavior near the fatigue threshold, although all of the near-threshold mechanisms are not incorporated into the model, the promoted crack-closure due to corrosion debris and surface roughness is introduced in terms of a material parameter CTOD∗. Under constant-amplitude loading, the results from the step-by-step analysis demonstrate that the closure level of the stress intensity considerably increases during the growth process due to accumulation of the residual strains. This may in part provide an explanation for the arrest a crack at a distance far from the root of a sharp notch, which is frequently observed in experiments. After a sufficiently large number of cycles, the crack-growth behavior from the step-by-step analysis reaches the steady state, and the results agree with the ones from the steady-state analysis. The crack growth law of second power is derived as a consequence of the assumed linear relation between Δa and ΔCTOD. The crack growth curves from the analysis agree with the ones obtained from experiments for Al 2024-T3 and Al 7075-T6. the ratio of Δa to ΔCTOD chosen for the calculation is in agreement with the direct, experimental measurement of Δa and ΔCTOD. With the presence of CTOD∗, the results show that the crack-growth behavior at low load changes drastically, and for each value of CTOD∗ a threshold exists below which no crack growth occurs at all. It is shown that under the action of variable-amplitude loading, the crack-growth, behavior from the model exhibits a transient behavior upon altering the loading condition. The study shows that the residual plastic deformation in the wake of an advancing crack has a large influence on the crack-growth behavior. The results also lend a theoretical support to the existence of the controversial plasticity-induced crack-closure under plane-strain conditions.

Fatigue crack propagation through residual stress fields with closure phenomena: Beghini, M. and Bertini, L. Eng. Fract. Mech. 1990 36 (3), 379–387

In the present study, an analytical model of fatigue crack growth based on the plastic-blunting mechanism is developed. Both the plastic and reversed plastic flow during the process of cycling are considered. In each loading cycle, the size of the plastic and reversed plastic zone, the crack-tip opening displacement, and the stress intensity factor at crack-closure are determined. With an assumption of a crack-extension law, the crack-growth behavior under arbitrary load history is traced by making a step-by-step analysis. For the crack growing under constant-amplitude loading, the analysis of the steady-state crack-growth behavior is carried out with a simplified formulation. Regarding the crack-growth behavior near the fatigue threshold, although all of the near-threshold mechanisms are not incorporated into the model, the promoted crack-closure due to corrosion debris and surface roughness is introduced in terms of a material parameter CTOD∗. Under constant-amplitude loading, the results from the step-by-step analysis demonstrate that the closure level of the stress intensity considerably increases during the growth process due to accumulation of the residual strains. This may in part provide an explanation for the arrest a crack at a distance far from the root of a sharp notch, which is frequently observed in experiments. After a sufficiently large number of cycles, the crack-growth behavior from the step-by-step analysis reaches the steady state, and the results agree with the ones from the steady-state analysis. The crack growth law of second power is derived as a consequence of the assumed linear relation between Δa and ΔCTOD. The crack growth curves from the analysis agree with the ones obtained from experiments for Al 2024-T3 and Al 7075-T6. the ratio of Δa to ΔCTOD chosen for the calculation is in agreement with the direct, experimental measurement of Δa and ΔCTOD. With the presence of CTOD∗, the results show that the crack-growth behavior at low load changes drastically, and for each value of CTOD∗ a threshold exists below which no crack growth occurs at all. It is shown that under the action of variable-amplitude loading, the crack-growth, behavior from the model exhibits a transient behavior upon altering the loading condition. The study shows that the residual plastic deformation in the wake of an advancing crack has a large influence on the crack-growth behavior. The results also lend a theoretical support to the existence of the controversial plasticity-induced crack-closure under plane-strain conditions.

Transient Brake Temperatures Found by use of Analytical Solutions for Finite Hollow Cylinders

A linear analytical model for calculating transient axisymmetric temperature distributions in finite hollow cylinders is established and adapted for practical use. A homogeneous isotropic material with temperature-independent thermal properties is assumed. Among possible applications are brake discs, brake drums and block braked railway wheels. The thermal power from braking is applied as prescribed heat influxes over parts of the lateral and radial surfaces of the cylinder. Bessel series solutions to the heat conduction equation are found where Newton's law for heat transfer is used in the boundary conditions. In addition to convection prescribed surface temperatures and insulated surfaces are also studied. Arbitrary thermal loading histories are treated by use of Duhamel's principle where step loadings are superposed. Convergence and limitations of the method and required computer times are discussed. Calculated numerical results for some typical braking operations are compared with those of other studies.

An analytical model of fatigue crack growth based on the crack-tip plasticity

In the present study, an analytical model of fatigue crack growth based on the plastic-blunting mechanism is developed. Both the plastic and reversed plastic flow during the process of cycling are considered. In each loading cycle, the size of the plastic and reversed plastic zone, the crack-tip opening displacement, and the stress intensity factor at crack-closure are determined. With an assumption of a crack-extension law, the crack-growth behavior under arbitrary load history is traced by making a step-by-step analysis. For the crack growing under constant-amplitude loading, the analysis of the steady-state crack-growth behavior is carried out with a simplified formulation. Regarding the crack-growth behavior near the fatigue threshold, although all of the near-threshold mechanisms are not incorporated into the model, the promoted crack-closure due to corrosion debris and surface roughness is introduced in terms of a material parameter CTOD ∗. Under constant-amplitude loading, the results from the step-by-step analysis demonstrate that the closure level of the stress intensity considerably increases during the growth process due to accumulation of the residual strains. This may in part provide an explanation for the arrest a crack at a distance far from the root of a sharp notch, which is frequently observed in experiments. After a sufficiently large number of cycles, the crack-growth behavior from the step-by-step analysis reaches the steady state, and the results agree with the ones from the steady-state analysis. The crack growth law of second power is derived as a consequence of the assumed linear relation between Δa and ΔCTOD. The crack growth curves from the analysis agree with the ones obtained from experiments for Al 2024-T3 and Al 7075-T6. the ratio of Δa to ΔCTOD chosen for the calculation is in agreement with the direct, experimental measurement of Δa and ΔCTOD. With the presence of CTOD ∗, the results show that the crack-growth behavior at low load changes drastically, and for each value of CTOD ∗ a threshold exists below which no crack growth occurs at all. It is shown that under the action of variable-amplitude loading, the crack-growth, behavior from the model exhibits a transient behavior upon altering the loading condition. The study shows that the residual plastic deformation in the wake of an advancing crack has a large influence on the crack-growth behavior. The results also lend a theoretical support to the existence of the controversial plasticity-induced crack-closure under plane-strain conditions.

Proposal of a New Crack Model Considering the Discontinuity in the Cracked Plane and Its Application to the Evaluation of Crack Parameter

A new crack model which enables us to evaluate crack parameters such as crack energy density and its distribution, COD and COA under arbitrary load history is proposed and the availability of the proposed model is demonstrated through finite element analyses of elasto-plastic crack. The contents are as follows ; (1) A crack model considering the discontinuity in the cracked plane is introduced and the constitutive equation for a plane with discontinuity is formulated on as elasto-plastic case. (2) The finite element formulation of the model is carried out by introducing a plane element. (3) An elasto-plastic crack expressed by the proposed model under monotonic loading is analyzed by finite element method and the availability of the model is verified through the evaluation of crack energy density.

Uniaxial Cyclic Stress‐Strain Behavior of Structural Steel

A simple mathematical model for the uniaxial cyclic stressstrain behavior of structural steel is proposed for arbitrary loading histories in the inelastic range. The model uses the monotonic and cy...

A cumulative damage model for bone fracture.

A mathematical model is presented to describe the combined time-dependent and cycle-dependent fracture characteristics of devitalized cortical bone. Failure is interpreted based on a linear-life fraction rule, which accounts for cumulative creep and fatigue damage under arbitrary loading histories. The model is successful in describing the influence of loading rate on monotonic tensile strength, the time to failure in constant stress creep-fracture tests, and bone fracture in zero-tension and tension-compression cyclic loading. The possible implications of the model to in vivo bone fracture, deformity, and remodelling in response to various loading histories are considered.

The initial boundary-value problem for geometrically nonlinear elastic-plastic shells

Starting from three-dimensional theory, a global minimum principle for stresses in elastic-plastic shells subjected to arbitrary conservative loading histories is presented. Finite displacements but small strains are considered. For numerical illustration, the stress state in an elastic-plastic cylindrical shell under nonproportional loading is calculated.

An approximation of uniaxial creep during alternating tension-compression step loading at constant temperature

In the present paper a new simple approximate method is introduced and compared with experimental results, which enables the estimation of the uniaxial creep behaviour of asphalt or asphalt-like materials under arbitrary alternating tension-compression loading histories with a comparatively small mathematical and experimental effort. The method is based on the assumption that the total creep deformation consists of two independent components: a visco-elastic one calculated by the conventional modified superposition principle and a viscoplastic one estimated by a new extension of the so-called ORNL strain hardening theory following the idea that during loading in one direction structural weakening arises which the material remembers explicitly if loaded in the opposite direction. With this new approach the creep behaviour of asphalt under arbitrary step loading histories can be predicted fairly well and as experiments showed with better results than with conventional theories ignoring structural weakening effects.

A Viscoelastic Correspondence Principle for Plane Membranes Subjected to Lateral Pressure

The classical Fo¨ppl equations, governing the deflection of plane membranes, constitute the first-order consistent approximation in the case of linear elastic material behavior. It is shown that despite the static and kinematic nonlinearities present, for arbitrary load histories a correspondence principle for viscoelastic material behavior exists if all relevant relaxation moduli are of uniform time dependence. Application of the principle is illustrated by means of a popular material model.

Cumulative damage in steel structures subjected to earthquake ground motions

Experimental data are presented from low cycle fatigue tests of structural steel components. In these tests the failure modes of local buckling in beam flanges and fracture at weldments were studied in detail. Cumulative damage models are proposed which permit a life prediction for arbitrary cyclic loading histories. For the local buckling failure mode a series of linear damage models is used to predict deterioration threshold and deterioration, whereas a single model is used for the fracture mode. Crack propagation at weldments is modeled with a crack growth rate model based on the plastic strain range. Adequate predictions of lives were obtained from the analytical models.