Time dependent damage in half-plane part 1: Displacement loading

Analyzed in this study is the time dependent damage of a half-plane subjected to a normal surface displacement over a segment at 0.5 mm/s. The material is made of 4340 steel and dissipates energy in an irreversible manner such that the stress and strain response in each local element can be different and change with time. The inherent coupling between mechanical and thermal effects are accounted for without letting the rate change of volume with surface for the element to vanish. This is in contrast to the assumption made in the classical theories of continuum mechanics.Results obtained from the more refined theory of surface/volume energy density differed qualitatively and quantitatively from those of plasticity. The differences are particularly significant in the vicinity of load application and/or boundary. Stress and strain behavior in the local elements exhibited both hardening and softening behavior while plasticity pre-fixes the monotonic behavior for every element at all times. Uniaxial data are related uniquely to those under multiaxial conditions by invoking the concept of a plane of homogeneity. This is done in general without making the restrictive assumptions in plasticity that the effective stress and effective strain coincide with the uniaxial data. By including the change of local strain rates and strain rate history, local elements are found to undergo cooling and heating during the initial stage of loading. Such non-equilibrium phenomenon has also been observed experimentally in uniaxial and pre-cracked specimens.     

Effect of strain rate on crack growth in aluminum alloy 1100-0

Stress and damage analysis are performed to analyze the Mode I crack growth behavior of a central crack panel made of aluminum alloy 1100-0. On account of the highly nonhomogeneous stress state, each material element would experience a different strain rate depending on the location and loading rate. A data bank of uniaxial stress and strain curves is provided to cover the range local strain rates depending on the load time history. Such a approach is referred to as the strain rate dependent model in contrast to plasticity that utilizes a single constitutive relation.The strain energy density criterion is applied to determine the onset of crack initiation, stable crack growth and final termination. A unique feature of the approach is that the same criterion could describe the foregoing three distinct events of fracture behavior. Results are obtained for applied loads with different strain rates and compared with those obtained from the classical theory of plasticity, which is unconservative.     

Thermal/mechanical interaction of subcritical crack growth in tensile specimen

The mutually interacting thermal/mechanical effects are accounted for by retaining the rate of change of volume with surface dV/dA, in the surface/volume energy density theory. An exchange between surface and volume energy would thus prevail in accordance with the rates at which the loads are transmitted throughout the system. Obtained are results for a center-cracked specimen subjected to monotonically rising tensile load at the rate of . Eight load steps are taken with an even increment increase of 69 MPa starting from 276 MPa. The temperature is found to oscillate about the ambient condition as the crack grew incrementally in a stable fashion. What differed considerably from the plasticity theory are the local stress and strain distribution and the crack growth characteristics. This is particularly pronounced in regions close to the crack tip where the local strain rates and strain rate history change from element to element, an affect that is not accounted for in plasticity.     

Mechanics and physics of energy density theory

As a departure from the classical continuum mechanics approach, irreversible material behavior is uniquely identified with the exchange of surface and volume energy density through the rate of change of volume with surface area. This provides an one-to-one correspondence between the uniaxial and multiaxial stress or energy state. Equivalent uniaxial stress and strain response can thus be determined for each material element that undergoes damage by permanent deformation and/or fracture. Discussed in detail are the applications of surface energy or volume energy with continuum mechanics theories for analyzing failure in contrast to the strain energy density that analyzes stress and failure simultaneously. In particular, the results for the progressive damage of a slowly moving crack are presented and compared with those obtained from the theory of plasticity. It is shown that the neglected of dilatational energy in plasticity results in inaccurate prediction of the state of affairs near the crack tip.Since the stresses and strains in the strain energy density theory are determined separately, nonlinear and finite strains may be easily included into dV/dA and incorporated into the formulation. This opens the door to a class of nonlinear problems that can be solved directly even for finite and large strains including energy dissipation.     

Thermal/mechanical damage of 6061-T6 aluminum tensile specimen

As a 6061-T6 aluminum coupon specimen is stretched, energy is being converted from mechanical work to heat. This irreversible process of material damage is detected experimentally by measuring the change in surface temperature. Contrary to the ordinary notion that the material would heat up when loaded, it actually cools before returning to the ambient condition. The recovery time was approximately 26 sec for a displacement rate of 8.467·10⁻⁵ m/sec and 200 sec when the displacement rate is reduced by one of magnitude. Cooling and heating is a rate dependent process. Three sets of temperature data were obtained for each of the displacement rates and they coincide with those prediced from the energy density theory that accounts for the nonhomogeneous dissipation of energy at every location in the specimen.Unlike any classical theories in mechanics, the energy density theory determines the stress and strain response of each element in the specimen only from a knowledge of the initial material stiffness and the displacement time rate. This is necessary because the local strain rates for elements near the center and edge of the specimen can differ by a wide margin. The so-called uniaxial stress and strain curve is then obtained by taking the average of all the elements. The results agreed extremely well with those measured experimentally for the 6061-T6 aluminum. Obtained analytically are also the thermal conductivity coefficients that are loading rate dependent and anisotropic in character due to stretching in the longitudinal direction. Their values tend to stabilize beyond the cooling/heating period.     

Anisotropic plasticity model coupled with Lode angle dependent strain-induced transformation kinetics law

A phenomenological macroscopic plasticity model is developed for steels that exhibit strain-induced austenite-to-martensite transformation. The model makes use of a stress-state dependent transformation kinetics law that accounts for both the effects of the stress triaxiality and the Lode angle on the rate of transformation. The macroscopic strain hardening is due to nonlinear kinematic hardening as well as isotropic hardening. The latter contribution is assumed to depend on the dislocation density as well as the current martensite volume fraction. The constitutive equations are embedded in the framework of finite strain isothermal rate-independent anisotropic plasticity. Experimental data for an anisotropic austenitic stainless steel 301LN is presented for uniaxial tension, uniaxial compression, transverse plane strain tension and pure shear. The model parameters are identified using a combined analytical–numerical approach. Numerical simulations are performed of all calibration experiments and excellent agreement is observed. Moreover, we make use of experimental data from ten combined tension and shear experiments to validate the proposed constitutive model. In addition, punch and notched tension tests are performed to evaluate the model performance in structural applications with heterogeneous stress and strain fields.     

Finite strain––isotropic hyperelasticity

This paper presents a strain energy density for isotropic hyperelastic materials. The strain energy density is decomposed into a compressible and incompressible component. The incompressible component is the same as the generalized Mooney expression while the compressible component is shown to be a function of the volume invariant J only. The strain energy density proposed is used to investigate problems involving incompressible isotropic materials such as rubber under homogeneous strain, compressible isotropic materials under high hydrostatic pressure and volume change under uniaxial tension. Comparison with experimental data is good. The formulation is also used to derive a strain energy density expression for compressible isotropic neo-Hookean materials. The constitutive relationship for the second Piola–Kirchhoff stress tensor and its physical counterpart, involves the contravariant Almansi strain tensor. The stress stretch relationship comprises of a component associated with volume constrained distortion and a hydrostatic pressure which results in volumetric dilation. An important property of this constitutive relationship is that the hydrostatic pressure component of the stress vector which is associated with volumetric dilation will have no shear component on any surface in any configuration. This same property is not true for a neo-Hookean Green’s strain–second Piola–Kirchhoff stress tensor formulation.     

Stress and energy density fields near a blunted crack front in power hardening material

Using Jaumann and Dienes rates of Euler stress in elastic-plastic constitutive equations of finite deformation, plane strain finite element analysis for a compact tension specimen with a blunted crack front is made. The Euler stress, Kirchhoff stress and volume strain energy density near a blunted crack tip are computed. Constitutive relations with different deformation rates affect the the near crack tip solution in a region within an order of magnitude of the crack opening displacement. The results differed from the corresponding solution of deformation plasticity (or nonlinear elasticity) with increasing deformation. They are smaller in a local region of about 2 to 10 times of the crack opening distance.The volume energy density near the crack tip is computed, the stationary values of which determine the locations of extensive yielding and possible sites of crack initiation. It remained nearly constant with increasing deformation. Such a character tends to support the volume energy density criterion as a means for quantifying the ductile fracture behavior of metals.     

Size dependent response of large shell elements under in-plane tensile loading

Large-scale thin-walled structures with a low weight-to-stiffness ratio provide the means for cost and energy efficiency in structural design. However, the design of such structures for crash and impact resistance requires reliable FE simulations. Large shell elements are used in those simulations. Simulations require the knowledge of the true stress–strain response of the material until fracture initiation. Because of the size effects, local material relation determined with experiments is not applicable to large shell elements. Therefore, a numerical method is outlined to determine the effect of element size on the macroscopic response of large structural shell elements until fracture initiation. Macroscopic response is determined by introducing averaging unit into the numerical model over which volume averaged equivalent stress and plastic strain are evaluated. Three different stress states are considered in this investigation: uniaxial, plane strain and equi-biaxial tension. The results demonstrate that fracture strain is highly sensitive to size effects in uniaxial tension whereas in plane strain or equi-biaxial tension size effects are much weaker. In uniaxial and plane strain tension the fracture strain for large shell elements approaches the Swift diffuse necking condition.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I

A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to account for effects of geometrically necessary dislocations (GNDs) only in work hardening. The specific lower-order gradient plasticity model chosen to substantiate this work requires one additional material parameter compared to its conventional continuum plasticity counterpart. The further addition of dislocation mechanics requires no additional material parameters. The model (a) retains the constitutive dependence of the free-energy only on elastic strain as in conventional continuum plasticity with no explicit dependence on dislocation density, (b) does not require higher-order stresses, and (c) does not require a constitutive specification of a ‘back-stress’ in the expression for average dislocation velocity/plastic strain rate. However, long-range stress effects of average dislocation distributions are predicted by the model in a mechanistically rigorous sense. Plausible boundary conditions (with obvious implication for corresponding interface conditions) are discussed in some detail from a physical point of view. Energetic and dissipative aspects of the model are also discussed. The developed framework is a continuous-time model of averaged dislocation plasticity, without having to rely on the notion of incremental work functions, their convexity properties, or their minimization. The tangent modulus relating stress rate and total strain rate in the model is the positive-definite tensor of linear elasticity, and this is not an impediment to the development of idealized microstructure in the theory and computations, even when such a convexity property is preserved in a computational scheme. A model of finite deformation, mesoscopic single crystal plasticity is also presented, motivated by the above considerations.Lower-order gradient plasticity appears as a constitutive limit of PMFDM, and the development suggests a plausible boundary condition on the plastic strain rate for this limit that is appropriate for the modeling of constrained plastic flow in three-dimensional situations.     

Finite element simulation of PSB macroband nucleation and propagation in single crystal nickel cycled at low plastic strain amplitudes

Substructure models for vein matrix and persistent slip band (PSB) structures are extracted from a uniaxial mixtures model that was developed to simulate cyclic loading experiments on nickel single crystals oriented for single slip. Reverse magnetostriction is included as well. These substructure models are implanted in a single crystal plasticity framework with fully anisotropic elasticity. The resulting constitutive models are incorporated in finite element models to simulate the process of PSB macroband formation and propagation. Perturbation elements (PEs), elements assigned with PSB properties, are used as the loci for PSB macroband nucleation. Transition of elements with vein matrix properties to elements with PSB properties is triggered at integration points by a shear stress criterion applied on slip systems. The resulting finite element models successfully demonstrate the process of PSB formation and propagation, and plastic strain amplitude partitioning between vein matrix and PSB macrobands. The effect of model boundary constraints, strain increment dependence, mesh sensitivity, PE distribution, specimen axis misorientation, and PSB volume fraction generated is examined.     

Rigid-plastic and damage behavior in metal forming: Part II—model prediction

Indeterminacy of the Lévy-Mises relations in plane strain rigid-plasticity is overcome by considering change in plastic strain rate due to triaxiality. Better agreement is achieved on the coincidence of uniaxial data with the effective stress and effective strain in addition to the removal of ill conditions associated with necking stresses derived from other theories of rigid plasticity. The formulation accounts for material compressibility and makes use of the variational principle when applied to discrete locations of the continuum. These physical refinements provide the necessary accuracy for predicting potential damage sites and establishing design limits on metal forming products. Developed are damage thresholds for defining the integrity of rigid-plastic materials with nonlinear behavior. Distortion and dilatation of the material elements are mutually interactive and their proportion changes with the local strain rates in the metal forming process. Only the stationary values of the volume energy density could automatically account for the nonlinear relationship between distortion and dilation. Completed in Part I is the development of a modified theory of rigid-plasticity that better describes the yield and fracture behavior of metals. Numerical results for the permanent deformation of sheet metals are provided in Part II; they are compared with those obtained from other theories of rigid-plasticity.     

Hysteresis loops predicted by isoenergy density theory for polycrystals. Part I: fundamentals of non-equilibrium thermal–mechanical coupling effects

When a polycrystal is stressed or strained at fifty percent of the corresponding yield value, damage will be inflicted non-homogeneously in the material due to the fact that the stress and/or strain distribution is non-uniform even if isotropy and homogeneity are assumed for the initial microstructure. This effect will be cumulated for each cycle of the load if the applied stress or strain is repeated continuously. Nucleation of microcracks can eventually lead to the propagation of a macrocrack.The process of damage accumulation in fatigue is defined to be sufficiently slow such that inhomogeneity of material behavior created by loading is a significant factor that can not be arbitrarily dismissed without a good reason. What this means specifically is that the difference of the stress and strain behavior at each point in a fatigue specimen must be accounted for in the analytical model in order to predict the correct cumulative effect. Such a requirement translates into a non-equilibrium formulation where the constitutive relations for each point and loading cycle must be determined separately. In this sense, the true problem of fatigue cannot be completely treated by the classical continuum mechanics approach that is limited to equilibrium mechanics for a closed system. Having said this, the isoenergy density theory will be applied to estimate the hysteresis loops of a hour-glass profile cylindrical bar specimen as recommended by the American Society for Testing and Materials (ASTM) for low-cycle fatigue.The work will be divided into two parts. Part I will cover the fundamentals of a non-equilibrium theory where the continuum elements are finite in size; they do not vanish in the limit. Therefore, size effects are immediately encountered as a function of time. General expressions for the rate change of volume of these elements with surface area are derived such that they can be computed from the nine displacement gradients. These elements can differ in size and must fit together without discontinuities or gaps to form the continuum. The condition of isoenergy energy density is invoked such that the size of these individual elements under large and finite deformation and rotation can be determined without loss in generality. The existence of such a space having the property of the same isoenergy density in all directions is thus proved. This enables the establishment of the one dimensional energy state with that in three dimensions without restriction, the absence of which has prevented the development of a complete non-linear theory of mechanics that can be solved in a direct fashion in contrast to the inverse method of assuming the displacement field. Illustration is provided for deriving the constitutive relation incrementally for a given location for the hour-glass specimen made of 6061-T6 aluminum. Once the specimen is loaded, each material point will follow a different stress and strain curve according to the local displacement rate. Hence, the method applies to material with non-homogeneous microstructure if their individual expressions can be assessed and fed into the computer.Part II computes for the non-equilibrium temperature and an entropy-like quantity that can be positive and negative. This implies that the system can absorb or dissipate energy with reference to the surrounding. Additional data for hysteresis loops are given for 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium. Accumulation of the local hysteresis energy per cycle is found to be the highest near the surface of the uniaxial specimen where load symmetry prevails. This is a consequence of the difference in accumulation of the energy density due to distortion in contrast to dilatation at the specimen center. This is why fatigue cracks tend to nucleate near the specimen surface, at a small distance towards the interior. Another distinct feature of fatigue is that the non-equilibrium temperature is found to oscillate about the ambient temperature while the local stress states fluctuate between tension and compression. This temperature reversal behavior is typical of non-equilibrium behavior and also occurs under monotonic loading. The space and time variations of the dissipated energy density for different materials are found to be related to the initial monotonic energy density or area under the true stress and true strain curve.What will be demonstrated is that no special consideration need to be made when applying the isoenergy density theory for analyzing the nucleation of micro and macrocracks in addition to failure of the specimen. Crack nucleation under fatigue is assumed to occur when the total hysteresis energy reaches a critical value. It is possible to establish a relation between the average hysteresis energy per cycle and the number of cycles to failure. The proposed method requires only a knowledge of the initial monotonic energy density curve for a given material. Predicted results for the fatigue of cylindrical bar specimens with hour-glass profile are given and they can be found in Part II of this work.     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.     

Thickness reduction rate of plastically compressed metal plate by hardened rollers

The rate at which a solid deforms permanently depends on the load history, geometry and material properties. When a metal plate is compressed between two hardened rollers, its thickness reduces continuously if the material elements are deformed beyond their elastic limits. Those near the region of contact will experience more distortion as compared with those interior to the plate. This effect is analyzed incrementally in time by the theory of plasticity coupled with the strain energy density criterion. Failure is examined by assuming that the location of crack initiation coincides with the maximum of the minimum strain energy density function, (dW/dV)_min^max, when reaching its critical value. This is found to occur at the center of the plate depending on the rate of deformation. An increase in plate thickness reduction without failure can be achieved by taking smaller loading steps. Displayed graphically are numerical results for five different load histories that provide useful insights into the rate dependent process of metal forming.     

Separation phenomena at the interface of a finitely deformed composite sphere

The problem of the finite deformation of a composite sphere subjected to a spherically symmetric dead load traction is revisited focusing on the formation of a cavity at the interface between a hyperelastic, incompressible matrix shell and a rigid inhomogeneity. Separation phenomena are assumed to be governed by a vanishingly thin interfacial cohesive zone characterized by uniform normal and tangential interface force–separation constitutive relations. Spherically symmetric cavity shapes (spheres) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation, the dead loading and the volume concentration of inhomogeneity. Spherically symmetric and non-symmetric bifurcations initiating from spherically symmetric equilibrium states are analyzed within the framework of infinitesimal strain superimposed on a given finite deformation. A simple formula for the dead load required to initiate the non-symmetrical rigid body mode is obtained and a detailed examination of a few special cases is provided. Explicit results are presented for the Mooney–Rivlin strain energy density and for an interface force–separation relation which allows for complete decohesion in normal separation.     

Springback in cold metal-forming of AOO---H steel: nonhomogeneous deformation

The incremental theory of plasticity is applied in conjunction with finite elements to obtain the state of affairs in metal-forming. The allowable overforming for a AOO---H steel sheet bent around a circular die is investigated numerically. For a 90° bend, the springback angle is found to be approximately 25%. That is, an overform of 25% is required if the sheet is to remain permanently deformed at 72°. The residual energy is found by comparing the energy states before and after spring back. The amount depends on the rate of forming.A modified theory of plasticity is also employed to find the nonhomogeneous character of the deformation field in the forming of metal sheets. Changes in the local strain rates and strain rate history are accounted for by deriving individual constitutive relations for each of the finite elements. Parameters assigned to describe material properties no longer remained constant but changed according to the loading rates.The strain rates at different locations were found to change by more than three orders of magnitude that could not have been adequately described by a single constitutive equation. Inhomogeneous deformation could play a significant role in metal-forming when local strain rates vary over a wide range, as demonstrated in this work.     

考虑损伤的内变量黏弹-黏塑性本构方程

基于Rice 不可逆内变量热力学框架,在约束构型空间中讨论材料的蠕变损伤问题. 通过给定具体的余能密度函数和内变量演化方程推导出考虑损伤的内变量黏弹-黏塑性本构方程. 通过模型相似材料单轴蠕变加卸载试验对一维情况下的本构方程进行参数辨识和模型验证,本构方程能很好地描述黏弹性变形和各蠕变阶段.不同的蠕变阶段具有不同的能量耗散特点. 受应力扰动后,不考虑损伤的材料系统能自发趋于热力学平衡态或稳定态. 在考虑损伤的整个蠕变过程中,材料系统先趋于平衡态再背离平衡态发展. 能量耗散率可作为材料系统热力学状态偏离平衡态的测度;能量耗散率的时间导数可用于表征系统的演化趋势;两者的域内积分值可作为结构长期稳定性的评价指标.      

Failure initiation in unnotched specimens subjected to monotonic and cyclic loading

Failure initiation in unnotched cylindrical bar specimens is predicted by application of the strain energy density theory. Maximum value of the local minimum strain energy density function is calculated, the critical value of which is assumed to coincide with failure by monotonic as well as cyclic uniaxial loading. Damage is accumulated in the specimen for each increment of monotonically rising load and each cycle of repeatedly applied load. Use is made of the incremental theory of plasticity to account for permanent deformation that is nonuniformly distributed throughout the cylindrical bar. Failure initiation site is found to occur at the center of the bar for monotonic loading where dilatation is dominant and near the specimen surface for fatigue loading where distortion is more significant. The results are consistent with the experimental observations without including microstructural effects. Nonhomogeneity caused by macro-dilatation and macro-distortion is also shown to play an important role in failure initation.     

Integral-based constitutive equation for rubber at high strain rates

High-speed experiments were conducted to characterize the deformation and failure of Styrene Butadiene Rubber at impact rates. Dynamic tensile stress–strain curves of uniaxial strip specimens and force–extension curves of thin sheets were obtained from a Charpy tensile impact apparatus. Results from the uniaxial tension tests indicated that although the rubber became stiffer with increasing strain rates, the stress–strain curves remained virtually the same above 280 s⁻¹. Above this critical strain rate, strength, fracture strain and toughness decreased with increasing strain rates. When strain rates were below 180 s⁻¹, the initial modulus, tensile strength and breaking extension increased as the strain rate increased. Between strain rates of 180 and 280 s⁻¹, the initial modulus and tensile strength increased with increasing strain rates but the extension at break decreased with increasing strain rates. A hyper-viscoelastic constitutive relation of integral form was used to describe the rate-dependent material behavior of the rubber. Two characteristic relaxation times, 5 ms and 0.25 ms, were needed to fit the proposed constitutive equation to the data. The proposed constitutive equation was implemented in ABAQUS Explicit via a user-defined subroutine and used to predict the dynamic response of the rubber sheets in the experiments. Numerical predictions for the transient deformation and failure of the rubber sheet were within 10% of experimental results.