Application of finite strain viscoplasticity to polymeric fiber composites

The tensile behavior of polymer matrix fiber composites has been described in terms of an anisotropic model of finite viscoplasticity. Our constitutive approach is based on the kinematics of the additive form of the deformation rate tensor D generalized by Mandel [Mandel, J., 1971, Plasticité classique et viscoplasticité. Courses and Lectures, No. 97, International Center for Mechanical Sciences/Springer, Udine/Wien-New York] and Dafalias [Dafalias, Y.F., 1985. The plastic spin. ASME J. Appl. Mech. 52, 865–871]. The constitutive laws for D^p and W^p were written in accordance with the material anisotropy, whereas the constitutive law of hypoelasticity has been accordingly written in its objective form. Moreover, a viscoplastic model has been applied to represent the non-linear rate dependence. Experimental results performed in a wide strain rate region and in a wide strain range were simulated in a very accurate way. Additionally, the model was proved to predict creep behavior of the same material type as well.     

An anisotropic micromechanics model for predicting the rafting direction in Ni-based single crystal superalloys

An anisotropic micromechanics model based on the equivalent inclusion method is developed to investigate the rafting direction of Ni-based single crystal superalloys. The micromechanical model considers actual cubic structure and orthogonal anisotropy properties. The von Mises stress, elastic strain energy density, and hydrostatic pressure in different inclusions of micromechanical model are calculated when applying a tensile or compressive loading along the [001] direction. The calculated results can successfully predict the rafting direction for alloys exhibiting a positive or a negative mismatch, which are in agreement with pervious experimental and theoretical studies. Moreover, the elastic constant differences and mismatch degree of the matrix and precipitate phases and their influences on the rafting direction are carefully discussed.     

A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded(FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton's principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.     

A dislocation density based strain gradient model

Strain gradients play a vital role in the prediction of size-effects in the deformation behavior of metals at the micrometer scale. At this scale the behavior of metals strongly depends on the dislocation distribution. In this paper, a dislocation density based strain gradient model is developed aiming at predictions of size-effects for structural components at this scale. For this model, the characteristic length is identified as the average distance of dislocation motion, which is deformation dependant and can be determined experimentally. The response of the model is compared to the strain gradient plasticity model of Huang et al. [Huang, Y., Qu, S., Hwang, K.C., Li, M., Gao, H., 2004. A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plasticity 20, 753–782]. It is shown that the present strain gradient model, which only requires a physically measurable length-scale, can successfully predict size effects for a bar with an applied body force and for void growth.     

Continuum slip viscoplasticity with the Haasen constitutive model: Application to CdTe single crystal inelasticity

Small strain constitutive equations are developed for the thermomechanical behavior of semiconductor single crystals, including dislocation density as an evolving parameter. The model of Haasen, Alexander and coworkers is modified (extended) to include evolution of coefficients in the definition of internal stress. These account for an evolving dislocation substructure. The resulting model is applied in a continuum slip framework to allow multiple slip orientations. Slip system interaction rules are adapted to include slip system interaction for multiple slip conditions and to suppress secondary slip and dislocation density generation for single slip orientations. The approach is discussed relative to other models for viscoplasticity of single crystals and is examined in the context of thermodynamics with internal state variables. The framework is used to correlate experimental data from compression tests of single crystals of the compound semiconductor CdTe from room temperature to near the melting point. Sensitivity of the model to uncertainties such as initial dislocation density is explored.     

A phase field model of electromechanical fracture

Structural reliability analyses of piezoelectric solids need the modeling of failure under coupled electromechanical actions. However, the numerical simulation of failure due to fracture based on sharp crack discontinuities may suffer in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. In this work, we develop a framework of diffusive fracture in piezoelectric solids. We start our investigation with the definition of a crack surface functional of the phase field that Γ-converges for vanishing length-scale parameter to a sharp crack topology. This functional provides the basis for the definition of suitable dissipation functions which govern the evolution of the crack phase field. Based on experimental results available in the literature, we suggest a non-associative dissipative framework where the fracture phase field is driven by the mechanical part of the coupled electromechanical driving force. This accounts for a hierarchical view that considers (i) the decrease of stiffness due to mechanical rupture as the primary action that is followed by (ii) the decrease of electric permittivity due to the generated free space. The proposed definition of mechanical and electrical parts of the fracture driving force follows in a natural format from a kinematic assumption, that decomposes the total strains and the total electric field into energy-enthalpy-producing parts and fracture parts, respectively. Such an approach allows the insertion of well-known anisotropic piezoelectric storage functions without change. We end up with a three-field-problem that couples the displacement with the electric potential and the fracture phase field. The latter is governed by a micro-balance equation, which appears in a very transparent form in terms of a history field containing a maximum fracture source obtained in the time history of the electromechanical process. This representation allows the construction of a very robust algorithmic treatment based on a operator split scheme, which successively updates in a typical time step the history field, the crack phase field and finally the two piezoelectric fields. The proposed model is considered to be the canonically simple scheme for the simulation of diffusive electromechanical crack propagation in solids. We demonstrate its modeling capacity by means of representative numerical examples.     

A dynamic flow rule for viscoplasticity in polycrystalline solids under high strain rates

For visco-plasticity in polycrystalline solids under high strain rates, we introduce a dynamic flow rule (also called the micro-force balance) that has a second order time derivative term in the form of micro-inertia. It is revealed that this term, whose physical origin is traced to dynamically evolving dislocations, has a profound effect on the macro-continuum plastic response. Based on energy equivalence between the micro-part of the kinetic energy and that associated with the fictive dislocation mass in the continuous dislocation distribution (CDD) theory, an explicit expression for the micro-inertial length scale is derived. The micro-force balance together with the classical momentum balance equations thus describes the viscoplastic response of the isotropic polycrystalline material. Using rational thermodynamics, we arrive at constitutive equations relating the thermodynamic forces (stresses) and fluxes. A consistent derivation of temperature evolution is also provided, thus replacing the empirical route. The micro-force balance, supplemented with the constitutive relations for the stresses, yields a locally hyperbolic flow rule owing to the micro-inertia term. The implication of micro-inertia on the continuum response is explicitly demonstrated by reproducing experimentally observed stress–strain responses under high strain-rate loadings and varying temperatures. An interesting finding is the identification of micro-inertia as the source of oscillations in the stress–strain response under high strain rates.     

Application of a kinematic hardening viscoplasticity model with thresholds to the residual stress relaxation

The life analysis of engine components needs to take into account the residual stress relaxation induced by cyclic service loads. The paper recalls a new class of constitutive equations for cyclic viscoplasticity, using a series of kinematic hardening models with thresholds. The equations are introduced within a recently enlarged thermodynamic framework. Some attention is focused to the relations with multisurface approaches and to a specific determination procedure of the model parameters. The new model is applied to the calculation of the near surface residual stress relaxation after shot peening, when the structure is submitted to cyclic service loads. The simulated stabilized residual stresses are in good accordance with experimental results obtained on an N18 disk alloy at 650°C. In comparison, the classical model without threshold predicts the complete vanishing of the residual stresses, which is not satisfactory.     

A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory

A size-dependent Kirchhoff micro-plate model is developed based on the strain gradient elasticity theory. The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. The static bending, instability and free vibration problems of a rectangular micro-plate with all edges simple supported are carried out to illustrate the applicability of the present size-dependent model. The results are compared with the reduced models. The present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.     

A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions

In this paper, a viscoplastic constitutive model previously proposed by the authors was extended to apply to the cyclic deformation analysis of the modified 9Cr-1Mo steel. A series of cyclic deformation tests were conducted on modified 9Cr-1Mo steel at various temperatures, including those under anisothermal conditions. Furthermore, cyclic evolution of state variables used in the authors' constitutive model was experimentally measured. Based on the test results, cyclic softening behavior depending on the temperature and its history was introduced into the constitutive model. The extended model was applied to simulations of inelastic deformation behavior under monotonic tension, stress relaxation, creep, isothermal cyclic deformations including stress relaxation and anisothermal cyclic deformations. It was found that the present constitutive model has a capability of describing the inelastic deformation behavior of modified 9Cr-1Mo steel adequately at various loading conditions.     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

A phase field model for failure in interconnect lines due to coupled diffusion mechanisms

We describe a diffuse interface, or phase field model for simulating electromigration and stress-induced void evolution and growth in interconnect lines. Microstructural evolution is tracked by defining an order parameter, which takes on distinct uniform values within solid material and voids, and varying rapidly from one to the other over narrow interfacial layers associated with the void surfaces. The order parameter is governed by a form of the Cahn-Hilliard equation. An asymptotic analysis demonstrates that the zero contour of order parameter tracks the motion of a void evolving by coupled surface and lattice diffusion, driven by stress, electron wind and vacancy concentration gradients. Efficient finite element schemes are described to solve the modified Cahn-Hilliard equation, as well as the equations associated with the accompanying mechanical, electrical and bulk diffusion problems. The accuracy and convergence of the numerical scheme is investigated by comparing results to known analytical solutions. The method is applied to solve various problems involving void growth and evolution in representative interconnect geometries.     

A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids

In order to simulate quasi-brittle failure in porous elastic solids, a continuum damage model has been developed within the framework of strain gradient elasticity. An essential ingredient of the continuum damage model is the local strain energy density for pure elastic response as a function of the void volume fraction, the local strains and the strain gradients, respectively. The model adopts Griffith’s approach, widely used in linear elastic fracture mechanics, for predicting the onset and the evolution of damage due to evolving micro-cracks. The effect of those micro-cracks on the local material stiffness is taken into account by defining an effective void volume fraction. Thermodynamic considerations are used to specify the evolution of the latter. The principal features of the model are demonstrated by means of a one-dimensional example. Key aspects are discussed using analytical results and numerical simulations. Contrary to other continuum damage models with similar objectives, the model proposed here includes the effect of the internal length parameter on the onset of damage evolution. Furthermore, it is able to account for boundary layer effects.     

Strain gradient effects on microscopic strain field in a metal matrix composite

The purpose of this paper is to demonstrate the improved modeling accuracy of a finite-deformation strain gradient crystal plasticity formulation over its classical counterpart by conducting a joint experimental and numerical investigation of the microscopic details of the deformation of a whisker-reinforced metal-matrix composite. The lattice rotation distribution around whiskers is obtained in thin foils using a TEM technique and is then correlated with numerical predictions based on finite element analyses of a unit-cell of a single crystal matrix containing a rigid whisker. The matrix material is first characterized by a classical, scale-independent crystal plasticity theory. It is found that the classical theory predicts a lattice rotation distribution with a spatial gradient much higher than experimentally measured. A strain gradient crystal plasticity formulation is then applied to model the matrix. The strain gradient formulation accounts for both strain hardening and strain gradient hardening. The deformation thus predicted exhibits a strong dependence on the size of the whisker. For a constitutive length scale comparable to the whisker diameter, the spatial gradient of the lattice rotation is several times lower than that predicted by the classical theory, and hence correlates significantly better with the experimental results.     

A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

A microscale nonlinear Bernoulli–Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton’s principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.     

Finite deformation analysis of mechanism-based strain gradient plasticity: torsion and crack tip field

A finite deformation theory of mechanism-based strain gradient (MSG) plasticity is developed in this paper based on the Taylor dislocation model. The theory ensures the proper decomposition of deformation in order to exclude the volumetric deformation from the strain gradient tensor since the latter represents the density of geometrically necessary dislocations. The solution for a thin cylinder under large torsion is obtained. The numerical method is used to investigate the finite deformation crack tip field in MSG plasticity. It is established that the stress level around a crack tip in MSG plasticity is significantly higher than its counterpart (i.e. HRR field) in classical plasticity.     

A model for brittle fracture based on the hybrid phase field model

We propose a phase field model for crack propagation based on the hybrid model and justify the model by constructing a family of asymptotic solutions.