Duality in constitutive formulation of finite-strain elastoplasticity based on F=FeFp and F=FF decompositions

A constitutive theory for large elastic–plastic deformations is presented by employing F=F^pF^e decomposition of the total deformation gradient. A duality in constitutive formulation based on this and the well-known Lee's decomposition F=F_eF_p is established for isotropic polycrystalline and single crystal plasticity.     

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

The Burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity

We consider finite plasticity based on the decomposition F=F^eF^p of the deformation gradient F into elastic and plastic distortions F^e and F^p. Within this framework the macroscopic Burgers vector may be characterized by the tensor field . We derive a natural convected rate for G associated with evolution of F^p and as our main result show that, for a single-crystal,

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temporal changes in G—as characterized by its convected time derivative—may be decomposed into temporal changes in distributions of screw and edge dislocations on the individual slip systems.

We discuss defect energies dependent on the densities of these distributions and show that corresponding thermodynamic forces are macroscopic counterparts of classical Peach-Koehler forces.     

Effects of Fuel Lewis Number on Localised Forced Ignition of Globally Stoichiometric Stratified Mixtures: a Numerical Investigation

The influences of fuel Lewis number Le_F on localised forced ignition of globally stoichiometric stratified mixtures have been analysed using three-dimensional compressible Direct Numerical Simulations (DNS) for cases with Le_F ranging from 0.8 to 1.2. The globally stoichiometric stratified mixtures with different values of root-mean-square (rms) equivalence ratio fluctuation (i.e. ?^′= 0.2, 0.4 and 0.6) and the Taylor micro-scale l_? of equivalence ratio ? variation (i.e. l_?/l_f= 2.1, 5.5 and 8.3 with l_f being the Zel’dovich flame thickness of the stoichiometric laminar premixed flame) have been considered for different initial rms values of turbulent velocity u^′. A pseudo-spectral method is used to initialise the equivalence ratio variation following a presumed bi-modal distribution for prescribed values of ?^′ and l_?/l_f for global mean equivalence ratio 〈?〉=1.0. The localised ignition is accounted for by a source term in the energy transport equation that deposits energy for a stipulated time interval. It has been observed that the maximum values of temperature and the fuel reaction rate magnitude increase with decreasing Le_F during the period of external energy deposition. The initial values of Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f have been found to have significant effects on the extent of burning of the stratified mixtures following localised ignition. For a given value of u^′/S_b(?=1), the extent of burning decreases with increasing Le_F. An increase in u^′ leads to a monotonic reduction in the burned gas mass for all values of Le_F in all stratified mixture cases but an opposite trend is observed for the Le_F=0.8 homogeneous mixture. It has been found that an increase in ?^′ has adverse effects on the burned gas mass, whereas the effects of l_?/l_f on the extent of burning are non-monotonic and dependent on ?^′ and Le_F. Detailed physical explanations have been provided for the observed Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f dependences.     

Polycrystalline kinematics: An extension of single crystal kinematics that incorporates initial microstructure

Single crystal F^eF^P kinematics are widely used as the basis for many crystal plasticity models. Within this kinematic framework, geometrically necessary dislocations (GNDs) initially do not exist and then they evolve as needed in the material. A shortcoming of this kinematic model is that there is no rigorous way to define the initial and evolving GND state in the same manner. By augmenting the single crystal F^eF^P kinematics with a geometric argument, a consistent methodology for determining the initial and evolving GND state has been derived. The augmented kinematics describe GND related microstructural features in the undeformed material like low angle sub-grain boundaries and high angle grain boundaries. Therefore these kinematics are particularly applicable to polycrystalline materials.     

Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects

Recently it has been demonstrated that, on the basis of the separation D=D^e+D^p arising from the split of the stress power and two consistency criteria for objective Eulerian rate formulations, it is possible to establish a consistent Eulerian rate formulation of finite elastoplasticity in terms of the Kirchhoff stress and the stretching, without involving additional deformation-like variables labelled “elastic” or “plastic”. It has further been demonstrated that this consistent formulation leads to a simple essential structure implied by the work postulate, namely, both the normality rule for plastic flow D^p and the convexity of the yield surface in Kirchhoff stress space. Here, we attempt to place such an Eulerian formulation on the thermodynamic grounds by extending it to a general case with thermal effects, where the consistency requirements are treated in a twofold sense. First, we propose a general constitutive formulation based on the foregoing separation as well as the two consistency criteria. This is accomplished by employing the corotational logarithmic rate and by incorporating an exactly integrable Eulerian rate equation for D^e for thermo-elastic behaviour. Then, we study the consistency of the formulation with thermodynamic laws. Towards this goal, simple forms of restrictions are derived, and consequences are discussed. It is shown that the proposed Eulerian formulation is free in the sense of thermodynamic consistency. Namely, a Helmholtz free energy function in explicit form may be found such that the restrictions from the thermodynamic laws can be fulfilled with positive internal dissipation for arbitrary forms of constitutive functions included in the constitutive formulation. In particular, that is the case for the foregoing essential constitutive structure in the purely mechanical case. These results eventually lead to a complete, explicit constitutive theory for coupled fields of deformation, stress and temperature in thermo-elastoplastic solids at finite deformations.     

A weakened form of Ilyushin's postulate and the structure of self-consistent Eulerian finite elastoplasticity

From a general standpoint in terms of internal variables, we formulate a general theory of self-consistent Eulerian finite elastoplasticity based on the additive decomposition of the Eulerian strain rate, i.e., D=D^e+D^p, as well as two consistency criteria. In this theory, the elastic behaviour is characterized by an exactly integrable elastic rate equation for D^e with a general form of complementary elastic potential. It is assumed that the yield function depends in a general manner on the Kirchhoff stress and the internal variables. Moreover, the plastic rate equation for D^p and the evolution equation for each internal variable are allowed to assume general forms relying on the just-mentioned variables and the stress rate. It is indicated that two consistency criteria, i.e., the self-consistency for the elastic rate equation and Prager's yielding stationarity, lead to the unique choice of objective rates, i.e., the logarithmic rate.The structure of the above theory is further studied and examined by virtue of a weakened form of Ilyushin's postulate. In a spinning frame defining the logarithmic rate, we introduce the notion of standard elastoplastic strain cycle, which starts at a point not on but inside a yield surface and incorporates only one infinitesimal plastic subpath. We show that this type of strain cycle is always possible. Then, by ruling out strain cycles starting at points on yield surfaces we propose a weakened form of Ilyushin's postulate, which says that the changing rate of the stress work done along every standard strain cycle should be non-negative, whenever the incorporated plastic subpath tends to vanish. By virtue of simple, rigorous procedures, we demonstrate that this weakened form of Ilyushin's postulate is adequate to ensure direct results concerning the normality rule and the convexity of the yield surface in the context of the foregoing Eulerian finite elastoplasticity theory. Specifically, with an exactly integrable elastic rate equation defining D^e, we prove that, in the space of the Kirchhoff stresses, the difference (D−D^e) is just the gradient of the yield function multiplied by a plastic multiplier, and thus bears the very kinematical and physical feature of plastic strain rate. Furthermore, we prove that, in the space of the Kirchhoff stresses, the elastic domain bounded by each yield surface should be convex. The main results are derived in a self-contained manner within the context of an Eulerian theory of finite elastoplasticity, without involving issues concerning how to define intermediate stress-free states and plastic strains, etc.     

A gradient based iterative algorithm for solving structural dynamics model updating problems

The procedure of updating an existing but inaccurate model is an essential step toward establishing an effective model. Updating damping and stiffness matrices simultaneously with measured modal data can be mathematically formulated as following two problems. Problem 1: Let M _a ∈SR ^n×n be the analytical mass matrix, and Λ=diag{λ ₁,…,λ _p }∈C ^p×p , X=[x ₁,…,x _p ]∈C ^n×p be the measured eigenvalue and eigenvector matrices, where rank(X)=p, p<n and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in\nobreak{\mathbf{C}} $ , $x_{2j} = \bar{x}_{2j-1} \in{\mathbf{C}}^{n} $ for j=1,…,l, and λ _k ∈R, x _k ∈R ⁿ for k=2l+1,…,p. Find real-valued symmetric matrices D and K such that M _a XΛ ²+DXΛ+KX=0. Problem 2: Let D _a ,K _a ∈SR ^n×n be the analytical damping and stiffness matrices. Find $(\hat{D}, \hat{K}) \in\mathbf{S}_{\mathbf{E}}$ such that $\| \hat{D}-D_{a} \|^{2}+\| \hat{K}-K_{a} \|^{2}= \min_{(D,K) \in \mathbf{S}_{\mathbf{E}}}(\| D-D_{a} \|^{2} +\|K-K_{a} \|^{2})$ , where S _E is the solution set of Problem 1 and ∥?∥ is the Frobenius norm. In this paper, a gradient based iterative (GI) algorithm is constructed to solve Problems 1 and 2. A sufficient condition for the convergence of the iterative method is derived and the range of the convergence factor is given to guarantee that the iterative solutions consistently converge to the unique minimum Frobenius norm symmetric solution of Problem 2 when a suitable initial symmetric matrix pair is chosen. The algorithm proposed requires less storage capacity than the existing numerical ones and is numerically reliable as only matrix manipulation is required. Two numerical examples show that the introduced iterative algorithm is quite efficient.     

The decomposition F = FeFp,material symmetry,and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous

This study develops a general framework for discussing both isotropic-viscoplastic materials and amorphous materials. The framework, which allows for large deformations, is based on the Kröner–Lee decomposition of the deformation gradient into elastic and inelastic parts, a system of microforces consistent with its own balance, and a mechanical version of the second law that includes, via the microforces, work performed during inelastic flow. The constitutive theory allows for dependences on the elastic and inelastic parts of the deformation gradient and on the inelastic stretch-rate, but dependences on the inelastic spin are not included. The constitutive equation for the microstress T^p conjugate to inelastic flow – suitably restricted by the second law – and the microforce balance are shown to be together equivalent to a flow rule that includes a back stress due to the variation in the free energy with inelastic deformation. The introduction of a concept of material microstability reduces this flow rule to one of classical Mises-type.In a theory based on the Kröner–Lee decomposition, there are two classes of symmetry transformations available: transformations of the reference configuration and transformations of the relaxed spaces. We discuss the notion of material symmetry for a general class of materials that includes, as special cases, isotropic-viscoplastic solids, and amorphous solids. Essential to this discussion of symmetry is a general constitutive relation for the microstress T^p.The symmetry-based framework allows us to show that for typical boundary-value problems involving isotropic, viscoplastic solids or amorphous solids, if a problem has a solution, then every time- and space-dependent rotation of the relaxed spaces also yields a solution, and it is possible to choose this rotation such that the transformed solution is inelastically spin-free: W^p ≡ 0. Thus, when discussing such materials, we may, without loss in generality, restrict attention to flow rules that are inelastically irrotational.     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat

The aim of this paper is to provide sufficient conditions for the existence of periodic solutions emerging from an upright position of small oscillations of a sleeping symmetrical gyrostat with equations of motion being α and β parameters satisfying Δ=α ²?4β>0 and $\beta-\frac{\alpha^{2}}{2}\pm \frac{\alpha \sqrt{\varDelta }}{2}<0$ , ε a small parameter and, F ₁ and F ₂ smooth periodic maps in the variable t in resonance p:q with some of the periodic solutions of the system for ε=0, where p and q are positive integers relatively prime. The main tool used is the averaging theory.     

Blasenbewegung aufgrund von Marangonikonvektion

This is an experimental study on gas bubble motion in a vertical temperature gradient. Surface tension convection (Marangoni convection) superimposing gravity is discussed with respect to the theory of Young et al. as another cause for bubble motion. The Marangoni numbers for the convection inside the bubble (N ⁱ _Ma) and outside the bubble in the medium (N ^e _Ma) are introduced as criterion for the applicability of the theory. It is shown by the experiments that the Young theory holds up to Marangoni numbersN ⁱ _Ma=10^?2 and N^e _Ma=2, respectively.     

Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F ^T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.     

Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity

The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which X_a⁰ and X_c⁰ play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector X_a:=(YQ_a^t,YQ_a⁰)^t is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime $\text{M}{}^{\mathrm{n+1}}$ on which the proper orthochronous Lorentz group SO_o(n,1) left acts. Then, constructed is a Poincaré group ISO_o(n,1) on space X:=X_a+X_b, of which X_b reflects the kinematic hardening rule in the model. We also find that the space (Q_a^t,q₀^a) is a Robertson–Walker spacetime, which is conformal to X_a through a factor Y, and conformal to X_c:=(ρQ_a^t,ρQ_a⁰)^t through a factor ρ as given by ρ(q₀^a)=Y(q₀^a)/[1−2ρ₀Q_a⁰(0)+2ρ₀Y(q₀^a)Q_a⁰(q₀^a)]. In the conformal spacetime the internal symmetry is a conformal group.     

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP ⁽¹⁾+μ² P ⁽²⁾+…, q=μq ⁽¹⁾+μ² q ⁽²⁾+…. The Burnett terms P ⁽²⁾, q ⁽²⁾ depend on 11 coefficients ω _i , 1≦i≦6, θ&; _i , 1≦i≦ 5. This paper shows that ω₃+ω₄+θ₃= 0.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

Dynamic Crack Growth Past a Stiff Inclusion: Optical Investigation of Inclusion Eccentricity and Inclusion-matrix Adhesion Strength

Interactions between a dynamically growing matrix crack and a stationary stiff cylindrical inclusion are studied optically. Test specimens with two different bond strengths (weak and strong) and three crack-inclusion eccentricities (e = 0, d/2 and 3d/4, d being inclusion diameter) are studied using reflection mode Coherent Gradient Sensing (CGS) and high-speed photography. These variants produce distinct dynamic crack trajectories and failure behaviors. A weaker inclusion-matrix interface attracts a propagating crack while a stronger one deflects the crack away. The former results in a propagating crack lodging (‘key-hole’) into the inclusion-matrix interface whereas in the latter the crack tends to circumvent the inclusion. When the inclusion is in the prospective crack path, the maximum attained crack speed is much higher in the weakly bonded inclusion cases relative to the strongly bonded counterparts. For a crack propagating towards a weakly bonded inclusion, the effective stress intensity factor (K _e) value remains constant for each inclusion eccentricity considered. But these constant K _e values increase with increasing eccentricity. A distinct drop in K _e occurs when the crack is near the inclusion. In strongly bonded inclusion cases, on the other hand, monotonically increasing K _e before the crack reaches the inclusion is observed. A drop in K _e is seen just before the crack reaches the inclusion. The mode-mixity estimates are of opposite signs for weakly and strongly bonded inclusions in case of the largest eccentricity studied, confirming the observed crack attraction and deflection mechanisms.

On the Transformation Property of the Deformation Gradient under a Change of Frame

If the deformation gradients are denoted by F and F ^* respectively before and after a change of frame, they are related by the transformation formula, F ^*=QF, where Q is the orthogonal transformation associated with the change of frame. Although it has been pointed out that this relation is valid “provided that the reference configuration be unaffected by the change of frame” (see p. 308 of [1]), this formula is found in most textbook of Continuum Mechanics, and is used, without further justification, in deriving the condition of material frame-indifference, ?(QF)=Q?(F)Q ^T for the constitutive function ? of the stress tensor of an elastic body. In this note, we shall analyze the effect of change of frame on the transformation property of the deformation gradient, and show that the above transformation formula is not valid in general. However, we shall confirm the validity of the above well-known condition of material frame-indifference without the assumption that the reference configuration be unaffected by the change of frame.