Constitutive Equations In Plasticity Research Articles

An Efficient Method for Integrating von-Mises Plasticity with Mixed Hardening

Integrating the constitutive equations in plasticity in order to update the stresses is the most important part of an elastoplastic finite element analysis. Here, a von-Mises plasticity model along with the isotropic and kinematic hardenings is taken into account in the small strain realm. An accurate integration is formulated by converting the differential constitutive equations into an augmented stress space and utilizing the return mapping algorithm. Subsequently, a broad range of numerical tests are performed to investigate the accuracy and performance of the proposed integration. The results demonstrate the robustness of the new integration scheme.

Диаграмма предельных деформаций при горячей листовой штамповке металлов. Обзор моделей материала, критериев вязкого разрушения и стандартных испытаний

Для обоснованного выбора определяющих уравнений материала при математическом моделировании процессов горячей и теплой обработки давлением тонколистовых металлических изделий с большой степенью вытяжки рассматриваются способы теоретического анализа и экспериментального подтверждения условий предельного деформирования материала. Внимание сконцентрировано на кривой предельного деформирования листового материала на плоскости главных деформаций (одна из которых соответствует растяжению, а вторая может задавать растяжение или сжатие), характеристике локального состояния материала, отвечающей критическому росту локализации деформации. Локализация здесь понимается как локальное утонение листа и соответствует диффузной форме локализации, другие дефекты (полосы сдвига, образование трещин) развиваются из данного предельного состояния либо (образование складок и морщин) не являются локальными и требуют полной постановки задачи. Данная кривая, определяющая условия реализации того или иного технологического процесса, может быть теоретически предсказана по заданным модели пластического течения и критерию вязкого разрушения материала и начальным несовершенствам. Для этого рассматриваются возможности схемы Марциньяка - Куцзинского (Marciniak - Kuczyński scheme), образец в рамках которой имеет две зоны однородной деформации и допускает аналитическое сведение задачи к системе нескольких обыкновенных дифференциальных уравнений, решаемых численно. Экспериментальный метод предусматривает испытание вдавливанием пуансона со сферическим или цилиндрическим наконечником в образец, вырубленный из листа, который в зависимости от глубины боковых вырезов может обеспечивать растяжение либо сжатие образца в поперечном направлении. Оба подхода анализируются в работе в качестве инструментов выбора и экспериментальной верификации модели материала и критерия предельного состояния, помогающих решению сложной методической проблемы идентификации математической модели по достаточно нетипичным для механики деформируемого твердого тела экспериментам, сопровождающимся локализацией деформации. С применением схемы Марциньяка - Куцзинского выполнен анализ ряда критериев текучести анизотропного листового материала, законов упрочнения и моделей повреждаемости, а также критериев предельного состояния на кривую предельного деформирования, для чего был разработан собственный алгоритм. Экспериментальные стандартные схемы испытания по методам Хасека (V. Hasek), Накадзимы (K. Nakajima) и Марциньяка (Z. Marciniak) были реализованы численно в пакете программ LS-DYNA, данные которых для сравнения также были нанесены на плоскость главных деформаций. Обсуждается возможность интегрирования в схему Марциньяка - Куцзинского для каждой базовой жестко-пластической (склерономной) модели зависимости от температуры, скорости деформации и микроструктуры. Отмечено существенное ограничение теоретической схемы Марциньяка - Куцзинского рамками пропорционального изменения главных деформаций в образце вне и внутри зоны локализации деформации, а также то, что она не приспособлена для определения предельных свойств металлов, деформируемых в условиях деформационного разупрочнения, демонстрируемого алюминиевыми и титановыми сплавами и некоторыми сталями при температурах динамической рекристаллизации. Для более широкого диапазона условий деформирования материала альтернативы упомянутому численному методу предсказания кривой предельного деформирования не выявлено. Отдельным открытым и актуальным вопросом остается описание эволюции анизотропных свойств пластичности и разрушения вследствие анизотропного накопления поврежденности.

Nonlinear elastoplastic analysis of pressure sensitive materials

When structures undergo extreme loading conditions, the materials pass the elastic limits. Therefore, to preserve economy as well as safety, it is essential to perform a realistic elastoplastic analysis using the constitutive equations in plasticity. On the other hand, computing the stress alongside its associated variables on Gauss points is a delicate process and virtually the most important part of these analyses. In this study, an efficient stress-updating technique is presented for the constitutive rate equations of the pressure sensitive materials such as concrete, rock, soil and some kind of metals. Accordingly, the Drucker–Prager plasticity is utilized to consider the hydrostatic pressure in addition to the J 2-invariant of the deviatoric stress. Moreover, the isotropic and kinematic hardenings are used to take into account more realistic behavior of the materials. Finally, a wide range of numerical tests is carried out to show the performance of the presented method together with the application of the suggested formulations in elastoplastic analysis of a gravity dam.

Rice’s Internal Variables Formalism and Its Implications for the Elastic and Conductive Properties of Cracked Materials, and for the Attempts to Relate Strength to Stiffness

Rice’s internal variables formalism [1975, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms,” in Constitutive Equations in Plasticity, edited by A. Argon, MIT Press, Cambridge, MA, pp. 23–75] is one of the basic tools in the micromechanics of materials. One of its implications is the possibility to relate the compliance/resistivity contributions of cracks—the key quantities in the problem of effective elastic/conductive properties—to the stress intensity factors (SIFs) and thus to utilize a large library of available solutions for SIFs. Examples include configurations that are common in materials science applications: branched and intersecting cracks, cracks with partial contact between crack faces, and cracks emanating from pores. The formalism also yields valuable physical insights of a qualitative character, such as the impossibility to correlate, in a quantitative way, the strength of microcracking materials and their stiffness reduction.

Strain rate dependence of hardness of AlN doped SiC

Tailored nanoindentation experiments were used to determine the strain rate dependence of the hardness of AlN doped SiC produced by spark plasma sintering. It is shown that in the nanoindentation regime, where cracking is limited, the hardness of SiC reduces by 0·8 GPa/decade reduction in strain rate. This measured variation in hardness is consistent with estimates for the strain rate dependence of the lattice resistance (Peierls stress) of SiC. These experiments therefore open up the possibility to determine real constitutive equations for plasticity in SiC and other armour ceramics.

Mroz' Local Response Moduli as a Means for Comparing Constitutive Equations in Plasticity

Linear and nonlinear relations of time independent plasticity are compared and their local response for varying stress or strain increment orientation is studied. The applicability of incremental relations to stability and post-critical response analysis can be assessed from such comparative study. In particular, the endochronic model providing nonlinear incremental relations is considered.

New constitutive equation for plasticity in high speed torsion tests of metals

Present work investigates a phenomenological plasticity equation for the plastic behavior of metals deformed in high speed torsion tests. The tests were carried out at room temperature in a laboratory torsion test equipment and also in an universal tensile test machine on annealed commercial pure copper and aluminum specimens. The tensile tests were performed at room temperature by an universal testing machine at low strain rate of 0.017/s. The experimental torsion tests were carried out at constant angular speed that imposed a constant shear strain rate to the specimen. In the tests, the rotation speed were set to 61 rpm and 200 rpm which imposed high strain rates of about 2/s and 6.5/s respectively in tubular specimens. The torsion test lasted between 0.5 s and 1 s. The experimental hardening curves of equivalent stress versus strain in torsion were sigmoidal type and were fitted by different constitutive equations for plasticity that takes into account the strain hardening and thermal softening effects due to local temperature increase as the Johnson-Cook model, the modified Voce equation and others. The specimen local temperature increase was calculated assuming adiabatic deformation process. It is proposed a new constitutive equation or modified Voce equation for the equivalent flow stress which considers the effects of strain hardening, strain rate hardening and thermal softening for the best fit to the present experimental data of annealed aluminum and copper.

Relationship between plastic and intrinsic dissipations

In this paper, the relationship between the plastic and intrinsic dissipations is addressed within the normality structure of [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455; Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT Press, Cambridge, MA, pp. 23–79.] It is shown that the plastic dissipation is generally not equal to the intrinsic dissipation. Within the normality structure, the microscale and macroscale thermodynamic fluxes and forces are related by the conditions of energy and dissipation equivalence. If the plastic dissipation is required to be equal to the intrinsic dissipation, J 2 potential and the Levy–Mises equation are recovered from the condition of dissipation equivalence for incompressible plastic flows.

Elastic compliances of non-flat cracks

Cracks of non-flat geometries are common in materials science applications. Their effect on the overall elastic properties is addressed. Applications of Rice’s [Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A. (Ed.), Constitutive Equations in Plasticity, MIT Press, pp. 23–75.] theorem that, in particular, relates stress intensity factors to crack compliances, are discussed and illustrated by a 2-D example of a circular arc crack. Then we conduct computational studies of several non-flat patterns and suggest a simple approximation for compliances of non-flat cracks. Implications for anisotropy due to non-random orientation distributions of cracks of non-flat geometries are discussed.

Multiscale Thermodynamic Significance of the Scale Invariance Approach in Continuum Inelasticity

AbstractIn this paper, the scale invariance approach from micro- to macro-plasticity by Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach,” ASME J. Eng. Mater., 117, pp. 352–355) and Ning and Aifantis (1996, “Anisotropic Yield and Plastic Flow of Polycristalline Solids,” Int. J. Plasticity, 12, pp. 1221–1240) is investigated within Rice’s normality structure (1971, “Inelastic Constitutive Relations for Solids: An Integral Variable Theory and its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455; 1975, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms,” Constitutive Equations in Plasticity, A. S. Argon, ed., MIT Press, Cambridge, MA, pp. 23–79). The normality structure provides a minimal framework of multiscale thermodynamics, and the dissipation equivalence between the microscale and macroscale is ensured by a variational equation which can be further formulated into principle of maximum equivalent dissipation. It is revealed in this paper that within the framework of normality structure, the so-called hypothesis of generalized scale invariance holds for the kinetic rate laws, flow rules, and orthogonality conditions in the sense of Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach”). Stemming from Rice’s kinetic rate laws, the generalized scale invariance reflects the inherent self-consistent character of the normality structure. If the plastic work rate is assumed to be equal to the intrinsic dissipation rate, the kinematic hardening plasticity as a demonstration of the scale invariance approach by Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach”), can be well accommodated within the framework of normality structure. Therefore, the scale invariance approach is justified from a multiscale thermodynamic viewpoint. It is further shown that the maximization procedure in this approach just corresponds to the principle of maximum equivalent dissipation.

On Microscopic Thermodynamic Mechanisms of Damage Evolution Laws

Most existing phenomenological damage evolution laws can be covered by phenomenological equations or linear irreversible thermodynamics. In this paper, general microscopic thermodynamic mechanisms leading to nonlinear phenomenological equations are explored within the framework of ‘normality structures’ by Rice (Rice, J.R. (1971). Inelastic Constitutive Relations for Solids: An Internal Variable Theory and its Application to Metal Plasticity, Journal of the Mechanics and Physics of Solids, 19: 433-455, Rice, J.R. (1975). Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms, In: Argon, A.S. (ed.), Constitutive Equations in Plasticity, MIT Press, Cambridge, MA, pp. 23-79.) at the level of microstructural rearrangements. Rice’s kinetic rate laws of local internal variables, with each rate being stress dependent only via its conjugate thermodynamic force, are cornerstones of the normality structure. It is revealed in this paper that nonlinear phenomenological equations and Onsager reciprocal relations emerge naturally from the normality structures if each rate is a homogeneous function of degree q in its conjugate force. Furthermore, the nonlinear phenomenological coefficient matrix is identical to the Hessian matrix of the flow potential function in conjugate forces scaled by q. Finally, as an application and demonstration, some fundamental issues on damage evolution laws for microcracked solids have been addressed based on the revealed remarkable properties. It is shown that the deduced flow potential functions of microcracked solids can be expressed in the forms of well-established Hill (Hill, R. (1950). The Mathematical Theory of Plasticity, Clarendon Press, Oxford.) anisotropic yield function and Karafillis and Boyce (Karafillis, A.P. and Boyce, D.B. (1993). A General Anisotropic Yield Criterion Using Bounds and a Transformation Weighting Tensor, Journal of the Mechanics and Physics of Solids, 46: 85-113.) isotropic yield surface.

Relationship between normality structure and orthogonality condition

The nonlinear generalization of the phenomenological equations along with Onsager [Reciprocal relations in irreversible processes, I, II. Physical Review 37 (1931) 405] reciprocal relations includes the normality structure of Rice [Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. Journal of the Mechanics and Physics of Solids 19 (1971) 433; Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT Press, Cambridge, MA, 1975, pp. 23–79] and the orthogonality condition of Ziegler [An Introduction to Thermomechanics. North-Holland, Amsterdam, 1977] and Ziegler and Wehrli [The derivation of constitutive relations from the free-energy and the dissipation function. Advances in Applied Mechanics 25 (1987a) 183–238; On a principle of maximal rate of entropy production. Journal of Non-Equilibrium Thermodynamics 12 (1987b) 229–243]. The normality structure is generally not consistent with the orthogonality condition due to the different potential functions. The aim of this paper is to examine whether or not the two theories conflict with each other and to establish the consistency condition between the two theories. It is shown in this paper that monotonic increasing and homogeneous kinetic rate laws of local internal variables are thermo-dynamically admissible within the normality structure. The homogeneity property of the kinetic rate laws is the necessary and sufficient consistency condition between the normality structure and orthogonality condition. The thermodynamic processes obeying the orthogonality condition belong to a subset of these processes covered by the normality structure, so the normality structure possesses much more generality than the orthogonality condition.

Microscopic thermodynamic basis of normality structure of inelastic constitutive relations

The essential content of plasticity theory is the normality of plastic strain increments to the yield or flow potential surface at smooth points. The corresponding microscopic thermodynamic mechanism has been explored by Rice [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. Journal of the Mechanics and Physics of Solids 19, 433; Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms, in: Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT Press, Cambridge, MA, p. 23] in his thermodynamic framework with internal variables. Rice’s kinetic rate laws of local internal variables, with each rate being stress dependent only via its conjugate thermodynamic force, directly lead to a unifying normality structure and represent a wide class of inelastic behaviors. The aim of this paper is to reveal the underlying physical mechanism and basis of Rice’s kinetic rate laws. It is shown that Onsager fluxes which satisfy the so-called non-linear Onsager reciprocal relations Edelen [Edelen, D.G.B., 1972. A non-linear Onsager theory of irreversibility. International Journal of Engineering Science 10, 481; 1973. Asymptotic stability, Onsager fluxes and reaction kinetics, International Journal of Engineering Science 11, 819], can ensure the existence of the flow potential function and lead to the normality structure, and Rice’s kinetic rate laws of local internal variables are just certain specific Onsager fluxes. It is also shown that the convexity of the entropy production rate can ensure the convexity of the flow potential function based on homogeneous Onsager fluxes.

Relationship between refined Griffith criterion and power laws for cracking

Rice [J. Mech. Phys. Solids 26 (1978) 61] proposes a refined Griffith criterion, (G−2γ) a ̇ ⩾0 at any local crack front, where G is the Irwin's energy release rate, γ is the surface free energy and ȧ is the rate of crack advance. The refined version implies that the entropy production inequality should holds locally rather than globally from the thermodynamic point of view. Within the irreversible thermodynamic framework developed by Rice [J. Mech. Phys. Solids 19 (1971) 433; Constitutive Equations in Plasticity, 1975, p. 23], it is revealed in this paper that the entropy production inequality holds for each internal variable if its rate is a homogeneous function in its conjugate force. It is further shown that widely-used power laws for crack growth are just certain homogeneous kinetic rate laws, so it is concluded that the power laws directly lead to the refined Griffith criterion.

Constitutive Modelling of Geomaterials

Constitutive Modelling of Geomaterials

A Study on Nonlinear Viscoelastic Viscoplastic Finite Element Method using Iterative Method.

In this paper, we propose an algorithm of FEM for stress analysis of the materials which have viscoelastic viscoplastic chatacteristics such as UHMW-PE. The constitutive equations for plasticity and viscoelasticity derived from the experimental results were used in the algorithm. The new displacement meter made from eddy current sensors, which was made to obtain more accurate test data at low strains, and the constitutive equations derived from the experimental results by the meter were shown. Finally, the comparisons of the numerical results and the experimental results of the load vs. displacement curves for the specimens with side groove were performed. The calculated values were in good agreement with the experimental results. It was confirmed that the algorithm of FEM and the constitutive equations proposed by the authors were valid for stress analysis of nonlinear viscoelastic viscoplastic problems.

Predictive Modeling of the Nonuniform Deformation of the Aluminum Alloy 5182

We pose an experimental model for hot deformation that, in complexity, falls between a homogeneous laboratory test and an industrial process. Our objective is to document a transition between two distinct modes of thermally-activated deformation: diffusion controlled solute drag and hardening with concurrent recovery of dislocations. We demonstrate that constitutive equations for plasticity, describing different regimes of dislocation kinetics and calibrated with a minimum of adjustable constants, can be incorporated into finite element analysis and used reliably to predict the mechanical response of a nonuniform body. [S0094-4289(00)00302-9]

Mechanical Property and Constitutive Equation of Ultra High Molecular Weight Polyethylene.

UHMW-PE has been used as a bearing material in the artificial knee joint. Stress analysis, using FEM, is generally performed to clarify the cause of in vivo wear and delamination. A test machine, of the displacement control type, was designed and built and equipped with a measuring system to obtain accurate test results and the constitutive equation of UHMW-PE. The maximum crosshead speed of the machine is 104 mm/min. Furthermore, it has the capability of determining stress relaxation and strain recovery behavior immediately after performing a tensile test. Interesting properties of UHMW-PE have been obtained using this machine. Also, the constitutive equations for plasticity and viscoelasticity were derived from the experimental results. The equations have strongly nonlinear properties in the mechanical model described in this work. Stress relaxation simulations based on the equation for viscoelasticity have been performed. From this, it was confirmed that the equation is valid.

Constitutive equations of plasticity of a two-phase hardening medium

Constitutive equations of plasticity of a two-phase hardening medium

Structure of constitutive equations in plasticity for different choices of state and control variables

The structure of incremental equations in plasticity is discussed with respect to the choice of control variables representing stress, strain or mixed control, while (quite independently) the yield surface may be represented in either stress, strain, or mixed space. It is concluded that the choice of state variables does not affect the analytical properties of the resulting equations, whereas the choice of control variables has a major influence on the uniqueness of the stress-strain behaviour. For an associated flow rule the most and least severe restrictions are placed by pure stress control and pure strain control, respectively.