Numerical Prediction of Macroscopic Material Failure

The aim of this contribution is the numerical determination of macroscopic material properties based on constitutive relationships characterising the microscale. A macroscopic failure criterion is computed using a three dimensional finite element formulation. The proposed finite element model implements the Strong Discontinuity Approach (SDA) in order to include the localised, fully nonlinear kinematics associated with the failure on the microscale. This numerical application exploits further the Enhanced–Assumed–Strain (EAS) concept to decompose additively the deformation gradient into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. This finite element formulation is then used for the modelling of the microscale and for the discretisation of a representative volume element (RVE). The macroscopic material behaviour results from numerical computations of the RVE. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A gradient model for finite strain elastoplasticity coupled with damage

This paper describes the formulation of an implicit gradient damage model for finite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the finite element formulation, in contrast with standard local material models. A 3D finite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations.

A numerical set of examples comparing the results of local and the gradient formulations relative to the mesh size influence is presented and some examples comparing results from other authors are also presented, illustrating the capabilities of the present proposal.     

A non-local damage approach for the boundary element method

The conventional (local) constitutive modelling of materials exhibiting strain softening behaviour is susceptive to a spurious mesh dependence caused by numerically induced strain localization. Also, for refined meshes, numerical instabilities may be verified, mainly if the simulations are performed by the boundary element method. An alternative to overcome such difficulties is the adoption of the so called non-local constitutive models. In these approaches, some internal variables of the constitutive model in a single point are averaged considering its values of the neighbouring points. In this paper, the implicit formulation of the boundary element method for physically non-linear problems in solid mechanics is used with a non-local isotropic damage model and a very simple averaging scheme, over internal cells, is introduced. It is shown that the analysis become more stable in comparison to the case of a local application of the same model and that the results recover the desired objectiveness to mesh refinement.     

Failure in anisotropic nonwoven materials at finite deformation

The current work proposes a finite deformation strong discontinuity approach based modeling of failure in anisotropic materials with reorientation based micromechanical network model for the bulk response. These materials consist of randomly cross-linked one dimensional filaments at their microstructure which undergo non-affine deformation as well as reorientation upon loading before undergoing complete failure. The computationally efficient strong discontinuity approach allows to capture the failure kinematics by introducing a local problem where a strong discontinuity exists, thereby leading to an enhanced deformation gradient. A precise evaluation of the bulk response is done by homogenizing the physical microscopic response of constituent filaments where reorientation is introduced with an initial straightening effect of fiber undulations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Locally Embedded Strong Discontinuities at finite strains – Numerical Implementation

In this paper, a geometrically nonlinear finite element approximation for highly localized deformation in structures undergoing material failure in the form of strain softening, is developed. The basis for its numerical implementation in this class of problems is defined through the elaboration of Strong Discontinuity Approach-fundamentals. Proposed numerical model uses an Enhanced Assumed Strain Concept for the additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, this part of the deformation mapping is condensed out at the material level, without the application of static condensation technique. The resulting set of constitutive equations is formally identical to that of standard plasticity and therefore, can be solved using the return-mapping algorithm. No assumptions regarding the interface law connecting the displacement discontinuity with the conjugate traction vector are made. As a result, the proposed numerical solution can be applied to a broad range of different mechanical problems including mode-I fracture in brittle materials or the analysis of shear bands. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

混凝土材料的粘塑性损伤本构模型

基于不可逆热力学,引入运动硬化、等向硬化和损伤内变量,构造了相应的自由能函数和流动势函数,推导出了混凝土材料的粘塑性损伤本构模型．数值模拟的结果表明,该模型能够避开屈服面和破坏准则的基本假设来描述混凝土材料的以下特性:压缩载荷作用下的体积膨胀现象;应变率敏感性;峰值后由损伤和破坏引起的应力软化和刚度退化现象A·D2由于此模型避开了根据各种变形阶段选择与其相应的本构模型的繁琐计算,因此更便于纳入复杂工况下应力分析有限元程序中．     

A higher-order Godunov method for modeling finite deformation in elastic-plastic solids

In this paper we develop a first-order system of conservation laws for finite deformation in solids, describe its characteristic structure, and use this analysis to develop a second-order numerical method for problems involving finite deformation and plasticity. The equations of mass, momentum, and energy conservation in Lagrangian and Eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy, as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient. Particular attention is paid to the influence of a curl constraint on the deformation gradient, so that the characteristic speeds transform properly between the two frames of reference. Next, we consider models in rate-form for isotropic elastic-plastic materials with work-hardening, and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion. In spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric, we still find that the characteristic speeds are always real for elastic behavior, and essentially always real for plastic response. These results allow us to construct a second-order Godunov method for the computation of three-dimensional displacement in a one-dimensional material viewed in the Lagrangian frame of reference. We also describe a technique for the approximate solution of Riemann problems in order to determine numerical fluxes in this algorithm. Finally, we present numerical examples of the results of the algorithm.     

Structural damage accumulation and stable postcritical deformation of composite materials

The macroscopic failure of composite materials is preceded by complex multilevel processes accompanied by accumulation and localization of damaged centers and formation of a failure cluster. Therefore, the study of these mechanisms is one of the basic problems for the mechanics of modern composite materials used in aerospace engineering. The formation of a theory of the stable postcritical deformation of the work-softening media is considered. The pseudo-plastic deformation affected by structural damage of granular composites is investigated within the framework of the considered two-level structurally phenomenological model of heterogeneous media. The stable evolution of the interconnected processes is accompanied by stress redistributions, partial or complete unloading, and strain or damage localization that are one of the main causes of implementation of the postcritical deformation stage. The numerical calculation results of inelastic deformation and failure of the periodic unidirectional fiber-reinforced composites are presented under conditions of the displacement-controlled transverse proportional loading mode. The main mechanisms of the work-softening behavior for the indicated type of materials are described in the macro-homogeneous stress-strain states. Macroscopically, the failure of heterogeneous media as a result of postcritical deformation and the loss of stability of damage accumulation depends on the stiffness of the loading system. When a deformable body is fixed on the closed surface with sufficiently but not infinitely large coefficients of stiffness, it is possible to observe the equilibrium development of the localized volumes of work-softening and damage. The constitutive equations for the work-softening isotropic, transverse isotropic, and orthotropic media are presented. The effect of the loading system on the stability of deformation, damage accumulation, and failure under monotone and nonmonotone triaxial loading was studied. The growth of failure strains with increase in stiffness of the loading system and unequal resistance of heterogeneous body are registered and investigated. A preventive unloading method is offered for the mathematical modeling of the damage accumulation during the testing of the materials on the servo-controlled systems. The displacement-controlled mode is simulated by a series of soft loading and unloading cycles. The detected phenomenon of failure where the unloading leads to stress-strain diagrams with a negative slope of the descending branch was not found either in the displacement or stress-controlled monotone loading mode.Submitted to the 10th International Conference on Mechanics of Composite Materials, April 20–23, 1998, Riga, Latvia.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 234–250, March–April, 1998.     

爆炸和冲击载荷下金属材料及结构的动态失效仿真

通过数值模拟研究爆炸冲击载荷下金属材料和结构的动态失效规律,对表征爆炸冲击毁伤效应及设计新型抗冲击结构有重要意义.强动载下金属材料失效涉及材料大变形、热力耦合、材料状态变化等多个复杂物理过程,给数值仿真带来了极大挑战,其中包括裂纹、剪切带等复杂失效模式的几何描述、动态失效准则的确定、塑性与损伤耦合演化的描述等问题.针对这些挑战性问题,基于能量变分建立描述金属动态失效过程的热弹塑性相场理论和计算模型,实现了断裂与剪切带失效模式的统一描述,并提出了其显式有限元高效求解策略.进一步将该模型应用于爆炸冲击载荷下金属脆韧失效模式转变、绝热剪切带(ASBs)自组织及冲击波作用下薄壁圆盘失效形式转变三个典型金属动态失效问题,验证了理论模型的准确性及计算模型的稳健性.该工作为后续开展基于仿真的爆炸冲击毁伤评估及防护结构设计研究奠定了基础.     

Modelling of composite plates including damage

The paper is concerned with the modelling and numerical simulation of fibre-composite plates in the nonlinear range due to large strains and damage. The layer-wise approach is applied. Each layer is treated as elastic-brittle and assumed to be orthotropic in the local material coordinate system. The appearance of damage is controlled according to the failure criteria [1,2,3,4]. When the failure condition is satisfied, the mechanical properties of the material are modified appropriately, depending on the type of damage (fibre breakage, matrix crack, fibre-matrix shear). We have programmed the model as a user subroutine within the ABAQUS environment and carried out a number of numerical simulations. The obtained numerical results are compared with the experimental data available in the literature [3]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A three-dimensional progressive damage model for fiber reinforced composites with an implicit-explicit integration scheme

This contribution proposes a fully three dimensional “continuum damage model” (CDM) to describe the interlaminar and intralaminar failure mechanisms of transversely isotropic elastic-brittle materials under static loading. The constitutive model is derived from an energy function with independent damage variables for each damage mode. The evolution law is based on energy dissipation within the damage process, taking into account the critical energy release rate to weaken the effect of mesh dependent outcome. The onset of damage can be predicted with Cuntze's failure mode concept [1] as well as with Hashin's failure criteria. In this model linear stress decreasing is assumed. In addition, an implicit-explicit integration scheme, first proposed by Oliver [3] for isotropic damage models, is adapted to increase the stability and robustness of numerical simulations and to decrease the computational cost of material failure analyses. By comparing the results from implicit-explicit integration schemes and standard implicit integration schemes, a high level of agreement is found. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale Modeling for the Simulation of Damage Processes at Refractory Materials under Thermal Shock

Refractory materials, for example ceramic materials, initially contain a multitude of defects such as voids, microcracks, grain boundaries etc. The deformation process and failure mechanisms due to thermal shock at high temperatures above 1000°C are going along with the creation of new micro defects as well as the growth and coalescence of cracks. A material damage model based on the theoretical concept of damage mechanics and the mechanics of microcracks is presented in this paper. Cell models are developed as representative volume elements (RVE) including crack initiation and growth as well as microstructural shielding effects. For simple configurations of the microstructure, the relation between stress, strain and temperature is derived from analytical considerations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

FE‐formulation for plate and beam elements to simulate onset and propagation of discrete cracks in solid structures

The aim of this paper is to present some numerical aspects related to the modeling both the formation and the propagation of discrete cracks in solid structures. The presented formulation corresponds to the concept of embedded discontinuities [1], and will be applied to a plate and to a beam element. The failure of solid structures is often triggered by a highly localized pattern of inelastic deformation in the form of narrow bands. Characteristic examples are shear bands in metals and soils, or localized bands of cracking in brittle materials, like concrete or rocks. A well known difficulty associated with classical (local, rate‐independent) continuum theories with strain softening attributes is that numerical solutions are found to lack invariance with respect to the choice of spatial discretization. For quasi‐static boundary problems, this mathematical inconsistency causes the loss of ellipticity for the governing equations (material instability). To regularize this inconsistency, several strategies have been applied. In the presented formulation, additional degrees of freedom are considered. Within the concept of embedded discontinuities, the regular displacements are enriched by discontinuities. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Iterative parameter identification methods for nonlinear functions

This paper considers identification problems of nonlinear functions fitting or nonlinear systems modelling. A gradient based iterative algorithm and a Newton iterative algorithm are presented to determine the parameters of a nonlinear system by using the negative gradient search method and Newton method. Furthermore, two model transformation based iterative methods are proposed in order to enhance computational efficiencies. By means of the model transformation, a simpler nonlinear model is achieved to simplify the computation. Finally, the proposed approaches are analyzed using a numerical example.     

Towards modelling skeletal muscle growth and adaptation

Despite an increasing interest in modelling skeletal muscles adaptation, models that address the phenomena within a continuum-mechanical framework using muscle-specific material models are rare in literature. This work focuses on modelling one form of skeletal musle adaptation, namely sarcomerogenesis. Sarcomerogenesis occurs when a given stretch is sustained over a period of time and the number of basic contractile units, which are the sarcomeres, increase. To model sarcomerogenesis within a continuum-mechanical setting, the growth framework based on a multiplicative split of the total deformation gradient is employed. An evolution equation that describes sarcomerogenesis is used and incorporated in a transversally isotropic material model that accounts for a skeletal muscle's active force production capabilities. The material tangent modulus is derived and implemented within the finite-element analysis software. Using this model, one sees that increased number of sarcomeres results in a decreased force response of the muscle tissue over time. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures

In order to design the microstructure of metamaterials showing high toughness in extension (property to be shared with muscles), it has been recently proposed (Dell’Isola et al. in Z Angew Math Phys 66(6):3473–3498, 2015) to consider pantographic structures. It is possible to model such structures at a suitably small length scale (resolving in detail the interconnecting pivots/cylinders) using a standard Cauchy first gradient theory. However, the computational costs for such modelling choice are not allowing for the study of more complex mechanical systems including for instance many pantographic substructures. The microscopic model considered here is a quadratic isotropic Saint-Venant first gradient continuum including geometric nonlinearities and characterized by two Lamé parameters. The introduced macroscopic two-dimensional model for pantographic sheets is characterized by a deformation energy quadratic both in the first and second gradient of placement. However, as underlined in Dell’Isola et al. (Proc R Soc Lond A 472(2185):20150790, 2016), it is needed that the second gradient stiffness depends on the first gradient of placement if large deformations and large displacements configurations must be described. The numerical identification procedure presented in this paper consists in fitting the macro-constitutive parameters using several numerical simulations performed with the micro-model. The parameters obtained by the best fit identification in few deformation problems fit very well also in many others, showing that the reduced proposed model is suitable to get an effective model at relevantly lower computational effort. The presented numerical evidences suggest that a rigorous mathematical homogenization result most likely holds.