Statistical damage model for quasi-brittle materials under uniaxial tension
来源期刊：中南大学学报(英文版)2009年第4期
论文作者：陈健云 白卫峰 范书立 林皋
文章页码：669 - 677
Key words：quasi-brittle material; damage mechanism; microstructure; tensile properties; fracture process zone
Abstract: Based on the parallel bar system, combining with the synergetic method, the catastrophe theory and the acoustic emission test, a new motivated statistical damage model for quasi-brittle solid was developed. Taking concrete for instances, the rationality and the flexibility of this model and its parameters-determining method were identified by the comparative analyses between theoretical and experimental curves. The results show that the model can simulate the whole damage and fracture process in the fracture process zone of material when the materials are exposed to quasi-static uniaxial tensile traction. The influence of the mesoscopic damage mechanism on the macroscopic mechanical properties of quasi-brittle materials is summarized into two aspects, rupture damage and yield damage. The whole damage course is divided into the statistical even damage phase and the local breach phase, corresponding to the two stages described by the catastrophe theory. The two characteristic states, the peak nominal stress state and the critical state are distinguished, and the critical state plays a key role during the whole damage evolution course.
基金信息：the National Natural Science Foundation of China
the Program for New Century Excellent Talents in University
J. Cent. South Univ. Technol. (2009) 16: 0669-0676
DOI: 10.1007/s11771-009-0111-6
CHEN Jian-yun(陈健云), BAI Wei-feng(白卫峰), FAN Shu-li(范书立), LIN Gao(林皋)
(School of Civil and Hydraulic Engineering, Dalian University of Technology, Dalian 116024, China)
Abstract: Based on the parallel bar system, combining with the synergetic method, the catastrophe theory and the acoustic emission test, a new motivated statistical damage model for quasi-brittle solid was developed. Taking concrete for instances, the rationality and the flexibility of this model and its parameters-determining method were identified by the comparative analyses between theoretical and experimental curves. The results show that the model can simulate the whole damage and fracture process in the fracture process zone of material when the materials are exposed to quasi-static uniaxial tensile traction. The influence of the mesoscopic damage mechanism on the macroscopic mechanical properties of quasi-brittle materials is summarized into two aspects, rupture damage and yield damage. The whole damage course is divided into the statistical even damage phase and the local breach phase, corresponding to the two stages described by the catastrophe theory. The two characteristic states, the peak nominal stress state and the critical state are distinguished, and the critical state plays a key role during the whole damage evolution course.
Key words: quasi-brittle material; damage mechanism; microstructure; tensile properties; fracture process zone
1 Introduction
Damage mechanics (DM) is a relatively new field studying the response and reliability of materials with countless randomly distributed irregular microcracks [1-3]. In recent years, great progress on simulating the whole damage and fracture process for quasi-brittle rock and concrete has been made by introducing the statistical damage theory. In this theory, the heterogeneity of material on meso scale has been introduced by the unequal strength of meso-cells that obeys a certain probability distribution form.
Since the failure of the quasi-brittle materials essentially attributes to the nucleation and growth of microcracks produced by local tensile strain, the uniaxial tension can be regarded as one of the most fundamental failure forms. In 1982, KRAJCINOVIC and SILVA [4] developed a physically motivated damage model, i.e. the bundle parallel bar system (PBS), to explore the damage mechanism of a quasi-brittle material under uniaxial tension. Subsequently, TANG and ZHU [5] introduced the Weibull distribution function to characterize the unequal strength of meso-cells, which led to a new path for the research on damage evolution model. LI [6] developed a new stochastic constitutive model that could explain the stress drop phenomenon under uniaxial tension. Based on the PBS model and the hypothesis of “equivalent strain”[7], CAO et al [8], XU and WEI [9] established the statistical damage constitutive models in compressive stress state by the hypothesis that the strength of meso-cells follows the Weibull or normal distribution function.
These statistical damage constitutive models can reflect the characteristics of strain softening in stress—strain curves under specific confining pressure to some extent. However, there is a great gap between theoretical and practical damage processes, as BAZANT and PANG [10] commented, “PBS had historically played a useful role in theoretical development. However, it represents an unacceptable model for a material that is brittle on a large scale and it is not suitable for describing damage mechanisms of real materials.” These kinds of statistical damage model only consider the rupture damage mode on meso-scale, and they are based on the hypothesis that the whole deformation and failure process of materials is a uniform damage evolutional course. However, according to the experimental phenomenon observed in Ref.[11], the damage process of quasi-brittle materials can be obviously divided into two damage phases, the homogeneous damage phase and the local breach phase; and there exists irreversible deformation during the damage process. The damage model that adopts a single damage variable cannot reflect the two different damage phases.
In recent years, the modern nonlinear science based on catastrophe, synergetics, self-organization and bifurcation, is introduced into the research of the damage mechanism of the quasi-brittle materials such as concrete and rock, and provides the new thought to re-recognize the physical phenomenon of material failure [12-13]. The introduction of the advanced detecting techniques such as acoustic emission (AE) and computerized tomography (CT), provides the new tools to reveal the mesoscopic damage mechanism of materials. Combining with the synergetic method, the catastrophe theory and the AE, the present study focuses on the establishment of simple statistical micromechanics models for the uniaxial tensile response of a gradually damaging structure of quasi-brittle engineering materials. Two more rational internal variables that can represent the mesomechanism of damage due to the heterogeneous in the microstructure of materials are introduced. The main purpose of this work is to promote new understanding of the physical background of the quasi-brittle material damage mechanism, and to establish relationship between the damage behavior and its underlying physical mechanism.
2 Basic hypotheses of material damage mechanism
The fundamental hypotheses of the proposed damage mechanism are given as follows.
(1) The essential damage and fracture process of a quasi-brittle material includes the nucleation, growth and coalescence of micro-cracks, interaction among micro-defects and irreversible rearrangement of the microstructure.
(2) The influence of these irreversible micro- changes on the material macroscopic mechanical properties can be addressed by two dominant aspects: the decrease of the effective cross-section and the degradation of elastic modulus in the effective region corresponding to the effective cross-section.
(3) The two damage modes can be simulated, respectively, by rupture and yield of the micro-element (bar); and the essence of failure can be interpreted as a continuous accumulation and evolution process of the two damage modes. It should be noted that the yield damage mode is almost neglected in traditional statistical damage models.
3 Characterization of quasi-brittle material failure process under uniaxial tension
A typical master stress—strain curve under uniaxial tension is shown in Fig.1. Signs A and B denote, respectively, the two damage characteristic states, i.e. the peak nominal stress state and the critical state at which macro-cracks start to develop in the fracture process zone (FPZ) of the test specimen and the damage process switches into local softening phase. Many experiments show that critical state B in a stress—strain curve lies in the softening region behind peak state A [14-15].
Fig.1 Typical nominal tensile stress—strain curve of quasi-brittle materials
Before state B, the material response is considered to be statistically homogeneous. The damage evolves only by nucleation and growth of micro-defects. The density of micro-defects is modest, retaining a dilute degree. After this threshold, the macroscopic response of the specimen strongly depends on the size of the largest crack with a preferential orientation in the FPZ. Local softening and fracture process (the decrease of the nominal stress with increase of the strain) occurs in the FPZ. The rest region enters the unloading process. This portion of the stress—strain curve becomes dependent on the specimen size, which cannot be treated as the mechanical behavior of a pure material.
According to the synergetic theory [13], the loaded material is regarded as an integrated, self-organization and motivated system; and the whole damage and fracture course is regarded a self-organization evolution process manifested as from random to concentrated, and from disorder to order.
According to the catastrophe theory [12], the damage evolution course of quasi-brittle materials can be divided into two phases, globally stable (GS) mode and evolution induced catastrophic (EIC) mode, namely the distributed damage phase and the local catastrophe phase. The critical state representing the transformation from GS to EIC plays a key role during the whole damage evolution course, and exhibits the characteristics of critical sensitivity, and many physical quantities exhibit abnormal behavior.
Acoustic emission phenomenon is close to the mesoscopic damage evolution process of material [13]. The acoustic emission signals are mostly due to the cracking and extending of the micro-cracks. The results show that the acoustic emission counting rate has direct correspondence with the amount of crack growth. The evolution of the acoustic emission counting rate has typical mutation characteristics. At a certain state after the peak nominal stress state, the acoustic emission rate reaches the sharp peak value.
So we have reasons to think, the critical state during the macro fracture process, the critical state in the catastrophe theory and the mutation state during the acoustic emission course represent the same physical state, lagging behind the peak nominal stress state, and playing a key role during the whole damage evolution course.
According to the basic hypotheses in this work, the rupture damage mode that represents the nucleation, the growth and the coalescence of countless micro-defects in the material microstructure reduces the effective cross-section. It has the same meso-damage evolution process with the acoustic emission rate. The yield damage mode that represents the mutual interaction of micro-defects and the irreversible rearrangement of the microstructure results in the degradation of the elastic modulus of the effective part. It also represents a certain regulating capacity of material’s self-organization behavior. The two damage modes will take place simultaneously once damage starts to occur in the specimen.
4 Improved parallel bar system (IPBS)
A physically motivated statistical damage model, i.e. the improved parallel bar system (IPBS), is proposed to reflect the two aspects of damage mentioned above, and to separate the two damage evolution processes of a quasi-brittle material under uniaxial tension. As shown in Fig.2, it is able to explore the essential damage mechanism in the FPZ of the material.
4.1 Characterization of physical model
Like the PBS [4], the IPBS model is also an approximation of a uniaxial link by a system with many vertical deformable bars connected by two rigid horizontal end-bars, which can move only in the direction of the imparted elongation. The representative volume element (RVE) in the FPZ of material, is assumed as a system composed of M (M→∞) parallel bars. The stiffness of the link k and effective cross-sectional area dA are identical and constant for all bars. The material disorder is introduced by the unequal rupture strain εR and yield strain εy to each bar, which are defined by the independent probability density function q(εR) and p(εy), as shown in Fig.2(b).
The IPBS is established based on the following fundamental recognitions.
(1) With the growth of the meso-defect density during the damage process, the effective cross-section of the material decreases. In other words, the actual cross-section dominated by the residual framework increases.
Fig.2 Improved parallel bar system model (IPBS) (a) and sketch maps for probability density functions of yield and rupture strain, p(εy) and q(εR), in IPBS and RVE, respectively (b)
(2) The meso-bars may have two kinds of fracture mode, as shown in Fig.3. The yielded bar i, after tensile strain ε exceeds the yield strain εyi, can also rupture when ε reaches the rupture strain εRi, if εyi＜εyi. The two events are not mutually exclusive. Therefore, we can adopt two independent probability functions q(εR) and p(εy) instead of the joint probability function p(fy, fR) in PBS.
Fig.3 Two fracture modes of mesoscopic bar element: (a) Brittle-fracture mode; (b) Yield-fracture made
The IPBS is a system composed of N (N→? and N/M→0) parallel bars, selected from M bars in the RVE. The following assumptions are employed for these N bars.
(1) The rupture strain εRi of each bar is large enough to neglect its rupture in the scope of simulation. Thus, only a random yield strain εyi needs to be attributed to each bar. The probability distribution of the N bars’ yield strains can represent the statistical mean property of the total M bars.
(2) As shown in Fig.2(a), the equation AE=NdA denotes the effective cross-section area of the N bars stays constant during the entire damage process. AN represents the nominal cross-section area controlled by the N bars, which increases with the damage growth.
(3) At any instant, the responses of the N bars corresponding to AE can represent the effective statistical mean property of the RVE (effective stress etc); and those to AN can represent the nominal statistical mean property of the RVE (nominal stress etc).
Here, N is defined as the minimum number of bars, corresponding to which the effective and nominal properties can represent the statistical properties of the RVE at any damage instant.
As shown in Fig.2(b), εymin=εRmin=0 denotes the minimum values of the bar yield and rupture strain (the elastic phase is neglected), respectively. εymax＜εRmax denotes the corresponding maximum values. We assume that εymax represents the strain of critical state B in Fig.1, and εRmax corresponds to the final failure state.
4.2 Formulae of governing equation in monotonic tensile process
The force—deflection relationships of the N bars displayed in the IPBS are
fi=ku (0≤u≤uyi, 
≥0)
(1)
fi=kuyi (u＞uyi, ≥0)
where u is the axial displacement of the bar system, uyi is the yield displacement of bar i corresponding to εyi and fi represents the tensile force on bar i.
The IPBS satisfies:
AN=AE+AD (2)
where AD is defined as the actual area controlled by the N bars in the IPBS expect AE due to the rupture of other bars in the RVE (they are not displayed in the IPBS). At the initial state, AD=0 and AN=AE=NdA.
For a large number of bars N (N→∞), two non-negative accumulated damage parameters are defined as
(3)
(4)
with 0≤Dy(ε)≤1 and 0≤DR(ε)≤1 as probability distribution functions of the yield and the rupture strains of the total M bars in the RVE, respectively.
The equilibrium condition satisfies
F=σEAE=σNAN (5)
where σE is the effective stress of the IPBS corresponding
to AE, and σN is the nominal stress corresponding to AN.
(1) Partial yield phase (0＜ε＜εymax)
The bar of the N bars will yield if εyi＜ε is satisfied. The rest bars in the IPBS remain elastic. The constitutive relation can be written as
(6)
(7)
(8)
where E0 is the initial elastic modulus; 
is defined as the accumulated damage variable of the IPBS elastic modulus due to the yield of the bars; the second term in Eqn.(8) denotes that the yielded bars still have contribution to the elastic modulus of the model; and DR is the accumulated damage variable of the RVE’s elastic modulus due to the rupture of the bars that are not displayed in the IPBS.
(2) Full yield phase (εymax≤ε＜εRmax)
In this phase, all N bars in the IPBS are yielded. The rupture damage continues to increase until ultimate failure. The effective stress σE stays constant with the maximum value. The expression can be written as
(9)
(10)
(11)
(12)
4.3 Formulation of governing equation in cyclic tensile loading process
The IPBS can also simulate the mechanical behavior of material subjected to unloading-reloading tension. Here we assume that each bar in the IPBS has identical yield strength in both tensile and compress states.
In Fig.4, εa is defined as the strain corresponding to
Fig.4 Sketch map for unloading-reloading process
the initial unloading point a, and εb is corresponding to the initial reloading point b. Assume that during the entire hysteretic process, DR(ε) keeps constant and equals the largest previously accumulated damage level DR(εa), neglecting the rupture damage accumulation.
(13)
During the hysteretic process, the formulae of σE and σN have the same forms with Eqns.(6) and (7). The form of the yield accumulative damage parameter 
is more complex. Here only the case satisfying σE≥0 is considered, i.e. the IPBS resides in the tensile state.
(1) ε0＜εa≤εymax
Here, ε0=εymin=εRmin=0 is defined as the smallest yield strain.
For unloading phase (
, εb≤ε＜εa), we have
(14)
For reloading phase (
, εb≤ε＜εa), we have
(15)
where σEb is the effective stress corresponding to reloading point b.
(2) εymax＜εa≤εRmax
takes the similar form as Eqn.(14) or (15) obtained by the same method.
5 Comparisons with hypothesis of equivalent strain and PBS model
LEMAITRE and DESMORAT [7] presented the hypothesis of “equivalent strain” (see Fig.5), which assumed the deformation behavior of a damaged material
Fig.5 Schematic diagram of hypothesis of equivalent strain
can be embodied by the effective stress acting on the pristine material. The constitutive relations can be written as
(16)
σE=E0ε (17)
σN=E0(1-D)ε (18)
(19)
where A and E denote the effective cross-section area and the elastic modulus of the damaged material, respectively; A0 and E0 represent the initial cross-section area and the initial elastic modulus of the pristine material, respectively; and D is the accumulated damage variable. According to Eqn.(19), it is assumed that the macroscopic effects of the decrease of the elastic modulus and the degradation of the effective cross-section on the damaged material are equivalent [16].
The constitutive relations established by the PBS can also be expressed by the same forms as Eqns.(16)-(19). Therefore, the two theories essentially embody the same damage mechanism. They ignore the degradation of the elastic modulus in the effective region due to interaction among microcracks and rearrangement of the effective framework in the microstructure. Unfortunately, for the quasi-brittle materials, the specimen will be close to failure (the macro-cracks occur in the FPZ) when the elastic module reduces in some sorts. The density of micro-crack and D maintain a certain dilute degree. So the rupture damage mode might not be the principal factor leading to the failure of materials.
The physical models proposed in this work can remedy the limitation of the two classic theories. The effect of the irreversible change on the micro-scale on macroscopic properties can be classified into the following two aspects: the decrease of effective cross-section area and the enervation of the effective framework’s elastic modulus. The constitutive relation can be expressed as Eqns.(6) and (7).
DR varies from 0 to 1 and has the same meaning with D in traditional damage models. The uniform damage phase can be measured by the yield damage variable varying from 0 to 
or by Dy varying from 0 to 1. The yield damage mode is regarded as the principal factor leading to the failure of the material. And or Dy represents the exerting degree of the regulating capacity of material’s self-organization behavior.
6 Examples and verification
In order to avoid unnecessary complication, the two
probability density functions, p(εy) and q(εR) are introduced with uniform distribution or triangular distribution. Here two examples are conducted for concrete.
In Example 1, the damage evolutions obey the forms shown in Fig.6(a). q(εymax) denotes the maximum value of the density function. The parameters are chosen as follows: E0=3.0×1010Pa, εymin=εRmin=0, εymax=1×10-4, p(εy)=1/εymax=const, DR(εymax)=0.2, εRmax= 5εymax.
Example 2 is used to make comparison of the stress—strain curves in uniaxial tension recommended by the Specification in Ref.[17]. The two probability density functions, p(εy) and q(εR) are both introduced as a triangular distribution shown in Fig.6(b).
The simulation results for Example 1 are shown in Figs.7-9. The effective stress—strain curve and the nominal stress—strain curve in the FPZ of concrete under uniaxial tension are modeled, as shown in Fig.6.
Before the critical state, σE increases monotonically, however, σN does not increase monotonically and reaches its maximum value at the peak nominal stress state. After the critical state, σE keeps constant at its maximum value, and σN continues to decrease with the growth of the microcrack density until ultimate fracture.
Fig.6 Selected probability density functions of yield and rupture strains, p(εy) and q(εR) in IPBS: (a) Example 1; (b) Example 2
Fig.7 Master stress—strain curves in IPBS
Fig.8 Effective stress—strain curves under cyclical tension in IPBS
Fig.9 Nominal stress—strain curves under cyclical tension in IPBS
The full yield phase corresponding to the softening curve after the threshold point satisfies
?σE?ε=0 (20)
thus this local softening damage phase in the FPZ satisfies the hypothesis of Drucker stability in the thermodynamic theory [18] from the viewpoint of effective stress.
The effective and nominal stress—strain curves under cyclic tension are illustrated in Figs.8 and 9, respectively. The stresses at the reloading points equal zero. They display evident hysteretic and irreversible deformation behaviors similar to the experimental observation.
The simulation results for Example 2 are shown in Fig.10, indicating the uniform damage phase of concrete with different tensile strengths (varying form 1 to 4 MPa) under uniaxial tension. Here, assume that the critical state corresponds to the location in the softening curves where the differential coefficient of second rank is zero. The computed results illustrate the flexibility of the suggested model in the application for a wide range of different materials. The model can also quantificationally describe the damage evolution process of the two damage modes by confirming the shape of the triangular distribution of the two probability density functions. The critical state is proposed as the final failure point in the constitutive model. Treated by this way, not only the ductibility of uniform damage phase of materials can be considered adequately, but also the size effect of constitutive model during the local breakage phase can be neglected.
Fig.10 Nominal stress—strain curves for concrete with different strengths in Example 2
7 Conclusions
(1) The material damage mechanism is classified into two dominant aspects: the decrease of effective cross-section area and degradation of elastic modulus of the effective framework. And they can be modeled by rupture and yield of meso-bars.
(2) A physically motivated damage model, i.e. IPBS is proposed, which can account for the two aspects of damage mentioned above. The character of heterogeneity is introduced by the unequal rupture and yield limits of bars defined by two independent probability distribution functions. The damage mechanism of the whole process in the FPZ can be comprehended by the equivalent relationship between nominal and effective stresses.
(3) The two characteristic states, i.e. the peak nominal stress state and the critical state, are distinguished. The entire deformation process under uniaxial tension is divided into a uniform damage phase and a local breach phase by the critical state. The transitional mechanism from the hardening to the softening phase is interpreted.
(4) The whole damage evolution process described by the IPBS is in accordance with the synergetic theory, the catastrophe theory and the acoustic emission phenomenon. The yield damage mode represents a certain regulating capacity of material’s self-organization behavior. εymax represents a certain intrinsic critical state, which is close to the damage localization and the mutation state of the acoustic emission test.
(5) A physical mechanism existing in the FPZ during the local breach phase is assumed. From the effective stress point of view, this local softening phase satisfies the principle of the minimum energy dissipation and the hypothesis of the Drucker stability in the thermodynamics theory. Therefore, it is a stable process.
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(Edited by CHEN Wei-ping)
Foundation item: Projects(90510018, 50679006) supported by the National Natural Science Foundation of China; Project(NCET-05-0413) support by the Program for New Century Excellent Talents in University
Received date: 2008-10-10; Accepted date: 2009-02-26
Corresponding author: BAI Wei-feng, PhD; Tel: +86-411-84708549; E-mail: yf9906@163.com
