Determining constitutive models for fracture prediction in ductile materials undergoing plastic deformation

Abstract

Described are methods, systems, and apparatus, including computer program products for determining material constitutive relationships of fracture in a ductile material undergoing plastic deformation. A material pressure function is determined that describes dependence of a damage rule of at least a portion of the ductile material on pressure. A material azimuthal angle function is determined that describes dependence of the damage rule of the portion of the ductile material on principle stress. The damage rule of the portion of the ductile material is determined based on the material pressure function and the material azimuthal angle function. A weakening function of the portion of the ductile material is determined based on the damage rule. Material constitutive relationships of fracture in the portion of the ductile material are determined based on the weakening function and a matrix property of the ductile material.

Description

GOVERNMENT SUPPORT

The present invention was supported by Grant No. 123163-01 from the United States Navy—Office of Naval Research. The Government has certain rights in the invention.

FIELD OF THE INVENTION

The present invention relates generally to computer-based methods and apparatuses, including computer program products, for predicting fracture in a ductile material undergoing plastic deformation.

BACKGROUND

The fracture of ductile metals is an important design consideration in many industrial applications. If excessive plastic deformation and/or fracture occur in machine parts or structural members, a system may fail to perform its intended functions and worse yet, catastrophic failure of a part or system may occur. Therefore, modes of fracture that lead to failure must be understood in order to establish suitable failure criteria in the design of systems.

Various models have been proposed to predict the initiation of fracture. The simplest theory, maximum-stress theory, states that failure occurs at a point in the structure when the maximum principal stress σ₁ reaches a critical stress in simple tension, such as the yield stress or the ultimate stress. The principle stress components are ordered such that σ₁≧σ₂≧σ₃. The failure criterion for yielding, for example, can be stated to occur when σ₁ reaches σ_ys (σ₁=σ_ys), the yield stress. Under maximum-stress theory, the onset of fracture is not affected by the level of the other principal stresses. Maximum-shear and octahedral shear stress theories, on the other hand, take into account the other principal stresses. Maximum-shear stress theory, or Tresca's theory, assumes that failure occurs in a body when the maximum shear stress at a point reaches the same value of stress for failure in a simple tension test. For example, maximum-shear stress theory states that yielding failure occurs when σ_max−σ_min=σ_ys. Octahedral shear stress theory, or von Mises' theory, states that failure at a particular location occurs when the energy of distortion reaches a critical value, such as the same energy for failure in simple tension. The failure criterion for yielding, for example, under the von Mises theory can be stated to occur when

(σ₁−σ₂)²+(σ₁−σ₃)²+(σ₂−σ₃)²=2σ_ys².  (1)

These simple threshold conditions work well with brittle materials that fail in the linear elastic range. For ductile metals, however, irreversible plastic deformation starts when yielding occurs. Therefore, more complicated criteria are required to better model the underlying physics of fracture in ductile materials.

The principal stress system can be equally represented in the Cartesian coordinate system (σ₁, σ₂, σ₃) or a cylindrical coordinate system (p, θ, σ), where p is the hydrostatic pressure, θ is the azimuthal angle on an octahedral plane, and σ is the equivalent stress. For isotropic materials, the azimuthal angle can be represented by the Lode angle θ_L, where −π/6≦θ_L≦π/6, because of permutation symmetry.

Both macroscopic and microscopic models have been proposed from different perspectives, such as fracture strain locus, energy absorption, and void growth, to predict failure modes of ductile materials. Macroscopic models generally assume homogeneous materials and are constructed based on macroscopic field variables. Unlike macroscopic homogeneous assumptions, microscopic models treat the materials as clusters of inhomogeneous cells. Microscopic models of metals are, in general, more complex and can include descriptions of, for example, the multiphase materials, such as grains, precipitates, voids, and their evolution.

One representative macroscopic fracture model is the Johnson-Cook model (G. R. Johnson, W. H. Cook, “Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures,” Engineering Fracture Mechanics, 21(1):31-48, 1985). The Johnson-Cook strength model is an empirical equation used in simulations for its consideration of plastic hardening, strain rate sensitivity and temperature softening. In the Johnson-Cook model, the equivalent stress, σ, is expressed as the following function of the equivalent plastic strain, ε_p, and the temperature, T, and the plastic strain rate, {dot over (ε)}_p:

$\begin{array}{cc}\stackrel{\_}{\sigma }=\left[A+B{\varepsilon }_p^n\right]\left[1+C\ln \left(\frac{{\stackrel{.}{\varepsilon }}_p}{{\stackrel{.}{\varepsilon }}_0}\right)\right]\left[1-{\left(\frac{T-T_{\mathrm{room}}}{T_{\mathrm{melt}}-T_{\mathrm{room}}}\right)}^q\right],& \left(2\right)\end{array}$

where ε₀ is a reference plastic strain rate, T_room and T_melt are the room temperature and the material melting temperature, respectively, and A, B, C, n, and q are five material constants. The Johnson-Cook model accounts for isotropic strain hardening, strain rate hardening, and temperature softening in the uncoupled form. The first term of the right hand side in Eq. (2) represents the quasi-static stress-strain relation at room temperature; the second term represents the strain-rate hardening; the third term represents the temperature dependence of the stress-strain relation. The fracture criterion used in the Johnson-Cook model is a weighted integral with respect to the effective strain:

$\begin{array}{cc}D={\int }_0^{{\varepsilon }_c}\frac{{\varepsilon }_p}{{\varepsilon }_f\left(\frac{{\sigma }_m}{\stackrel{\_}{\sigma }},{\stackrel{.}{\varepsilon }}_p,T\right)},& \left(3\right)\end{array}$

where

${\varepsilon }_f\left(\frac{{\sigma }_m}{\stackrel{\_}{\sigma }},{\stackrel{.}{\varepsilon }}_p,T\right)$

describes the fracture envelope, σ_m is the mean stress, and ε_c is the plastic strain when fracture occurs.

The functional form of the weighting function is the reciprocal of the effective failure strain, which is treated as a function of the stress triaxiality status, the strain rate and the temperature:

$\begin{array}{cc}{\varepsilon }_f\hspace{1em}\left(\frac{{\sigma }_m}{\stackrel{\_}{\sigma }},{\stackrel{.}{\varepsilon }}_p,T\right)=\left[D_1+D_2\exp \left(D_3\frac{{\sigma }_m}{\stackrel{\_}{\sigma }}\right)\right]\left[1+D_4\ln \left(\frac{{\stackrel{.}{\varepsilon }}_p}{{\stackrel{.}{\varepsilon }}_0}\right)\right]\hspace{1em}\hspace{1em}\left[1-D_5\frac{T-T_{\mathrm{room}}}{T_{\mathrm{melt}}-T_{\mathrm{room}}}\right].& \left(4\right)\end{array}$

where D₁, D₂, D₃, D₄, and D₅ are material constants. The Johnson-Cook model does not take into account any dependence of the fracture criterion on the Lode angle (or the third stress invariant).

Another representative macroscopic model is the Wilkins model (M. L. Wilkins, et al., “Cumulative Strain-Damage Model of Ductile Fracture,” Technical Report UCRL-53058, Lawrence Livermore National Laboratory, October 1980). Wilkins et al. proposed the failure strain envelop as a product of two parts: the hydrostatic tension and the stress asymmetry. The original form of Wilkins model is

$\begin{array}{cc}D={\int }_0^{{\varepsilon }_c}w_1w_2{\varepsilon }_p=1,& \left(5\right)\end{array}$

where w₁=1/(1+ap)^α, w₂=(2−A)^β, A=max{s₂/s₁,s₂/s₃}, and s₁, s₂, and s₃ are the ordered principal deviatoric stress components. The material constants are a, α, and β. The pressure dependence is taken into account in w₁, while w₂ accounts for the Lode angle dependence. The Wilkins damage model is linear with respect to the plastic strain. The Wilkins model does not use a weakening or softening model in its fracture criterion.

A representative microscopic model was proposed by Lemaitre in “A Continuous Damage Mechanics Model for Ductile Fracture,” Journal of Engineering Materials and Technology, 107:83-89, 1985. Lemaitre's proposed model differs from void nucleation growth coalescence (VNGC) models in that the micro-mechanism of the growth of individual voids and their interactions are depicted in a phenomenological way. Macroscopic parameters and equations are used in Lemaitre's model to describe the aggregative responses of the body of solids. Therefore, the constitutive and damage model of the material is directly based on the externally observed behavior of the material and no microscopic interpretation of the material structure is required. Continuum damage mechanics considers the material property as the combination of the matrix material and damage accumulation. Lemaitre determined damage accumulation experimentally for a light alloy. Their results show an increasing tendency of damage accumulation. The law of elasticity coupled with damage is σ_ij=C_ijklε_kl^e(1−D), where C_ijkl is the fourth order elasticity stiffness tensor. Here, the damage variable D has a meaning of material deterioration of the stiffness. Lemaitre's model does not take into account any dependence of the fracture criterion on the Lode angle.

Another representative microscopic model is the Gurson-Tvergaard-Needleman model, the so-called GTN model, which takes into account the details of void-nucleation-growth and coalescence. Gurson derived the pressure dependent yield function for voids containing cell structure. See, A. L. Gurson, “Continuum theory of ductile rupture by void nucleation and growth: Part I Yield criteria and flow rules for porous ductile media,” Journal of Engineering Materials and Technology, 99:2-15, 1977. Tvergaard and Needlemen further extended this model to include the void nucleation and coalescence process. The GTN model has been used to predict several important fracture modes, such as the cup-cone fracture of a tensile round bar. See, V. Tvergaard, “Influence of voids on shear band instabilities under plane strain conditions,” International Journal of Fracture, 17:389-407, 1981, and V. Tvergaard and A. Needleman, “Analysis of the cup-cone fracture in a round tensile bar,” Acta Metallurgica, 32(1):157-169, 1984. The GTN model does not take into account any dependence of the fracture criterion on the Lode angle.

SUMMARY OF THE INVENTION

One approach to predicting fracture in a ductile material is to determine material constitutive relationships of fracture in the ductile material undergoing plastic deformation. In one aspect, there is a method of determining material constitutive relationships of fracture in the ductile material undergoing plastic deformation. The method includes determining a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure. The method also includes determining a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress. The method further includes determining a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function. The method also includes determining a weakening function of the portion of the ductile material based on the damage rule. The method also includes determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

In another aspect, there is a method of determining material constitutive relationships of fracture in the ductile material undergoing plastic deformation. The method includes determining a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure. The method also includes determining a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress. The method further includes determining a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function. The method also includes determining a weakening function of the portion of the ductile material based on the damage rule. The weakening function is a function of damage accumulation and the rate of change of the weakening function with respect to damage accumulation is dependent on damage accumulation. The method also includes determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

In another aspect, there is a computer program product. The computer program product is tangibly embodied in an information carrier, the computer program product including instructions being operable to cause a data processing apparatus to determine a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure. The computer program product also including instructions being operable to determine a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress. The computer program product also including instructions being operable to determine a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function. The computer program product also including instructions being operable to determine a weakening function of the portion of the ductile material based on the damage rule. The computer program product also including instructions being operable to determine material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

Another approach to predicting fracture in a ductile material is to determine material constitutive relationships of fracture in the ductile material undergoing plastic deformation. In one aspect, there is a method of determining material constitutive relationships of fracture in the ductile material undergoing plastic deformation. The method includes determining a material pressure function describing dependence of a damage rule of at least a portion of the ductile material on pressure. The method also includes determining a material azimuthal angle function describing dependence of the damage rule of the portion of the ductile material on principle stress. The method also includes determining a nonlinear damage accumulation rule for the portion of the ductile material based on the material pressure function and the material azimuthal angle function. The method can include determining a weakening function of the portion of the ductile material based on the nonlinear damage accumulation rule. The method can also include determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

Another approach to predicting fracture in a ductile material is to determine material constitutive relationships of fracture in the ductile material undergoing plastic deformation. In one aspect, there is a method of determining damage accumulation characteristics of material constitutive relationships of fracture in a ductile material undergoing plastic deformation. The method includes determining a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure. The method also includes determining a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress. The method also includes determining a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function. The method also includes determining a damage accumulation in an integral form based on the material damage rule, the pressure function, and the material azimuthal angle function.

In other examples or embodiments, any of the aspects above can include one or more of the following features. The method can also include determining an initiation of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material. The material pressure function can include a logarithmic function of pressure. The material pressure function can be μ_p(p)=1−q log(1−p/p_lim). When a relative ratio of principal stresses is between about 0 and about 0.5, the material azimuthal angle function can be μ_θ=√{square root over (χ²−χ+1)}/(1+(√{square root over (3)}/γ−2)χ), and when the relative ratio of principal deviatoric stresses is between about 0.5 and about 1, the material azimuthal angle function can be μ_θ=√{square root over (χ²−χ+1)}/(1+(√{square root over (3)}/γ−2)(1−χ)). The material azimuthal angle function can be a material Lode angle function. The material Lode angle function can be μ₀=γ+(1−γ)(6|θ_L|/π)^k. The material azimuthal angle function can be μ₀=1−(1−γ)(6|θ|/π)^k for 0≦θ≦π/6. The material azimuthal angle function can be μ₀=1−(1−γ)(6|π/3−θ|/π)^k for 0≦θ≦π/3. The damage rule can be a function of the ratio of a plastic strain, ε_p, to a fracture strain, ε_f. The damage rule can be D=(ε_p/ε_f)^m. The damage rule can be D=(exp[λ(ε_p/ε_f)^m]−1)/(exp[λ]−1). The damage rule can be a function of the ratio of a plastic distortion ε_d to a fracture strain ε_p. The damage rule can be D=(ε_d/ε_p)^m. The damage rule can be D=(exp[λ(ε_d/ε_f)^m]−1)/(exp[λ]−1). The weakening function can be a linear function of damage accumulation. The weakening function can be w(D)=(1−D^β)^1/η. β can be greater than one. η can be greater than one. An equivalent stress can be used to determine material constitutive relationships of fracture. The equivalent stress can be based on the weakening function and a matrix property of the ductile material. The equivalent stress can be σ=w(D)σ_M.

Any of the above implementations can realize one or more of the following advantages. By incorporating both the pressure dependence and the azimuthal angle dependence into damage and weakening rules, numerical fracture prediction codes can provide simulated results with improved accuracy with respect to the physical systems they are modeling. By incorporating a weakening function and a damage rule, numerical simulations can maintain a continuum assumption that allows for a simplified model. Furthermore, numerical simulations incorporating a weakening function and a damage rule based on pressure and azimuthal angle can provide accurate simulations of ductile materials that have gone beyond moderate plastic deformation. Under this model, fracture simulations can be used by a wide range of industrial users (e.g., designers of bridges, ships, vehicles, airplanes, buildings, consumer appliances, etc.) to improve the safety of products, lower development costs, and/or speed up the time frame of research and design.

Other aspects and advantages of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, illustrating the principles of the invention by way of example only.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features, and advantages of the present invention, as well as the invention itself, will be more fully understood from the following description of various embodiments, when read together with the accompanying drawings.

FIG. 1A is a flowchart depicting the determination of material constitutive relationships of fracture in a ductile material undergoing plastic deformation.

FIG. 1B is a flowchart depicting the determination of material constitutive relationships of fracture in a ductile material undergoing plastic deformation.

FIG. 2 illustrates a load displacement curve illustrating the sequences leading to fracture in a load carrying material.

FIG. 3 illustrates a fracture envelope represented in the three dimensional space of plastic strain and the hydrostatic tension.

FIG. 4 illustrates a hydrostatic pressure function used to fit with experimental data.

FIG. 5 illustrates the Lode angle in the octahedral plane.

FIG. 6 illustrates a Lode angle dependence function of the first kind, which is a linear function in the plastic strain plane.

FIG. 7 illustrates a Lode angle dependence function of the second kind, which is a curvilinear function in the plastic strain plane.

FIG. 8 illustrates a fracture envelope in the principal stress space with the combined effects of the hydrostatic pressure and the deviatoric state.

FIGS. 9A-9C illustrates the displacement history of a repeated loading path, the total equivalent plastic strain, and the equivalent plastic distortion and the damage accumulation using equivalent plastic strain and equivalent plastic distortion for a strain controlled fracture loading.

FIG. 10 illustrates the damage rule functions to the ratio of the plastic strain to the fracture strain with damage exponent m=2 and the weakening function for several weakening exponents β and η.

FIG. 11 illustrates a hypothetical matrix stress-strain curve and the weakened material stress-strain curve for several weakening exponent β values with η=1.

DETAILED DESCRIPTION

The term “ductile fracture” can have multiple meanings. “Ductile fracture” can be referred to as the void nucleation-growth-coalescence (VNGC) type of fracture observed, which is usually rough. In a more general sense, “ductile fracture” refers to fractures where the material experiences large plastic deformation while not related to a specific type of fracture. Ductile fracture can result from the accumulation of plastic damage.

The problem of fracture initiation in large structures involves both material and geometrical nonlinearities. Therefore, numerical approaches to fracture problems are well suited for many engineering applications. The present invention relates generally to a new method using a constitutive material property model for ductile materials that includes four effects: the pressure sensitivity, the azimuthal angle dependence (or, equivalently, the relative ratio of the principle stresses, and/or the Lode angle), a damage rule, and a material weakening effect.

FIG. 1A illustrates a flowchart 100 depicting the determination of material constitutive relationships of fracture in a ductile material undergoing plastic deformation. The determined material constitutive relationships can be used in numerical simulations to model arbitrary ductile materials for design and analysis purposes, including determination of fracture criterion of parts or systems. A material pressure function is determined (101), which can incorporate material ductility and/or fracture initiation dependence on pressure. This pressure effect on the fracture strain can be the result of, for example, the suppression of the initiation and propagation of micro cracks and voids. The material pressure function can be determined, for example, by fitting nonlinear equations to experimental data. A material azimuthal angle function is determined (102), which can incorporate material ductility and/or fracture initiation dependence on the azimuthal angle. The azimuthal angle, which can be the Lode angle, can be one-to-one mapped with a relative ratio of the principal stress components, so that the relative ratio of the principal stresses can be used instead of the azimuthal angle. A damage rule is determined (103), which can be based on a restricted loading path, the material pressure function, and/or the material azimuthal angle function. The damage rule can also be based on the ratio of the plastic strain to the fracture strain. The damage rule, for example, can be based on a damage integral function. For example, incremental damage can be determined by comparing the current plastic strain to the fracture strain at the current stress state. A weakening function is determined (104), which can be based on a damage rule and/or damage state. The weakening function can account for material deterioration while maintaining a continuum assumption of the material. Material constitutive relationships of fracture are determined (105), wherein the strength of the matrix material can be assumed to be a basic property of the material, while the macroscopic behavior of the material can be affected by a weakening factor that accounts for the microscopic damage.

FIG. 1B illustrates a flowchart 150 depicting an embodiment of determining material constitutive relationships of fracture in a ductile material undergoing plastic deformation. A material pressure function is determined (151), which can incorporate material ductility and/or fracture initiation dependence on pressure. A material azimuthal angle function is determined (152), which can incorporate material ductility and/or fracture initiation dependence on azimuthal angle. A damage rule is determined (153), which can be based on a restricted loading path, the material pressure function, and/or the material azimuthal angle function. The damage rule can also be based on the ratio of the plastic strain to the fracture strain. Damage accumulation is determined (161) in an integral fashion by combining the damage rule with the pressure function and the azimuthal angle function. A matrix strength of the material is determined (162), wherein the strength of the matrix material can be assumed to be a basic property of the material, while the macroscopic behavior of the material can be affected by a weakening factor that accounts for the microscopic damage. A weakening function is determined (163), which can be based on a damage rule and/or damage state. Mechanical properties of the material are determined (170), which can be based on the damage accumulation, the material matrix strength, and the material weakening function.

Plasticity Model

FIG. 2 illustrates a load displacement curve 200 that shows the sequences leading to fracture in a load carrying material. Material fracture can be characterized, for example, by a complete loss of its load carrying capacity or deformability. Usually, a material displays some elasticity which is fully recoverable when the load is removed. However, beyond a certain limit, often called the elastic limit, a part of the deformation is unrecoverable. The elastic limit occurs at the yield strength 210. The region above the yield strength 210 is often called plasticity. Due to strain hardening, deformation can still be stable until the ultimate strength 220 is reached. Beyond the ultimate strength 220, the deformation is unstable and fracture 230 can soon result. It can be assumed that the matrix stress and the plastic strain curve is a basic material property,

σ_M=σ_M(ε_p),  (6)

where σ_M is the matrix stress and ε_p is the plastic strain. Material deterioration can be characterized by a weakening factor, which can be a function of the damage, e.g.,

σ=w(D)σ_M,  (7)

where σ is the applied stress and w(D) is a weakening or softening function defined on the damage variable D.

Up to moderate plastic deformation, the microstructure of materials can be viewed as being undamaged. The strength of material is often simplified as a one-dimensional problem where the resistance of material is characterized as a function of the equivalent plastic strain only. Classical plasticity theory, or conventional continuum mechanics, is acceptable in practical problems where only small ductile damage is involved. However, these theories are not sufficient in dealing with many engineering applications where deformation goes beyond moderate plasticity, when changes in microstructure may no longer be insignificant and cannot be ignored in order to obtain accurate fracture prediction. In such conditions, the ignorance of the change of microstructure is over simplified. A damage rule and a damage induced weakening effect have to be included in the model to characterize the material deterioration.

From experimental observations, ductile failure of structures comprises three phases: (1) accumulation of damage, (2) initiation of fracture, and (3) crack propagation. Fracture initiation can be modeled as the result of the accumulation of the ductile plastic damage. Microscopically, such damage can be associated with void nucleation, growth and coalescence, shear band movement, propagation of micro-cracks, and/or the like. Macroscopically, the degradation of the material can be exhibited as a decrease in the material stiffness, material strength, and/or a reduction of the remaining ductility. These physical changes can be used as an indicator to predict the onset of fracture, either based on the current value or in a cumulative fashion.

Various experiments have shown that the damage process that eventually leads to the fracture of a material exhibits strong dependence on the loading history of the material (e.g., the damage). Therefore, the model includes a damage variable that takes into account the effect of a material's loading history. The damage evolves when the solid material is subjected to plastic loadings, which can be treated in the three dimensional space of the principal stresses. The principal stresses can be represented in the cylindrical coordinate system denoted by (p, θ, σ), where p is the hydrostatic pressure, θ is the azimuthal angle on the octahedral plane, and u is the von Mises equivalent stress. On an octahedral plane, the pressure is fixed. The aximuthal angle can be fixed on the octahedral plane, which represents a class of deviatorically proportional loading path. The damage evolution can then depend only on the ratio of the plastic strain to the fracture strain for monotonic loadings, i.e., D=f(ε_p/ε_f), where D is the accumulated damage on the path, ε_p is the plastic strain and ε_f is the fracture strain on that path. This path can be called the “restricted loading path” for convenience. The rate form can be represented as

$\begin{array}{cc}\stackrel{.}{D}=\frac{\partial f\left(\frac{{\varepsilon }_p}{{\varepsilon }_f}\right)}{\partial \left(\frac{{\varepsilon }_p}{{\varepsilon }_f}\right)}\frac{{\stackrel{.}{\varepsilon }}_p}{{\varepsilon }_f}.& \left(8\right)\end{array}$

In Eq. (8), the normalization denominator ε_f is defined by the given pressure and the azimuthal angle θ, i.e., ε_f=ε_f(p,θ). Assuming the effects of the hydrostatic pressure and the azimuthal angle are independent of each other, the equivalent fracture strain can be written as

ε_f=ε_f0μ_p(p)μ_θ(θ),  (9)

where ε_f0 is a reference fracture strain, μ_p(p) represents the effect of the hydrostatic pressure, and μ_θ(θ) represents the deviatoric state. When both μ_p(p) and μ_θ(θ) are unity, the model degenerates to the constant failure strain criterion. The fracture envelope defined by ε_f shows the relative extent of ductility for the given pressure and the azimuthal angle. At a more accurate level, the actual damage is an accrued effect of the loading history of varying pressure and azimuthal angle, which can be described by a history variable.

Based on these assumptions, the constitutive relationship of the material can be described by a set of four equation, Eqs. (6)-(9). These equations can form a theoretical basis of a new damage plasticity model. The internal variables are the plastic strain and the damage. The input functions for the material are the five curves: σ_M(ε_p)=the stress-strain curve for the matrix material, w(D)=weakening function, D(ε_p/ε_f)=damage rule, μ_p(p)=material pressure function, and μ_θ(θ)=material azimuthal angle function. This method to calculate the damage is called “cylindrical decomposition.”

In this application, “cylindrical decomposition” refers to the method of using the pressure, the aximuthal angle (or, equivalently, the Lode angle), and the plastic strain, which forms a cylindrical coordinate system in a three-dimensional space. The pressure sensitivity and the azimuthal angle dependence of fracture are used to construct a fracture surface oriented in the hydrostatic axis in the principal stress space. The fracture surface can be equally represented in the space of plastic strain and the hydrostatic tension, as illustrated in FIG. 3. A fracture surface can be determined based on the hydrostatic pressure p and the relative ratio of the principal deviatoric stresses: χ=(s₂−s₃)/(s₁−s₃), which can be used interchangeably with the Lode angle. The principal deviatoric stresses, s_i, can be defined as s_i=σ_i−σ_m=σ_i−(σ₁+σ₂+σ₃)/3, where σ_i are the principal stresses and i=1, 2, 3.

Pressure Sensitivity

One approach to constructing the fracture surface includes using experimental data to determine pressure dependence and azimuthal angle dependence on fracture. From the orthogonality of p and χ, it can be approximated that the effects of p and χ are independent of each other. Thus, the effects of p and χ can be uncoupled to a first approximation and can be quantified separately.

Experimental data used to determine the pressure and the azimuthal angle dependence on the fracture surface can be obtained by adjusting the pressure and the azimuthal angle separately, such that two series of tests can be conducted at constant p and constant χ, respectively. For example, some experiments show that materials can become more ductile as they experience higher compressive pressures. For some metals, the ductility can be an order higher under high compressive pressure than its ductility at atmospheric pressure. For example, zinc exhibits a sudden transition from brittle to ductile at a critical pressure of about 70 MPa. One test, for example, that can be used to determine experimental data is the unidirectional tension test with confining pressure.

Round bars can be used as specimens and pulled in a pressure chamber to test the effect of hydrostatic pressure on the material fracture strain. The round bar specimens can be viewed as being in a uniaxial tension condition superimposed by a hydrostatic pressure up to a compressure pressure of about, for example, 30 kbar. Because the two lateral principle stress components remain identical at the symmetric line for the axisymmetric specimen, the ratio of the principal stresses remains essentially zero at the center of the neck throughout these experiments. The ratio of the cross-sectional area at the neck at fracture to the initial cross-sectional area usually decreases with respect to the lateral confining pressure.

A series of experimental tests have been performed on tensile round bars on armor steels using a pressure chamber (Bridgman, P. W., Studies in large plastic flow and fracture. McGraw-Hill Inc., 1952.). From the test results, the relationship of the fracture strain and the confining pressure can be expressed as

$\begin{array}{cc}\frac{A_f}{A_0}=\frac{A_{f0}}{A_0}{\left(1-\frac{p_{\mathrm{conf}}-p_{\mathrm{atm}}}{p_{\lim }-p_{\mathrm{atm}}}\right)}^{\stackrel{\_}{q}},& \left(10\right)\end{array}$

where q is a material constant that fits experimental data, A₀ is the original cross-sectional area, A_f is the cross-sectional area after fracture at the neck, A_f0 is the cross-sectional area after fracture at the neck at atmospheric pressure, p_atm is the atmospheric pressure, and p_lim is a limiting pressure beyond which the material will not fail in the uniaxial tensile condition. The limiting pressure has been estimated, for example, to be between about 1 GPa to about 2 GPa for some steels.

Assuming p_atm<<p_lim and taking the logarithm on both sides of Eq. (10), the following equation can be obtained

$\begin{array}{cc}{\varepsilon }_f={\varepsilon }_{f0}\left[1-q\log \left(1-\frac{p}{p_{\lim }}\right)\right],& \left(11\right)\end{array}$

where ε_f is the fracture strain at the confining pressure p, ε_f0 is a reference failure strain at zero mean stress, ε_f0=log(A₀/A_fo) is the uniaxial tensile failure strain without confining pressure, q=is a shape parameter, and p_lim is a limiting pressure beyond which no damage occurs. Therefore, the pressure dependence function μ_p(p) becomes

$\begin{array}{cc}{\mu }_p\left(p\right)=1-q\log \left(1-\frac{p}{p_{\lim }}\right).& \left(12\right)\end{array}$

For numerical implementation, the confining pressure can be replaced by the hydrostatic pressure and the form of Eq. (12) is retained, except that the material constants can be re-calibrated for the hydrostatic pressure.

The average hydrostatic pressure 410 experienced in the course of a pulling test can be illustrated as in FIG. 4. The pulling test starts at 420 with no deformation. The equivalent stress path on the σ−p plane is shown as the thick solid line 430. Line 440 is the fracture loci of σ with respect to the hydrostatic pressure. The mean hydrostatic pressure along the entire loading path can be estimated to be p_ave=p_conf−σ_flow/6. As illustrated in FIG. 4, the pressure at the fracture point is p_conf−σ_flow/3.

Azimuthal Angle Dependence

From the pressure point of view, the simple shear is zero mean stress and the simple tension is positive mean stress. Experiments have shown that simple shear (or torsion) fracture strain can be less than the simple tension fracture strain for at least some materials. This experimental result can not be explained by the pressure dependence. Therefore, an azimuthal dependence function characterizing the azimuthal variation of the fracture locus on the same pressure can be necessary in modeling ductile fracture.

For isotropic materials, the principal stresses are interchangeable to reflect the independence of damage to the observation frame. The azimuthal angle dependence function for damage evolution can have permutation symmetry in all three principal planes. Therefore, on an octahedral plane, the azimuth angle can be divided into six parts that have the same weighting function. Each part covers the complete range of the relative ratio of the principal stresses from 0 to 1. Assuming that the forward motion and the backward motion induce the same amount of damage, an additional symmetry can be introduced to the octahedral plane—i.e., the shape of the twelve segments are the same (apart from the reflections).

For isotropic materials, the azimuth angle can be characterized by the Lode angle, which can be defined as

$\begin{array}{cc}{\theta }_L={\tan }^{-1}\left\{\frac{1}{\sqrt{3}}\left[2\left(\frac{s_2-s_3}{s_1-s_3}\right)-1\right]\right\},& \left(13\right)\end{array}$

where s₁, s₂, and s₃ are the maximum, intermediate and minimum principal deviatoric stresses respectively. FIG. 5 illustrates the Lode angle. For a given principal deviatoric stress represented by vector O'A, the Lode angle is represented by θ_L.

The relative ratio of the principal deviatoric stresses, χ, can be defined as χ=(s₂−s₃)/(s₁−s₃). To recognize the difference in plane strain and triaxial tension, a new material parameter, γ, can be defined as the ratio of the fracture strain at the plane strain condition to that at the triaxial tension (or compression) condition at the same constant hydrostatic pressure, i.e.

$\begin{array}{cc}\gamma =\frac{{\varepsilon }_f\left(\chi =0.5\right)}{{\varepsilon }_f\left(\chi =0\right)}.& \left(14\right)\end{array}$

FIG. 6 illustrates an example of a Lode dependence function that is a first order linear relationship in the plane of plastic strain components. The fracture point representing the fracture strain at the triaxial tension or compression is connected to that of the plane strain condition by a straight line and, thus, forms a polygon on the strain plane. Using the relative ratio of the stress deviators, the “six point star” illustrated in FIG. 6 can be represented by the function

$\begin{array}{cc}{\mu }_{\theta }=\{\begin{array}{cc}\frac{\sqrt{{\chi }^2-\chi +1}}{1+\left(\frac{\sqrt{3}}{\gamma }-2\right)\chi },& 0\le \chi \le 0.5;\\ \frac{\sqrt{{\chi }^2-\chi +1}}{1+\left(\frac{\sqrt{3}}{\gamma }-2\right)\left(1-\chi \right)},& 0.5\le \chi \le 1;\end{array},& \left(15\right)\end{array}$

which is symmetric with respect to χ=0.5. Therefore, a fracture envelope defining a region of material properties in which fracture will occur can be constructed for each of the twelve pie slices on the octahedral plane, which are identified by either 0≦χ≦0.5 or 0.5≦χ≦1. The first kind of the Lode dependence function reduces to a right hexagon when γ=√{square root over (3)}/2. The axes of the right hexagon is a measurement of strain.

Experimental data shows that for many materials, the material parameter γ which governs the difference in ductility between shear and tension at the same pressure is less than unity. Consequently, the fracture favors a shear mode. This is consistent with experimental data that shows the formation of a shear lip at the edge of an otherwise tensile specimen.

FIG. 7 illustrates another example of an azimuthal angle dependence function, which is a curvilinear model defined by

$\begin{array}{cc}{\mu }_{\theta }=\{\begin{array}{cc}1-\left(1-\gamma \right){\left(\frac{6\theta }{\pi }\right)}^k,& 0\le \theta \le \frac{\pi }{6};\\ 1-\left(1-\gamma \right){\left(\frac{6\pi /3-\theta }{\pi }\right)}^k,& \frac{\pi }{6}\le \theta \le \frac{\pi }{3};\end{array},\mathrm{or}{\mu }_{\theta }=\gamma +\left(1-\gamma \right){\left(\frac{6{\theta }_L}{\pi }\right)}^k,& \left(16\right)\end{array}$

where θ is the azimuthal angle with respect to the s₁ axis, θ_L is the Lode angle, k is the Lode dependence exponent, γ=ε_f(θ=π/6)/ε_f(θ=0) is the same material constant as in the polygon model, and β is the Lode dependence exponent, or shape parameter. When k=1 in Eq. (16), the curvilinear Lode dependence function is an Archimedes' spiral in the polar coordinate system, which is linear with respect to the azimuth angle on the triaxial plastic strain plane. When γ=1, the curvilinear function degenerates to a perfect circle. Other nonlinear forms may be obtained by fitting the experimental data.

FIG. 8 illustrates the fracture envelope in the principal stress space representing the entire family of the restricted loading paths with the combined effects of the hydrostatic pressure and the deviatoric state. The limiting hydrostatic pressure plane 80 can correspond to experimental data that suggests that no damage occurs when the hydrostatic pressure is sufficiently high. For example, when the hydrostatic pressure is above the hydrostatic pressure plane 80, new bonds can be created at room temperature. The cutoff pressure 81 represents the point in which the equivalent fracture strain ε_fo is zero. As illustrated in FIG. 8, the fracture locus can shrink to a single point 81 at the triad axis. Line 83 represents the fracture loci of equivalent stress at uniaxial tensile condition with respect to hydrostatic pressure. For example, line 85 represents a reference failure stress on the π-plane 84 under uniaxial tensile condition. Polygon 89 illustrates the failure loci on the π-plane 84. Line 86 represents another deviatoric state under uniaxial tensile condition on a different constant pressure plane 87. Polygon 88 illustrates, for example, the failure strain at different deviatoric states on the constant pressure plane 87.

Damage Rule

As described above, in continuum damage mechanics, material deterioration can be described by an internal damage variable. Damage can be modeled as an anisotropic quantity by a tensor in general. However, damage can also be assumed to be an isotropic process and treated as a scalar. In many applications, modeling damage as an isotropic process still provides for accurate predictions. To utilize cumulative damage as a criterion to predict the onset of fracture, the relationship of damage with respect to one or more measurable quantities can be established based on measurable field variables.

The physical properties of a material (e.g., the elastic modulus, the ductility, the local mass density, and/or the like) can change with the accumulation of the damage. In this sense, there can be multiple ways of measuring the quantity of damage. One of these ways can be viewed from the point-of-view of the relative deformability of the material, which can be obtained by comparing the deformability of the current state of the material with a prior state of the material, such as the initial undamaged state. In this way, ductile damage can be defined as the relative loss of deformability of the material. The relative loss of deformability can be expressed formally, in one example, by D=1/N*100%, where N is the number of times that the material can survive the same loading. For example, if a material fractures after 10 times of repeated loading, the material loses 10% of its initial deformability after the each loading.

A macroscopic crack can result from damage accumulation. Unlike the observation of crack initiation, damage accumulation may not be easily tracked along a deformation path. By conventional means, a single test gives only one point of the ultimate fracture in the damage accumulation graph with respect to the plastic strain, i.e. D=1 at ε_p=ε_f.

The loss of relative deformability can also be viewed by using low-cycle-fatigue test results. For materials under strain reversals, the total plastic strain at fracture can be greater than that of monotonic loading condition. This observation suggests relatively small damage at low plastic distortion for the same amount of incremental equivalent strain.

Reversed deviatoric loading can induce substantially the same amount of damage as forward loading. Reverse deviatoric loading means that the load is at the opposite direction of the forward loading or the azimuthal angle is at 180° from forward loading, while the pressure remains the same. For example, reverse loading for uniaxial tension σ₁=σ₀>0, σ₂=σ₃=0 is σ₁=σ₂=2σ₀/3>0, σ₃=−σ₀/3<0. Reverse loading for simple shear σ₁=σ₀>0, σ₂=0, σ₃=−σ₀<0 is the same as its forward loading, i.e. σ₁=σ₀>0, σ₂=0, σ₃=−σ₀<0. Here, σ₀ is a positive scalar.

Assuming that reversed deviatoric loading induces the same amount of damage as forward loading, the fracture locus on a hydrostatic plane has reverse symmetry in addition to permutation symmetry. The fracture strain for triaxial tension can be the same as the triaxial compression for a monotonic loading at identical constant pressure. Here, the terms triaxial tension (i.e., θ_L=−30°) and triaxial compression (i.e., θ_L=30°) should be distinguished from tension and compression, which are used for “simple tension” and “simple compression” sometimes in the literature. This results in an identical twelve pieces in the azimuthal direction on a hydrostatic plane. Any point of fracture loci (except the origin and those on the triad axes) has eleven image points.

The repeated plastic loading path (or low-cycle-fatigue test) reverses the direction of straining at intervals. This type of loading can be characterized by the ratio of the minimum and the maximum strain, i.e. R=ε_min/ε_max. For one type of reversed loading path, R=0. A full cycle can be viewed as two branches: a loading branch and a reversed loading branch, between which a strain reversal occurs. The plastic strain on the current branch can be denoted by the plastic distortion ε_d to distinguish from the total plastic strain ε_p.

The plastic strain is a history variable which can be defined as

${\varepsilon }_p=\sqrt{\frac{2}{3}}\int \sqrt{{\left({\varepsilon }_1\right)}^2+{\left({\varepsilon }_2\right)}^2+{\left({\varepsilon }_3\right)}^2},$

where dε₁, dε₂, and dε₃ are the principal plastic strain increments. The plastic distortion is a state variable which can be defined as

${\varepsilon }_d=\sqrt{\frac{2}{3}}\sqrt{{\left({\varepsilon }_1\right)}^2+{\left({\varepsilon }_2\right)}^2+{\left({\varepsilon }_3\right)}^2},$

where ε₁, υ₂, and ε₃ are the current principal plastic strain components. FIG. 9B illustrates the difference between the plastic strain and the plastic distortion.

FIGS. 9A, 9B, and 9C illustrate the time dependence of the rotational angle, the total equivalent plastic strain and the equivalent plastic distortion, respectively. To remove the pressure dependence, a torsion test, where pressure is constant zero, can be used to demonstrate the evolution of damage using the plastic strain and the plastic distortion. FIGS. 9A-9C show two identical cycles. FIG. 9A illustrates a rotational repeated plastic loading path with R=0. A loading branch 910 changes the rotational angle from zero to Δθ_max and a reverse loading branch 920 returns back to zero. FIG. 9B shows the evolution of the plastic strain 930 and the plastic distortion 940. Assuming a power law damage rule with damage exponent m=2, the damage accumulation using the plastic strain can be plotted as the solid line 950. The damage accumulation using the plastic distortion can be plotted as the dash-dot line 960. For this example, the predicted total fracture strain using these two methods can be as much as twice for repeated loading conditions, as shown in FIG. 9C.

For a low cycle fatigue test, a particular form of the damage rule can be derived from the Manson-Coffin's relationship. Experiments show that low-cycle fatigue is plasticity dominated phenomenon and appears to be dependent mainly upon the ductility of metals. Using the Palmgren-Miner rule, which assumes a linear relationship of damage accumulation of low-cycle fatigue. Therefore, damage is only related to the plastic deformation on the current branch.

Generally speaking, the number of cycles for low-cycle fatigue can be less than 10000 cycles. The relationship between the applied plastic strain and the number of cycles to failure (Δε_p−N curve) can be described by the Manson-Coffin relationship for a number of materials, i.e., Δε_p·N^k′=C, where C and k′ are material constants. The Manson-Coffin relationship appears to be linear on a log-log scale plot of Δε_p and N. Experimental data from push-pull tests and torsion tests of numerous materials such as bars of aluminum alloy, copper, and/or carbon steel show good agreement with this linear relationship.

In each loading branch, the loading is monotonic, dΔε_p is the same as dε_p. The relationship of incremental damage with respect to incremental equivalent plastic strain is

$\begin{array}{cc}\frac{D}{{\varepsilon }_p}=\frac{1}{2k^{\prime }C^{\left(1/k^{\prime }\right)}{\varepsilon }_p^{\left(1-1/k^{\prime }\right)}}.& \left(17\right)\end{array}$

For N=½, the material fails at a plastic strain of ε_f from a monotonic test. Thus, C=ε_f/2^k′.

Letting m=1/k′, Eq. (17) can be rewritten as

$\begin{array}{cc}\mathrm{dD}={m\left(\frac{{\varepsilon }_p}{{\varepsilon }_f}\right)}^{\left(m-1\right)}\frac{1}{{\varepsilon }_f}d{\varepsilon }_p.& \left(18\right)\end{array}$

Eq. (18) can be used in an integral calculation to determine damage.

“Damage accumulation” can refer to the integral equation, for example

$\begin{array}{cc}D={\int }_0^{{\varepsilon }_c}{m\left(\frac{{\varepsilon }_p}{{\varepsilon }_f}\right)}^{\left(m-1\right)}{\varepsilon }_p,& \left(19\right)\end{array}$

where ε_c denotes a critical plastic strain at the onset of ductile fracture. A “damage rule” can refer to the integrated function, for example, D(ε_p/ε_f)=(ε_p/ε_f)_m.

For monotonic proportional plastic loading paths, the plastic strain and the plastic distortions are identical. For repeated loading conditions, as illustrated in FIG. 9, the plastic distortion is used in the damage accumulation function,

$\begin{array}{cc}D={\int }_0^{{\varepsilon }_c}{m\left(\frac{{\varepsilon }_d}{{\varepsilon }_f}\right)}^{\left(m-1\right)}{\varepsilon }_d.& \left(20\right)\end{array}$

to depict the actual damage rate. The power law damage rule function is illustrated in FIG. 10 for m=2 along with the weakening function for several weakening exponents β. Another damage rule function is the exponential damage rule function. By more careful examination of the Δε_p−N curve on the log-log scale plot, the curve is concave toward the Δε_p axis. This damage rule can be more accurately represented by

$\begin{array}{cc}D=\frac{\exp \left[{\lambda \left(\frac{{\varepsilon }_p}{{\varepsilon }_f}\right)}^m\right]-1}{\exp \left[\lambda \right]-1},\mathrm{or}D=\frac{\exp \left[{\lambda \left(\frac{{\varepsilon }_d}{{\varepsilon }_f}\right)}^m\right]-1}{\exp \left[\lambda \right]-1}.& \left(21\right)\end{array}$

where λ is a material constant.

It is postulated that the damage potential is a fundamental material behavior. The hydrostatic pressure and the relative ratio of the stress deviators enter into the damage potential implicitly through the equivalent fracture strain and do not change the damage potential in other form.

Material Deterioration

In continuum damage mechanics (CDM), damage is one of the constitutive variables that takes into account the degradation and loss of load carrying capacity of materials. Considering a unitary reference volume element, the effective load carrying area can be viewed as decreasing as damage accumulates. The loss of the effective resisting area can be used in continuum damage mechanics to describe the material deterioration. The loss of effective resisting area can be defined as

$\begin{array}{cc}{D}_s=1-\frac{A_{\mathrm{eff}}}{A_0},& \left(22\right)\end{array}$

where A₀ is the nominal sectional area of the reference element and A_eff is the effective resisting area. The concept of A_eff is useful when considering microstructural change in the solid.

For an undamaged material, no defects can be assumed to exist and, therefore, D_s=0. At a fully damaged state, there is no resistance area, i.e. A_eff=0, or D_s=1. Thus, by using the strain equivalence hypothesis, the macroscopic behavior of the solid can be calculated from the matrix strength from the force balance of the unitary element, i.e.

$\begin{array}{cc}\stackrel{\_}{\sigma }=\frac{A_{\mathrm{eff}}}{A_0}{\sigma }_M.& \left(23\right)\end{array}$

Applying Eq. (22) to Eq. (23), the equivalent strength of the macroscopic solid is

σ=(1−D_s)σ_M.  (24)

where a weakening factor w(D)=1−D_s is applied on the matrix strength. A relationship between D and D_s can be established to couple the damage accumulation and the material deterioration. FIG. 11 illustrates a hypothetical matrix stress-strain curve and the weakened material stress-strain curve for several weakening exponent β values. The other exponent η is assumed to be unity here, i.e., η=1, and the pressure effect is neglected. The equivalent fracture strain for this hypothetical material is 1.

The damage induced material weakening can be introduced into the constitutive model by coupling of the yield function and the associated flow rule with the damage. The material deterioration can be considered at a material volume level. Following the continuum damage mechanics, the constitutive equation of the damaged material can be derived from the modified yield function, viz.

Φ= σ²−[w(D_s)σ_M]².  (25)

The strain for a damaged material can be represented by constitutive equations of the undamaged material in the yield function of which the stress is simply replaced by the effective stress. The material hardening is described by the hardening behavior of the matrix material, which is represented by σ_M in Eq. (25), while the material softening can be represented by the multiplication factor of w(D).

The stress-strain relationship of the matrix material can be assumed to be a basic material property, i.e., it does not change with the loading condition and other internal state variables, such as, for example, the damage, the pressure and the azimuthal angle. Upon accumulation of the micro structural change along with the plastic deformation, the external behavior the solid that contains the matrix material and the micro structural damage also changes—often, for example, in a weakened manner. Assuming the yielding of the damaged material follows von Mises type of yield criterion, the material weakening can refer to, for example, the macroscopic material yield strength, i.e. σ≦σ_M.

Because the damage can be evaluated in various ways (e.g., reduction of fracture strain, reduction of stiffness, increase of porosity, etc.), in general, the accumulation of these different types of damage may not necessarily be the same. For example, the damage associated with the reduction of the effective load carrying area may not necessarily be the same as the damage that addresses the reduction of the deformability, i.e. D_s<D or D_s>D.

The undamaged state can be assumed to have D=0 and a fully damage state of D=1. The undamaged state can have full strength of the matrix material, therefore, w(D=0)=1. On the other hand, the fully damage material no long carries any loading, therefore, w(D=1)=0. These two equations can be the end conditions that a weakening function has to satisfy.

For example, a class of the weakening function can be assumed to be w(D)=(1−D^β)^1/η, where β and η are weakening exponents. It follows that w(D)^η+D^β=1. It can be verified that this class of weakening function satisfies the end conditions. FIG. 10 illustrates weakening functions associated with the ductile damage. Other nonlinear relationships that satisfy the end condition and link the reduction of the effective loading carrying area to a general quantity of damage to replace the multiplication factor (1−D_s) are also possible.

When β=1 and η=1, this class of weakening function reduces to a special case of linear weakening function with respect to the ductile damage. The stiffness damage and the ductile damage can be approximate to be the same a first order accuracy. i.e. D_s=D. Experiments show that the stiffness damage accumulate can lag behind the accumulation of ductile damage. Since Dε[0,1] and D_sε[0,1], then D_s≦D when β>1 and η>1.

The above-described techniques can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations of them. The implementation can be as a computer program product, i.e., a computer program tangibly embodied in an information carrier, e.g., in a machine-readable storage device or in a propagated signal, for execution by, or to control the operation of, data processing apparatus, e.g., a programmable processor, a computer, or multiple computers. A computer program can be written in any form of programming language, including compiled or interpreted languages, and the computer program can be deployed in any form, including as a stand-alone program or as a subroutine, element, or other unit suitable for use in a computing environment. A computer program can be deployed to be executed on one computer or on multiple computers at one site. The above-described techniques can be incorporated in a numerical simulation software package using a finite-element method solver. Software packages can include, for example, LS-DYNA and ABAQUS.

Method steps can be performed by one or more programmable processors executing a computer program to perform functions of the invention by operating on input data and generating output. Method steps can also be performed by, and an apparatus can be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application-specific integrated circuit). Subroutines can refer to portions of the computer program and/or the processor/special circuitry that implements that functionality.

Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor receives instructions and data from a read-only memory or a random access memory or both. The essential elements of a computer are a processor for executing instructions and one or more memory devices for storing instructions and data. Generally, a computer also includes, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, or optical disks. Data transmission and instructions can also occur over a communications network. Information carriers suitable for embodying computer program instructions and data include all forms of non-volatile memory, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in special purpose logic circuitry.

To provide for interaction with a user, the above described techniques can be implemented on a computer having a display device, e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor, for displaying information to the user and a keyboard and a pointing device, e.g., a mouse or a trackball, by which the user can provide input to the computer (e.g., interact with a user interface element). Other kinds of devices can be used to provide for interaction with a user as well; for example, feedback provided to the user can be any form of sensory feedback, e.g., visual feedback, auditory feedback, or tactile feedback; and input from the user can be received in any form, including acoustic, speech, or tactile input.

The above described techniques can be implemented in a distributed computing system that includes a back-end component, e.g., as a data server, and/or a middleware component, e.g., an application server, and/or a front-end component, e.g., a client computer having a graphical user interface and/or a Web browser through which a user can interact with an example implementation, or any combination of such back-end, middleware, or front-end components. The components of the system can be interconnected by any form or medium of digital data communication, e.g., a communication network. Examples of communication networks include a local area network (“LAN”) and a wide area network (“WAN”), e.g., the Internet, and include both wired and wireless networks.

The computing system can include clients and servers. A client and a server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other.

One skilled in the art will realize the invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The foregoing embodiments are therefore to be considered in all respects illustrative rather than limiting of the invention described herein. Scope of the invention is thus indicated by the appended claims, rather than by the foregoing description, and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.

Claims

1. A method of determining material constitutive relationships of fracture in a ductile material undergoing plastic deformation, the method comprising:

determining a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure;

determining a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress;

determining a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function;

determining a weakening function of the portion of the ductile material based on the damage rule; and

determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

2. The method of claim 1 further comprising determining an initiation of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

3. The method of claim 1 wherein the material pressure function includes a logarithmic function of pressure.

4. The method of claim 3 wherein the material pressure function is: μ p  ( p ) = 1 - q   log  ( 1 - p p lim ).

5. The method of claim 1 wherein when a relative ratio of principal stresses is between about 0 and about 0.5, the material azimuthal angle function is: μ θ = χ 2 - χ + 1 1 + ( 3 γ - 2 )  χ; and when the relative ratio of principal deviatoric stresses is between about 0.5 and about 1, the material azimuthal angle function is: μ θ = χ 2 - χ + 1 1 + ( 3 γ - 2 )  ( 1 - χ ).

6. The method of claim 1 wherein the material azimuthal angle function is a material Lode angle function.

7. The method of claim 6 wherein the material Lode angle function is: μ 0 = γ + ( 1 - γ )  ( 6   θ L  π ) k.

8. The method of claim 1 wherein the material azimuthal angle function is: μ 0 = 1 - ( 1 - γ )  ( 6   θ  π ) k   for   0 ≤ θ ≤ π 6.

9. The method of claim 1 wherein the material azimuthal angle function is: μ 0 = 1 - ( 1 - γ )  ( 6   π / 3 - θ  π ) k   for   π 6 < θ ≤ π 3.

10. The method of claim 1 wherein the damage rule is a function of the ratio of a plastic strain, εp, to a fracture strain, εf.

11. The method of claim 10 wherein the damage rule is D = ( ɛ p ɛ f ) m.

12. The method of claim 10 wherein the damage rule is D = exp  [ λ  ( ɛ p ɛ f ) m ] - 1 exp  [ λ ] - 1.

13. The method of claim 1 wherein the damage rule is a function of the ratio of a plastic distortion εd to a fracture strain εp.

14. The method of claim 13 wherein the damage rule is D = ( ɛ d ɛ p ) m.

15. The method of claim 13 wherein the damage rule is D = exp  [ λ  ( ɛ p ɛ f ) m ] - 1 exp  [ λ ] - 1.

16. The method of claim 1 wherein the weakening function is a linear function of the damage rule.

17. The method of claim 1 wherein the weakening function is:

w(D)=(1−Dβ)1/η.

18. The method of claim 17 wherein β is greater than one.

19. The method of claim 17 wherein η is greater than one.

20. The method of claim 1 wherein an equivalent stress is used to determine material constitutive relationships of fracture, the equivalent stress based on the weakening function and a matrix property of the ductile material.

21. The method of claim 20 wherein the equivalent stress is:

σ=w(D)σM.

22. A computer program product, tangibly embodied in an information carrier, the computer program product including instructions being operable to cause a data processing apparatus to:

determine a material pressure function describing dependence of a damage rule of at least a portion of a ductile material on pressure;

determine a material azimuthal angle function describing dependence of the damage rule of the portion of the ductile material on principle stress;

determine the damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function;

determine a weakening function of the portion of the ductile material based on the damage rule; and

determine material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

23. The computer program product of claim 22 wherein the instructions are incorporated in at least one of: a software or hardware package.

24. The computer program product of claim 23 wherein the software package is LS-DYNA or ABAQUS.

25. A method of determining material constitutive relationships of damage accumulation in a ductile material undergoing plastic deformation, the method comprising:

determining a material pressure function describing dependence of a damage rule of at least a portion of the ductile material on pressure;

determining a material azimuthal angle function describing dependence of the damage rule of the portion of the ductile material on principle stress; and

determining a nonlinear damage accumulation rule for the portion of the ductile material based on the material pressure function and the material azimuthal angle function.

26. The method of claim 25 further comprising:

determining a weakening function of the portion of the ductile material based on the nonlinear damage accumulation rule; and

determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

27. A method of determining damage accumulation characteristics of material constitutive relationships of fracture in a ductile material undergoing plastic deformation, the method comprising:

determining a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure;

determining a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress;

determining a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function;

determining a damage accumulation in an integral form based on the material damage rule, the pressure function, and the material azimuthal angle function.

28. The method of claim 27 further comprising:

determining a weakening function of the portion of the ductile material based on the nonlinear damage accumulation rule; and

determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

29. A method of determining material constitutive relationships of fracture in a ductile material undergoing plastic deformation, the method comprising:

means for determining a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure;

means for determining a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress;

means for determining a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function;

means for determining a weakening function of the portion of the ductile material based on the damage rule; and

means for determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.

30. A method of determining material constitutive relationships of fracture in a ductile material undergoing plastic deformation, the method comprising:

determining a material pressure function describing dependence of damage accumulation of at least a portion of the ductile material on pressure;

determining a material azimuthal angle function describing dependence of damage accumulation of the portion of the ductile material on principle stress;

determining a damage rule of the portion of the ductile material based on the material pressure function and the material azimuthal angle function;

determining a weakening function of the portion of the ductile material based on the damage rule, wherein the weakening function is a function of damage accumulation and the rate of change of the weakening function with respect to damage accumulation is dependent on damage accumulation; and

determining material constitutive relationships of fracture in the portion of the ductile material based on the weakening function and a matrix property of the ductile material.