Qualitative Behaviour of Elastic-Plastic Solutions for a Class of Damage Mechanics Models Near Bimaterial Interfaces: A Simple Analytical Example

Abstract

The paper presents an exact analytic solution for a class of elastic-plastic models with damage evolution. The boundary value problem consists of a planar deformation comprising the simultaneous shearing and expansion of a hollow cylindrical specimen of material and involves a bimaterial interface at which the materials stick to each other. With no loss of generality for understanding the qualitative behaviour of the solution near the bimaterial interface, an extreme case when the hard material is rigid is considered. The solution is reduced to a transcendental equation for the value of the equivalent plastic strain at the bimaterial interface. This equation predicts that the equivalent plastic strain attains a maximum under certain conditions. The existence of the solution of the boundary value problem depends on the value of the damage parameter at fracture, which is a material constant. In particular, if this value is larger than the value of the damage parameter at the bimaterial interface corresponding to the maximum possible value of the equivalent strain at this interface, then no solution exists. Experimental data available in the literature are used to assess whether Lemaitre’s model is applicable.

1. Introduction

The ductile damage evolution equations of continuum mechanics can be conveniently divided into three groups: (i) uncoupled models, (ii) partly coupled models, and (iii) coupled models. In the case of the models of group (i) the damage evolution equation is solved having found the solution for the system of equations of plasticity theory. A recent review of models of this group is provided in [1]. The models of group (iii) include the fully coupled system of equations of plasticity theory and the damage evolution equation. In particular, the material is plastically compressible. A recent review of models of this group is provided in [2]. In the present paper, a wide class of models of group (ii) for elastic-plastic materials is adopted. These models containing one scalar measure of damage are based on the pressure-independent yield criterion and the associated flow rule for plastic strain rates. The evolution of the damage parameter is controlled by an additional equation. Such models are presented, for example, in [3–7]. The yield criterion and elastic moduli depend on the damage parameter. Many structures and machine parts are composed of two or more different materials and, therefore, contain bimaterial interfaces. Solution behaviour in the vicinity of such interfaces may reveal some qualitative features that provide an assessment of the model adopted to find the solution. For example, using a rigid perfectly plastic material model, it has been shown in [8] that the velocity field is singular in the vicinity of some bimaterial interfaces. This singularity in the velocity field is not compatible with the class of damage evolution models reviewed in [9]. In order to assess the mathematical model, it is possible to study any boundary value problem, even if its practical application is not feasible. It is more important to find the exact solution of the problem without relaxing the boundary conditions and field equations. Such a boundary value problem has been proposed and solved for a rigid plastic linear hardening material model in [10]. Then, this boundary value problem has been successfully used in [11, 12] to study various qualitative features of solutions for several material models. However, all of these solutions are for rigid plastic materials with no damage evolution. In the present paper, the same boundary value problem is combined with a class of elastic-plastic models with damage evolution at small strains. The solution is reduced to a transcendental equation for the value of the equivalent plastic strain at the bimaterial interface. The solution to this equation predicts that this value attains a maximum under certain conditions. The existence of the solution of the boundary value problem depends on the value of the damage parameter at fracture, which is supposed to be a constitutive parameter. In particular, if the value of the damage parameter at the bimaterial interface corresponding to the maximum possible value of the equivalent plastic strain at this interface is smaller than the damage parameter at fracture, then no solution exists. The model proposed by Lemaitre [3] is studied in some detail. In particular, experimental data provided in [13] are used to assess whether the model for this specific set of constitutive parameters is applicable (in the sense that the solution exists).

2. Statement of the Problem

Consider an infinite elastic-plastic circular cylinder of internal radius 𝑟₀ and external radius 𝑅₀ subject to the action of tangential stresses uniformly distributed over its external radius. It is assumed that a rigid rod of radius 𝑟₀ is inserted into the hole of the cylinder. The rod is motionless and fixed to the cylinder. Due to this system of loading and constraints, the cylinder is twisted. It is convenient to introduce a cylindrical polar coordinate system (𝑟,𝜃,𝑧) with its z-axis coinciding with the axis of symmetry of the cylinder. The boundary conditions are 𝑢_𝜃=0 (1) for 𝑟=𝑟₀ and 𝑢_𝜃=𝜔𝑅₀ (2) for 𝑟=𝑅₀. Here 𝑢_𝜃 is the circumferential displacement and 𝜔 is the angular displacement of the external surface of the cylinder.

A typical dependence of the shear yield stress, 𝑘, on the equivalent plastic strain, 𝜀^𝑝_eq, and damage parameter, 𝐷, is [3, 4, 7] 𝑘=𝑘₀1+𝑓𝜀^𝑝_eq(1−𝐷). (3) Here 𝑘₀ is the initial shear yield stress when 𝜀^𝑝_eq=0 and 𝐷=0. The function 𝑓(𝜀^𝑝_eq) satisfies the conditions 𝑓(0)=0 and 𝑑𝑓/𝑑𝜀^𝑝_eq>0 for any 𝜀^𝑝_eq. The equivalent plastic strain is defined by the equation 𝑑𝜀^𝑝_eq𝑑𝑡=𝜉^𝑝_eq. (4) Here 𝑡 is the time and 𝑑/𝑑𝑡 is the convected derivative. The equivalent plastic strain rate, 𝜉^𝑝_eq, is defined by 𝜉^𝑝_eq=23𝜉^𝑝_𝑖𝑗𝜉^𝑝_𝑖𝑗, (5) where 𝜉^𝑝_𝑖𝑗 are the plastic strain rate components. The yield criterion is 𝜏_𝑖𝑗𝜏_𝑖𝑗=2𝑘², (6) where 𝜏_𝑖𝑗 are the deviatoric components of the stress tensor. The flow rule associated with (6) is 𝜉^𝑝_𝑖𝑗=𝜆𝜏_𝑖𝑗, (7) where 𝜆 is a scalar factor of proportionality. The elastic strains, 𝜀^𝑒_𝑖𝑗, obey Hooke’s law. The total strains are 𝜀_𝑖𝑗=𝜀^𝑒_𝑖𝑗+𝜀^𝑝_𝑖𝑗. (8) In the case of isotropic models with a single scalar measure of damage typical damage evolution equations are [3, 4, 7] 𝑑𝐷𝜎𝑑𝑡=𝐹√𝜉3𝑘,𝐷^𝑝_eqif𝜀^𝑝_eq≥𝜀₀,𝑑𝐷𝑑𝑡=0if𝜀^𝑝_eq<𝜀₀,𝑑𝐷𝜎𝑑𝑡=𝐹√𝜀3𝑘,𝐷^𝑝_eq^1/𝑀𝜉^𝑝_eqif𝜀^𝑝_eq≥𝜀₀,𝑑𝐷𝑑𝑡=0if𝜀^𝑝_eq<𝜀₀. (9)(10) Here 𝜎 is the hydrostatic stress, 𝜀₀ is the threshold equivalent plastic strain for the initiation of damage, 𝑀 is constant, and 𝐹 is a given function of its arguments. The fracture criterion is 𝐷=𝐷_𝑐, (11) where 𝐷_𝑐 is a material constant whose value for metallic materials lies in the range 0.2<𝐷_𝑐<0.8 [7]. Equations (3) to (11) along with Hooke’s law represent a typical isotropic elastic-plastic material model with damage evolution at small strains. These equations should be supplemented with the equilibrium equations.

3. Solution

In this section, an exact solution of the equations of the previous section is derived in the cylindrical coordinates assuming that the solution is independent of 𝜃 and z. It is then demonstrated that the solution is compatible with the boundary conditions (1) and (2). Suppose that all components of the stress tensor vanish except for the shear component 𝜏_𝑟𝜃. The yield criterion (6) is satisfied if 𝜏_𝑟𝜃=𝑘. (12) The only nontrivial equilibrium equation is 𝜕𝜏_𝑟𝜃+𝜕𝑟2𝜏_𝑟𝜃𝑟=0. (13) Denote the radius of the elastic-plastic boundary by 𝑅, 𝑟₀≤𝑅≤𝑅₀. The material just on the elastic side of the boundary must be on the point of yielding. Therefore, 𝜏_𝑟𝜃=𝑘=𝑘₀ at 𝑟=𝑅. Integrating (13) with the use of this condition gives 𝜏_𝑟𝜃𝑘₀=𝑅𝑟². (14) It follows from Hooke’s law that 2𝐺𝜀^𝑒_𝑟𝜃=𝜏_𝑟𝜃, (15) where 𝐺 is the modulus of shear.

Consider the elastic region, 𝑅≤𝑟≤𝑅₀. In this region, 𝜀^𝑒_𝑖𝑗=𝜀_𝑖𝑗 and 𝐺=𝐺₀, where 𝐺₀ is the shear modulus of undamaged material. Therefore, substituting (14) into (15) yields 𝜀_𝑟𝜃=𝑘₀2𝐺₀𝑅𝑟². (16)

In the plastic region, 𝑟₀≤𝑟≤𝑅, combining (12) and (14) gives 𝑘𝑘₀=𝑅𝑟². (17) A typical assumption is that 𝐺=𝐺₀(1−𝐷) [3, 4, 7]. Then, substituting (12) into (15) results in 𝜀^𝑒_𝑟𝜃=𝑘2𝐺₀().1−𝐷 (18) Since the only nonzero stress component is 𝜏_𝑟𝜃, it follows from (7) that the only nonzero plastic strain rate component is 𝜉^𝑝_𝑟𝜃. Then, (5) reduces to 𝜉^𝑝_eq=2√3𝜉^𝑝_𝑟𝜃. (19) It has been taken into account here that 𝜏_𝑟𝜃>0, as follows from (12). Since strains are small, 𝜉^𝑝_eq=𝜕𝜀^𝑝_eq/𝜕𝑡 and 𝜉^𝑝_𝑟𝜃=𝜕𝜀^𝑝_𝑟𝜃/𝜕𝑡. Therefore, integrating (19) with respect to 𝑡 gives 𝜀^𝑝_eq=2√3𝜀^𝑝_𝑟𝜃. (20) The total shear strain in the plastic region is found from (8), (18), and (20) as 𝜀_𝑟𝜃=𝑘2𝐺₀()+√1−𝐷32𝜀^𝑝_eq. (21) Since 𝜎=0, (9) becomes 𝑑𝐷𝑑𝑡=𝐹₁(𝐷)𝜉^𝑝_eqif𝜀^𝑝_eq≥𝜀₀,𝑑𝐷𝑑𝑡=0if𝜀^𝑝_eq<𝜀₀, (22) where 𝐹₁(𝐷)=𝐹(0,𝐷). Assuming that 𝐷=0 at the initial instant, (22) can be integrated to give Ω(𝐷)=𝜀^𝑝_eq−𝜀₀if𝜀^𝑝_eq≥𝜀₀𝐷=0if𝜀^𝑝_eq<𝜀₀. (23) Here Ω(𝐷)=^𝐷₀𝑑𝑥𝐹₁(𝑥), (24) where 𝑥 is a dummy variable of integration. It follows from (3), (17), and (23) that 𝑅𝑟²=1+𝑓𝜀^𝑝_eq1−Ω⁻¹𝜀^𝑝_eq−𝜀₀, (25) where Ω⁻¹ is the function inverse to Ω. It has been assumed here that 𝜀^𝑝_eq≥𝜀₀. Substituting (3) into (21) yields 𝜀_𝑟𝜃=𝑘₀1+𝑓𝜀^𝑝_eq2𝐺₀+√32𝜀^𝑝_eq. (26) For the problem under consideration, the total shear strain is defined by 𝜀_𝑟𝜃=12𝜕𝑢_𝜃−𝑢𝜕𝑟𝑟^𝜃=𝑟2𝜕𝑢_𝜃/𝑟.𝜕𝑟 (27) Combining (26) and (27) gives 𝑟𝜕𝑢_𝜃/𝑟=𝑘𝜕𝑟⁰1+𝑓𝜀^𝑝_eq𝐺₀+√3𝜀^𝑝_eq. (28) It is convenient to introduce new independent variables as 𝑅𝜌=𝑟𝑅,𝑠=𝑅₀. (29) Then, 𝜕𝜌𝜕𝑟=−²𝑅𝜕,𝜕𝜕𝜌=𝑅̇𝜕𝑡𝑅𝜌𝜕𝜕𝜕𝜌+𝑠,𝜕𝑠 (30) where 𝑅̇≡𝑑𝑅/𝑑𝑡. Using (29), (30), (25) and (28) transforms to 𝜌²=1+𝑓𝜀^𝑝_eq1−Ω⁻¹𝜀^𝑝_eq−𝜀₀≡Λ𝜀^𝑝_eq−𝑢,𝜌𝜕_𝜃/𝑟=𝑘𝜕𝜌⁰1+𝑓𝜀^𝑝_eq𝐺₀+√3𝜀^𝑝_eq≡Φ𝜀^𝑝_eq, (31)(32) respectively. Eliminating 𝜌 between (31) and (32) leads to 𝜕𝑢_𝜃/𝑟𝜕𝜀^𝑝_eqΦ𝜀=−^𝑝eqΛ^′𝜀^𝑝_eq2Λ𝜀^𝑝_eq, (33) where Λ^′(𝜀^𝑝_eq)≡𝑑Λ(𝜀^𝑝_eq)/𝑑𝜀^𝑝_eq. Let 𝑢_𝑅 be the value of 𝑢_𝜃 at 𝑟=𝑅. Since 𝜀^𝑝_eq=0 at 𝑟=𝑅, the solution to (33) is 𝑢_𝜃𝑟1=−2^𝜀𝑝eq₀Φ(𝑥)Λ^′(𝑥)Λ(𝑥)𝑢𝑑𝑥+^𝑅𝑅. (34) The value of 𝑢_𝑅 should be found using the solution in the elastic region. Substituting (27) into (16) and using (29) and (30) give 𝜕𝑢_𝜃/𝑟𝑘𝜕𝜌=−𝜌⁰𝐺₀. (35) The solution of this equation satisfying the boundary condition (2) is 𝑢_𝜃𝑟=𝑘₀2𝐺₀𝑠²−𝜌²+𝜔. (36) It follows from (29) that 𝜌=1 if 𝑟=𝑅. Therefore, the value of 𝑢_𝑅 is determined from (36) as 𝑢_𝑅𝑅=𝑘₀2𝐺₀𝑠²−1+𝜔. (37) Substituting (37) into (34) gives 𝑢_𝜃𝑟1=−2^𝜀𝑝eq₀Φ(𝑥)Λ^′(𝑥)Λ(𝑥)𝑘𝑑𝑥+⁰2𝐺₀𝑠²−1+𝜔. (38) Let 𝜀_𝑠 be the value of 𝜀^𝑝_eq at 𝑟=𝑟₀. Then, combining (38) and the boundary condition (1) results in ^𝜀𝑠₀Φ𝜀^𝑝_eqΛ^′𝜀^𝑝_eqΛ𝜀^𝑝_eq𝑑𝜀^𝑝_eq=𝑘₀𝐺₀𝑠²−1+2𝜔. (39) It follows from (29) that 𝜌=𝑅/𝑟₀ at 𝑟=𝑟₀, and, therefore, 𝜌=𝑠𝑅₀/𝑟₀ at 𝑟=𝑟₀. Then, (31) shows that 𝑠²𝑅²₀/𝑟²₀=Λ(𝜀_𝑠). Using this equation to eliminate 𝑠 in (39) yields ^𝜀𝑠₀Φ𝜀^𝑝_eqΛ^′𝜀^𝑝_eqΛ𝜀^𝑝_eq𝑑𝜀^𝑝_eq−𝑘₀𝐺₀𝑟²₀𝑅²₀Λ𝜀_𝑠−1=2𝜔. (40) It follows from (23) that the value of 𝐷 at 𝑟=𝑟₀ is given by 𝐷_𝑠=Ω⁻¹𝜀_𝑠−𝜀₀. (41) It has been assumed here that 𝜀_𝑠≥𝜀₀.

4. Fracture Initiation and Nonexistence of the Solution

Equation (40) determines 𝜀_𝑠 as a function of 𝜔. It follows from this equation that 2𝑑𝜔𝑑𝜀_𝑠=Φ𝜀_𝑠Λ𝜀_𝑠−𝑘₀𝐺₀𝑟²₀𝑅²₀Λ^′𝜀_𝑠. (42) The right hand side of this equation vanishes if Φ𝜀_𝑠Λ𝜀_𝑠=𝑘₀𝐺₀𝑟²₀𝑅²₀ (43) or Λ^′𝜀_𝑠=0. (44) Using (31) and (32), the left hand side of (43) can be represented in the form Φ𝜀_𝑠Λ𝜀_𝑠=𝑘₀𝐺₀𝐴+√3𝜀_𝑠𝐺₀/𝑘₀𝐴1−Ω⁻¹𝜀_𝑠−𝜀₀, (45) where 𝐴=1+𝑓(𝜀_𝑠). Consider the coefficient of 𝑘₀/𝐺₀ in (45). It follows from the definition for the function Ω(𝐷) that Ω⁻¹(𝜀_𝑠−𝜀₀) is equal to the value of 𝐷 at 𝑟=𝑟₀. Therefore, 0<1−Ω⁻¹(𝜀_𝑠−𝜀₀)<1 and the denominator of the coefficient of 𝑘₀/𝐺₀ in (45) is smaller than 𝐴. On the other hand, it is evident that its numerator is larger than 𝐴. Therefore, the coefficient is larger than 1. However, the coefficient of 𝑘₀/𝐺₀ in (43) is smaller than 1. Hence (43) has no solution. Let 𝜀_𝑚 be the solution to (44). The corresponding values of 𝜔=𝜔_𝑚 and 𝐷_𝑠=𝐷_𝑚 are determined from (40) and (41) at 𝜀_𝑠=𝜀_𝑚. Expanding the left hand side of (42) in a series in the vicinity of 𝜀_𝑠=𝜀_𝑚 gives 𝑑𝜔/𝑑𝜀_𝑠=𝛽(𝜀_𝑠−𝜀_𝑚) to leading order. Here 𝛽 is constant. Integrating this equation with the use of the condition 𝜀_𝑠=𝜀_𝑚 at 𝜔=𝜔_𝑚 results in 2(𝜔−𝜔_𝑚)=𝛽(𝜀_𝑠−𝜀_𝑚)² to leading order. This solution can be rewritten in the form 𝜀_𝑠=𝜀_𝑚−2𝛽⁻¹(𝜔−𝜔_𝑚). It is evident from this expression that 𝜀_𝑠 cannot exceed the value of 𝜀_𝑚. Thus if (43) has a solution, then the solution given in the previous section does not exist for 𝜔≥𝜔_𝑚. On the other hand, according to the fracture criterion (11), the initiation of fracture at the bimaterial interface occurs if 𝐷_𝑠=𝐷_𝑐. The corresponding value of 𝜀_𝑠=𝜀_𝑐 is determined from (41) as 𝜀_𝑐=Ω(𝐷_𝑐)+𝜀₀. Hence the solution to the boundary value problem of the previous section, including the fracture criterion, exists if and only if 𝜀_𝑐≤𝜀_𝑚or𝐷_𝑐≤𝐷_𝑚. (46) In particular, the solution predicts the initiation of fracture at the bimaterial interface at 𝜀_𝑠=𝜀_𝑐. It will be shown in the next section that the inequality (46) is not satisfied for some specific damage evolution laws and constitutive parameters used in applications.

5. Illustrative Example

In the case under consideration the damage evolution model [3, 4] is obtained if 𝐹₁(𝐷)=𝛼 where 𝛼 is constant. Therefore, evaluating the integral in (24) gives Ω(𝐷)=𝐷𝛼. (47) Then, it follows from (23) that Ω⁻¹𝜀^𝑝_eq−𝜀₀=𝛼𝜀^𝑝_eq−𝜀₀. (48) The equation of the strain hardening curve for the steel tested in [13], is 𝑓𝜀^𝑝_eq=2.78𝜀^𝑝_eq^0.32. (49) Using experimental data provided in [13] it is possible to find that 𝐷_𝑐=0.24,𝜀₀=0.023,𝛼=0.6. (50) Substituting (48)–(50) into (31) gives Λ(𝜀^𝑝_eq)=[1+2.78(𝜀^𝑝_eq)^0.32][1−0.6(𝜀^𝑝_eq−0.023)]. Hence, (44) becomes 0.89𝜀_𝑚^−0.68[1−0.6(𝜀_𝑚−0.023)]−0.6(1+2.78𝜀_𝑚^0.32)=0. Its solution is 𝜀_𝑚=0.29. Substituting this value of 𝜀^𝑝_eq=𝜀_𝑚 into (41) with the use of (48) gives 𝐷_𝑚=0.16. This means that the solution to the boundary value problem without using the fracture criterion breaks down when 𝐷_𝑠=0.16. On the other hand, the fracture criterion (11) is not satisfied as follows from (50). Therefore, it is necessary to conclude that the damage evolution model adopted is not compatible with the constitutive parameters recommended in [13].

6. Conclusions

An exact analytic solution for an elastic-plastic model coupled with a damage evolution equation has been found. Special attention has been devoted to the qualitative behavior of the solution in the vicinity of the bimaterial interface. Even though the boundary value problem solved is not feasible for practical applications, the main result obtained is useful for assessing damage evolution models by mathematical methods. In particular, it has been shown that the boundary value problem may have no solution because the equation relating the equivalent plastic strain at the bimaterial interface and one of the boundary conditions has no solution. This conclusion is not valid if the initiation of fracture occurs at a lower value of this strain. Therefore, having specific damage evolution equation and constitutive parameters, it is possible to verify that the model is compatible with conditions at bimaterial interfaces where shear straining prevails. As an example, the constitutive equations proposed in [13] have been used, and it has been demonstrated that the aforementioned equation has no solution. This means that some boundary value problems may have no solution for this model.

Even though (9) has been considered explicitly, it is evident that (10) can be treated in a similar manner. It is just necessary to replace (23) with Ω(𝐷)=𝐸^𝑝_eq−𝐸₀if𝜀^𝑝_eq≥𝜀₀,𝐷=0if𝜀^𝑝_eq<𝜀₀, (51) where 𝐸^𝑝_eq=𝑚⁻¹(𝜀^𝑝_eq)^𝑚, 𝐸₀=𝑚⁻¹𝜀^𝑚₀, and 𝑚=1+𝑀⁻¹.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research described in this paper has been supported by the Grants RFBR-11-01-00987 and NSH-3842.2012.1.