International Journal of Fracture 128: 81–93, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Thermomechanics of brittle fracture I.M. DUNAEV1 and V.I. DUNAEV2
1 Materials Strength Department, Kuban State Technology University, Moskovskaya street 2, Krasnodar, 350072,
Russia 2 Applied Mathematics Department, Kuban State University, Stavropolskaya street 149, Krasnodar, 350040,
Russia
Abstract. On the basis of the thermodynamics of thermoelastic deformation we propose an energy condition of the Griffith type for brittle fracture of solids under single loading. An analysis is given of the proposed condition under the plane stressed (strained) state for the two known models of an isolated defect. In the first model, stresses on the external (distant) surface of a solid with a defect are prescribed the same as in a similar solid without a defect. In the second model, displacements are prescribed on the external surface of a solid with a defect, which correspond to the load applied to the solid, but before the defect or crack was formed. Stresses on the surface of the defect in both models are equal to zero. It is shown that the first model in the isothermal case of deformation leads to the Griffith condition. The second model meets the energy condition of the Griffith type from which, under additional assumptions concerning the shape of the defect, and from conditions of isotropy and convexity, we obtain a curve of fracture (macroscopic criterion of fracture) in an ellipse form in the space of principal stresses. The orientation of the crack was determined. Coefficients of the curve of fracture obviously depend upon elastic constants, temperature, linear coefficient of thermal expansion and crack dimensions.
1. Introduction A. Griffith (1925), (Sih and Liebowitz, 1967) has proposed an energy condition dUp + γp d = dA
(1)
for determining a relation between critical stresses and critical dimensions of the defect or crack under single loading of solids. The following conventions are introduced in the condition (1): Up = Up(0) − Up(1) is the change of a potential energy caused by the defect formation; Up(0) and Up(1) are potential energies of a solid without and with defect, respectively; γp is the work necessary for the formation of a unit area of defect surface ; A is the change of the work of external forces during defect formation. Griffith also assumed that γp = const represents a certain physical parameter of the material and that its value, in the case of brittle fracture, equals the surface tension of the material analogously to the surface tension of fluids. Condition (1) for the problems of material fracture under uniaxial and biaxial tension (compression) of a plate with a defect (Sih and Liebowitz, 1967) leads to temperature-independent and identical-in-absolute-value critical stresses, which contradicts experimental data for practically all known materials. In order to determine the critical stress at compression, Griffith (Paul, 1968) considered the problem of a biaxial compression by the stresses of an elastic plate, reduced by an elliptic cavity, the major axis of which makes an angle with the direction of one of the stresses. When solving the problem, Griffith assumed that the real defect is represented in the form of a ‘narrow’ ellipse with a half-axis ratio much less than one; fracture under
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compression occurs at a point on the boundary of the defect, where normal tensile stress, directed along the ellipse boundary, reaches a critical value at tension. After computing, Griffith obtained a ratio between critical stresses under compression and tension, which equals eight in absolute value for all materials, and which also, in a general case, contradicts the experimental data. Numerous investigations and generalizations have not changed qualitatively the results which follow from the condition (1) and publications (Griffith, 1925; Sih and Liebowitz, 1967; Paul, 1968). In this paper, on the basis of thermodynamics laws for thermoelastic deformation of solids (Section 2), there is formulated the necessary energy condition of fracture during formation of a critical dimension defect. On the basis of analyses of this condition for the two known models of a solid with a defect (Section 3), it is shown that one of the models leads to a Griffith type energy condition of fracture, which, unlike condition (1), contains an increment of an entropy component of the internal energy, which is non-zero in a general case. Another model of a solid with a defect leads to a thermodynamically incomplete (increment of the entropy component of the internal energy is zero) condition (1) and, therefore, to physically groundless and experimentally unsupported results. A Griffith type energy condition is used (Section 6) to derive a macroscopic criterion for brittle fracture of isotropic materials at a plane stressed (strained) state. 2. Thermodynamic formulation of an energy condition of fracture Let us introduce (Nowacki, 1970) the specific free energy = (εij , T ) such that ψ = u − ηT
(2)
Here u is the specific internal energy, η is the specific entropy, T is the absolute temperature, 1 ∂ui ∂uj + i, j = 1, 2, 3 (3) εij = 2 ∂xj ∂xi are components of the strain tensor, ui are components of the displacement vector, xi are Cartesian coordinates of the solid points. The free energy for linear thermoelastic isotropic solids may be presented in the form (Nowacki, 1970): λ 2 T (4) − (T − T0 ) , ψ = µεij εij + θ − 3α0 K0 θ (T − T0 ) − cε T ln 2 T0 where the summation is performed over repeating indices i, j, θ = εij δij is the first invariant of the strain tensor, δij is Kronecker’s delta, α0 is the linear coefficient of thermal expansion, cε is the specific heat at constant strain, µ and λ are Lame’s coefficients, µ=
E , 2(1 + ν)
λ=
νE , (1 + ν)(1 − 2ν)
E 2 K0 = λ + µ = , 3 3(1 − 2ν)
(5)
E is Young’s modulus, ν is Poisson’s ratio. Using the general relationships σij =
∂ψ , ∂εij
η=−
∂ψ ∂T
and expressions (2), (4), we obtain: λ u = µεij εij + θ 2 + 3αK0 T0 θ + cε (T − T0 ) 2
(6)
Thermomechanics of brittle fracture η = 3α0 K0 θ + cε ln
T ≈ 3αK0 T0 θ + cε (T − T0 ) /T0 T0
83 (7)
(at T − T0 )/T0 1) and Hooke’s law on account of temperature σij = 2µεij + λθδij − 3αK0 (T − T0 ) δij ,
(8)
where σij are components of stress tensor. Let us consider a linear thermoelastic solid, which is in the state of a static equilibrium before and after formation of a defect, under the influence of prescribed external forces (displacements) on the surface and stationary temperature field. Taking into consideration expressions (2)–(8), let us introduce appropriate denotations: u(0) , (0) , V0 is a volume, S0 is surface area for the solid without a defect, η(0) , ψ (0) , εij(0) , σij(0), u(0) i ,T (1) (1) (1) (1) (1) (1) u , η , ψ , εij , σij , ui , T (1) , V1 is a volume, S1 = S0 + is the surface area for the solid with a defect, is the surface area of the defect. Using the integral form of the first and second laws of thermodynamics (Il’yshin, 1990), let us write the energy conditions of a solid transition from the state of equilibrium without a defect into the state of equilibrium with a defect dU + dU ∗ = dA,
(9)
dS + dS ∗ = 0
(10)
U =U
(0)
−U
(1)
,U
(q)
=
U∗ =
u dV , S = S
− S ,S = η(q) dV , V q u∗ ds = γ , S ∗ = η∗ ds, q = 0, 1, (q)
Vq
(0)
(1)
(q)
(11)
where A = A(0) − A(1) is the work of external forces, u∗ is a specific internal energy, η∗ is the specific entropy of the formation of a unit area of the defect surface . Investigating static equilibrium of the defect in expressions (9), (10) instead of differentiation with respect to time, we differentiate with respect to representative dimension of the defect or crack length. With admissible limitations for the properties of the defect surface (Fihtengoltz, 1969), we may reduce first a surface integral in relationships (11) to an ordinary double integral and obtain an expression ∗ u∗ (x1 , x2 , x3 )ds = γ = Nγ0 U =
on the basis of a generalized mean-value theorem. Here m γ M, m = inf {u∗ }, M = sup {u∗ }, u∗ > 0, γ0 is a value of specific energy of bond rupture on a surface unit, N is an effective number of ruptured bonds necessary for a defect formation. Number N may considerably exceed the number of ruptured bonds for a unit of defect surface at the close packing, on account of the bonds which have ‘prepared’ the formation of the defect. Satisfying the Second law (10) of thermodynamics at S ∗ = −S and taking into consideration expression (9), we obtain the necessary energy condition of a static equilibrium at defect formation dU + γ d = dA,
(12)
where U and U ∗ are defined by relations (6), (11). Condition (12) must be complemented by geometrical, physical and mathematical models of the defect and also by the condition which
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determines the relative position of the defect. Assuming in (12) that the entropic component (11) of the internal energy (6) is zero at T (0) = T (1) = T0 , we obtain the Griffith criterion (1). In this case the constant γp is equal to the specific work of internal stresses spent for bonds rupture, and formation of the defect in a solid, in a general case, takes place without changing its entropy. 3. Energy condition of fracture for two models of an isolated defect Let us consider in detail (Dunaev and Dunaev, 2000) the energy condition (12) at plane stressed (strained) state in isothermal conditions T (0) = T (1) = T0 for the two known models (Goodier, 1968) of an isolated defect. In model (A) the same stresses are prescribed on the external surface of the solid S0 before and after formation of the defect, and stresses on the surface of the defect are equal to zero. External forces in this model produce work on the external surface S0 on displacements, caused by formation of the defect. In model (B), on the external surface of the solid S0 before and after formation of the defect, displacements which correspond to the applied load are fixed, but before the defect was formed. Stresses are also zero on the surface of the defect in model (B). Work of external forces on the external surface of the solid S0 during formation of the defect is dA = 0 as displacements are fixed. Integrals of the internal energy (11) taking into consideration (6)–(8) at T (0) = T (1) = T0 may be written (0) (0) (1) 1 (0) (0) (1) (1) (0) (1) (1) σ ε dV − σij εij dV + U = U −U = Up −Up +T0 S −S = 2 V0 ij ij V1 (13) (0) (1) εij δij dV − εij δij dV i, j = 1, 2, +α0 T0 k1 V0
V1
where Up = Up(0) − Up(1) and S = S (0) − S (1) are an increment of a potential energy and an increment of the entropic component of the internal energy, respectively; k1 = E/(1 − ν) is for the plane stressed state, k1 = E/(1 − 2ν) is for the plane deformation. Let us compute the increment of the total energy W = U − A+ γ
(14)
For model (A), using relations (13), (8), Betti’s reciprocal theorem (Nowacki, 1970) σij(0)εij(1) = σij(1)εij(0)
(15)
and Airy stress function F (q)
σ11 =
∂ 2 F (q) , ∂x22
(q)
σ22 =
∂ 2 F (q) , q = 0, 1 ∂x12
and considering the work of external forces
(1) σij(0) u(0) − u nj ds, A= i i S0
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85
we obtain
1 1 (0) (1) (0) (1) σ (0)ε (0) dV − σ + σij εij − εij dV + W= 2 V1 ij 2 V ij ij 2 (0)
∂ 2 F (0) ∂ F (1) σij(0) u(0) − u ds + α T χ + n dV − − j 0 0 i i ∂x12 ∂x22 S0 V0 2 (1) ∂ 2 F (1) ∂ F + dV + γ − ∂x12 ∂x22 V1
(16)
Here V = V0 − V1 , χ = 1 and χ = 1 + ν for plane stressed state and plane deformation, respectively, nj are direction cosines of the external normal n to the solid boundary. Next, let use (Nowacki, 1970) on boundaries S0 and S0 + expressions s s ∂F (q) ∂F (q) (q) (q) (q) (q) =− σnx2 (s)ds + Aq , Fx2 (s) = = σnx (s)ds + Bq , (17) Fx1 (s) = 1 ∂x1 ∂x2 0 0 where (q)
(q)
(q)
(q) = σ11 n1 + σ12 n2 , σnx 1
(q)
(q) σnx = σ12 n1 + σ22 n2 , 2
q = 0, 1
(18)
are boundary conditions in stresses. If boundary S0 limits the simply connected domain, then constants A0 and B0 may be assumed to zero. Using Green’s formula, equations of equilibrium ∂σij(0) ∂xi
= 0,
∂σij(1) ∂xi
= 0,
i, j = 1, 2
(19)
and relations (17)–(18), the expression (16) may be written
1 1 (1) − u ds + σ (0)u(0) nj ds− σij(0) + σij(1) u(0) n W= j i i 2 S0 + 2 ij i s
(1) (1) (0) σij(0) u(0) − u ds + α T χ − σ σ ds dx1 + n − j 0 0 nx1 nx1 i i S0
s
+ 0
(1) (σnx 2
−
(0) σnx )ds 2
S0
s
dx2 +
0
(1) σnx ds 1
0
s
dx1 + 0
(1) σnx ds 2
(20)
dx2 + γ
Here also Aq dx1 + Bq dx2 = 0
So far, because boundary conditions in a solid with and without the defect in model (A) are (0) (1) (0) (1) = σnx , σnx = σnx , (x1 , x2 ) ∈ S0 ; σnx 1 1 2 2
(1) (1) σnx = σnx = 0, (x1 , x2 ) ∈ 1 2
(21)
then the last two integrals in the expression (20) are equal to zero. Therefore, the entropic component of the internal energy (13) also is equal to zero. Then, on account of boundary conditions (21), we finally obtain 1 σ (0)u(1) nj ds + γ . (22) W = 2 ij i
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Substituting expression (22) into the energy condition (12), we obtain the Griffith criterion (1) in the following form 1 (0) (1) σij ui nj ds + γ d = 0 (23) d 2 Let us compute the increment of the total energy (14) for model (B). In this case the work of external forces is dA = 0. Using relations (13), (15), Green’s formula and equations (19) for the expression (14), we obtain
1 1 (0) (1) (0) (1) W = U + γ = σ (0)u(0) nj ds+ σ + σij ui − ui nj ds + 2 S0 + ij 2 ij i (24)
(0) (1) (0) ui δij nj ds + γ ui − ui δij nj ds + +α0 T0 k1 S0 +
For the model (B) on the boundary S0 of a solid with and without a defect (1) u(0) i = ui
(x1 , x2 ) ∈ S0 ,
(25)
and on the boundary of the defect (1) (1) = σnx = 0, σnx 1 2
(x1 , x2 ) ∈
Then, on account of boundary conditions (25), (26), (18) reduce expression (24) to 1 σij(0)u(1) n ds + α T k u(1) W = U + γ = j 0 0 1 i i δij nj ds + γ 2 Substituting expression (27) into the energy condition (12) obtains 1 (0) (1) (1) σ u nj ds + α0 T0 k1 ui δij nj ds + γ d = 0 d 2 ij i
(26)
(27)
(28)
Therefore, the model of the defect (B) leads to the condition (28) in which the increment of the entropy component of the internal energy in a general case is not zero. This is the main essential difference of the proposed condition (28) from the criterion (23). 4. Test problem Let us consider a problem of determination of critical stresses and critical dimensions of a round shaped defect with radius a at tension (compression) of a round plate with radius b under symmetrical load P in the case of a plane stressed state. The solution of the problem of elasticity theory is (0) (0) (0) (0) (0) (0) (0) u(0) 1 = P (1 − ν)r/E, u2 = 0, σ11 = σr = σ22 = σϑ = P , σ12 = σrϑ = 0,
C2 (1) , u(1) u(1) 2 = 0, 12 = σrϑ = 0, r E E E E (1) C1 − C1 + = C2 , σ22 = σϑ(1) = C2 2 1−ν (1 + ν)r 1−ν (1 + ν)r 2
u(1) 1 = C1 r + (1) = σr(1) σ11
(29)
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87
where r, ϑ are polar coordinates of the solid points, C1 , C2 are constants of integration, which we determine from the boundary conditions in models (A) and (B). For model (A), using (1) (1) and r = b, σ11 = P , solution (29), we obtain boundary conditions at r = a, σ11 C1 (A) =
1 P (1 − ν) · , E (1 − a 2 /b2 )
C2 (A) =
a2 P (1 + ν) · E (1 − a 2 /b2 )
(30)
Substituting solution (29) and (30) at n1 = −1, n2 = 0, = 2π a into the expression (22) after integration, we calculate W = −2π E −1 (1 − a 2 /b2 )−1 P 2 a 2 + 2π γ a Then, from the Griffith condition (23), we obtain equal (in absolute value) critical stresses at tension P + and compression P −
P ± = ± γ E(1 − a 2 /b2 )2 /2a (31) which neither quantitatively nor qualitatively correspond to experimental data. For model (B), (1) (0) = 0 and at r = b, u(1) using boundary conditions at r = a, σ11 1 = u1 and also solution (29), we obtain −1 −1 1 + ν a2 1 + ν a2 P (1 − ν) P (1 + ν)a 2 · 2 , C2 (B) = 1 + · 2 (32) C1 (B) = 1 + 1−ν b E 1−ν b E Substituting solution (29), (32) into (27), we compute −1 2 1 + ν a2 2 · E −1 + 2π aγ W = −2π a P + 2a0 T0 k1 P 1 + 1 − ν b2 Differentiating this expression with respect to a from (28), we obtain the equation 2 1 + ν a2 γE 2 1+ · =0 P + 2α0 T0 k1 P − 2a 1 − ν b2 from which we find, unlike the condition (31), the critical stresses 1 + ν a2 2 γE ± 2 1+ · P = −α0 T0 k1 ± (α0 T0 k1 ) + 2a 1 − ν b2
(33)
(1) At b → ∞C1 (A) = C1 (B), C2 (A) = C2 (B), u(1) 1 (A) = u1 (B),
2π a 2 P 2 4π a 2 α0 T0 k1 P − + 2π aγ (34) E E which also follows from the condition of uniqueness of the solution of the problem of the elasticity theory for the plane with a round hole at prescribed conditions on infinity (Muskhelishvili, 1958). In this case formula (33) takes on a simple form
(35) P ± = −α0 T0 k1 ± (α0 T0 k1 )2 + γ E/2a W =−
This allows one to use the solution of the problem of the elasticity theory for an infinite plate to calculate directly the energy integrals. Thus, the entropic component of the internal energy for the defect model (B) is not zero, and thereby formulas (33), (35) give qualitatively correct
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values of critical stresses under symmetrical tension P + , compression P − depending upon the physical and mechanical constants of the material (5), the linear coefficient of thermal expansion α0 , the temperature T0 , and the dimension of the defect a. 5. Complex presentation of the integral of increment of internal energy Components of stresses and displacements in plane problems of the elasticity theory are determined by two complex functions ϕ(z) and ψ(z) of the complex variable z = x1 + ix2 and their derivatives (Muskhelishvili, 1958) σ11 + σ22 = 2 ϕ (z) + ϕ (z) (36) σ22 − σ11 + 2iσ12 = 2 zϕ (z) + ψ (z) 2µ (u1 + iu2 ) = æϕ(z) − zϕ (z) − ψ(z) where æ = 3 − 4ν, æ = (3 − ν)/(1 + ν) for plain deformation and plane stressed state, respectively, where a prime denotes derivative of the functions with respect to z. Let a superimposed bar indicate the complex conjugate of a complex quantity. Considering relations n1 ds = −dx2 , n2 ds = dx1 , as the external normal to the boundary is directed into the defect or crack, we write the integral of increment of the internal energy (27) in the form
1 (0) (1) (0) (1) (0) (1) (0) (1) (1) u1 +σ22 u2 dx1 − σ11 u1 +σ21 u2 dx2 +α0 T0 k1 u(1) σ12 U= 2 dx1 − u1 dx2 (37) 2 Using the presentation of the integral of the function of a complex variable f = u + iν via curvilinear integrals of the real functions (Privalov, 1984) f (z)dz = udx1 − νdx2 + i νdx1 + udx2
we introduce two new functions into the integral (37)
(0) (1) (0) (1) (0) (1) (0) (1) u1 + σ22 u2 + σ11 u1 + σ21 u2 g1 (z) = σ12
(1) g2 (z) = u(1) 2 + iu1
After elementary transformations we obtain 1 U = Re g1 (z)dz+α0 T0 k1 Re g2 (z)dz = 2
1 (0) (1) (1) (0) (1) (0) (1) (1) +iu σ12 u1 +iu2 +σ22 u2 +iσ11 u1 dz+α0 T0 k1 Re u(1) dz = Re 2 1 2 Substituting into the last expression the equations
(1) (1) (1) (1) (1) (1) (1) u(1) u + iu + + iu + iu − + iu u u 1 2 1 2 1 2 1 2 ; u(1) u(1) 1 = 2 = 2 2i we find finally (Dunaev and Dunaev, 2000)
1 1 (1) (1) (0) (0) (0) (1) (1) (0) (0) σ22 −σ11 +2iσ12 − u1 +iu2 σ11 +σ22 dz + u1 +iu2 U = Re 4 i
(1) (1) (38) u1 + iu2 dz +α0 T0 k1 Re i
Thermomechanics of brittle fracture Considering the expressions (36), the integral (38) reads 1 1 Re æϕ1 (z) − zϕ1 (z) − ψ1 (z) zϕ0 (z) + ψ0 (z) − U= 4µ i − æϕ1 (z) − zϕ1 (z) − ψ1 (z) ϕ0 (z) + ϕ0 (z) dz +
89
(39)
α0 T0 k1 æϕ1 (z) − zϕ1 (z) − ψ1 (z) dz Re i + 2µ 6. Macroscopic criterion of brittle facture Consider a problem of fracture of a plate (Dunaev and Dunaev, 2000) during the defect formation having as cross-section an ellipse form with semi axes a, b, b = b(a), a b under the influence of principle tensile and (or) compressive stresses P1 , P2 . The stress P1 direction makes an angle α = α(a) with the axis x1 O, and the stress P2 direction makes an angle α1 = α + π/2 with the axis x1 O. In the case of a plane (Muskhelishvili, 1958) under uniform stressed-strained state P1 + P2 P1 − P2 −2iα z, ψ0 (z) = − ze (40) ϕ0 (z) = 4 4 and a boundary condition on the contour of the ellipse under z → t ∈ reads as ϕ1 (z) + zϕ1 (z) + ψ1 (z) = 0
(41)
or a respective expression in a complex conjugated form ϕ1 (z) + zϕ1 (z) + ψ1 (z) = 0
(42)
Substituting expressions (40), (41), (42) into the integral (39) after elementary transformations, we obtain α T k (æ+1) æ+1 0 0 1 Re i ϕ1 (z) (P1 +P2 )+ϕ1 (z) (P1 −P2 ) e−2iα dz + Re i ϕ1 (z)dz U= 8µ 2µ (43) Let us introduce the function a+b a−b m ,c = ,m = z = ω(ξ ) = c ξ + ξ 2 a+b
(44)
which maps the viewed plane with an elliptic hole to an infinite plane |ξ | > 1 with a round hole. Changing over in the integral (43) to the variable ξ according to the formula (44), we obtain æ+1 −2iα Re i ϕ1 (ξ ) (P1 + P2 ) + ϕ1 (ξ ) (P1 − P2 ) e ω (ξ )dξ + U= 8µ |ξ |=1 (45) α0 T0 k1 (æ + 1) Re i ϕ1 (ξ )ω (ξ )dξ + 2µ |ξ |=1
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Using Muskhelisvili’s solution (Muskhelishvili, 1958), we find the function ϕ1 (ξ ), which, in case of an infinite plate, for boundary problems corresponding to models (A) and (B), is the same and equals 2e2iα − m P2 c 2e2iα + m P1 c ξ+ + ξ− = ϕ1 (ξ ) = 4 ξ 4 ξ (46) m (P1 − P2 ) c e2iα (P1 + P2 ) c ξ− + · = 4 ξ 2 ξ Taking into consideration the fact that dA = 0 in model (B), also π/2 1/2 dβ, q 2 = 1 − b2 /a 2 (47) 1 − q 2 sin2 β γ = 4γ aE(q), E(q) = 0
and integrating (45) along the circle |ξ | = 1 at subscribed function ϕ1 (46) and (47), we obtain for the increment of total energy (14) the expression (Dunaev and Dunaev, 2000) W = W [a, b(a), α(a)] = U + γ = 2 (æ + 1)π c2 (P1 + P2 )2 (P1 − P2 )2 2 2 1 + m − P1 − P2 m cos 2α + − =− 4µ 4 2 α0 T0 k1 (æ + 1)π c2 (P1 + P2 ) 1 + m2 − (P1 + P2 ) m cos 2α + 4γ aE(q) − 2µ 2 Using formulas (44) let us write the expression (48) in the equivalent form (æ + 1)π 2 2 P1 a + ab + b2 − a 2 − b2 cos 2α − W =− 16µ 2 2 +P2 a + ab + b2 + a 2 − b2 cos 2α − 2P1 P2 ab − α0 T0 k1 (æ + 1)π 2 P1 a + b2 − a 2 − b2 cos 2α + − 8µ 2 +P2 a + b2 + a 2 − b2 cos 2α + 4γ aE(q)
(48)
(49)
It is obvious that at, a = b, P1 = P2 , æ = (3 − ν)/(1 + ν) the expression for energy (49) has the form (34). Substituting the energy increment (49) into the necessary condition (28), we obtain an expression for a macroscopic criterion of fracture (æ + 1)π a 1 db b db b db b ∂W 2 =− P1 + + − 1− cos 2α+ 1+ ∂a 16µ 2a 2 da a da a da 1 db b db b db dα b b2 2 + P2 + + + 1− cos 2α− 1+ +a 1 − 2 sin 2α a da 2a 2 da a da a da α0 T0 k1 (æ + 1)π a dα b db b2 (50) − P1 P2 + − × −a 1 − 2 sin 2α a da a da 8µ b db b2 b db dα − 1− cos 2α + a 1 − 2 sin 2α + × P1 1 + a da a da a da b db b2 d b db dα + 1− cos 2α − a 1 − 2 sin 2α + 2γ [aE(q)] = 0 +P2 1 + a da a da a da da
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91
In the considered case for isotropic materials the criterion of fracture (50) represents a curve of second degree in the space of variables P1 , P2 , which must be symmetrical relative to the axis P1 , P2 . This is possible under the following condition b db dα b2 = 1− cos 2α (51) a 1 − 2 sin 2α a da a da On account of Equation (51), after elementary transformations the criterion (50) may be written as b 1 db b db 2 (æ+1)π a 1+ + + P1 +P22 − 2ν∗ P1 P2 +2α0 T0 k1 (1−ν∗ ) (P1 +P2 ) + − 16µ 2a 2 da a da +2γ Here
d [aE(q)] = 0 da
1 db b db b + + 2ν∗ = 1 + 2a 2 da a da
(52)
−1
b db + a da
(53)
For further identification of the criterion (52) and condition (51) it is necessary to compute or assume the dependence b = b(a). In paper √ (Sih and Liebowitz, 1968), G.C. Sih and H.Liebovitz have proposed the dependence b = b0 a/c0 that proceeded from the assumption that the defect curvature radius ρ = b2 /a at the apex of the ellipse main axis remains constant ρ = b02 /c0 at the change of axis length. Then, according to formula (53) we obtain 2 2 b/a = 3(1 − ν∗ ) − 9(1 − ν∗ ) − 32ν∗ · (4ν∗ )−1 Therefore, at ν∗ ∈ [0, 1/3] there exists b/a ∈[0,1]. Introducing a more general dependence b = b0 (a/c0 )ε , ε 0,
a/c0 < 1
(54)
into the formula (53) we get b/a = (ε + 1)(1 − ν∗ ) − (ε + 1)2 (1 − ν∗ )2 − 16εν∗2 · (4εν∗ )−1
(55)
Using natural bounds (ε + 1)2 (1 − ν∗ )2 − 16εν∗2 0,
b/a ∈ [0, 1]
(56)
and expressions (54), (55) we find that ν∗ ∈ [0, 1]. Hence, for any ν∗ ∈ [0, 1] value there exists a finite ε0 , that at ε ε0 conditions (56) will be met. In the limit, as ε → ∞, on account of expression (55) we have 2ν∗ b b db = 0, b → 0, lim = lim ε = , ε→∞ a ε→∞ da ε→∞ a 1 − ν∗ 1 1 b 1 db b db + + = lim 1 + ε→∞ 2 a 2 da a da 1 − ν∗ lim
lim ε
ε→∞
b2 = 0, a2
(57)
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I.M. Dunaev and V.I. Dunaev
Substituting expressions (57) into the criterion (52) we find P12 + P22 − 2ν∗ P1 P2 + 2α0 T0 k1 (1 − ν∗ ) (P1 + P2 ) =
32µ(1 − ν∗ ) γ , (æ + 1)π a∗
(58)
where a∗ is a crack critical dimension. Insofar as, according to A.A. Il’yushin and D.C. Drucker postulate, the curve of the fracture must be convex (Ogibalov et al., 1975), we obtain ν∗ ∈ (−1, 1) from the expression (58). With these ν∗ values curve (58) is an ellipse in the space of principle stresses P1 , P2 . Criterion (58) satisfies the general requirements formulated for phenomenological strength criteria (Ogibalov et al., 1975) and allows computing of the critical dimension of crack a∗ depending upon physical and mechanical parameters of the material E, ν, α0 , γ and temperature T0 in a general case of a stressed-strained state. Thus, in the case of a uniaxial tension (compression) at P1 = P , P2 = 0 from criterion (58) for critical stresses at extension P + and compression P − , we calculate ± (59) P = −α0 T0 E(1 − ν∗ )k1 ± [α0 T0 E(1 − ν∗ )k1 ]2 + 32µ(1 − ν∗ )γ [π(æ + 1)a∗ ]−1 and the relations P + + P − = −2α0 T0 E(1 − ν∗ )k1 , P + · P − = −32µ(1 − ν∗ )γ [π(æ + 1)a∗ ]−1
(60)
The first of relation (60) allows us to compute ν∗ if P + and P − are defined experimentally and the second relation gives value γ /a∗ . Taking into consideration expressions (57), (51) for the case cos 2α = 0, we obtain the equation tg2αdα = da/a from the solution of which we find | cos 2α| = c02 /a 2 ,
c02 /a 2 ∈ (0, 1]
(61)
In order to determine the arbitrary constant in equation (61) we use the condition α = π/2 at a = a∗ , P1 = P = P + , P2 = 0, as crack under uniaxial tension (59) propagates perpendicularly to the tensile stress. Then, c0 = a∗ and solution (61) takes the form | cos 2α| = a∗2 /a 2
(62)
One more the value a = 0 follows from equation (62) at a = a∗ . In this case the crack is formed along the stress P1 action. Value α = 0 corresponds to the case P1 = P − , P2 = 0 from equation (59), physical sense and experimental data. Therefore, at uniaxial compression a crack of a critical length a∗ lays along the compressive stress action. As it follows from equation (62), at a = a∗ and at any relations of critical stresses, corresponding to ‘points’ lying on the curve of fracture, the crack will be oriented perpendicular to tensile or along compressive stresses. The crack orientation in every specific case may be chosen on account of the impossibility of joining of the crack’s faces, which follows from boundary condition (26) on the contour of the crack. Then, in the first quadrant of the plane OP1 P2 for P1+ 0, P2+ 0, the crack is oriented perpendicular to the highest of these tensile stresses. In the second quadrant, P2+ 0, P1− 0 the crack is oriented perpendicular to the stress P2+ direction. In the third quadrant, for P1− 0, P2− 0 the crack is oriented along the highest of the compressive stresses. In the fourth quadrant, P1+ 0, P2− 0 the crack is oriented perpendicular to the stress P1+ direction. Crack orientation in the first and third quadrants at P1 = P2 is not determined unequivocally.
Thermomechanics of brittle fracture
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7. Conclusion The work proposes an energy condition of brittle fracture of solids on the basis of the thermodynamics of a thermoelastic deformation. A curve of fracture in the form of an ellipse in the space of principle stresses has been obtained with additional limitations imposed upon the physical and mathematical models of the defect, and the conditions of isotropy and convexity. The orientation of the crack was determined. Parameters of the curve of fracture are implicitly dependent upon physical and mechanical characteristics of the material, the linear coefficient of thermal expansion, the temperature, and the dimensions of a crack. An efficient way of computing the integral of the internal energy of a linear thermoelastic body spent for defect formation has been proposed. References Dunaev, I.M. and Dunaev, V.I. (2000). On the Energy Condition for Fracture of Solids. J. Doklady Physics 45(5), 213–215. Fihtengoltz, G.M. (1969). Differential and Integral Calculations [in Russian] (vol. 3, Nauka, Moscow). Griffith, A.A. (1925). The theory of rupture, Proceedings of the 1st Inter. Congress for Applied Mechanics, Delft: J. Waltman, 55–63. Goodier, J. (1968). Mathematical theory of equilibrium cracks Fracture, Vol. 2 (Edited by H. Liebowitz), Math. Fundamentals, Acad. Press, New York and London. Il’yshin, A.A. (1990). Continuum Mechanics [in Russian] (Mosk. Gos. Univ., Moscow). Muskhelishvili, N.I. (1958). Some Basic Problems of Mathematical Theory of Elasticity (Van Nostrand, Princeton, N.J.). Nowacki, W. (1970). Teoria Sprezystosci [in Russian translation] (PWN, Warszawa). Ogibalov, P.M., Lomakin, V.A. and Kishkin, B.P. (1975). Polymer mechanics [in Russian] (Mosk. Gos. Univ., Moscow). Paul, B. (1968). Macroscopic criterion of plastic flow and brittle fracture Fracture, Vol. 2, (Edited by H. Liebowitz), Math. Fundamentals, Acad. Press, New York and London. Privalov, I.I. (1984). Introduction to the theory of complex variable functions [in Russian] (Nauka, Moscow). Sih, G.C. and Liebowitz, H. (1967). On the Griffith energy criterion for brittle fracture, Int. J. Solids Structures 3, 1–22. Sih, G.C. and Liebowitz, H. (1968). Mathematical theory of brittle fracture Fracture, Vol. 2 (Edited by H. Liebowitz), Math. Fundamentals, Acad. Press, New York and London)
