Topological sensitivity derivative with respect to area, shape and orientation of an elliptic hole in a plate

Consider the functional in the form where F is a function of strain vector \(\upvarepsilon =\left[ {\upvarepsilon _{11} ,\upvarepsilon_{22} ,2\upvarepsilon_{12} } \right]^{{\rm \thinspace }T}\), f and g are functions of displacements u and A _ξ denotes the plate domain with the elliptical hole, whose size is defined by the parameter ξ. The integrand functions are assumed to be differentiable with respect to their arguments.

It can be demonstrated from the asymptotic analysis (cf. Gao 1996), that where u∣  _ξ = 0 and \(\boldsymbol\upvarepsilon \vert _{\xi =0}\) denote the displacement and strain states for the plate with elliptic hole for ξ → 0, and u ⁽⁰⁾, \(\boldsymbol\upvarepsilon ^{(0)}\) are the states for a homogeneous plate. In view of (3) there is where denotes a value of the functional for the homogeneous plate.

Thus, the continuity of functional (2) occurs for ξ → 0 and the derivative (1) with respect to hole area for ξ = 0 can be calculated assuming nucleation and growth of hole of fixed geometric and orientation parameters a, b, η and α.

The variational approach is applied to specify the sensitivity derivative, and the first variation of functional (2) in the hole expansion process at the material point x is expressed as follows where \({\rm {\bf n}}=\left[ {n_1 ,{\rm \thinspace \thinspace }n_2 } \right]^T\) is the unit vector normal to the hole boundary Γ _ξ , \(\delta \boldsymbol\Psi =\left[ {\delta \boldsymbol\Psi_1 ,{\rm \thinspace \thinspace }\delta \boldsymbol\Psi_2 } \right]^T\) is the hole boundary incremental transformation vector and (·) denotes the scalar product. Following the previous derivation for plates cf. Bojczuk and Mróz (2009) and the general methodology of sensitivity analysis cf. Dems and Mróz (1984), the variations of state fields can be eliminated by introducing an adjoint plate structure of the same form, as the primary plate, but with induced initial stress and body force fields, namely and satisfying the following boundary conditions Here \(\sigma^{a}\), u ^a , \(\boldsymbol\upvarepsilon ^{a}\) are the state vector fields in the adjoint structure. The stress field \(\sigma^a=\left[ {\sigma_{11}^a ,\sigma_{22}^a ,\sigma_{12}^a } \right]^{{\rm \thinspace }T}\) is a following sum where \(\boldsymbol\sigma^{ar}=\boldsymbol\sigma^a-\boldsymbol\sigma^{ai}=E\left( {\boldsymbol\upvarepsilon ^a-\boldsymbol\upvarepsilon^{ai}} \right)\) is the elastic stress field and then \(T_i^{a0} =\sigma_{ij}^{ar} n_j^\Gamma ,{\rm \thinspace \thinspace }i,j=1,2.\) Now, the equilibrium equations can be written in the form \(\sigma _{ij,j}^{ar} +p_i^{a0} =0,{\rm \thinspace \thinspace }i,j=1,2\). In view of (7) and (8) the functional variation (6) can be rewritten as follows The virtual work equation for the adjoint structure can be written in the form since \(\sigma ^{ar}\) satisfies the equilibrium equations and boundary conditions of the adjoint structure and u(x), \(\boldsymbol\upvarepsilon \)(x) are the kinematically admissible displacement and strain fields.

The virtual work equation for the stress increment \(\delta \sigma \) of the primary structure and the kinematically admissible displacement field of the adjoint structure u ^a can be written as follows Accounting for (9), (11) and (12), the following equality can be derived In view of (13), the expression for the functional variation can be derived in terms of primary, adjoint fields and normal transformation of the hole boundary Γ _ξ , namely

Introduce the local coordinate system x ₁₀, x ₂₀ with axes coinciding with elliptic hole semi-axes ξa and ξb, Fig. 2. The coordinates of boundary points and their increments are The unit normal vector to the boundary Γ _ξ can be expressed as follows and the length element of the hole perimeter is The transformation function specifying the radial boundary evolution now takes the form and its variation is So, in view of (16), (17) and (19), the variation of the functional G expressed by (14), can be presented in the form The sensitivity derivative of G with respect to size parameter is and it vanishes for ξ = 0. To express sensitivity derivative with respect to the hole area A = πξ ² ab, the following formula is obtained using the relation (21) and the incremental form \(dA=2\pi \xi ab\;d\xi \), thus (cf. Novotny et al. 2003) To express integrand of (22) in terms of the local stress components, introduce the coordinate system s, n by the unit tangent and normal vectors to the hole boundary, Fig. 2. Then on the boundary Γ _ξ , we have and the non-vanishing terms are \(\sigma _{ss}\) and \(\sigma_{ss}^a \). We then get where \(\boldsymbol\upvarepsilon_{ss}^a =\left( {1 \mathord{\left/ {\vphantom {1 E}} \right. \kern-\nulldelimiterspace} E} \right)\sigma_{ss}^a \) and E denotes Young modulus. Further, assume the function F(\(\boldsymbol\upvarepsilon \)) in the form where the matrix S is assumed as follows and p, l, m are the selected parameters. In particular, the parameters l and m can correspond to elastic moduli λ, μ of the linear isotropic material. Referring the local stress state \(\sigma_{ss} \) \(\sigma_{nn} = \sigma_{sn}=0\) to the reference system x ₁₀, x ₂₀ coinciding with the principal stress axes at x for the homogeneous plate, we have where v is the Poisson’s ratio and \(G=E \mathord{\left/ {\vphantom {E {\left[ {2\left( {1+\nu } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {2\left( {1+\nu } \right)} \right]}\) denotes the Kirchhoff modulus. From the transformation formulae for the stress state on the hole boundary, we get where φ is the angle between x ₁₀ and s axes (Fig. 2). Now, the function F can be expressed in terms of stress component \(\sigma_{ss} \) as follows In view of the equality formulae (24), (29) and assuming p = 1, the sensitivity derivative can be expressed as follows Consider now the elliptical hole in the primary plate with its major axis oriented at angle α to the principal stress \(\sigma_1 \) acting along the coordinate axis x ₁, Fig. 3a. The tangential stress distribution at the hole perimeter is specified from the elastic solution for an infinite plate with an elliptic hole, subjected to biaxial loading, cf. Gao (1996) or Timoshenko and Goodier (1951), thus where \(\sigma_1 \), \(\sigma_2 \) are the principal stresses at the point x of the domain A without the elliptical hole, the angle θ specifies the position on the perimeter with respect to the major ellipse axis, Fig. 2, and The alternative form of (32) in terms of the shape parameter η is It is important to notice that determination of \(\sigma_{ss} \) from the solution for an infinite plate under uniform stress state is an approximation and range of its validity was discussed by Silva et al. (2010). However, the expression (31) is still presented in general form and can be applied to cases of non-uniform stress states in structural elements.

Next, consider the adjoint structure, where the principal stress \(\sigma _1^a \) acting along the coordinate axis \(x_1^a \) is oriented at the angle β to the x ₁ axis and inclined at the angle α − β to the major ellipse axis, Fig. 3b. Now, the distribution of the adjoint stress \(\sigma_{ss}^a \) on the elliptic hole boundary is expressed similarly as in the primary structure, namely or The integrals occurring in (31) can be presented in the form and Accounting for (37), (38), the topological sensitivity derivative (31) can be expressed analytically as follows The considered here case p = 1 corresponds to the quadratic strain (or stress) integral function (25). The cases \(p \ne 1\) can be treated numerically.

The particular case occurs when f(u) = 0 on Γ _T , g(u) in A, p ⁰ = 0 at the point x and the integrand function of (2) is the strain energy function (\( F \boldsymbol\upvarepsilon = U \boldsymbol\upvarepsilon \)). This case corresponds to the strain or complementary energy variations. The self-adjoint sensitivity problem now is obtained and then The topological sensitivity derivative then equals The diagrams of topological derivative variation of the strain energy at arbitrary point x of the infinite plate in biaxial uniform stress state, in function of shape parameters η = b/a, 1/η = a/b, for different values of stress ratio \(\zeta = \sigma _{2}/\sigma _{1}\) and orientation angle α, are presented in Fig. 4. Let us note, that in the case α = 0 and for ζ ∈(0;1\(\rangle \), topological derivative attains minimum for the shape parameter η = ζ. In the case α = π/4 the minimum always occurs for η = 1.

The topological derivative can also be expressed in terms of components \(\sigma_{11}, \sigma_{22} , \sigma _{12}\) of the stress field in the coordinates x ₁₀, x ₂₀ located along ellipse axes. In this case, using the stress transformation formulae, we have

Now, let us consider the following functional of stress state and boundary tractions on Γ _u where H(\(\boldsymbol\sigma \)) is the function of stress state \(\sigma = [ \sigma_{11}, \sigma_{22}, \sigma_{12}]^{T} \) and h(T) denotes function of reaction tractions acting on the displacement controlled boundary Γ _u . From the asymptotic analysis, it is seen that where \(\sigma \)∣  _ξ = 0 and T∣  _ξ = 0 denote the stress state and reactions for the plate with elliptic hole for ξ → 0, and \(\sigma \) ⁽⁰⁾, T ⁽⁰⁾ are the states for a homogeneous plate. Similarly as for the functional (2), the relation (4) holds, where in this case The adjoint structure now can be defined as follows where \(\boldsymbol\upvarepsilon^{ai} \) are the initial strains. The field of initial strains induces the corresponding field of the total strains in the form \(\boldsymbol\upvarepsilon^a=\boldsymbol\upvarepsilon^{ai}+\boldsymbol\upvarepsilon^{ar}\), where \(\boldsymbol\upvarepsilon^{ar} \) is the field of the elastic strains. Now, in view of (46), the first variation of functional (43) with respect to hole expansion process at the material point x, can be expressed analogously to (6), namely So, in view of the virtual work equation and the complementary virtual work equation the first variation of the functional (43) takes the form Next, assume the function H(\(\sigma \)) in a homogeneous form of stress state of degree 2p, thus where S is expressed by (26). Further, using the approach analogous to the analysis described by (15)–(30) and assuming p = 1, we get Accounting for (37), (38), the topological sensitivity derivative (52) can be expressed analytically as follows

The particular case occurs when h(T) = 0 on Γ _u , p ⁰ = 0 at the point x and the integrand function of (43) is the specific stress energy function, \(H(\sigma )=W\)(\(\sigma \)). In this case the self-adjoint sensitivity problem is obtained and then The topological sensitivity derivative with respect to hole area now has the form which is identical to (41).

At first, consider the shape variation of the elliptical hole corresponding to length variation of the semi-axis a. The shape transformation function (18) and its variation for ξ = 1 now are The unit normal vector n to the hole perimeter is specified by (16) and the length element of boundary Γ _ξ equals The variation (14) of the functional G now takes the form Assuming the uniform stress states in the hole vicinity of primary and adjoint plates with the principal stresses \(\sigma_1 \) ≥ \(\sigma_2 \) and \(\sigma _1^a \ge \sigma_2^a \), and the integrand function F(\(\boldsymbol\upvarepsilon \)) of the form (25) to be quadratic, p = 1, the integrals of (58) can be analytically expressed after substituting stress states (34) and (36), namely and The variation of G specified by (58) can now be analytically expressed as follows The self-adjoint case occurs when \(F(\boldsymbol\upvarepsilon)= U(\boldsymbol\upvarepsilon)\)), where U(\(\boldsymbol\upvarepsilon \)) is the specific strain energy. Assuming that f(u) = 0, g(u) = 0 and the body forces p ⁰ = 0 in neighborhood of the elliptical hole, we have \(\sigma_{ss} =\sigma_{ss}^a \) and the functional variation or the sensitivity derivative is It is seen that the sensitivity derivative with respect to a does not vanish for finite lengths values of ellipse semi-axes. For the specified ellipse shape, η = const, it is a linear function of the size parameter a and b.

Next, consider the shape variation of the elliptical hole corresponding to length variation of the semi-axis b. In this case, the shape transformation function (18) and its variation for ξ = 1 now are and variation (14) of the functional G takes the form Using the same approach and the same assumptions as in the case of variation of a, finally the variation of G specified by (64) can now be analytically expressed as follows When the functional G represents the global strain energy and f(u) = 0, g(u) = 0, p ⁰ = 0 in neighborhood of the elliptical hole, the problem becomes self-adjoint and the sensitivity derivative is

Consider now the case of rotation of the elliptical hole with respect to the principal stress axes, specified by the varying angle α, Fig. 1. The reference system x ₁,x ₂ follows the principal stress axes and the system x ₁₀, x ₂₀ follows the ellipse axis. Assuming ξ = 1, the hole boundary point coordinates in the local system x ₁₀, x ₂₀ are and in the global system x ₁, x ₂ these coordinates are The hole boundary transformation vector due to rotation is now specified as follows and the unit normal vector and length element are In view of (69), (70), the variation (10) of the functional is now expressed as follows and for the quadratic function F(\(\boldsymbol\upvarepsilon \)) of the form (25), p = 1, we obtain assuming the uniformly loaded plate where A = πab. For the particular case, when F(\(\boldsymbol\upvarepsilon \)) = U(\(\boldsymbol\upvarepsilon \)), and f(u) = 0, g(u) = 0, p ⁰ = 0 in neighborhood of the elliptical hole, the self-adjoint problem occurs. Then, we have

Consider the shape variation of the elliptical hole by assuming constant area A = πab = const. For varying lengths of semi-axes a and b the area constraint provides the relation Then, the analyzed shape sensitivity derivative takes the form and, in view of (61) and (65), it can be expressed as follows For the particular case, when the functional G represents the global strain energy and f(u) = 0, g(u) = 0, p ⁰ = 0 in neighborhood of the elliptical hole, the problem becomes self-adjoint and the shape sensitivity is

Consider the problem of determination of the topological derivative with respect to introduction of the infinitesimally small elliptic hole. We have the size growth relations where a ^(mod), b ^(mod) are the semi-axes lengths controlled by the expansion parameter ξ. The topological derivative can be calculated as the derivative with respect to the expansion parameter ξ for ξ = 0. Thus, in view of (22), we have Taking into account, that \(\partial G\)/\(\partial a^{(\rm mod)}\), \(\partial G\)/\(\partial b^{(\rm mod)}\) are specified by (61) and (65), and the topological derivative is finally expressed by the same formula, namely by (39), as previously.

Consider a general optimization problem defined as follows where G(p _i ) is the objective functional (2), C(p _i ) denotes the global cost and C ₀ is the upper bound on the global cost. Introducing the Lagrangian functional the stationary conditions are where λ is the Lagrange multiplier and p _i , i = 1,2,...n are the design parameters. The sensitivity derivatives \(\partial G\)/\(\partial p_{i}\) are specified by the formulae derived in the paper, where the size parameter ξ, shape parameters a and b, or orientation parameter α are to be determined. The cost derivatives \(\partial C\)/\(\partial p_{i}\) can easily be derived for the assumed cost function expressed in terms of design parameters. The optimal values of the parameters p _i and the multiplier λ are determined in the incremental process of gradient optimization.

Consider a simple problem when the plate is subjected to the uniform remote principal stresses \(\sigma _{1}\), \(\sigma _{2}\) \((\vert \sigma _{1}\vert \ge \vert \sigma _{2}\vert )\), following the x ₁,x ₂ coordinate axes, Fig. 5a. The optimal design problem is aimed to specify lengths of semi-axes a, b, and orientation angle α of the hole assuming its area A ₀ = πab to be fixed. The optimization problem is to maximize global plate stiffness expressed by the potential energy Π, or to minimize global compliance measured by the strain energy, thus Using the sensitivity derivatives (77) and (73), the stationary conditions can easily be stated in the form For η \(\ne \) 1 and \(\sigma\) \(_{1}\ne\) \(\sigma_2 \) the optimal orientation of ellipse axes is α = 0 or α = π/2. The maximal stiffness design is obtained by setting and the maximal compliance design corresponds to the parameters The optimal designs are shown in Fig. 5b, c. It is seen that the optimal orientation of the ellipse corresponds to coaxiality of ellipse and principal stress axes.

In order to find location of crack initiation on the boundary of the elliptic hole, the maximal stress criterion can be used. The distribution of the stresses on the rim of the hole is expressed by (34). Position of the maximal stress \(\sigma _{ss}\) determined by the angle θ (Fig. 2) in function of stress ratio \(\sigma _{2}/\sigma _{1}\) for selected values of the shape parameter η and for the orientation angles α = π/6, α = π/4, α = π/3, α = − π/4 is presented respectively in Fig. 6a, b, c, d.

Let us note that in order to specify position of a point on the ellipse boundary, instead of the angle θ also angles β and γ can be used (Fig. 2), where the relations between these angles are as follows and γ specifies the orientation of crack growth in normal direction to the hole perimeter.

Consider now the case of a plane crack of length 2a in a uniformly loaded infinite plate, inclined at the angle α to the major tensile stress \(\sigma_1 \) of the primary state (Fig. 7) and at the angle α–β to the major stress \(\sigma _1^a \) of the adjoint state.

The crack can be treated as the limiting case of the elliptical hole by setting b = 0, 1/η = ∞ in the derived formulae. The sensitivity derivative with respect to the hole area is now infinite but the derivative with respect to length of ellipse semi-axis is finite and depends linearly on the crack length 2a. Consider first the self-adjoint case for which the sensitivity derivative of the strain plate energy U = U(u,a), or the potential plate energy Π = Π(u,a), is specified. From (62) for b = 0, 1/η = ∞ it follows that For α = 0 the crack is oriented along the \(\sigma_1 \)- axis and then we have Similarly, for α = π/4 and \(\sigma _{1} = -\sigma _{2} = \sigma _{12}\), the plane crack is subjected to shear stress and there is Here K _I and K _II denote the stress intensity factors of tension and shear modes specifying the singular asymptotic stress fields at the crack tip expressed in the local polar coordinate system (r,θ), so Referring the stress field to the coordinate axes x ₁₀, x ₂₀ coaxial and normal to the crack, Fig. 7b, in view of the stress transformation formulae, the sensitivity derivative (89) takes the form well familiar in fracture mechanics (cf. Miannay 1998), thus and it expresses the release of potential energy due to crack growth.

In the general case of an arbitrary functional G, its sensitivity in view of (61), assuming f(u) = 0, p ⁰ = 0, is expressed as follows Referring to the coordinate system x ₁₀, x ₂₀ the sensitivity expression (94) can be presented in the form where \(K_I^a \), \(K_{II}^a \) are the stress intensity factors associated with the adjoint stress state. The sensitivity derivatives (93) and (95) are linearly related to the crack length 2a and vanish for a = 0. The Griffith crack generation governed by the critical value of the potential energy release therefore cannot occur. However, the crack generation of finite length can be assumed as physically admissible.

Consider, for instance a plane crack in a wide plate under uniaxial tension, Fig. 8. The presence of crack induces the release of potential energy ΔΠ _l and growth of surface or dissipated energy ΔΠ _s thus the total energy variation is where γ is the specific surface energy per unit crack length (cf. Bojczuk and Mróz 2007). The critical crack length is specified from the condition Let us note that for a > a _c there is unstable crack growth at constant stress with associated decrease of the total energy. The other length is obtained from the constant energy transformation ΔΠ = 0 providing the crack length The finite topology transformation then occurs along the path OB, Fig. 8b.

The concept of finite topology variation was discussed by Bojczuk and Mróz (2003, 2009) and applied in optimal design of discrete supports in beam structures and of plate structures.

Let us also note that the transition from an ellipse to a crack cannot be always conducted within the regular perturbation theory, as it was shown by Il’in and Gadyl’shin (2001).