Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate ˇk P. Baˇ Jan Vorel1 , Zdene zant2 and Mahendra Gattu3 Abstract: Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress rate the Jaumann rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the the Truesdell objective stress rate, which is work-conjugate to the GreenLagrangian finite strain tensor. The commercial codes should switch to this rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell rate and Green-Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell rate. Comparisons of the errors of various approaches are tabulated.
1
Motivation and Nature of Problem
This paper is motivated by the design of large foam-core sandwich panels, intended for the cladding of a ribbed hull of light long ships with superior maneuverability and fuel-efficiency. In a laboratory test, one such panel failed at one third of the axial compressive load predicted by a standard commercial finite element program. Although the main cause of this gross underestimation of strength is probably the neglect of size effect due to cohesive delamination fracture, which was previously identified for cylindrical buckling of sandwich plates [7] and is the subject of a separate study, an important additional cause to be studied here appears to lie in two long-ignored flaws [2, 5] in the handling of finite strain by standard commercial codes [1, 15, 16, e.g.]: 1) One flaw of these code is the use of objective stress rates that are not work-conjugate to any finite strain tensor [2]. In the implicit updated Lagrangian analysis, it is the Jaumann rate 1
Visiting Scholar, Northwestern University; Assistant Professor on leave from Czech Technical University in Prague. 2 McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, Illinois 60208;
[email protected] (corresponding author) 3 Graduate Research Assistant, Northwestern University.
of Cauchy stress and, in the explicit analysis, the use of the Green-Naghdi rate. 2) Another flaw is that the bifurcation analysis uses the Jaumann rate of Kirchhoff stress. Although this rate is work-conjugate to the Hencky (or logarithmic) strain tensor, it cannot correctly capture the work of initial in-plane stresses in soft-in-shear highly orthotropic structures compressed in the strong direction. This work can be captured correctly only by the Truesdell objective stress rate [3, 4]. The former flaw can lead to major errors in volume changes of polymeric, ceramic and metallic foams, fiber-reinforced foams and other highly compressible porous materials such as loess, silt, tuff, snow, under-consolidated granular materials, light wood, honeycomb, osteoporotic bones and various biological tissues. But it is unimportant for elastic buckling of a sandwich, which is the only case to be studied here. The reason is that even though the foam core is highly compressible, its hydrostatic stress is negligible (except for inelastic buckling with delamination, or for indentation [6]). This study will focus on the latter flaw. Its seriousness has already been demonstrated for sandwich columns [3, 4] and for highly orthotropic columns [12, 13], but not for sandwich plates. The energetically correct form of the differential equations of equilibrium of sandwich plates will also be identified, and their solution will be compared to finite element analysis of two kinds–a two-dimensional analysis with sandwich-type elements having a linear strain profile across the core, and three-dimensional analysis in which the core thickness is subdivided into several elements. In contrast to finite elements for the entire cross section of sandwich column, this subdivision has already been shown to yield correct results regardless of the choice of objective stress rate; see [4] (and for elastomeric bearings see [28]). It will be verified whether this is also true for sandwich plates. A salient characteristic of sandwich plates is that the shear strain in a soft core is important for buckling. The shear buckling is a problem with a hundred-year controversial history. It requires using the stability criteria for a three-dimensional continuum, which were for half a century a subject of polemics. Although the polemics were resolved four decades ago, some authors still dispute various aspects. All the historical controversies can be traced to the arbitrariness in choosing the finite strain measure and to inattention to the work-conjugacy requirement, which means that the (doubly contracted) product of the incremental objective stress tensor with the incremental finite strain tensor must give a correct expression for the second-order work [5, ch.11]. How does the choice of strain measure affect the differential equations of equilibrium and the eigenvalues of compressed sandwich plates? And which choice is correct? These questions will be addressed first analytically, and then in the context of finite element analysis, with and without subdividing the core thickness into several layers of elements. The aim is to appraise the magnitude of errors and choose the best practical approach.
2
Review of Objective Stress Rates and Their Energy-Variational Basis
A broad class of equally admissible finite strain measures which comprises virtually all of those ever used is represented by the Doyle-Ericksen tensors (m) = (U m − I) /m, where m is a real parameter, I = unit tensor and U = right-stretch tensor [5]. The second-order approximation of these tensors is (m)
ij = eij + 21 uk,i uk,j − αeki ekj ,
eki = 12 (uk,i + ui,k ) , 2
α = 1 − 21 m
(1)
where eij refers to the small (linearized) strain tensor, ui are the displacement components and the superscripts refer to Cartesian coordinates xi (i = 1, 2, 3). The repetition of tensorial subscripts implies summation. Note that m = 2 gives the Green-Lagrangian strain tensor and m = 0 the Hencky (logarithmic) strain tensor. It has been commonplace, especially in finite element texts, to derive the objective stress rate by tensorial coordinate transformations. However, such an approach does not guarantee work-conjugacy with the adopted finite strain measure. To guarantee it, the variational energy approach is inevitable, and is in fact also much simpler. The derivation, given in [2], may be briefly sketched as follows (for a detailed exposition see [5, ch.11]). Consider the work δW done at small deformations of a unit material element starting from (m) (m) an initial state under initial Cauchy (or true) stress Sij . One may write δW = (Sij + σij )δij (m) (m) where δij is an arbitrary strain variation and σij is a small stress increment that is symmetric and objective with respect to coordinate transformations (and represents an increment of the second Piola-Kirchhoff stress referred to the stressed initial state). (m) But this work may also be expressed as δW = (Sij +τij )δui,j where ui,j is the displacement gradient and τij is a small Lagrangian (or first Piola-Kirchhoff) stress increment, which is nonsymmetric. Next, set these two expressions for δW equal for any δui,j . Then substitute (m) (m) (m) Sij δui,j = Sij δeij = Sij e˙ ij δt (by virtue of symmetry of Sij ), σij δij ≈ σij e˙ ij δt (which suffices (m) (m) (m) (m) for second-order work accuracy in eij ), Sij δij = Spq (∂pq /∂ui,j )vi,j δt and σij = Sˆij δt (where vi,j δt = δui,j , vi = u˙ i , δt = small increment of time t, and superior dots stand for ∂/∂t). Thus one can obtain the following general expression for the objective stress rate associated with m: (m) ∂ ∂(pq − epq ) (m) Sˆij = T˙ij − Spq (2) ∂t ∂ui,j T˙ij = S˙ ij − Sik vj,k + Sij vk,k
(3)
in which T˙ij = ∂Tij /∂t, S˙ ij = ∂Sij /∂t = material rate of Cauchy stress (rate per constant mass in Lagrangian coordinates of the initial stressed state). Evaluating Eq. (2) for a general m and for m = 2, one gets (m) Sˆij = Sˆij + 12 (2 − m)(Sik e˙ kj + Sjk e˙ ki )
(4)
where Sˆij = objective stress rate associated with the Green-Lagrangian strain (m = 2), as the reference case. In particular, (2) Sˆij = Sˆij = S˙ ij − Skj vi,k − Ski vj,k − Sij vk,k (Truesdell stress rate) (0) Sˆij = S˙ ij − Skj ω˙ ik + Sik ω˙ kj + Sij vk,k (Jaumann rate of Kirchhoff stress)
(5) (6)
where ω˙ jk = 12 (vi,j − vj,i ). Furthermore, m = 1 gives the Biot stress rate, and m = −2 gives the Lie derivative of Kirchhoff stress. Equation (2) is the key equation which must be satisfied for an objective stress rate to be work-conjugate to the incremental finite strain tensor [2, 5]. Another rate, widely used in commercial codes, is SˆijJ = S˙ ij − Skj ω˙ ik + Sik ω˙ kj
(Jaumann, or corotational, rate of Cauchy stress)
(7)
There exists no finite strain tensor for which Eq. (7) would yield this rate. Therefore, the use of this rate violates energy balance (or the first law of thermodynamics), and so 21 SˆijJ e˙ ij ∂t2 is 3
generally not the correct expression for the second-order work of stress increments. Nevertheless, the use of this rate is not objectionable for incompressible or nearly incompressible materials, for which the volumetric term Sij vk,k , representing the work-conjugacy correction, is negligibly small. However, for foams (as well as other highly compressible materials such as honeycomb, loess, silt, clay, loose sand, organic soils, pumice, tuff, corral, soft wood, osteoporotic bone and various biologic tissues), the term Sij vk,k is not negligible and then the lack of energy balance can be serious (a parallel study [6] shows that, in the case of indentation of a foam-core sandwich, the use of SˆijJ for incremental analysis in ABAQUS, LS-DYNA, ANSYS and some versions of NASTRAN can cause errors greater than 28%). Aside from SˆijJ , the violation of work-conjugacy according to Eq. (2) also afflicts the Oldroyd rate, Cotter-Rivlin and GreenNaghdi rates [8, 9, 20] (the first would correspond to m = 2 and the second to m = −2 if the term Sij vk,k were added). Correcting the Cotter-Rivlin rate by adding the term Sij vk,k yields the Lie derivative (m = −2). (m) (m) The tangential stress-strain relation has generally the form Sˆij = Cijkl e˙ kl , and so the (m)
tangential moduli Cijkl must be different for different choices of m. Based on the fact that the second-order work δ 2 W = 12 T˙ij vi,j δt2 must be the same for all m, it can be shown [2, 5] that (m)
(2)
Cijkl = Cijkl + (2 − m) [Sik δjl ]sym
(8)
[Si kδjl]sym = 14 (Sik δjl + Sjk δil + Sil δjk + Sjl δik ).
(9)
where (2)
Here, Cijkl are the tangential moduli associated with the Green-Lagrangian strain (m = 2) and Sij is the current Cauchy stress. When the loading step is finite but small, one gets the first-order (or second-order) accuracy when Sij is evaluated for the beginning (or the middle) of the step. Because of missing the volumetric term, the Jaumann rate of Cauchy stress, Eq. (7), is not associated by work with any finite strain tensor. This deficiency, which was shown to cause errors up to 28% in type of loading of foam [6], can be corrected based on Eq. 19c in [2] or Eq. 11.4.6 in [5], which reads: (0) JC Cijkl = Cijkl − Sij δkl (10) If this transformation is made in a user material subroutine at each integration point of each finite element in each time step of a commercial code with this rate, one can obtain the solution for the Jaumann rate of Kirchhoff stress, which is work-conjugate for m = 0. Note that the last term in Eq. (10) does not allow interchanging subscripts ij and kl. This means that major JC symmetry, and thus the existence of an incremental potential, cannot characterize both Cijkl (0) and Cijkl . Foam core sandwich plates typically reach the critical load of buckling when the transverse shear stresses in the foam are much less than the shear strength of the foam and the in-plane normal stresses in the laminate skins (or faces) are much less than the the compressive strength of the skins. Thus one might think that constant and the same elastic moduli should be used for the core and the skins. But then the critical loads would be different for different choices of m (as shown for the special case of orthotropic or sandwich column buckling in [2, 5, 3, 4]). This is, of course, impossible. So one must conclude that, even though all the material is well within the linear elastic range, constant moduli can be used for one m value only. But for which value? 4
This question was answered briefly in [3] and in full detail in [4], with the result that, for flexural-shear buckling, the elastic moduli in the linear elastic range can be kept constant if and only if m = 2 (associated with Green-Lagrangian tensor) (11) One can, of course, use a formulation for another m value, e.g. for m = 0 (which is the case in most commercial finite element programs). But then the tangential moduli must be varied with the in-plane stresses in the plate according to Eq. (8) even if all the material is in the linear elastic range.
3
Analytical Formulations of Sandwich Plate Buckling for Different Choices of Finite Strain Measure
In the theory of sandwich plates with thin skins and soft core (Fig. 1), it is generally assumed that the skins carry all the bending and twisting moments and the core carries the transverse shear forces but no in-plane stresses [22, 3, 4]. Within the skins as well as the core, the normals to the middle surface are assumed to remain straight. For each skin, the initial normals are further assumed to remain normal to its middle surface, but not for the soft core. This means that rotations ϕ and ψ of the normals in planes (x1 , x3 ) and (x2 , x3 ) are independent of the slopes w,1 and w,2 of the deflection surface w(x1 , x2 ) (x1 , x2 are the in-plane Cartesian coordinates of the plate, and the subscripts preceded by a comma denote partial derivatives). The differential equations of equilibrium of orthotropic sandwich plates and ordinary plates are the same. They read: Q1 = −M11,1 − M12,2 , Q2 = −M22,2 − M12,1 V1,1 + V2,2 = Q1,1 + Q2,2 + Nij w,ij = −p
(12) (13)
where p = p(x1 , x2 ) = transverse distributed load; M11 , M22 = bending moments; M12 = twisting moments; N11 , N22 , N12 = in-plane forces. The special case of these equations for cylindrical bending in the x1 direction gives the differential equations of equilibrium of a sandwich column loaded only by the axial force, P = −N11 (p = 0): Q = −M,x , V,x = Q,x − P w,x (14) where x ≡ x1 , Q = Q1 , V = V1 and M = M11 . Because Q = GAγ where GA = shear stiffness and γ = w,1 − ϕ = shear angle, these equations lead to Engesser’s formula. This formula is one of two well-known formulas for shear buckling and is energetically associated with the GreenLagrangian tensor, m = 2 [2]. The other is the Haringx formula, known to be associated with m = −2 (Sec. 1.7 and 11.6 in [5]). The critical loads predicted by these formulas for practical sandwich columns can differ by 100% or more if the same shear modulus value is used. On the other hand, these formulas are equivalent if the shear modulus is transformed according to Eq. (8). The same must, of course, be expected for the sandwich plates. Since it is impossible for a special case of the plate equilibrium equations to be associated with one m and the general case with a different m, we must conclude that Eqs. (12)–(13) correspond to m = 2 provided that, as usual, the transverse shear strains, given by γ1 = w,1 − ϕ and γ2 = w,2 − ψ, are considered to be proportional to Q1 and Q2 , i.e., (2)
(2)
Q1 = C1 (w,1 − ϕ),
Q2 = C2 (w,2 − ψ) 5
(15)
(m)
(m)
Here we affix superscript (m) to the shear stiffnesses C1 and C2 , to make it clear that they are associated with m = 2. The conclusion that Eqs. (12)–(13) are associated with the Green-Lagrangian strain tensor can, of course, be also established by a variational derivation from the potential energy of the plate; see Appendix A. Note that it is the inclusion of the strain energy of transverse shear deformations which causes the result to depend on m. Without them, the differential equations of equilibrium are the same regardless of m, which is the case of ordinary plates. The differential equations of sandwich plate for a general m value are apparently unavailable in the literature. They are of interest as a check that finite element programs, most of which correspond to m 6= 2, converge upon mesh refinement. To obtain them, it suffices to make in Eq. (15) the substitutions (2)
(m)
C1 = C1
(2)
− 14 (2 − m)N11 ,
(m)
C 2 = C2
− 41 (2 − m)N22
(16)
which are a special case of Eq. (8). Most commercial programs use for incremental (or Riks) analysis the Jaumann rate of Cauchy stress, which corresponds to m = 0, and so the sandwich finite elements in these programs must converge at mesh refinement to a different solution, corresponding to the substitution: (2)
(0)
(2)
C1 = C1 − 21 N11 , (0)
(0)
C2 = C2 − 21 N22
(17)
(0)
with C1 , C2 being the transverse shear stiffnesses associated with m = 0. For the rotations ϕ and ψ, it is convenient to use their effective values determined as the rotations corresponding to the in-plane displacements at mid-thickness of skins in the x1 and x2 directions, respectively. Because the skins are much thinner than the core, these rotations differ only slightly from the actual rotations ϕc , ψc of the initial normals of the core (Fig. 1(b)). The effective shear angle γ1 in direction x1 (Fig. 1) is related to the actual shear angle γc1 in the core by hγ1 = tc γc1 , where tc , ts = thicknesses of the core and the skins, h = tc + ts = distance between the skin centroids, and ts tc is assumed. Equating the shear strain energies expressed (2) 2 , where Gc in terms of γ1 and γc1 , we have, per unit width of the plate, 21 G(2) hγ12 = 12 Gc (2) tc γc1 is the actual shear modulus of the core, and G(2) is the effective modulus corresponding to h. This yields for the x1 and x2 directions the equivalent shear stiffnesses (for isotropic core): (2)
(2)
C1 = C2 = G(2) h = Gc (2) h2 /tc
(18)
Based on the hypotheses of the linear sandwich plate theory(Fig. 1(b)), the stiffness constants have the form [22]: D11
E1s I , = s s 1 − ν12 ν21
H12 = Gs12 I,
(2)
D22
C1 =
E2s I = , s s 1 − ν12 ν21
h2 (2) Gc , tc
(2)
C2 =
s s D12 = D21 = D11 ν21 = D22 ν12
h2 (2) Gc tc
(19) (20)
assuming that the x1 , x2 directions coincide with the principal axes of orthotropy, and I=
ts h2 t3s + 2 6
(21)
(Fig. 1(a)), where E = Young’s modulus, G = shear modulus, ν12 , ν21 = Poisson’s ratios, ν12 /E1 = ν21 /E2 , and superscripts or subscripts c and s refer to the core and the skins. 6
(a)
(b)
Figure 1: Element of sandwich panel: a) Dimensions, b) rotation angles and shear forces
According to Eq. (8) and assuming that only in-plane forces N11 , N22 are present, the bending stiffnesses D11 , D22 and D12 = D21 and the torsional stiffness H12 depend on the choice of m as (2) (m) (2) (m) (2) (m) follows: D11 = D11 − (2 − m)S11 I, D22 = D22 − (2 − m)S22 I, H12 = H12 − 41 (2 − m)(S11 + S22 )I, in which I = moment of inertia, S11 = N11 tc /h2 , S22 = N22 tc /h2 . If the in-plane shear forces N12 are considered as well, similar expressions, resulting in the differential equations (m) for an anisotropic plate, can be obtained. However, since typically |(2 − m)S11 I| D11 , (m) (m) |(2 − m)S22 I| D22 and | 14 (2 − m)(S11 + S22 )I| H12 , the correction due to different m can here be neglected. So, one may set (m)
(2)
D11 = D11 = D11 ,
(m)
(2)
D22 = D22 = D22 ,
(m)
(2)
H12 = H12 = H12
(22)
Accordingly, the bending moments M11 , M22 and twisting moments M12 (= M21 ) have the same relationship to plate curvatures for all m values, as in the classical theory ([22]; also Sec. 7.8 in [5]): M11 = D11 ϕ,1 + D12 ψ,2 ,
M22 = D21 ϕ,1 + D22 ψ,2 ,
M12 = H12 (ϕ,2 + ψ,1 )
(23)
For a finite strain measure with any parameter m, the complete set of partial differential equations for a sandwich plate with initial in-plane forces consists of Eqs. (12),(13),(15),(16) and (19)–(23).
4
Analytical Solution of Critical Load for Buckling of Rectangular Sandwich Plate
Consider a sandwich plate, shown in Fig. 2, which is assumed to buckle while both the core and the skins are in their small-strain linear range. In spite of that, as explained, a constant shear modulus G of the core can be assumed only for m = 2 (i.e., a formulation associated with the Green-Lagrangian strain and Truesdell stress rate). However, since most commercial finite element codes use the Jaumann (corotational) rate of Cauchy stress, which corresponds to m = 0 (or to Hencky strain), we will also give the solution for m = 0, to assess the error when a constant G is used in these codes. In the case of rectangular sandwich plates (e.g., Fig. 2) it is easy to carry out an exact analytical solution. For an eigenvalue problem we consider p = 0. Although generalization would be easy, we consider the longer sides of the plate, of length a, to be simply supported and the shorter sides, of length b, to be clamped (Fig. 2) and the plate to be under unidirectional (m) (2) in-plane load N11 = −P while N22 = N12 = 0 (i.e., C2 = C2 = C2 ). In that case, the lowest critical load must occur for a single sinusoidal half-wave in the direction of the short sides of length b, and so the solution may be sought in the form: πx2 (24) w(x1 , x2 ) = W (x1 ) sin b πx2 ϕ(x1 , x2 ) = Φ(x1 ) sin (25) b πx2 ψ(x1 , x2 ) = Ψ(x1 ) cos (26) b 7
where W (x1 ), Φ(x1 ), Ψ(x1 ) are unknown complex functions. After substituting these expressions into Eqs. (15),(23) and then into Eqs. (12),(13), we obtain π2 π 1 (2) (2) C1 + H12 2 Φ + (D12 + H12 ) Ψ,1 − C1 W,1 (27) Φ,11 = D11 b b 1 π π2 π Ψ,11 = − (D12 + H12 ) Φ,1 + C2 + D22 2 Ψ − C2 W (28) H12 b b b 1 π π2 (2) W,11 = C1 Φ,1 − C2 Ψ + C2 2 W (29) (2) b b N11 + C1 This system of differential equations Φ,1 Φ ,1 Ψ,1 = Ψ,1 W,1 W ,1 or r ,1 =
can be 0 A1 0 0 0 0
rewritten in the matrix form: Φ 1 0 0 0 0 Φ 0 0 A2 0 A3 0 0 1 0 0 Ψ A4 A5 0 A6 0 Ψ 0 0 0 0 1 W A7 A8 0 A9 0 W
(30)
Ar
(31)
in which Φ,1 = Φ, Φ,11 = Φ,1 , with similar definitions for Ψ and W , and A1 A4
(2) 1 π C π2 (2) = C1 + H12 2 , A2 = (D12 + H12 ) , A3 = − 1 D11 b bD11 D11 2 π 1 π πC2 = − (D12 + H12 ) , A5 = C2 + D22 2 , A6 = − bH12 H12 b bH12 (2)
A7 =
C1
, (2)
N11 + C1
πC2 , A8 = − (2) b N11 + C1
π 2 C2
A9 =
(2)
b2 N11 + C1
(32) (33) (34)
Looking for a solution in the form keλx1 we may write Φ = k1 eλx1 , Φ = k2 eλx1 , Ψ = k3 eλx1 , Ψ = k4 eλx1 , W = k5 eλx1 , W = k6 eλx1 (35) Φ,1 = λk1 eλx1 , Φ,1 = λk2 eλx1 , Ψ,1 = λk3 eλx1 , Ψ,1 = λk4 eλx1 , W,1 = λk5 eλx1 , W ,1 = λk6 eλx1 (36) After substituting Eqs. (35)–(36) into Eq. (31), this leads to the linear homogeneous matrix equation (A − λI) k = 0 (37) where I = identity matrix. The non-trivial solution is obtained when det (A − λI) = 0. This (1) (6) equation gives 6 roots λ1 , λ2 , ..., λ6 with 6 eigenvectors k(1) , ...k(6) of components kn , ...kn . If all the 6 roots are different, the general solution may be written as un =
6 X
λ i x1 Bi k(i) n e
(n = 1, 2, ...6)
(38)
i=1
The solution for complex conjugate roots can be found, e.g., in [23, e.g]. If there are multiple roots, slightly different roots apply, but multiple roots have not been obtained in numerical 8
Figure 2: Plate analyzed: both edges perpendicular to the axis x1 are clamped (C) and the longer edges are simply supported (SS)
Table 1: Boundary conditions (Fig. 2) clamped edge (c) w=ϕ=ψ=0 simply supported edge (ss) w = M11 = M12 = 0 simulations. Subsequently, the boundary conditions must be imposed on Eqs. (38) (see Table 1 and Fig. 1, [22]). This yields another system of six homogeneous linear algebraic equations for unknowns B1 , B2 , ..., B6 for which the determinant must vanish. The critical load solution requires solving two eigenvalue problems given by Eqs. (30) and Eq. (38) with the boundary conditions. Obviously, to evaluate the coefficients of these equations, the lowest (first) critical value of N11 must be known. An iterative procedure must therefore be used. The commercial software MATLAB [18] has been utilized to calculate the critical load and the corresponding buckling mode. An estimate of N11 is made first. Then the coefficients of Eq. (30) are evaluated and six eigenvalues with six eigenvectors are solved by the computer from the first eigenvalue problem. The second eigenvalue problem (Eq. (38) with the boundary conditions) has a non-trivial solution if and only if the determinant of the equation system is zero. The zero value is sought by updating the N11 value. The standard minimization MATLAB function “fminbnd ” is utilized in this approach. Once the N11 -value corresponding to the zero determinant is found, the buckling mode for the lowest eigenvalue can be determined. (2) Considering that the shear modulus, C1 , of the core is constant, one gets the critical load corresponding to m = 2 (i.e., to the Green-Lagrangian strain and Truesdell rate), which is the only correct result. To check the error in critical load when a formulation for a different m is (2) used, one needs to substitute expression (17) for C1 into Eqs. (32) and (34), and then proceed with the solution in the same way.
5
Numerical Analysis of Buckling of Rectangular Sandwich Plate
A rectangular sandwich plate with a light PVC Divinycell H200 foam core and skins made of carbon fiber reinforced polymer (CFRP) has been analyzed by finite elements. The core is isotropic and the skins quasi-isotropic (Table 2). The material stiffness data considered are partly based on experiments, partly taken from the literature, and partly estimated by the MoriTanaka homogenization method [26, 27]. The plate sides are a = 3, 380 mm and b = 2, 540 mm. For layer thicknesses, see Table 2.
Table 2: Material properties (m - measured, c - calculated, l - lower bounds from technical specifications [10]) E [GPa] ν [-] G [GPa] t [mm] m m CFRP in-plane 46 0.3 (ν12 ) 17.7m c c (skins) transversal 5.7 0.24 (ν13 ) 2.0c 9.6m H200 (core) 0.23l 0.353l 0.085l 150m 9
Two different discretizations of sandwich panel are used in computations: • The plate is homogenized through its whole thickness and uniform effective material properties for the combined thickness of the core and skins are used, which defines a homogeneous highly orthotropic plate [22]; or • the skins and the core are discretized separately, each of them by several layers of finite elements. The eigenvalue and nonlinear incremental (Riks) analyses are performed using ABAQUS. While an eigenvalue buckling analysis leads to the critical load of a plate without imperfections directly, the nonlinear incremental analysis considers a plate with small imperfections and the critical load value is approached by increasing the in-plane load in small steps until a sudden deflection growth is detected. The imperfection ordinates x3 are considered to have the shape of the first buckling mode obtained in the aforementioned analytical solution (Figs. 4(a),6(a)), with the maximum ordinate x3 = 1.0 mm. In ABAQUS, the eigenvalue buckling analysis and Riks (nonlinear) analysis are based on the Jaumann rate of Kirchhoff stress or Cauchy stress, respectively. This corresponds to m = 0 and, as already pointed out, is not correct for shear buckling of columns and plates [3, 4] (m) (correctly, m = 2 must be used, unless variable moduli Cijkl , transformed according to Eq. (8), are introduced). For comparison, another nonlinear incremental analysis is performed using the open source code OOFEM [21], which is based on the Truesdell stress rate, corresponding to m = 2 (since OOFEM does not support support the eigenvalue analysis for the given type of elements, the eigenvalues were calculated by nonlinear nonlinear incremental analysis of imperfect structure). (2) In this case, constant moduli Cijkl should be used [5, 4]. The critical buckling load is determined by the method of “sharp-break plot”, in which the applied load is plotted versus the average of the in-plane strains at opposite faces of the plate [25], cf. Figs. 4(b), 6(b) (the Southwell plot was tried, too, but was found less effective). The plate obtained by homogenizing the sandwich plate through the whole thickness has the following highly orthotropic effective properties [3, 14, 19]: E1ef = E1s I/I ef ,
E2ef = E2s I/I ef ,
E3ef =
hef 2ts /E3s + tc /E c
c ef c ef s ef Gef Gef Gef 13 = G tc /κh , 23 = G tc /κh , 12 = G12 I/I ef ef ef s s s = 2ts ν23 /tef , ν13 = 2ts ν13 /tef , ν23 ν12 = ν12
(39)
where I ef = centroidal moment of inertia per unit width, and hef = thickness of the homogenized plate; I is given by Eq. (21) and κ is the shear reduction coefficient for the homogenized crosssection (κ ' 0.8 for a rectangular cross section). Different discretizations and element types have been used to perform the numerical simulations; see Table 3. Since OOFEM contains no linear 8-node finite elements with reduced integration, quadratic 20-node elements are chosen to discretize the skins and the core. The large deformation option is employed for all element types.
6
Verification by Special Case of Cylindrical Buckling
To verify the present formulation and check the accuracy of the homogenized sandwich approximation, the special case of cylindrical buckling is analyzed. The simply supported edges 10
Table 3: Element types and characteristics [1, 21] package elem. name description ABAQUS C3D8R Linear 3D 8-node finite element with reduced integration S4R 4-node shell element with reduced integration OOFEM LSpace Linear 3D 8-node finite element QSpace Quadratic 3D 20-node finite element (a)
(b)
Figure 3: Cylindrical buckling: a) Evolution of normalized critical buckling load of rectangular plates of different span ratios a/b, for fixed b = 2, 540 mm (Engesser, Hencky, Haringx formulas given by Eq. (50), cases of m = 0, ±2 given by Eq. (48)), b) deflection curve for the first buckling mode for m = 2, a = 3, 380 mm
along the longer sides of length a are replaced by free edges and both short sides are considered clamped. Various length-thickness ratios a/h are considered. The results are compared to the critical loads from the well-known formulas for column buckling with shear [5, e.g.], shown in [2] and reviewed briefly in Appendix B (in more detail, see [5, 4]). Three different strain measures are considered, i.e. m = 0 and ±2. As seen in Fig. 3(a) and Table 4, if the transformation given by Eq. (8) is not made, the use of the same constant elastic moduli for different strain measures causes a major discrepancy among the solutions for different m, with differences up to about 100%, and can lead to dangerous overestimation of the buckling load. However, this discrepancy becomes less than about 1% when the both the core and the skins of the sandwich are discretized separately through the thickness. The last observation was previously made for shear buckling of highly orthotropic columns [3, 4]. Fig. 3(b) shows the correct buckling mode, which is obtained if the C (2) is assumed to be constant. For cylindrical buckling, there also exists solid experimental verification. Comparisons with the tests of laminate sandwich columns made in [3, 4] clearly show that a good agreement with test results is achieved for Engesser’s formula with a constant shear modulus, which corresponds to m = 2, while Haringx’s formula (m = −2) can deviate from these tests by up to about 100%.
Table 4: Cylindrical buckling: Comparison of dimensionless critical buckling loads |Ncr |(100/2ts E1s (2) ) for a = 3, 380 mm # of FE layers analysis skin core eigenvalue nonlinear Analytical sol. m=0 1.16 m=2 0.98 ABAQUS, m = 0 homogenized 6 (C3D8R) 1.12 1.11 skins and core 2 (C3D8R) 4 (C3D8R) 0.99 1.00 OOFEM, m = 2 homogenized 6 (LSpace) 0.99 skins and core 2 (QSpace) 2 (QSpace) 0.97
11
(a)
(b)
Figure 4: Cylindrical buckling: Homogenized plate simulation by OOFEM: a) buckling mode, b) sharp-break plot
7
Numerical Results for Buckling of Rectangular Sandwich Plates
The previous numerical studies of finite strain effects in the buckling of sandwich plates were confined to cylindrical bending or to columns [12, 3]. Finite element computations of the critical loads have now been run for rectangular plates supported on all the four edges, with the boundary conditions shown in Fig. 2. Different choices of finite strain measures have been made in these computations. Table 5 compares the finite element results to the analytical solutions of the critical load. The use of incorrect finite strain formulation with the Jaumann rate of Cauchy stress, which corresponds to m = 0, causes the critical load to be overestimated if the effective sandwich plate properties are used. For the typical sandwich plate, as considered for naval applications, the differences between the critical values for strain measures corresponding to m = 2 and m = 0 are seen in Fig. 5 and in Table 5 and are up to 40%. Therefore, either a switch to the correct strain measure must be made, or the tangential stiffnesses in each loading step must be transformed according to Eq. (8) if the effective properties of sandwich structures are used. On the other hand, as expected, there is only a negligible difference between the results of the eigenvalue and the nonlinear (Riks) analyses performed by ABAQUS, which both correspond to m = 0. The deficiency of ABAQUS, in which the finite strain measure with m = 0 is employed, can be easily overcome if the tangent moduli transformation derived by [11] and by variational energy approach in [2] is implemented in the user’s material subroutine UMAT in ABAQUS. To take into account the co-rotational formulation employed in UMAT, the tangential moduli read UMAT Cijkl =
1 (2) Fip Fjq Fkr Fls Cijkl − Sij δkl + (2 − 0) [Sik δjl ]sym J
(40)
where deformation gradient Fij = ∂xi /∂Xj and J = det (F ). This transformation ensures not only the m-value to be correct but also the energy balance to be satisfied. The results obtained upon implementing this transformation are also displayed in Table 5 (section ABAQUS, m = 2 transformation). They confirm strong dependence on the choice of the finite strain measure. To explore the effect of finer discretization, finite element analysis is carried out for various numbers of element layers in the core. First, the solid elements are utilized for both the skins and the core (Table 5). Representing each skin by only one layer of C3D8R elements leads to unrealistic buckling modes and overestimates the critical load by more than 50 %. So there is no point to include these results in Table 5. On the other hand, correct results are obtained if each skin is represented by at least two layers of C3D8R elements. Another discretization studied uses shell elements (S4R, cf. Table 3) for the skins while the core is meshed with 1, 2 or 4 layers of C3D8R solid elements. This discretization provides reasonable results, which are only slightly lower than the analytical predictions. The shell elements have nodes at the mid-planes of both skins. These nodes must be shared with the solid elements of the core, and so the material of the core is imagined to be extended up to the mid-planes of the skins. Such representation, of course, does not accurately represent the 12
Table 5: Rectangular plate: Comparison of normalized critical buckling loads |Ncr |(100/2ts E1s (2) ) (a = 3, 380 mm, b = 2, 540 mm, Fig. 2) # of FE layers analysis skin core eigenvalue nonlinear Analytical sol. m=0 2.22 m=2 1.58 ABAQUS, m = 0 homogenized 6 (C3D8R) 2.17 2.17 skins and core 2 (C3D8R) 2 (C3D8R) 1.61 1.60 2 (C3D8R) 4 (C3D8R) 1.60 1.60 1 (S4R) 1 (C3D8R) 1.50 1.46 1 (S4R) 2 (C3D8R) 1.50 1.44 1 (S4R) 4 (C3D8R) 1.50 1.44 ABAQUS, m = 2 transformation homogenized 6 (C3D8R) 1.60 skins and core 2 (C3D8R) 4 (C3D8R) 1.49 OOFEM, m = 2 homogenized 6 (LSpace) 1.58 skins and core 2 (QSpace) 2 (QSpace) 1.50 (a)
(b)
Figure 5: Analytical solution of homogenized plate: a) normalized critical buckling load vs. span (0) (2) ratio a/b, b = 2, 540 mm, b) error when m = 0 is used, err = (Pcr /Pcr − 1) · 100%
actual structure. Nevertheless, since the skins are quite thin, the effect on the final results is small. For a more accurate representation, shell elements with an offset could be employed [24]. The finite element solution of critical load for m = 0 and for the multilayer subdivisions of core and skins is very close to the solution for m = 2. This confirms that, for these finer subdivisions, the effect of different choices of m is very small. Note that what makes the use of the effective homogenized material properties possible is the assumption that both the core and the skins are elastic, and that the skins undergo no local buckling, i.e., no skin wrinkling. Otherwise, and especially in the case of delamination fracture, the sandwich plate must be subdivided into many element layers. (a)
(b)
Figure 6: Discretized simulation of rectangular plate by OOFEM (last line in Table 5): a) buckling mode, b) sharp-break plot
13
Discussion of Likely Questions 1. Isn’t it immediately clear from Eq. (8) that the differences between various objective stress rates can be significant only if some of the tangential elastic moduli are of the same order of magnitude as some of the stresses, Sij ? It is. But this observation does not apply to kinematically linked constituents of a composite. In the sandwich plate theory, the stiff skins are linked with the soft core by the constraint of a linear strain profile. This is the reason why it can happen that, at bifurcation buckling, the normal stress in the skins is of the same order of magnitude as the shear modulus of the core even when both the skin and the core are loaded well within the linear elastic range. 2. Why does the choice of m not matter when the kinematic constraint of skin and core is omitted and the core thickness is subdivided into several elements? The reason is that the stress in the skins becomes irrelevant for the elements of the core. What matters is only that the in-plane normal stress in the elements of the core cannot be of the same order of magnitude as the elastic moduli of the core, which means that the objective stress rates for different m are equivalent. 3. Since today it is no problem to run three-dimensional finite element analysis with a subdivided core, why do we bother to discuss the homogenized sandwich plate or a plate with the kinematic constraint? We do, because the use of commercial programs with homogenized plate subroutines is simpler for the user, because the transformation from m = 0 to m = 2 is easy to implement, and mainly because there are applications where the homogenization may still be either inevitable or much less demanding—e.g., for simulating the whole ship hull with hundreds of sandwich panels. 4. There are new types of sandwich plates in which the foam core is reinforced by in-plane fibers. Would element subdivision of the core eliminate the problem? It would not, since transformation to m = 2 is then necessary even for the elements of the core. 5. Should the preference for m = 2 be limited to sandwich plates? No; m = 2 is needed for all situations where the tangential moduli are highly orthotropic and the dominant compressive principal stress has the direction of strong orthotropy. Thus, e.g., m = 2 needs to be used for polymers reinforced by unidirectional or bidirectional stiff fibers (but note that when the maximum compressive stress is normal to the strong orthotropy directions as, e.g., for elastomeric bridge or seismic isolation bearings, the correct choice is m = −2,S and for other principal stress ratios it is an m-value between −2 and 2 given by Eq. 29 in [4]). 6. Is the choice m = 2 necessary when the material is in the nonlinear range and the tangential moduli vary with the stress? It is, because what matters is only the tangential moduli at the current stress level, and not how they evolve before. Thus, for example, m = 2 should be used for bifurcation analysis of concrete or rock with a constitutive model in which compression damage is modelled by a system of smeared parallel splitting cracks. In such a model, the shear stiffness is much smaller than the crack-parallel stiffness. 7. Although irreversible unloading can occur at large postcritical deflections, it has no effect on the present critical load analysis. Thus the constitutive law could be considered as hyperelastic and the potential energy could be formulated. So, wouldn’t the minimization of potential energy of the whole plate give different results than the present incremental formulation? It would not. But if the constitutive law were not transformed from one m to another, the potential energy would be different for different m. Of course, only the results for the work-conjugate objective stress rates could be matched by the potential energy approach. 14
8. Wouldn’t smaller imperfections and smaller load steps eliminate the discrepancies? They would not. It has been verified that refining the load step or mesh size would not be detectable in graphical plots. The numerical simulations appear to converge at decreasing load step and the analytical critical load is approached at decreasing imperfection.
8
Conclusions 1. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core have a form that corresponds to choosing the GreenLagrangian strain as the finite strain measure and the Truesdell rate as the objective stress rate. This form is energy conserving. 2. Even though sandwich structures often buckle while both the core and the skins are still in their small-strain linear range, the shear modulus of the core appearing in the differential equations for sandwich buckling must be considered to depend on the stress in the skins when other finite strain measures (m 6= 2) are used. This is especially true for the widely used Hencky finite strain measure and the Jaumann rate of Cauchy stress, used in most commercial finite element programs, e.g., ABAQUS, LS-DYNA, ANSYS or NASTRAN. 3. By stress-dependent transformation of elastic stiffnesses, the classical differential equations can be converted to those compatible with any choice of the finite strain tensor and the corresponding objective stress rate. In particular, the differential equations for sandwich buckling compatible with the Hencky strain tensor and the associated Jaumann rate of Kirchhoff stress can thus be obtained. 4. The open-source code OOFEM and the commercial code ATENA [17] need no such transformation because they use the Green-Lagrangian strain and the Truesdell objective stress rate. 5. If the elastic moduli are constant in both aforementioned formulations, and if either the normals of the core and the skins are assumed to remain straight, or if the classical assumption of an equivalent homogenized orthotropic plate with normals remaining straight is adopted, the critical load calculated in the present example on the basis of the Jaumann rate of Cauchy stress and the Hencky tensor overestimates the correct critical load by up to 40%. 6. The error of 40% is of the same order as the previously demonstrated difference between the Engesser and Haringx formulas for sandwich columns. These formulas, which correspond to the special case of cylindrical bending, received solid experimental verification [3, 4]. 7. If the core and the skins are each subdivided into several layers of finite elements, the numerical solution converges to virtually the same correct critical load regardless of the choice of finite strain tensor and the objective stress rate. 8. Commercial finite element codes should switch to using the Green-Lagrangian strain tensor and the Truesdell stress rate. Otherwise, stress-dependent transformation of the tangential moduli may have to be implemented in the user’s material subroutine.
Acknowledgment: Financial support under Grant N00014-11-1-0155 from the Office of Naval Research to Northwestern University, monitored by Dr. Roshdy Barsoum, is gratefully appreciated.
15
Appendix A: Variational Derivation of Differential Equations of Equilibrium from Potential Energy of Plate To obtain the potential energy Π for small transverse deflections w(x1 , x2 ) of a plate under initial in-plane forces, the in-plane strains produced at the middle surface are expressed in terms w(x1 , x2 ) [5]. These strains are obtained from the three-dimensional finite strain tensor (2) ij in Eq. (1) as follows: (2)
2 11 = 21 w,1 ,
(2)
2 22 = 12 w,2 ,
(2)
12 = 12 w,1 w,2
(41)
Consider now a plate that is initially in equilibrium at w(x1 , x2 ) = 0. The energy due to the in-plane displacement u, v at the middle surface during bifurcation is higher-order small and may be neglected. So, Π = U1 + U2 + U3 − W (42) where U1 = U2 = U3 = W =
ZZ h i (2) 2 (2) 2 (2) (2) 2 D11 ϕ,1 + D22 ψ,2 + 2D12 ϕ,1 ψ,2 + H12 (ϕ,2 + ψ,1 ) dA Z ZA h i (2) (2) 2 2 1 C (w − ϕ) + C (w − ψ) dA ,1 ,2 1 2 2 A ZZ ZZ 2 2 1 1 Nij w,i w,j dA = 2 N11 w,1 + N22 w,2 + 2N12 w,1 w,2 dA 2 A ZZ A pw dA 1 2
(43)
A
Here A = area of plate and the symmetries N12 = N21 , D12 = D21 are taken into account; U1 = strain energy of bending; U2 = strain energy of shear deformations; U3 = strain energy of in-plane deformations due to the deflection; W = work of transverse distributed loads p. The equilibrium condition can be expressed variationally as δΠ = δU1 +δU2 +δU3 −δW = 0. After some algabra, Eqs. (43) thus yield: ZZ h i (2) (2) (2) (2) −C1 (w,1 − ϕ) − D11 ϕ,11 + D12 ψ,12 − H12 (ϕ,22 − ψ,12 ) δϕ dA + Z ZA h i (2) (2) (2) (2) + −C2 (w,2 − ψ) − D12 ϕ,12 + D22 ψ,22 − H12 (ϕ,12 − ψ,11 ) δψ dA + Z ZA h i (2) (2) + −C1 (w,11 − ϕ,1 ) − C2 (w,22 − ψ,2 ) − Nij w,ij + Fi w,i − p δw dA + Z hA i (2) (2) (2) + n1 D11 ϕ,1 + D12 ψ,2 + n2 H12 (ϕ,2 + ψ,1 ) δϕ ds + Zs h i (2) (2) (2) + n2 D12 ϕ,1 + D22 ψ,2 + n1 H12 (ϕ,2 + ψ,1 ) δψ ds + Zs h i (2) (2) + n1 C1 (w,1 − ϕ) + n2 C2 (w,2 − ψ) + ni Nij w,j δw ds = 0 (44) s
Here Fi = −Nij,j denotes the in-plane distributed loads in the initial state and ni is the cosine of the angle formed by normal n with the axis xi . Since the variational condition must be
16
fulfilled for any variations δϕ, δψ and δw, the integrands must vanish at each point of the plate. This yields the following system of differential equations: (2) (2) (2) (2) C1 (w,1 − ϕ) = − D11 ϕ,11 + D12 ψ,12 − H12 (ϕ,22 − ψ,12 ) (45) (2) (2) (2) (2) C2 (w,2 − ψ) = − D12 ϕ,12 + D22 ψ,22 − H12 (ϕ,12 − ψ,11 ) (46) (2)
(2)
C1 (w,11 − ϕ,1 ) + C2 (w,22 − ψ,2 ) = −p − Nij w,ij + Fi w,i
(47)
The in-plane loads Fi are often usually insignificant and may be neglected. As expected, the same differential equations are obtained by combining the Eqs. (12),(13),(15) and (23). For explanation of the boundary curve integrals in Eq. (44) see, e.g., [5].
Appendix B: Cylindrical Buckling of Sandwich Columns In the special case of cylindrical buckling, which is equivalent to a cylindrical column, we use Eqs. (12),(13),(15) and (23) with the conditions ψ = ϕ,2 = w,2 = 0 to obtain the following system of two simultaneous linear ordinary differential equations (equilibrium conditions): (2)
D11 ϕ,111 − N11 w,11 = 0 (2)
(2)
(2)
D11 ϕ,11 − C1 ϕ + C1 w,1 = 0
(48)
The general solution takes the form w = B1 +h B2 x1 + B3 cos kx1i + B4 sin kx1 , ϕ = B2 − (2) (2) (2) and β = 1 + N11 /C1 . βB3 k sin kx1 + βB4 k cos kx1 where k 2 = −N11 / D11 (1 + N11 /C1 Constants B1 , ..., B4 are determined by introducing the boundary conditions (Table 1) into the general solution. Similar to plates, this formulation again corresponds to m = 2. For simply supported edges (x1 = 0 and x1 = a), the boundary conditions read w = 0 and (2) M11 = 0, i.e. ψ,1 = 0. The critical load is then given by ka = nπ, where n = 1, 2, 3, . . .. The smallest critical load occurs for n = 1 and is (2)
Pcr(2) (2)
= −N11 =
PE
(2)
(2)
(49)
1 + PE /C1
(2)
where PE = π 2 D11 /a2 . Eq. (49) is the same as the Engesser formula derived on the basis of energy formulation with the Green-Lagrangian finite strain [5]. The general quadratic equation for the critical load of a perfect column obtained by variational energy analysis is ! !" # (m) (m) (m) 2 2 − m P π 2 + m P P cr cr cr Pcr(m) 1 + − 2 E (m) I 1 − 1 − (m − 1) (m) = 0 (50) 4 G(m) A L 4 G(m) A E A where L = effective buckling length, A = area of the cross-section per unit width and I = moment of inertia. For cylindrical buckling, one substitutes the values E (m) I and G(m) A for (m) (m) D11 and C1 , respectively. Now note that shear is important for buckling analysis only if G(m) E (m) , and that the ax(m) (m) ial stress Pcr /A cannot have a greater magnitude than G(m) . Consequently (m − 1) Pcr /E (m) A 1. Therefore, this term can be neglected, and Eq. (50) then yields Engesser’s and Haringx’s formulas when m = 2 and m = −2, respectively. 17
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