Mechanics Analysis Composite Materials Episode 5 pot

To describe nonlinear elastic-plastic behavior of metal layers, we should use constitutive equations of the theory of plasticity.. According to the deformation theory of plasticity, the

The name “complementary” becomes clear if we consider a bar in Fig 1.1 and the

corresponding stress-strain curve in Fig 4.8 The area OBC below the curve represents U in accordance with the first equation in Eqs (4.8), while the area OAB above the curve is equal to U, As was shown in Section 2.9, dU in Eqs (4.8) is an

exact differential To prove the same for dU,, consider the following sum:

which is obviously an exact differential Since dU in this sum is also an exact differential, dU, should have the same property and can be expressed as

Comparing this result with Eq (4.9), we arrive at Castigliano’s formulas

(4.10)

which are valid for any elastic solid (for a linear elastic solid, U, = U)

Complementary potential, U,, in general, depends on stresses, but for an isotropic

material, Eq (4.10) should yield invariant constitutive equations that do not depend

on the direction of coordinate axes This means that U, should depend on stress

invariants 1 1 ~ 1 2 , I3 in Eqs (2.13) Assuming different approximations for function Uc(Zl, Z2, I3)we can construct different classes of nonlinear elastic models Existing

experimental verification of such models shows that dependence U, on 13 can be

neglected Thus, we can present complementary potential in a simplified form

U, (ZI, Z2)and expand this function into the Taylor series as

Chapter 4 Mechanics of a composite layer 127

u,= c0 +c ~+~-clzif I ~+-cI31; +-cI4if +

+ C 2 l i 2 +- c 2 & 1 + ??I; 1 +-c,4i; I +

Constitutive equations follow from Eq (4.10) and can be written in the form

au, ai, au, aiz

E

Assuming that for zero stresses U, = 0 and cij = 0 we should take co = 0 and C I I = 0

8.r = c12(o.v +ox)-C,Iql., 4;= c12(o., + oy)- c2lo.r’ y.vj = 2C21Z,,

These equations coincide with the corresponding equations in Eqs (4.6) if we take

Then, Eqs (4.12) yield the following cubic constitutive law:

The corresponding approximation is shown in Fig 4.6 with a solid line Retaining more higher-order terms in Eq (4.1l), we can describe nonlinear behavior of any isotropic polymeric material

To describe nonlinear elastic-plastic behavior of metal layers, we should use constitutive equations of the theory of plasticity As known, there exist two basic versions of this theory -the deformation theory and the flow theory that are briefly described below

According to the deformation theory of plasticity, the strains are decomposed into two components - elastic strains (with superscript 'e') and plastic strains (superscript 'p'), i.e.,

We again use the tensor notations of strains and stresses (Le., cij and ou)introduced

in Section 2.9 Elastic strains are linked with stresses by Hooke's law, Eqs (4.1), which can be written with the aid of Eq (4.10) in the form

(4.16)

where U, is the elastic potential that for the linear elastic solid coincides with

complementary potential U,in Eq (4.10) Explicit expression for U,can be obtained from Eq (2.5 1) if we change strains for stresses with the aid of Hooke's law, i.e.,

Now present plastic strains in Eqs (4.15) in the form similar to Eq (4.16):

(4.17)

(4.18)

where Up is the plastic potential To approximate dependence of Up on stresses,

a special generalized stress characteristic, i.e., the so-called stress intensity 0, is introduced in classical theory of plasticity as

Chapter 4 Mechanics of a composize layer I29

Transforming Eq (4.19) with the aid of Eqs (2.13) we can reduce it to the following form:

This means that 0 is an invariant characteristic of a stress state, i.e., that it does

not depend on position of a coordinate frame For a unidirectional tension as

in Fig 1.1, we have only one nonzero stress, e.g., 011.Then Eq (4.19) yields

Eqs (4.19) and (4.20) have an important physical meaning As known from

experiments, metals do not demonstrate plastic properties under loading with stresses 0, = or= 0: = 00 resulting only in the change of material volume Under such loading, materials exhibit only elastic volume deformation specified by

Eq (4.2) Plastic strains occur in metals if we change material shape For a linear

elastic material, elastic potential U in Eq (2.51) can be reduced after rather

cumbersome transformation with the aid of Eqs (4.3), (4.4) and (4.19), (4.20) to the following form:

characteristics associated with the change of material shape under which it demonstrates the plastic behavior

In the theory of plasticity, plastic potential Up is assumed to be a function of

stress intensity a, and according to Eqs (4.18), plastic strains are

(4.23)

Consider further a plane stress state with stresses a,, a),,and zxv in Fig 4.5 For this

case, Eq (4.19) acquires the form

To find ~ ( a ) ,we need to specify dependence of U, on a The most simple and

suitable for practical applications is the power approximation

where C and n are some experimental constants As a result, Eq (4.26) yields

To determine coefficients C and n we introduce the basic assumption of the plasticity theory concerning the existence of the universal stress-strain diagram (master curve) According to this assumption, for any particular material there exists the dependence between stress and strain intensities, i.e., a = ( P ( E ) (or E =f(a)), that is

one and the same for all the loading cases This fact enables us to find coefficients C and n from the test under uniaxial tension and extend thus obtained results to an

arbitrary state of stress

Chapter Mechanics of a composite layer 131

Indeed, consider a uniaxial tension as in Fig 1.1 with stress 6 1 For this case,

a = and Eqs (4.25) yield

(4.32)

To determine E,(o)= a/E, we need to plot the universal stress-strain curve For this purpose, we can use an experimental diagram o,(c,) for the case of uniaxial tension, e.g., the one shown in Fig 4.9 for an aluminum alloy with a solid line To plot the universal curve o ( E ) ,we should put 6 = a, and change the scale on the strain axis in

Fig 4.9 Experimental stress-strain diagram for an aluminum alloy under uniaxial tension (solid line),

the universal stress-strain curve (broken line) and its power approximation (circles)

accordance with Eq (4.21).To do this, we need to know the plastic Poisson's ratio

vp that can be found as vp = -E,,/&, Using Eqs (4.29) and (4.30) we arrive at

As follows from this equation vp = v if E, = E and vp + 112 for E, +0 Dependencies of Es and vp on E for the aluminum alloy under consideration are

presented in Fig 4.10 With the aid of this figure and Eq (4.21) in which we should

take 81 I = E,, we can calculate E and plot the universal curve shown in Fig 4.9 with

a broken line As can be seen, this curve is slightly different from the diagram

corresponding to a uniaxial tension For the power approximation in Eq (4.27), from Eqs (4.26) and (4.32) we get

correspond to E = 71.4 GPa, n = 6, and C = 6.23 x

Thus, constitutive equations of the deformation theory of plasticity are specified

by Eqs (4.25) and (4.32).These equations are valid only for active loading that can

Fig 4.10 Dependencies of the secant modulus (Es),tangent modulus (Et), and the plastic Poisson's ratio

(v,) on strain for an aluminum alloy

Chapter 4 Mechanics of a composite layer 133

be identified by the condition do > 0 Being applied for unloading (i.e., for d o < 0),

Eqs (4.25) correspond to nonlinear elastic material with stress-strain diagram shown in Fig 1.2 For elastic-plastic material (see Fig 1.5), unloading diagram

is linear So, if we reduce the stresses by some increments AO.~,Abv, AT^?, the corresponding increments of strains will be

Direct application of nonlinear equations (4.25) substantially hinders the problem

of stress-strain analysis because these equations include function o(0) in Eq (4.32)

which, in turn, contains secant modulus E,(a) For the power approximation

corresponding to Eq (4.33), E, can be expressed analytically, i.e.,

There exist several methods of such linearization that will be demonstrated using the first equation in Eqs (4.25), i.e.,

(4.34)

In the method of elastic solutions (Ilyushin, 1948), Eq (4.34) is used in the following form:

(4.35) where s is the number of the iteration step and

For the first step (s = l), we take qo = 0 and solve the problem of linear elasticity

with Eq (4.35) in the form

Finding the stresses, we calculate y l and write Eq (4.35) as

(4.36)

where the first term is linear, while the second term is a known function of coordinates Thus, we have another linear problem resolving which we find stresses,

calculate q2 and switch to the third step This process is continued until the strains

corresponding to some step become close within the given accuracy to the results found at the previous step

Thus, the method of elastic solutions reduces the initial nonlinear problem to a sequence of linear problems of the theory of elasticityfor the same material but with some initial strains that can be transformed into initial stresses or additional loads This method readily provides a nonlinear solution for any problem that has a linear solution, analytical or numerical The main shortcoming of the method is its poor convergence Graphical interpretation of this process for the case of uniaxial tension with stress (r is presented in Fig 4.1 la This figure shows a simple way to improve the convergence of the process If we need to find strain at the point of the curve that

is close to point A, it is not necessary to start the process with initial modulus E Taking E' < E in Eq (4.36)we can reach the result with much less number of steps According to the method of elastic variables (Birger, 1951), we should present

Eq (4.34) as

(4.37)

Fig 4.1 1 Geometric interpretation of (a) the method of elastic solutions, (b) the method of variable

elasticity parameters, (c) Newton's method, and (d) method of successive loading

Chapter 4 Mechanics of a composite layer 135

In contrast to Eq (4.39, stresses d, and v;.in the second term correspond to the

current step rather than to the previous one This enables us to write Eq (4.37) in

the form analogous to Hooke's law, i.e.,

method has got its name This method can be efficiently applied in conjunction with the finite element method according to which the structure is simulated with the system of elements with constant stiffness coefficients Being calculated for each step with the aid of Eqs (4.39), these stiffnesses will change only with transition from

one element to another, and it practically does not hinder the finite element method calculation procedure

The iteration process having the best convergence is provided by the classical Newton's method requiring the following form of Eq (4.34):

&; = c-1 + c;;'(o-; -a + qg(a;,-CT:') +c?;;yT& - ?;;I) ] (4.40)

where

Because coefficients c are known from the previous step (s -I ) , Eq (4.40) is linear

with respect to stresses and strains corresponding to step number s Graphical interpretation of this method is presented in Fig 4.1 IC In contrast to the methods discussed above, Newton's method has no physical interpretation and being characterized with very high convergence, is rather cumbersome for practical applications

Iteration methods discussed above are used to solve the direct problems of stress analysis, i.e., to find stresses and strains induced by a given load However, there exists another class of problems requiring us to evaluate the load carrying capacity

of the structure To solve these problems, we need to trace the evolution of stresses while the load increases from zero to some ultimate value To do this, we can use the method of successive loading According to this method, the load is applied with some increments, and for each s-step of loading the strain is determined as

(4.41)

where ES-l and v,-l are specified by Eqs (4.39) and correspond to the previous loading step Graphical interpretation of this method is presented in Fig 4.1 Id To obtain reliable results, the load increments should be as small as possible, because the error of calculation is accumulated in this method To avoid this effect, method

of successive loading can be used in conjunction with the method of elastic variables Being applied after several loading steps (black circles in Fig 4.1 Id) the

latter method allows us to eliminate the accumulated error and to start again the

process of loading from a proper initial state (light circles in Fig 4.1 Id)

Returning to constitutive equations of the deformation theory of plasticity,

Eq (4.25), it is important to note that these equations are algebraic This means that strains corresponding to some combination of loads are determined by the stresses induced by these loads and do not depend on the history of loading, i.e., on what happened to the material before this combination of loads was reached

However, existing experimental data show that, in generaI, strains should depend on the history of loading This means that constitutive equations should

be differential rather than algebraic as they are in the deformation theory Such

equations are provided by the flow theory of plasticity According to this theory,

decomposition in Eq (4.15) is used for infinitesimal increments of stresses, Le.,

Here, increments of elastic strains are linked with the increments of stresses by Hooke’s law, e.g., for the plane stress state

while increments of plastic strains

(4.43)

are expressed in the form of Eqs (4.18) but include parameter A which characterizes the loading process

Chapter 4 Mechanics of a composite layer 137

Assuming that Up = U,,(o), where o is the stress intensity specified by Eqs (4.19)

To determine parameter I-, assume that plastic potential Upbeing on the one hand a

function of o,can be treated as the work performed by stresses on plastic strains, Le.,

Substituting strain increments from Eqs (4.44) and taking into account Eq (4.24)

for o we get

With due regard to Eq (4.45) we arrive at the simple and natural relationship

di = do/o Thus, Eq (4.45) acquires the form

As can be seen, in contrast to the deformation theory, stresses govern the increments

of plastic strains rather than the strains themselves

In the general case, irrespective of any particular approximation of plastic potential U p ,we can obtain for function d o ( o ) in Eqs (4.47) the expression similar

to Eq (4.32) Consider a uniaxial tension for which Eqs (4.47) yield

Repeating the derivation of Eq (4.32) we finally get

d o ( o ) = * (- 1 -k) ,

0

(4.48)

where E, (0)= da/de is the tangent modulus introduced in Section 1.1 (see Fig 1.4)

Dependence of Et on strain for an aluminum alloy is shown in Fig 4.10 For the

power approximation of plastic potential

This result coincides with Eq (4.33) within the accuracy of coefficients C and B

As in the theory of deformation, Eq (4.50) can be used to approximate the

experimental stress-strain curve and to determine coefficients B and n Thus,

constitutive equations of the flow theory of plasticity are specified with Eqs (4.47) and (4.48)

For a plane stress state, introduce the stress space shown in Fig 4.12 and referred

to Cartesian coordinate frame with stresses as coordinates In this space, any loading can be presented as a curve specified by parametric equations a, = o,(p),

4.= o~,,(p), T ~ )= ~,~~(p),where p is the loading parameter To find strains

corresponding to point A on the curve, we should integrate Eqs (4.47) along this

curve thus taking into account the whole history of loading In the general case, the obtained result will be different from what follows from Eqs (4.25) of the deformation theory for point A However, there exists one loading path (the straight

line OA in Fig 4.12) that is completely determined by the location of its final point

A This is the so-called proportional loading during which the stresses increase in proportion to parameter p , i.e.,

Chapter 4 Mechanics of a composite layer 139

Fig 4.12 Loading path (OA) in the stress space

where, stresses with superscript ‘0’ can depend on coordinates only For such

loading, CT = cop, do = oodp, and Eqs (4.46) and (4.49) yield

Consider, for example, the first equation of Eqs (4.47) Substituting Eqs (4.51) and (4.52) we get

This equation can be integrated with respect top Using again Eqs (4.5 1) we arrive

at the constitutive equation of the deformation theory

Thus, for a proportional loading, the flow theory reduces to the deformation theory

of plasticity Unfortunately, before the problem is solved and the stresses are found

we do not know whether the loading is proportional or not and what particular theory of plasticity should be used There exists a theorem of proportional loading (Ilyushin, 1948) according to which the stresses increase proportionally and the deformation theory can be used if:

(1) external loads increase in proportion to one loading parameter,

( 2 ) material is incompressible and its hardening can be described with the power

In practice, both conditions of this theorem are rarely met However, existing experience shows that the second condition is not very important and that the deformation theory of plasticity can be reliably (but approximately) applied if all the loads acting on the structure increase in proportion to one parameter

law CT = Se”

4.2 Unidirectional orthotropic layer

A composite layer with the simplest structure consists of unidirectional plies

whose material coordinates, I , 2, 3, coincide with coordinates of the layer, x, y , z, as

in Fig 4.13 An example of such a layer is presented in Fig 4.14 -principal material axes of an outside circumferential unidirectional layer of a pressure vessel coincide with global (axial and circumferential) coordinates of the vessel

4.2.1 Linear elastic model

For the layer under study, constitutive equations, Eqs (2.48) and (2.53), yield

Chapter 4 Mechanics of a composite layer

and

where

(4.55)

(4.56)

Constitutive equations presented above include elastic constants of a layer that are

determined experimentally For in-plane characteristics E l , E?, GI?, and ~ 1 2 ,the corresponding test methods were discussed in Chapter 3 Transverse modulus E3 is

usually found testing the layer under compression in the z-direction Transverse shear moduli G13 and G23 can be obtained by different methods, e.g., by inducing

pure shear in two symmetric specimens shown in Fig 4.15 and calculating shear

modulus as G13 =P/(2Ay),where A is the in-plane area of the specimen

For unidirectional composites, G13 = Gl2 (see Table 3.5) while typical values of

G23 are listed in Table 4.1 (Herakovich ,1998)

Poisson’s ratios v31 and ~ 3 2can be determined measuring the change of the layer thickness under in-plane tension in directions 1 and 2

4.2.2 Nonlinear models

Consider Figs 3.40-3.43 showing typical stress-strain diagrams for

unidirection-al advanced composites As can be seen, materiunidirection-als demonstrate linear behavior only

under tension The curves corresponding to compression are slightly nonlinear, while the shear curves are definitely nonlinear It should be emphasized that this does not mean that linear constitutive equations presented in Section 4.2.1 are not valid for these materials First, it should be taken into account that the deformations of properly designed composite materials are controlled by the fibers, and they do not allow the shear strain to reach the values at which the shear stress-strain curve is strongly nonlinear Second, the shear stiffness is usually very small in comparison with the longitudinal one, and such is its contribution to the apparent material stiffness Material behavior is usually close to linear even if the shear deformation is nonlinear Thus, a linear elastic model provides, as a rule, a reasonable approximation to the actual material behavior However, there exist problems, to solve which we should allow for material nonlinearity and apply one

of nonlinear constitutive theories discussed below

First, note that material behavior under elementary loading (pure tension, compression, and shear) is specified by experimental stress-strain diagrams of the

type shown in Figs 3.40-3.43, and we do not need any theory The necessity of the

Fig 4.15 A test to determine transverse shear modulus

Table 4.1

Transverse shear moduli of unidirectional composites (Herakovich, 1998)

Material Glass-epoxy Carbon-poxy Aramid-epoxy Boron-AI