To solve the problems of thermal elastic-plastic deformation of structural elements, a method of using the deformation theory of plasticity, generalized to the possibility of taking into account unloading with the development of plastic deformations during unloading, or repeated loading with the development of repeated plastic deformations, is proposed. Dependencies between the intensities of excess (differences between current values and the corresponding values recorded at the moment of unloading) stresses and excess deformations are built on the basis of Mazing's principle. An algorithm for solving problems based on the method of successive loads has been developed. The method of additional deformations was used to linearize the indicated stress-strain dependences..
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To simulate the processes of elastic-plastic deformation we have suggested to use the theory of small thermal elastic-plastic deformations generalized for the case of unloading taken into account [6].
Stress-strain relations of small elastic-plastic deformations can be written as [6] where here coefficient of thermal linear expansion of the material, – components of stresses and deformations deviators which were reached in the specified point of the environment at the moment of unloading start. The last values are equal to zero, if any unloading wasn’t observed in the specified point.
is the value of temperature which was fixed in the specified point at the
(
σ 2 G ( 1+ v ) by the dependence ε0= 0 + εT, where K =
K ( 1−2 v )
E 2( 1+ v ) – shear modulus, εT =αt T ¿=αt ( T −T 0 ) – average temperature deformation, αtare the components of stresses and deformations
connected – volume compression modulus , moment of unloading start. Values calculated by formulae і are the stresses intensities and , which are Apparently, if any unloading isn’t observed in the specifiedpoint yet, the values and are transformed into ordinary intensities of the stresses and
is the parameter of plasticity determined by formula (
At the stage of initial deformation from stress-free and deformation-free state in the points where some active loading is taking place the intensity of total deformation is equal to the sum of intensities of elastic and plastic deformations components εi=εip+ εie.
Stress-strain relations (
The relationship between the intensities of stresses and deformations in this stage for the most structural materials can be written as relationship can be obtained on the basis of Mazing principle [7], if it is generalized on isothermal processes of deformation. After such generalization we have found [8] ,
Stress-strain relations (
~ −ψ
G σ (ij1)−σ ij− or relative to the stress tensor component
G G
1 σ ij= σ(ij1)− ~ [ ε(ij1)−εij+ 2 G ψ ~ ψ (1+ ν ) 1−2 ν
δij [( ε(01)−ε0)−( εT (
Here the plastic deformation can be determined by formulae
~
εipj=εipj (
ψ
Now we use the same symbols for ordinary stress and deformation tensors component which have been introduced earlier for deviator components
Then the dependence (10) is written as ~ σ ij=
G G 1
σ (ij1)−σ ij , ~εij=ε(ij1)−εij , ~ σ 0=
G G 1
σ (01)−σ 0, ~ε0=ε(01)−ε0, ~εT =εT (
~ ij
ψ
We can also show that
(9)
(10)
(11)
(12)
(13)
(14)
(15)
~eiej= 21G ~sij ,
(16)
where ~eiej=ee(
ij
To linearize the specified stress-strain relations the method of additional deformation (MAD) is used. We will demonstrate this method for the case when the unloading is taking place with the development of plastic deformation. ij∗¿e(k)+~eipj(k−1)=21G ~si¿j(k)+~εipj(k−1)¿ (17) solution ~e(ijk ) ,
Having solved the problem under stress-strain relations conditions (17), we will find the ~σ¿ij(k ) which point P corresponds to on fig.1. By the known values of component ~(k ) we have calculated the intensity of total deformations ~ε(i k ). Using the surface equation eij ~σi=~Φ( ~εi , T ) for the specified temperature valueT for the specified stage and value ~ε(i k ) we have found the intensity of stresses ~σ(k ) (point N on the figure). It has enabled us to find by i ~ε(k ) formula ~ψ(k )=3 G ~σ(k ) the value of plasticity parameter ~ψ(k ) for the specified approximation, i i and by the formulae ~eipj (k )= ~ψ~(ψk )(−k) 1 ~e(ijk ) we have calculated the components of plastic residual deformation of this k approximation which can be used in formulae of the method of additional deformations (17) in the next approximation.
The formulae of the method of additional deformations in this case can be written as ~ε(ijk )= 1 (~σi¿j(k )− 13+vv δij ~σ¿0(k ))+ δij ~εT + ~εipj (k−1), 2 G ~εp(k )= ~ψ(k )−1 ( ~ε(ijk )−δij ~ε(0k )) , ij ~ψ(k )
~ε(k ) ~ψ(k )=3 G ~σi(k ) .
i (18) (19) (20)
The whole process of loading (heating, cooling) is divided into separate stages. Specifying the values of the deformation plasticity component for zero approximation in (18) equal to these components which were reached for the previous stage of loading (at deformation from initial undeformed state they are accepted as zero) the elastic problem with additional deformations is solved. according to the found total deformations in approximation the intensities and
are calculated. Then according to the formula (20) for each point of the structure approximation of the plasticity parameter is calculated and by formula (20) the components of the deformation plasticity which will be further used in formulae (18) in the next approximation to find the component . Iteration process lasts till its complete coincidence, after that the transition to the next stage of loading is taking place.
We must admit that initially on every iteration in each point of the structure the abovementioned deformation has been assumed that occurred in it during the previous stage of loading, i.e. initial elastic or plastic deformation, elastic unloading or unloading with the development of further plastic deformation. After coincidence of the iteration process the examination is conducted in every point of the structure to find out if such deformation was taking place in fact. If in some points the deformation behavior does not correspond the accepted one on the basis of information from the previous stage of loading, then the stage is fully recalculated with the previous replacement of the deformation behavior to the opposite one in such points.
Due to the above-mentioned technique a number of practical problems have been solved, namely the welding of thin-walled structural parts, building-up welding aimed at strengthening or restoring the operational characteristics.
In this way, for instance, the problem of welding procedure simulation of two cylindrical shells by circular joint providing the welding is taking place along the whole length of the welding seam simultaneously [9]. The obtained results have completely correlated with the similar results found in the paper [5] using more complicated theory of plastic flow. The results of modelling have made possible to find the fields of residual stresses, deformations and displacements, study the kinetics of stress-and-strain state of the welding process, study the diagram of deformation in different points of the structure. As an example, we will show the distribution of residual welding stresses, elastic deformations, and also residual deflection of a shell.
−¿ is a dimensionless coordinate along the length of the shell, the
It is shown that to solve the complex problems of thermal elastic-plastic deformation of structural elements, it is possible to use the theory of the deformation theory of thermal plasticity deformation, which is much simpler than the flow theory, generalized to consider the possibility of unloading with the development of plastic deformations, or repeated loading with the development of repeated plastic deformations. [8] 8. Yu.M. Shevchenko Thermoplasticity under variable loads. -K.: Naukova dumka, 1970.287 p. [9] 9. Mykhailyshyn M. Problems of formation of residual stresses and deformations during welding. Bulletin of the Ternopil State Technical University. – Ternopil.: Volume 9, No. 2, 2004 – P. 19-26.