Constitutive modelling is an essential part of structural engineering and a key component of analytical, numerical and design models. Research into the accurate description of the stress-strain response of a range of metallic materials has been carried out in our research group in the last few years. An overview of the research, together with the key references, is given below.
The stress-strain curve of hot-rolled carbon steels typically exhibits a linear path up to a distinct yield point, followed by a region of plastic flow at an approximately constant stress before rising smoothly up to a peak stress due to strain hardening, after which there is a reduction in applied stress due to local necking as the specimen becomes local unstable and finally fracture occurs. Based upon and calibrated against data from over 500 experimental stress-strain curves collected from the global literature, two new standardised models to represent the yield plateau and strain hardening behaviour of hot-rolled steels has been proposed : a quad-linear stress-strain model (Fig. 1) and a bi-linear plus nonlinear hardening model to capture the gradual loss of stiffness in the strain hardening regime (Fig. 2). The models utilise three basic material parameters, Young’s modulus E, yield stress fy and ultimate stress fu, which are readily available to engineers in material standards, as well as additional material parameters, for which predictive expressions or values have been developed. Full details of the models are described in .
Cold-formed steel, stainless steel and aluminium
Metallic materials such as cold-formed steel, stainless steel and aluminium are characterized by a nonlinear stress-strain response, which differs from that typically exhibited by hot-rolled carbon steel. The most widely used constitutive material model to capture this behaviour is based on the general expression originally proposed by Ramberg and Osgood  and modified by Hill . A number of developments and extensions to the Ramberg-Osgood model, particularly in a two-stage form, have been made within the steel structures research group. The research and findings are reported in References [4-8] and an example is shown in Fig. 3 for stainless steel at elevated temperatures. The influence of cold-work of the stress-strain characteristics of metallic materials has been reported in [9-10], while a statistical evaluation of the key material properties of structural stainless steel is set out in .
 Yun X. and Gardner L. Stress-strain curves for hot-rolled steel. Journal of Constructional Steel Research (2017) 133: 36-46. DOI: 10.1016/j.jcsr.2017.01.024
 Ramberg W. and Osgood W.R. Description of stress-strain curves by three parameters, Technical Note No. 902, National Advisory Committee for Aeronautics, Washington, D.C., USA, 1943 (1943).
 Hill H.N. Deformation of stress-stain relations from “offset” yield strength values, Technical Note No. 927, 1944 (1944).
 Gardner L. and Ashraf M. Structural design for non-linear metallic materials. Engineering Structures (2006) 28(6): 926-934. DOI: 10.1016/j.engstruct.2005.11.001
 Arrayago I., Real E. and Gardner L. Description of stress–strain curves for stainless steel alloys. Materials and Design (2015) 87:540-552. DOI: 10.1016/j.matdes.2015.08.001
 Gardner L., Insausti A., Ng K.T. and Ashraf M. Elevated temperature material properties of stainless steel alloys. Journal of Constructional Steel Research (2010) 66(5):634-647. DOI: 10.1016/j.jcsr.2009.12.016
 Gardner L., Bu Y., Francis P., Baddoo N.R., Cashell K.A. and McCann F. Elevated temperature material properties of stainless steel reinforcing bar. Construction and Building Materials (2016) 114:977-997. DOI: 10.1016/j.conbuildmat.2016.04.009
 Gardner, L. and Yun, X. (2018). Description of stress-strain curves for cold- formed steels. Construction and Building Materials. 189, 527–538.
 Afshan S., Rossi B. and Gardner L. Strength enhancements in cold-formed structural sections—Part I: Material testing. Journal of Constructional Steel Research (2013) 83:177-188. DOI: 10.1016/j.jcsr.2012.12.008
 Rossi B., Afshan S. and Gardner L. Strength enhancements in cold-formed structural sections—Part II: Predictive models. Journal of Constructional Steel Research (2013) 83:189-196. DOI: 10.1016/j.jcsr.2012.12.008
 Afshan S., Francis P., Baddoo N.R. and Gardner L. Reliability analysis of structural stainless steel design provisions. Journal of Constructional Steel Research (2015) 114:293-304. DOI: 10.1016/j.jcsr.2015.08.012